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Instituto Mediterr´ aneo de Estudios Avanzados IMEDEA (CSIC-UIB)

Departamento de Tecnolog´ıas Marinas, Oceanograf´ıa Operacional y Sostenibilidad (TMOOS)

A Boussinesq-type model for wave propagation in deep and shallow waters and boundary layer considerations

A Doctoral Thesis Presented to the Department of Civil Engineering of Universidad de Castilla La-Mancha as Partial Fulfillment of Requeriments for the Degree of Doctor of Civil Engineering by ´ Alvaro Gal´ an Alguacil

PhD supervisors: Dr. Alejandro Orfila F¨ orster

Dr. Gonzalo Simarro Grande

Esporles, July 2011


Agradecimientos En primer lugar me gustar´ıa agradecer a mis directores de Tesis, Alejandro Orfila y Gonzalo Simarro, su gran ayuda en todo momento, la paciencia y entusiasmo mostrada en cada paso del camino, y el apoyo que han supuesto para mi en los momentos m´ as dif´ıciles. Darles las gracias por el tiempo empleado, por compartir sus conocimientos, por ser una fuente constante de ideas, por ser algo m´ as que simples directores de tesis. Esta Tesis no ser´ıa posible sin ellos. Agradecer a todas aquellas personas e instituciones que de una u otra manera han colaborado en el trabajo realizado, en especial al Consejo Superior de Investigaciones Cient´ıficas (CSIC) por el soporte econ´omico prestado para la consecuci´on de esta Tesis mediante su programa JAE (Junta de Ampliaci´on de Estudios) y por las ayudas para realizaci´on de estancias breves. Mostrar mi m´ as profunda gratitud al profesor Philip L.F. Liu por brindarme la oportunidad de pasar unos meses junto a ´el en la Universidad de Cornell durante la primavera de 2010 y a Yong Sung Park por su ayuda en la realizaci´on de los ensayos. Gracias a Jos´e Mar´ıa y Jorge por esas largas charlas en las que la condici´ on inf-sup estaba siempre presente, y como no, a Lara y Vera, mis compa˜ neras futboleras de mundial. Gracias a todos ellos por hacerme la estancia en Ithaca mucho m´ as agradable.

v


Desear´ıa agradecer a todas las personas con las que he compartido experiencias durante esta etapa de mi vida, a Javier Gonz´alez, quien me brind´o la oportunidad de comenzar en este mundo de la investigaci´on; al personal del TMOOS y del Laboratorio de Hidr´ aulica de la Escuela de Caminos de Ciudad Real; a Mar´ıa por su ayuda en todo momento; a Alejandro y Fernando por abrirme las puertas de su casa; a Mat´ıas, Tolo, Marian, Chema, etc. Quiero agradecer a mi familia el apoyo y cari˜ no recibido: a mis padres, Valent´ın y Luisa, un apoyo constante e incondicional imprescindible para llegar a buen puerto; a mis hermanos, C´esar y Gema; a mis sobrinos, Daniel y Alba, un manantial de alegr´ıa permanente; y por supuesto a mis abuelos. A mis amigos, en especial a Vir y Jaime por estar siempre ah´ı, por esas largas noches de “insomnio”, por todas esas risas sin motivo, por todos esos momentos inolvidables vividos y por los que a´ un nos quedan por vivir, en definitiva, por hacerme m´as feliz; a Rober, Miren, Felipe, Mau, Val, Tania, Jorge, Julio, Lourdes, Luci y a otros tantos que hab´eis aportado vuestro granito de arena. No puedo terminar estos agradecimientos sin acordarme de una de las personas m´as especiales e importantes en mi vida, Esther. Durante estos largos cuatro a˜ nos hemos compartido buenos y malos momentos, momentos de euforia por el trabajo bien hecho y momentos de decepci´on cuando las cosas no marchaban tan bien. Lo m´ as importante es que siempre tuve claro durante este periodo que, al llegar a casa, siempre encontrar´ıa a alguien con quien compartir la experiencia vivida y un apoyo constante para afrontar el d´ıa siguiente.

´ Alvaro Gal´an Alguacil Julio, 2011

vi


Resumen En esta Tesis se deriva un nuevo conjunto de ecuaciones tipo Boussinesq para la propagaci´ on de oleaje en aguas profundas y someras. Se trata de un nuevo conjunto de ecuaciones totalmente no lineal con propiedades dispersivas mejoradas respecto a los sistemas previos. Las nuevas ecuaciones son exactas hasta O (kh)2 . Se emplea un m´etodo de optimizaci´on para determinar el valor de los coeficientes introducidos en las nuevas ecuaciones propuestas con el objetivo de minimizar las diferencias entre el modelo y las teor´ıas de Airy (dispersi´on lineal y asomeramiento) y de Stokes (transferencia de energ´ıa d´ebilmente no lineal). Se muestra que con la adecuada elecci´on de estos coeficientes el modelo es aplicable hasta valores de kh = 20 con un error relativo menor del 1% en dispersi´ on lineal. En esta Tesis se presenta un nuevo esquema num´erico expl´ıcito de cuarto orden para resolver y verificar el nuevo conjunto de ecuaciones. Adem´ as, se ha llevado a cabo un an´alisis lineal de estabilidad para obtener una condici´ on tipo CFL para el paso de tiempo. La integraci´on temporal se lleva a cabo empleando un esquema Runge-Kutta de 4o orden. El oleaje se genera internamente en el dominio por medio de una funci´on fuente. El comportamiento tanto de las nuevas ecuaciones como del esquema num´erico es validado empleando experimentos y soluciones anal´ıticas en 1D y 2D. El comportamiento bidimensional del modelo se valida comprobando la evoluci´on temporal de una onda inicialmente gaussiana en un dominio cerrado cuadrado. Adem´as se han simulado num´ericamente dos experimentos para llevar a cabo vii


la validaci´on del nuevo modelo derivado. El primero de ellos es la barra 1D sumergida de Dingemans y el segundo es la batimetr´ıa 2D de Vincent y Briggs. Los resultados num´ericos obtenidos muestran buen ajuste para todos los casos. Para profundizar en el conocimiento de los procesos que se dan en la capa l´ımite de fondo, se ha dise˜ nado un nuevo instrumento para la medida directa de la tensi´on de fondo bajo distintas condiciones de oleaje peri´odico. Se presentan tanto casos monocrom´ aticos como ondas en forma de diente de sierra, comparando los resultados experimentales con un modelo num´erico. El objetivo final de este an´ alisis es contar con una metodolog´ıa para la inclusi´on de la tensi´on de fondo bajo oleajes fuertemente no lineales.

viii


Abstract In this Thesis, a new set of Boussinesq-type of equations is derived for water wave propagation in deep and shallow waters. The new set of equations are fully nonlinear and the dispersive properties are improved relative to previous systems. The model equations are accurate to O (kh)2 . An optimization method is used to determine the weighting coefficients employed in the proposed equations so as to minimize the differences between the model equations and the theories by Airy (linear dispersion and shoaling) and Stokes (weakly nonlinear energy transfer). It is shown that with the proposed choice of weighting coefficients, the model is applicable up to kh = 20 with 1% relative errors in linear frequency dispersion. A new explicit and fourth order numerical scheme is presented in this Thesis to solve and verify the new set of equations. Besides, a linear stability analysis is performed to obtain a CFL-type condition for the time step. The time integration is performed using a 4th order Runge-Kutta scheme. Waves within the numerical domain are generated by means of an internal source generation function. The features of the new equations as well as the numerical scheme are tested against experiments and analytical solutions in both 1D and 2D cases. The linear 2D performance of the equations is tested modelling the evolution of an initially gaussian wave in a squared domain. Besides, two experiments have been simulated to test the performance of the new model equations. The first one is the Dingemans bar (1D) and the second is the Vincent and Briggs shoal ix


(2D). The numerical results show very good agreement in all cases. In order to get a deeper knowledge in those processes occurring in the bottom boundary layer, a new instrument built to measure directly the bottom shear stress under dierent conditions of periodic waves is presented. Both monochromatic and sawtooth shaped waves are tested and experimental shear stress compared against a numerical model. The final goal of this analysis is to have a methodology to further include bottom shear stress under highly nonlinear waves.

x


To my family

xi


xii


List of Figures 2.1

Schematic picture for the range of application of different models: (a) [Peregrine 1967], denoted by P67, is valid for weakly nonlinear and weakly dispersive waves over arbitrary slopes; (b) [Nwogu 1993], denoted by N93, improves the linear frequency dispersion performance over flat beds into deeper waters; (c) [Wei et al. 1995], W95 here, extends N93 to fully nonlinear waves for the weakly dispersive case, and [Madsen & Schaffer 1998], denoted as M98, increases the applicability to deeper water; (d) new model equations. . . . . . . . . . . . . . .

2.2

13

Linear dispersion: capp /cAiry − 1 for sets A corresponding to κmax = 5 (�); κmax = 10 (�); κmax = 20 (�) in Table 2.3. W95 refers to Wei et al. (1995) and M98 to Madsen & Schaffer (1998) equations. . . . .

2.3

18

Frequency dispersion: relative errors for the phase speed using Peregrine (1967) (denoted as P67), [2/2] Pad´e, Nwogu (1993) (denoted as N93) and [4/4] Pad´e. . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

19

Airy Linear shoaling: Aapp − 1 for the coefficient sets A correspondη /Aη

ing to κmax = 5 (�); κmax = 10 (�); κmax = 20 (�) in Table 2.3. W95 refers to Wei et al. (1995) and M98 to Madsen & Schaffer (1998) equations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

. . . . . . . . . . . . . . .

26

2.5

Eigenvalues for the truncated matrices.

2.6

Function ν∗ = f (Π, s) . The different symbols stand for different values of Π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

28


2.7

Stokes Weakly nonlinear propagation: Gapp −1 for set A and κmax = 5. 31 ± /G±

3.1

Stability regions for third order Adams–Bashforth (AB3), fourth order Adams-Moulton (AM4) and fourth order Runge-Kutta (RK4) time integration schemes. . . . . . . . . . . . . . . . . . . . . . . .

3.2

47

General behaviour of function fx using coefficient from Table 2.3 for κmax = 5 (big symbols) and κmax = 10 (small symbols). Spatial derivatives of second order (triangles) and of fourth order (circles). .

48

3.3

Analysis of the order of convergence for a 1D linear case. . . . . . .

52

3.4

Monochromatic linear 1D propagation for h = 25 m and T = 4.5 s. Numerical results (circles) and Airy’s analytical solution (line). . . .

3.5

Monochromatic linear 1D propagation for T = 4.5 s.

54

Numerical

results for the new equations (circles), Madsen & Schaffer (1998) (squares), Wei et al. (1995) (crosses) and Airy’s analytical solution (line) at time t = 10T. . . . . . . . . . . . . . . . . . . . . . . . .

3.6

55

Experimental set-up (top panel) and free surface time histories at #A (left) and #B (right) for periods T = 0.45s (top), T = 0.55s (middle) and T = 0.55s (bottom). Experimental data (stars) and numerical results (line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

57

Free surface histories at the corner (top) and center (bottom) in a squared basin. Analytical solution for Airy’s theory (line) and numerical solution for the new Boussinesq-type equations (circles).

3.8

59

Dingemans’ experiments. Case A. Numerical results (lines) and experimental data (stars) for free surface elevation.

3.9

. .

. . . . . . . . . .

61

Dingemans’ experiments. Case C. Numerical results (lines) and experimental data (stars) for free surface elevation.

. . . . . . . . . .

62

3.10 Dingemans’ experiments. Case A. Comparison at sections #7 and #8 using proposed coefficients (circles), Madsen & Schaffer (1998) (squares) and Wei et al. (1995) (crosses) with experimental data (line). 63

xiv


3.11 Bathymetry and location of gages in Vincent and Briggs’ experiment (top) and comparison between experimental (stars) and numerical (line) significant wave heights. . . . . . . . . . . . . . . . . . . . .

65

3.12 Snapshot for free surface elevation at time t = 20s in Vincent and Briggs’ experiment. Values of η/η0 .

4.1

. . . . . . . . . . . . . . . . .

General overview of the new instrument designed to measure directly bed shear stress. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

66

68

Velocity profile in the laminar case for a steady (ub (m/s) = 1) and unsteady (ub (m/s) = cos (ωt) with ω = π/5 s−1 ) cases. In the latter case, snapshots correspond to t = 0 (—), t = T/8 (− −), t = T/4

(−·−) and t = T/2 (· · · ), where T = 2π/ω = 10 s is the period. . . .

4.3

Near bed velocity and bed shear stress time series in the laminar case for a periodic motion ub (m/s) = cos (ωt) with ω = π/5 s−1 . . . . . .

4.4

72

Function fw (Re, ε) according to the original model II in Simarro et al. (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

71

75

Comparison between the turbulent boundary layer models and experimental data in Kamphuis (1975) (+), Sleath (1988) (�), Jensen (1989) (◦) and Simons et al. (1992) (�). Root Mean Square (RMS) error takes values of 0.276, 0.185 and 0.215 respectively. . . . . . . .

76

4.6

Monochromatic and sawtooth time histories. . . . . . . . . . . . . .

79

4.7

Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2 ) (×, experimental; full line, model II) time histories for smooth tests in Table 4.1. Monochromatic waves (top panels) and sawtooth waves (bottom panels). . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

82

Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2 ) (×, experimental; full line, model II) time histories for smooth tests in Table 4.2. Monochromatic waves only. . . . . . . . . . . . . . . . .

4.9

85

Experimental and analytical values for fw : new data (black figures) and data available in the literature (white figures).

xv

. . . . . . . . .

86


4.10 Velocity ub (m/s) (dashed line) and shear stress τb (N/m2 ) (×, experimental; full line, model II) time histories for smooth tests in Table 4.2. Sawtooth waves only. . . . . . . . . . . . . . . . . . . . . . .

87

4.11 Shear stress τb (N/m2 ) time histories for tests mc T08 u08 k25 (monochromatic) and st T08 u08 k25 (sawtooth). Numerical results (lines) and experimental data (symbols). . . . . . . . . . . . . . . . . . . . . .

88

A.1 Weakly nonlinear propagation: GStokes . . . . . . . . . . . . . . . . 103 ± Stokes A.2 Weakly nonlinear propagation: Gapp − 1 for W95. . . . . . . 103 ± /G±

xvi


List of Tables 2.1

Coefficients for Wei et al. (1995) and Madsen & Schaffer (1998).

. .

12

2.2

Coefficients corresponding to [4/4] Pad´e approximant. . . . . . . . .

15

2.3

Sets A. Simple optimization. . . . . . . . . . . . . . . . . . . . . .

16

2.4

Sets B: joint optimization.

(∗)

correspond to joint optimization en-

suring stability for s = 5. . . . . . . . . . . . . . . . . . . . . . . .

20

2.5

Sets C. Linearly stable for smax � 5. . . . . . . . . . . . . . . . . .

29

3.1

Analysis of the order of convergence for a 1D linear case. . . . . . .

52

3.2

Dingemans’ experiments setup. . . . . . . . . . . . . . . . . . . . .

60

4.1

Smooth test cases for sinusoidal and sawtooth waves. . . . . . . . .

80

4.2

Rough test cases for sinusoidal and sawtooth shaped waves. . . . . .

81

4.3

Monochromatic tests errors with the three models. For each case, the model with minimum � is shown in bold font. . . . . . . . . . . . .

4.4

84

Sawtooth tests errors with the three models. For each case, the model with minimum � is shown in bold font. . . . . . . . . . . . . . . . .

xvii

84


xviii


Contents Agradecimientos

v

Resumen

vii

Abstract

ix

List of Figures

xiii

List of Tables

xvii

Contents

xix

1 Introduction

1

1.1

Boussinesq-type models . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Bottom shear stress . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Aims and motivation . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . .

6

2 An improved Boussinesq-type model for wave propagation

7

2.1

New set of wave propagation equations . . . . . . . . . . . . . .

7

2.2

Improving the linear propagation . . . . . . . . . . . . . . . . .

14

2.3

Linear stability considerations . . . . . . . . . . . . . . . . . . .

21

2.3.1

Flat bed case . . . . . . . . . . . . . . . . . . . . . . . .

21

2.3.2

Uneven bed case . . . . . . . . . . . . . . . . . . . . . .

24

xix


2.4

Improving the weakly nonlinear propagation . . . . . . . . . . .

29

2.5

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

30

3 An explicit numerical scheme for the new model 3.1

33

Source function for wave generation and sponge layers . . . . .

33

3.1.1

Source function . . . . . . . . . . . . . . . . . . . . . . .

34

3.1.2

Sponge layers . . . . . . . . . . . . . . . . . . . . . . . .

39

Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.2.1

Spatial discretization: matrix notation . . . . . . . . . .

41

3.2.2

Time integration: fourth-order Runge-Kutta scheme . .

44

3.3

Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4

Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.4.1

Linear tests over flat bed . . . . . . . . . . . . . . . . .

50

3.4.2

Nonlinear tests . . . . . . . . . . . . . . . . . . . . . . .

58

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

64

3.2

3.5

4 Turbulent bed shear stress 4.1

67

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4.1.1

Laminar flow . . . . . . . . . . . . . . . . . . . . . . . .

68

4.1.2

Turbulent flow . . . . . . . . . . . . . . . . . . . . . . .

72

4.2

Experimental facility and setup . . . . . . . . . . . . . . . . . .

77

4.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.4

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .

89

5 General conclusions and future work

91

A Features of the new equations

95

A.1 Linear propagation . . . . . . . . . . . . . . . . . . . . . . . . .

95

A.1.1 Linear frequency dispersion . . . . . . . . . . . . . . . .

97

A.1.2 Linear wave shoaling . . . . . . . . . . . . . . . . . . . .

98

A.2 Weakly nonlinear propagation . . . . . . . . . . . . . . . . . . . 100 xx


B Stability analysis for sinusoidal bathymetries

105

References

107

Additional works

115

xxi


xxii


Chapter 1

Introduction Water wave propagation from deep to shallow water is a crucial issue for many scientific and engineering activities in coastal regions. Accurate modelling of wave transformation is mandatory to study a wide range of processes such as sediment transport or coastal flooding. Besides, any wave model for relatively shallow waters propagation needs to properly represent not only wave shoaling, refraction and diffraction, but also nonlinearity. Early attempts to model large coastal areas were based on the wave ray theory. This approximation does not allow wave energy to cross the rays and, in fact, ray models are restricted to the study of small amplitude waves –linear theory– over slowly varying bathymetry and in situations where diffraction effects are not important. A different family of wave propagation models are obtained by depth averaging the vertical distribution of the velocity field. The simplest models within this category are the shallow water equations, valid if kh � 1 (where h is the water depth and k = 2π/λ is the wavenumber). Shallow water equations allow

to account for nonlinear effects, so that no restriction is made on a/h (where a is the wave height), which is the parameter characterizing the nonlinearity of the waves. The Boussinesq-type equations depart from the shallow water equations to improve their range of applicability to higher kh and they re1


2

Introduction

produce wave propagation processes such as shoaling, diffraction, refraction, wave-wave interaction and nonlinear transformation.

1.1

Boussinesq-type models

The traditional Boussinesq equations were originally derived for weakly nonlinear and weakly dispersive water waves, so the accuracy of the traditional Boussinesq equations is up to O (a/h) and O ( (kh) 2 ) [Peregrine 1967]. Madsen et al. (1991) showed that to keep the errors of the wave celerity, estimated

by the best form of the linearised Boussinesq equations (using the depthaveraged horizontal velocity in the wave equations), within 1% of that of Airy waves, the water depth must be less than one-fifth of the wavelength, i.e., kh < 1.1. On the other hand, as the water depth decreases, wave amplitude increases and so does the nonlinearity parameter a/h. Therefore, the traditional Boussinesq equations, strictly speaking, can only be used in a small region in coastal zones where both frequency dispersion and nonlinearity are relatively weak. In the last twenty years, much of the development of wave propagation modelling has been focused on extending the applicability of the depth-integrated Boussinesq wave equations into deeper water (or shorter waves). Nwogu (1993) derived his weakly nonlinear Boussinesq-type wave equations in terms of the horizontal velocity on a specified elevation from the still water level, zα . He showed that the frequency dispersion characteristics of his equations strongly depend on the choice of zα . Nwogu suggested that, if the velocity close to the mid-depth is used, namely zα ≈ −0.53 h, the practical limit of the resulting Boussinesq-type equations is roughly increased to kh ≈ 3.3 for 1% relative error in the wave celerity, which is a significant improvement over the traditional

Boussinesq equations (i.e., kh ≈ 1.1). Departing from Nwogu’s approach, Wei

et al. (1995) obtained the corresponding set of fully nonlinear Boussinesq-type 2


1.1 Boussinesq-type models

3

� � equations by including terms of O �µ2 , �2 , �3 . Other researchers have recently proposed different Boussinesq-type models that further improve the characteristics in deep waters by including highly dispersive terms. Perhaps the most straightforward approach is the one by Gobbi et al. (2000), which kept the higher order terms O ((a/h)2 ) and O ( (kh) 4 )

in their derivation of the depth-integrated wave equations. By doing so, they increased the practical application limit to kh ≈ 6 for the linear case. The

shortcoming of the higher order Boussinesq equations is that fifth order derivatives appear in the wave equations and, therefore, numerical schemes for solving these wave equations also become more complex and computationally demanding.

Based on the method of Agnon et al. (1999), Madsen et al. (2002) developed a fully nonlinear model, which is accurate in very deep waters (kh ≈ 40 for the linear dispersion). Their model requires more differential equations to be solved, compared to other higher order models such as that by Gobbi & Kirby (1999). The highest order of derivatives in the model is also fifth. Madsen, et al. (2003) presented a simplified version of their original model, where the highest order of derivative is reduced to three. However, the range of application is also reduced to kh < 10.

Most recently, Lynett & Liu (2002, 2004) introduced a multi-layer Boussinesqtype wave model. In their model, the water column is divided into several layers, in which the Boussinesq approximation is employed. More specifically, within each layer the horizontal velocity is assumed to be a quadratic polynomial and the matching conditions are required along the interface of two adjacent layers. Invoking the conservation of mass and momentum in each layer yields a set of model equations where the highest order of derivatives remains three. Lynett & Liu (2004) showed that the limits of application are 3


4

Introduction

kh ≈ 8, 17, 30 for 2, 3, and 4 layer model, respectively. In summary, there already exist several Boussinesq-type wave models that can be applied from very deep to shallow waters. However, these models contain either more unknowns or higher order derivatives than the lower order (up to O (kh) 2 ) Boussinesq-type wave models, are computationally demanding and

require artificial boundary conditions. In fact lower order equations by Wei et al. (1995) are the most popular nowadays. Additionally, to solve numerically the Boussinesq-type equations, implicit schemes to step in time are usually considered [Wei & Kirby 1995]. Further, waves are usually generated within the integration domain by means of a “wave-source” function [Wei et al. 1999]. Albeit the schemes are implicit in time, numerical instabilities have been reported in the literature, and the use of numerical filters is a common practice.

1.2

Bottom shear stress

The analysis of the bottom boundary layer is crucial at least for two reasons. In the one hand, the inclusion of the boundary layer in wave propagation models allows to capture the wave damping due to the bed friction [Keulegan 1949, Liu & Orfila 2004, Liu et al. 2006]. In the other, a good understanding of the boundary layer allows to properly represent the shear stress transmitted to the bottom, which is a crucial aspect in sediment transport, either suspended or bed load [Nielsen 1992, Fredsoe & Deigaard 1992] and, thus, in morphological problems. Despite the importance of the wave induced boundary layer, the knowledge of the physical processes occurring there is far from being complete, in part due to the difficulty to perform measurements [Kushnir 2005]. Some remarkable 4


1.3 Aims and motivation

5

experimental data for monochromatic oscillatory flows are those by Riedel (1972), Kamphuis (1975), Jonsson & Carlsen (1976), Jensen (1989) or Mirfenderesk & Young (2003). However, data for non-monochromatic boundary layers is scarce [Suntoyo et al. 2008]. Shear stress measurements under water wave fields are usually classified in two diďŹ&#x20AC;erent groups, depending on the use of direct or indirect techniques [Sheplak et al. 2004, You & Yin 2005]. Indirect methods are supported by the theoretical correlation between velocity field near the bottom and bed shear stress [Jonsson 1966, Sleath 1987, Justesen 1988, Suntoyo et al. 2008]. The quality of these methods relies on the validity of the turbulence model used to correlate the velocity and the shear stress (i.e., the law of the wall), and on the quality of the measurements of the velocity. On the other hand, direct methods usually employ a shear plate placed at the bottom of a wave flume to measure the horizontal force that the fluid exerts on it. The force measured by the shear plate consists on the addition of the integral of the wave bottom shear stress and the wave pressure gradient at the ends of the plate, which might provide inaccurate estimations if the pressure gradient is not well estimated as mentioned in Rankin & Hires (2000) and in You & Yin (2007). A full reference of modern measurement techniques, mostly for aerodynamics, is found in Naughton & Sheplak (2002).

1.3

Aims and motivation

Against this background, the aim of this Thesis is to derive a set of low order and one layer fully nonlinear Boussinesq-type equations for wave propagation with an improved range of applicability and the development of an eďŹ&#x192;cient algorithm to solve the model. The scientific motivation is twofold: first, to develop a numerical model able to propagate waves in areas where non linear eďŹ&#x20AC;ects are negligible but dispersion are important and second, to develop a 5


6

Introduction

model which should be used for boundary layer and sediment transport studies. Additionally, the Thesis pretends to get a deep insight on the wave generated boundary layer by the measurement of the bottom shear stress.

1.4

Overview of the Thesis

The present Thesis is structured as follows. In Chapter 2, a new set of fully nonlinear Boussinesq-type equations for wave propagation with an improved range of applicability from deep water to shallow water is presented. A 4th order Runge-Kutta explicit scheme including a source function for internal wave generation and a sponge layer to deal with wave radiation is presented in Chapter 3. The model is validated with analytical and laboratory benchmark problems. An special treatment of the linear stability condition is performed along the Chapter. In Chapter 4, a new instrument designed to measure directly the bottom shear stress under symmetric and asymmetric waves is presented. The new device is employed to validate an analytical turbulent boundary layer model that computes the bed shear stress transmitted to the bottom under diďŹ&#x20AC;erent conditions of wave and wave-current. Finally, Chapter 5 concludes the work and outline the future areas of research related with the Thesis.

6


Chapter 2

An improved Boussinesq-type model for wave propagation In this Chapter, we present a new set of low order Boussinesq-type fully non� � linear equations with improved dispersive properties accurate up to O (kh)2 ,

with k the characteristic wave number and h the water depth. An optimization method is used to determine the weighting coefficients employed in the proposed equations so as to minimize the differences, in terms of linear frequency dispersion and shoaling and weakly nonlinear energy transfer, between the model equations and Airy and Stokes theories in deep waters. It is shown that, with the proposed choice of weighting coefficients, the model is applicable up to kh = 20 with 1.00% relative errors in linear frequency dispersion. The results of this Chapter are under review in Journal of Waterway, Port, Coastal, and Ocean Engineering.

2.1

New set of wave propagation equations

Dimensional and dimensionless expressions will be used at convenience throughout this Chapter. The dimensional variables are denoted with primes, while the dimensionless variables are unprimed. In our wave propagation problem we consider k0� , h�0 and a�0 as the characteristic values for the wave number, 7


8

An improved Boussinesq-type model for wave propagation

the water depth and wave amplitude respectively. Dimensionless variables are defined as k≡

k� , k0�

{x, y} ≡ k0� {x� , y � } ,

{z, h} ≡

{z � , h� } , h�0

η≡

η� , a�0

with k the wave number, x and y the two horizontal coordinates, z the upward vertical coordinate with z = 0 being the still water level, h the local water depth and η the free surface elevation. We further define t ≡ k0�

� g � h�0 t� ,

u≡

h�0 u� � , a�0 g � h�0

{c� , c�g } {c, cg } ≡ � � , g � h0

as the corresponding dimensionless time, horizontal fluid particle velocity vector, wave celerity and group velocity, respectively. We also define here ��0 as the characteristic horizontal length for the bathymetric changes, i.e., so that O (∇� h� ) = O (h�0 /��0 ) . For later use �≡

a�0 , h�0

µ ≡ k0� h�0 ,

σ≡

(2.1)

1 , k0� ��0

(2.2)

are the dimensionless parameters characterizing, respectively, nonlinearity, dispersion and bed slope influence on the wave propagation. Note that, for given k0� and a�0 , if h�0 increases into deep water, the dispersive parameter µ increases while the nonlinear parameter � decreases. Also, for given k0� and a fixed bed slope h�0 /��0 , as h�0 increases into deep water, ��0 will increase and σ ≡ (k0� ��0 ) −1

will decrease, and this will be interpreted below as the bed slope being negligible in deep water. Defining u as the velocity at z = αh, the equations by Wei et al. (1995) can be written as W951 = O (µ4 ) , W952 = O (µ4 ) , 8


2.1 New set of wave propagation equations

9

where W951 ≡ X − X∗ + µ2 ∇· [d1α h2 ∇X + d2α h3 ∇Y] + �� � � � �η � �2 η 2 2 2 + �µ ∇· c1α h − η∇X + c2α h − η∇Y , (2.3a) 2 6 and W952 ≡ Z − Z∗ + µ2 [c1α h∇∇· (hZ) + c2α h2 ∇∇·Z] − � � �η 2 2 − �µ ∇ η∇· (hZ) + ∇·Z + 2 � � � � �2 η 2 (X + �ηY) 2 2 2 + �µ ∇ (c1α h − �η) u·∇X + c2α h − u·∇Y + , 2 2 (2.3b) in which Y ≡ ∇·u, X ≡ ∇· (hu) ,

Z ≡ ut , � Z∗ ≡ − ∇ (u·u) − g∇η. 2

X∗ ≡ −ηt − �∇· (ηu) , and α≡

zα , h

c1α ≡ α,

c2α ≡

α2 , 2

1 d1α ≡ α + , 2

d2α ≡

α2 1 − , 2 6

(2.4)

are dimensionless coefficients determined by the choice of α ≡ zα /h. The

subscript “t” above indicates that this variable appears derived respect to the time. For future use we also define here cα ≡ c1α + c2α ,

and

dα ≡ d1α + d2α {= cα + 1/3} .

(2.5)

In the definition of Z∗ above, g = 1. This variable g is introduced so that dimensionless equations can be easily converted into dimensional equations; the dimensional equations are recovered from the above equations by setting � = µ = 1 and g = 9.81 m/s2 and considering the rest of unprimed variables 9


10

An improved Boussinesq-type model for wave propagation

as the dimensional ones. The range of applicability of equations (2.3) is shown in Figure 2.1(c). The only differences respect the equations presented by Nwogu (1993), as shown � � in Figure 2.1, is the inclusion of terms of O �µ2 , �2 , �3 . Besides, the enhancement procedure by Madsen & Schaffer (1998) leads to the following expressions M981 = O (µ4 ) , M982 = O (µ4 ) , where M981 ≡ W951 + µ2 [ (δ − δσ ) ∇· (h2 ∇ (X − X∗ ) ) + δσ ∇2 (h2 (X − X∗ ) ) ] ,

(2.6a)

and M982 ≡ W952 + µ2 [ (γ − γσ ) h2 ∇∇· (Z − Z∗ ) + γσ h∇∇· (h (Z − Z∗ ) ) ] .

(2.6b)

Taking into account that, O (X − X∗ ) = O (Z − Z∗ ) = O (µ2 ) according to expressions (2.3), the equations (2.6) add terms of O (µ4 ) to (2.3), and there-

fore remain the same order of accuracy. However, they show an improvement of the behaviour of celerity if the coefficients δ, δσ , γ and γσ are carefully chosen (Figure 2.1). In fact, in Madsen & Schaffer (1998) the coefficients are chosen to mimic [4/4] Pad´e approximant for the dispersion expression. For convenience, we have renamed the original variables α1 , α2 , β1 , β2 in Madsen & Schaffer (1998) as β1 ↔ −δ,

β2 ↔ −δσ ,

α1 ↔ −γ

α2 ↔ −γσ , 10


2.1 New set of wave propagation equations

11

and we also note that α in this work by Madsen & Schaffer corresponds to our cα . The expression for M981 above introduces two new terms with respect to W951 , which are µ2 (δ − δσ ) ∇· (h2 ∇ (X − X∗ ) ) ,

and

µ2 δσ ∇2 (h2 (X − X∗ ) ) ,

that in the flat bed case reduce to µ2 δ∇· (h2 ∇ (X − X∗ ) ) . Thus, δ is introduced to improve the linear frequency dispersion and δσ to improve the linear

shoaling. We note also that the new terms have the same structure of the linear nondispersive term in W951 µ2 ∇· [d1α h2 ∇X] , but changing the constant and replacing X with X − X∗ , which is O (µ2 ) . The same arguments hold for M982 .

We will here extend the above arguments to introduce new nonlinear terms. Because we are to compare the new results with the weakly nonlinear theory over flat beds, we will only introduce one more term in each equation, and not two as for the linear case. Further, the terms to be added to M981 and M982 have the same structure of terms of O (�µ2 ) in Wei et al. (1995), i.e. �µ2 ∇· [hη∇X] ,

and

− �µ2 ∇ [η∇· (hZ)] ,

but, again, we will substitute X and Z by X − X∗ and Z − Z∗ respectively. In summary, the equations proposed here are

M981 + �µ2 δ� ∇· [hη∇ (X − X∗ ) ] = O (µ4 ) ,

M982 − �µ2 γ� ∇ [η∇· (h (Z − Z∗ ) )] = O (µ4 ) ,

(2.7a) (2.7b)

where the coefficients δ� and γ� will allow us to improve the nonlinear performance.

11


12

An improved Boussinesq-type model for wave propagation Wei et al. (1995)

Madsen & Schaffer (1998)

−0.53096

−0.54122

γ

0

δσ

0

−0.01052

γσ

0

δ�

0

γ�

0

α δ

−0.03917

0

−0.14453 −0.02153 0 0

Table 2.1: Coefficients for Wei et al. (1995) and Madsen & Schaffer (1998).

Obviously, the expressions (2.3) and (2.6) are particular cases of equations (2.7). Table 2.1 shows the coefficients corresponding to those expressions.

We are concerned in finding the free coefficients α, δ, γ, δσ , γσ , δ� and γ� to improve the performance of the equations (2.7) within the range described in Figure 2.1(d). The equations are derived using an asymptotic expansion in the dispersive parameter µ and, regardless of the choice of the above coefficients, their behaviour converges to the exact solution in shallow waters for arbitrarily high nonlinear and slope effects. We will find the parameters so as to improve, in deep water, its performance regarding linear dispersion over flat bed, linear shoaling over mild slopes and weakly nonlinear behaviour over flat beds.

Hereinafter the truncation error O (µ4 ) will not be written out, and µ is considered to be arbitrarily large.

12


2.1 New set of wave propagation equations

arbitrary slope

(a)

(b)

N93

fully dispersive

fully nonlinear

(c)

P67

13

W95

(d)

M98

Figure 2.1: Schematic picture for the range of application of diďŹ&#x20AC;erent models: (a) [Peregrine 1967], denoted by P67, is valid for weakly nonlinear and weakly dispersive waves over arbitrary slopes; (b) [Nwogu 1993], denoted by N93, improves the linear frequency dispersion performance over flat beds into deeper waters; (c) [Wei et al. 1995], W95 here, extends N93 to fully nonlinear waves for the weakly dispersive case, and [Madsen & SchaďŹ&#x20AC;er 1998], denoted as M98, increases the applicability to deeper water; (d) new model equations.

13


14

2.2

An improved Boussinesq-type model for wave propagation

Improving the linear propagation

In this Section we will focus on the improvement of the linear dispersion behaviour to very deep waters, namely k � h� � 20. The linear (and nonlinear) features of the new equations, which are to be used in this Section, are fully described in Appendix A. Hereafter, the results using Airy’s theory are denoted with “Airy”, while those from the Boussinesq-type equations (2.7) with “app” (from approximation). Linear frequency dispersion: α, δ and γ From expression (A.10) in Appendix A, the celerity corresponding to our linearised Boussinesq-type equations is given by c2app 1 − (mcη + mdu ) ξ 2 + mdu mcη ξ 4 = , gh 1 − (mdη + mcu ) ξ 2 + mcu mdη ξ 4

(2.8)

where ξ ≡ k � h� = µkh and the coefficients mdu , mdη , mcu and mcη , which

are defined in expression (A.2), depend exclusively on α, δ and γ. Comparing the above expression to the [4/4] Pad´e approximation of the Airy expression, which is [Dingemans 1997] c2Airy h

=

1 1 + 19 ξ 2 + 945 ξ4 10 1 4 + O (ξ ) , 1 + 49 ξ 2 + 63 ξ

(2.9)

one finds out the corresponding values for mdu , mdη , mcu and mcη . As already well known [Madsen & Schaffer 1998], the solutions are the four possible combinations of mdu mcu

√ 1 805 =− ± , 18 630 √ 4 10 133 =− ± , 18 630

4 mdη = −mcu − , 9 1 mcη = −mdu − . 9

i.e., the results for α, δ and γ in Table 2.2.

14

(2.10a) (2.10b)


2.2 Improving the linear propagation

α δ γ

15

I

II

III

IV

−0.54122

−0.37500

−0.02907

0.05965

−0.01052

−0.10059

−0.01052

−0.03917

−0.03917

−0.40528

−0.40528 −0.10059

Table 2.2: Coefficients corresponding to [4/4] Pad´e approximant. The four solutions correspond to a basic solution, namely solution I, which is mdu = −0.10059,

mdη = −0.03917,

mcu = −0.40528,

mcη = −0.01052,

and the combinations obtained by switching mdu ↔ mcη and mcu ↔ mdη which, according to equation (2.8), will give the same expression.

Madsen & Schaffer (1998) noted that the solutions II and III are bad conditioned to nonlinear behaviour. Further, solutions III and IV largely differ from the coefficients in Wei et al. (1995) (α = −0.531, δ = γ = 0) while solutions

II and, particularly, I, are close to them. We remark that Madsen & Schaffer (1998) considered the solution I. Here, we try to further reduce the error in linear frequency dispersion, relative to the Airy theory, by finding other values for α, δ and γ. We note that the comparisons should avoid the use of the variable ξ ≡ µkh to measure the

water depth because k itself, and consequently ξ, depends on whether we use the approximate expression (2.8) or the exact one (Airy’s expression (A.11) in Appendix A). Instead, we favour the use of a k-independent dimensionless variable, κ, defined as [Nwogu 1993] µ2 ω 2 h κ≡ g

� � c2 2 = ξ , gh

(2.11)

which, in the case of Airy’s theory, is κ = ξ tanh ξ, so that κ ≈ ξ for κ � 3, i.e., in deep waters.

15


16

An improved Boussinesq-type model for wave propagation κmax α

10

20

−0.54663

−0.54587

−0.53523

−0.00732

−0.00439

−0.00211

−0.03270

δ γ

−0.15138

δσ

−0.07523

γσ δ� γ� εs (κ�max

= 2)

smax

−0.02411 0.10169

−0.01502 0.08463

0.01919

0.01907

−0.26154

−0.34459

−0.47055

0.0080%

0.17%

1.0%

1.2%

12%

19%

15%

14%

13%

2.9

4.6

4.6

0.11840

εc ε�

5

0.13106

0.14687

Table 2.3: Sets A. Simple optimization. We now will minimize, for a given range of κ, 0 ≤ κ ≤ κmax , the error in celerity, εc , defined as

�� �� � capp � � ε ≡ max − 1�� . � 0�κ�κmax cAiry c

(2.12)

Table 2.3 provides three sets of coefficients (Sets A), α, δ and γ, by minimizing the error εc for different values of κmax . The errors εc are also included in the lower part of the Table. Analogously to the [4/4] Pad´e coefficients, for each κmax there are three other possible solutions yielding the same frequency dispersion. Following similar arguments as those by Madsen & Schaffer (1998), and also for stability considerations, we consider only the sets in Table 2.3, which are small perturbations from Pad´e’s solution I or those in Wei et al. (1995). Figure 2.2 shows capp /cAiry − 1 as a function of κ for the three κmax in set A, together with the results for Wei et al. (1995) and Madsen & Schaffer 16


2.2 Improving the linear propagation

17

(1998). The new sets allow a substantial reduction of the errors. We note that the errors for Madsen and Shaffer’s equations (2.6) at κ = 5, κ = 10 and κ = 20 are, respectively, 0.73%, 8.7% and 33% (for Wei et al.’s equations the errors are 9.2%, 36% and 81%). Only for very small values of κ, [4/4] Pad´e approximant [Madsen & Schaffer 1998] beats the three proposed sets, just the same way [2/2] Pad´e is better, for very small values of κ, than Nwogu’s choice (Figure 2.3). However, the new coefficients allow to reduce the error to just 0.008% for κ � 5. At this point, we remark that if the maximum expected value of κ in a particular problem is, e.g., κ ≈ 3, one should consider the coefficients for κmax = 5 , and not those for κmax = 20 (which should be used only when very high values of κ are expected). Linear wave shoaling: δσ and γσ To optimize the shoaling behaviour we consider the minimization of the error εs , given by

�� �� � Aapp � � η � ε ≡ max � Airy − 1� , 0�κ�κmax � A � η s

(2.13)

where Aη is the propagated wave amplitude. According to the results in Appendix A, depends on α, δ, γ, already determined by optimizing the linear frequency dispersion, but also on δσ and γσ . Table 2.3 shows the values of δσ and γσ minimizing the linear shoaling error in equation (2.13) for different values of κmax , assuming in each case the corresponding values of α, γ and δ. We remark here that the linear frequency dispersion behaviour depends only on α, δ and γ, and is not affected by δσ and γσ . Airy Figure 2.4 shows Aapp − 1 for all three sets in Table 2.3, together with η /Aη

the results from expressions (2.3) and (2.6). Specially remarkable is the fact that, for values of κ > 5, both Wei et al. (1995) and Madsen & Schaffer (1998) 17


18

An improved Boussinesq-type model for wave propagation

capp/cAiry!1

0.2

0.1

W95

M98

0

!0.1 0

5

10 !

15

20

0.01

capp/cAiry!1

zoom in

W95 M98

0

!0.01 0

1

2

!

3

4

5

Figure 2.2: Linear dispersion: capp /cAiry − 1 for sets A corresponding to κmax = 5 (�); κmax = 10 (�); κmax = 20 (�) in Table 2.3. W95 refers to Wei et al. (1995) and M98 to Madsen & Schaffer (1998) equations.

18


2.2 Improving the linear propagation

19

0.15

c/cAiry!1

0.1 N93

[2/2]

0.05

[4/4] +0.01

0 !0.01

!0.05 0

P67 5 kh

10

Figure 2.3: Frequency dispersion: relative errors for the phase speed using Peregrine (1967) (denoted as P67), [2/2] Pad´e, Nwogu (1993) (denoted as N93) and [4/4] Pad´e.

M98

0.2

"

0

"

Aapp/AAiry!1

W95

0.1

!0.1 !0.2 0

5

10 !

15

20

Airy Figure 2.4: Linear shoaling: Aapp − 1 for the coefficient sets A corresponding η /Aη

to κmax = 5 (�); κmax = 10 (�); κmax = 20 (�) in Table 2.3. W95 refers to Wei et al. (1995) and M98 to Madsen & Schaffer (1998) equations.

19


20

An improved Boussinesq-type model for wave propagation κmax α

5(∗)

10

20

−0.55590

−0.56209

−0.58520

−0.58347

−0.00260

−0.00407

−0.00542

−0.00166

−0.01071

δ γ

−0.07368

δσ

−0.04447

γσ γ� εs (κ�max

smax

= 2)

−0.03380 −0.03124

−0.03687 0.07650

−0.01945 0.02705 0.00349

−0.29739

−0.15052

−0.33143

0.29%

1.3%

2.4%

7.5%

0.29%

1.3%

2.4%

7.5%

13%

15%

17%

15%

3.5

5.0

10

25

0.14924

εc

−0.02726

0.01235

−0.51184

δ�

ε�

5

Table 2.4: Sets B: joint optimization.

0.13304

(∗)

0.13304

0.13063

correspond to joint optimization ensuring

stability for s = 5.

yield to errors in linear shoaling tend to 99% asymptotically, while, with the new equations, they are considerably reduced.

Because the errors εc and εs are unbalanced in Table 2.3, i.e., εs � εc , we secondly consider the joint search of α, δ, γ, δσ and γσ that minimize

max {εc (κmax ) , εs (κmax ) } ,

The results, Sets B, shown in Table 2.4, are again small perturbations around Wei et al. (1995). Compared to those in Table 2.3, the new set sacrifices dispersion performance in order to improve the shoaling.

20


2.3 Linear stability considerations

2.3

21

Linear stability considerations

So far, different values for coefficients α, δ, γ, δσ and δσ have been found, and those for δ� and γ� , which appear only in the nonlinear terms, are still required. Before proceeding with the nonlinear optimization, it is worth to mention that not all the combinations of α, δ, γ, δσ and δσ give useful expressions. Some of them, as we will show below for the linearised problem, can have instabilities. We remark that in the linear case, the new proposed equations (2.7) coincide with those in Madsen & Schaffer (1998). The linear 1D versions of equations (2.7) read (hu) x + ηt + µ2 [d1α h2 (hu) xx + d2α h3 uxx ] x + + µ2 [ (δ − δσ ) (h2 ( (hu) x + ηt ) x ) x + δσ (h2 ( (hu) x + ηt ) ) xx ] = 0, (2.14a) and ut + gηx + µ2 [c1α h (hut ) xx + c2α h2 uxxt ] + + µ2 [ (γ − γσ ) h2 (ut + gηx ) xx + γσ h (h (ut + gηx ) ) xx ] = 0, (2.14b) where subscript “x” indicate the spatial derivative of variables.

2.3.1

Flat bed case

In the flat bed case, the above equations reduce to ηt + hux + µ2 h2 (mdu huxxx + mdη ηxxt ) = 0,

(2.15a)

ut + gηx + µ2 h2 (mcu uxxt + gmcη ηxxx ) = 0.

(2.15b)

Now, let us consider a spatially periodic problem, where 2π/k is the principal � � length, so that η = ηn exp (inkx) and u = un exp (inkx) . The equations (2.15) read

ηn,t un,t

= A·

ηn un

A ≡ −i

, 21

0

a12

a21

0

,

(2.16)


22

An improved Boussinesq-type model for wave propagation

where, being ξn ≡ µknh, a12 ≡ knh

1 − mdu ξn2 , 1 − mdη ξn2

a21 ≡ gkn

1 − mcη ξn2 . 1 − mcu ξn2

The solutions for ηn and un can be straightforwardly obtained from equation (2.16) by diagonalizing the matrix A, being the behaviour of the solution governed by the eigenvalues of A, which are � √ (1 − mdu ξn2 ) (1 − mcη ξn2 ) νn,± = ± −a12 a21 = ±ikn gh . (1 − mcu ξn2 ) (1 − mdη ξn2 )

(2.17)

The argument in the square root is, recalling the expression (2.8), the squared celerity c2app for the wave number kn, and it must be a positive value for any n. Otherwise, there would be positive real eigenvalues, indicating the amplification of some harmonics. Therefore, we must require (1 − mdu ξn2 ) (1 − mcη ξn2 ) > 0, (1 − mcu ξn2 ) (1 − mdη ξn2 )

for any ξn .

(2.18)

It can be easily shown that max {mdu , mcu , mcη , mdη } � 0,

(2.19)

is a necessary and sufficient condition for the numerator and denominator in equation (2.18) to be positive, therefore ensuring the stability for the flat bed case. Recalling the definitions of mdu , mdη , mcu and mcη in equation (A.2), Appendix A, we remark that all the sets A obtained above (see Table 2.3) verify the stability condition for flat beds. In fact, they verify max {mdu , mcu , mcη , mdη } < 0, so that lim νn,±

n→∞

� √ mdu mcη = O (n h) . = ±ikn gh mcu mdη

For future use, we note that the characteristic value for νn,± is, according to expression (2.17) ν0� = k0�

� g � h�0 ,

22


2.3 Linear stability considerations and we define

23

� mdu mcη Ωn (h) = kn gh , mcu mdη

(2.20)

which in turn is the frequency associate to the nth component for a given depth for big values of n. In the 2D case the linearised equations (2.7) reduce, for the flat bed case, to ηt + h∇ · u + µ2 h2 {mdu h∇2 ∇ · u + mdη ∇2 ηt } = O (µ4 ) , ut + g∇η + µ2 h2 {mcu ∇∇ · ut + gmcη ∇∇2 η} = O (µ4 ) ,

(2.21a) (2.21b)

so that, proceeding as in the 1D case, now with � η= ηnm exp (inkx x) exp (imky y) , � u= unm exp (inkx x) exp (imky y) , the equations  b11 0   0 b22  0 b32

read, u and v being the x    0 ηnm,t     unm,t  = −i  b23     b33 vnm,t

and y components of the velocity,    0 a12 a13 ηnm     unm  , (2.22) · a21 0 0     a31 0 0 vnm

2 ≡ ξ2 + ξ2 , b 2 where ξn ≡ µkx nh, ξm ≡ µky mh, ξnm n m 11 ≡ 1 − mdη ξnm ,

b22 ≡ 1 − mcu ξn2 ,

2 b33 ≡ 1 − mcu ξm ,

b23 = b32 ≡ −mcu ξn ξm ,

and 2 a12 = hnkx (1 − mdu ξnm ),

2 a13 = hmky (1 − mdu ξnm ),

2 a21 ≡ gnkx (1 − mcη ξnm ),

2 a31 ≡ gmky (1 − mcη ξnm ).

The three eigenvalues of the system (2.22) are � � 2 ) (1 − m ξ 2 ) (1 − mdu ξnm cη nm νnm,± = ±i kx2 n2 + ky2 m2 gh , 2 ) (1 − m ξ 2 ) (1 − mcu ξnm dη nm and zero. Therefore, the stability condition remains the same. 23

(2.23)


24

An improved Boussinesq-type model for wave propagation

2.3.2

Uneven bed case

In general, no analytical solution can be found for arbitrary bathymetries, even in the simplest 1D linearised problem. In order to show the above anticipated stability problems, let us consider the simple sinusoidal bathymetry h = hc + h1 exp (+ikx) + h1 exp (−ikx) = hc + 2h1 cos (kx) . Assuming periodic solutions in space � η= ηn exp (inkx) and

u=

equations (2.14) read

2 �

j=−2 2 �

3 �

aη,j ηn+j,t =

(2.24)

un exp (inkx) ,

bu,j un+j ,

(2.25a)

bη,j ηn+j ,

(2.25b)

j=−3

au,j un+j,t =

j=−2

2 �

j=−2

for −∞ < n < +∞. The coefficients aη,j , bu,j , au,j and bη,j are in Appendix

B. The above equations provide the information of the interaction between harmonics due to the uneven bathymetry. Denoting here η and u as the column vectors including all Fourier components ηn and un for −∞ < n < ∞, the above equations (2.25) can be written as � � � � � � � � L11 0 ηt 0 R12 η · = · , (2.26) 0 L22 ut R21 0 u where L11 , L22 , R21 and R12 are banded matrices with the bathymetric information. For the flat bed case the matrices become diagonal, recovering the results presented above for flat bed case. The system in equation (2.26) can be rewritten as � � � � ηt η = A· . ut u 24


2.3 Linear stability considerations

25

To study the stability of the problem, the eigenvalues νj of A have to be analyzed. The above matrices L11 , L22 , R12 and R21 are infinite and will be truncated to finite squared matrices for the analysis. Let m be the maximum harmonic of the Fourier components considered in the column vectors, so that the size of these matrices is (2m + 1) × (2m + 1) . The Figure 2.5 displays the eigenvalues of A for the coefficients corresponding to κmax = 10 in Table 2.3 with k� =

2π , 250m

h�c = 500m,

h�1 = 225m,

i.e., a sinusoidal bathymetry ranging between 50 m and 950 m with a wavelength �� = 250 m. According to the expression (2.24), the maximum slope is 2k � h�1 ≈ 11.3, a huge value. Two different values for the truncation m are considered in Figure 2.5: m = 50 and m = 100. As depicted from the Figure, the higher the number of harmonics considered, the higher frequencies are achieved. This is an expectable result since higher wavenumbers have higher corresponding frequencies, as shown in equation (2.20). In fact, as shown in Figure 2.5, the maximum values of the imaginary parts of the eigenvalues of A are controlled by Ω�m (h� ) defined in equation (2.20), for the given coefficients and the maximum depth, in this case h� = 950 m. For the lower frequencies, which will be shown to be the most important to analyze stability, the eigenvalues for m = 50 and m = 100 coincide, showing that the truncation procedure does not destroy the analysis. Besides, from Figure 2.5, for these coefficients and bathymetry, the eigenvalues νj have real positive parts for low frequencies, thus indicating instability of the equations. For high frequencies (wavenumbers) the wave does not feel bathymetric changes and, provided that condition (2.19) is satisfied, the eigenvalues are all pure imaginaries. The existence of positive real parts is to be avoided, since they destroy the 25


An improved Boussinesq-type model for wave propagation

50

m = 50 m = 100

#"100(h"=950m)

Imag{!"} (Hz)

#"50(h"=950m) 0

!50 !1

2 Imag{!"} (Hz)

26

!0.5

0 Real{!"} (Hz)

m = 50 m = 100

0.5

1

zoom in

1 0 !1 !2 !1

!0.5

0 Real{!"} (Hz)

0.5

1

Figure 2.5: Eigenvalues for the truncated matrices.

26


2.3 Linear stability considerations

27

solution. Amongst the different sets of coefficients presented so far (Wei et al. (1995), Madsen & Schaffer (1998) and set A for the new equations in Table 2.3), only coefficients for Wei et al. has shown to give pure imaginary eigenvalues even for the most steep sinusoidal bathymetries. For the rest of sets, the trend is that the equations are stable for “mild” slopes and become unstable for “steep” ones. Thinking specifically in a dimensional way, given a set of coefficients, the maximum real part of the eigenvalues of A depends on g � , k � , h�c and h�1 , provided the influence of m is null for m sufficiently large. Applying dimensional analysis {ν∗ ≡}

maxj (� {νj� } ) � = f (Π, s) , k � g � h�c

Π≡

h�1 , h�c

s ≡ 2k � h�1 ,

(2.27)

where Π, which satisfies 0 � Π < 1/2 according to equation (2.24), represents the ratio of the maximum to minimum depth and s is the maximum slope of the bathymetry. The above function has been evaluated numerically, and its behaviour is shown in Figure 2.6 for the coefficients corresponding to κmax = 5 in the Table 2.3. For the given set of coefficients, ν∗ = 0 for small slopes, while ν∗ > 0 if a given value of s, herein smax , is surpassed. The value of smax will depend on Π, but the influence of Π has shown to be rather small, having most unfavourable values of smax for Π in its middle values (Figure 2.6). The trend has shown to be similar for the rest of coefficient sets, except for Wei et al.’s, for which ν∗ = 0 for any pair {Π, s} , i.e., smax → ∞. For the computation of the

maximum admissible slope, smax , we always used Π = 0.25 and m = 50, and the variable ν∗ is considered positive when ν∗ > 10−8 . The values of smax for sets A are included in Table 2.3. From Table 2.3, smax decreases for sets with smaller κmax to smax ≈ 2.9, which might be an insufficient value. Despite, for Madsen & Schaffer (1998), smax ≈ 2.5 {≤ 2.9}. 27


28

An improved Boussinesq-type model for wave propagation 6 5

!*

4 3

0.05 0.10 0.20 0.30 0.40 0.45 0.49

2 1 0 0

2

4 s

6

8

Figure 2.6: Function ν∗ = f (Π, s) . The different symbols stand for different values of Π.

In order to ensure linear stability for steeper slopes, i.e., to increase smax , we consider here two different options. We first propose to keep the coefficients α, δ and γ as in Table 2.3 and recompute δσ and γσ to optimize the linear shoaling performance but so that smax � 5, since s = 5 is a sufficiently steep slope to be considered (Sets C). The results are shown in Table 2.5. From Tables 2.3 and 2.5, it can be seen that we are now sacrificing shoaling performance to obtain stability. However, the errors in shoaling are still reduced relative to works by Wei et al. (1995) and Madsen & Schaffer (1998). Further, the new values for δσ and γσ are closer to zero, and the overall coefficients are very close to those in Wei et al. (1995), which has shown to be very stable. Following the same argument given above about the unbalanced errors in shoaling and linear dispersion we consider the joint search of α, δ, γ, δσ and 28


2.4 Improving the weakly nonlinear propagation κmax α

5

10

20

−0.54663

−0.54587

−0.53523

−0.00732

−0.00439

−0.00211

−0.03270

δ γ

−0.03334

δσ

−0.03754

γσ δ� γ� εs ε�

(κ�max

−0.02411 0.09314

−0.01502 0.07096

0.01630

0.01419

−0.26154

−0.34459

−0.47055

0.0080%

0.17%

1.0%

2.6%

12%

20%

15%

14%

13%

0.11840

εc

29

= 2)

0.13106

0.14687

Table 2.5: Sets C. Linearly stable for smax � 5. γσ ensuring stability for s = 5. The results are shown in Table 2.4 for κmax = 5, the only one which yields to values of smax < 5 with joint optimization procedure is not used.

2.4

Improving the weakly nonlinear propagation

Taking into account the weakly nonlinear properties of the equations, fully described in Appendix A, to obtain the coefficients δ� and γ� we will minimize the error

� � app �� �� app � G+ � � G− � � � � � , ε = max � − 1 , − 1 � � � � 0�κi ,κj �κmax GStokes GStokes �

(2.28)

+

where G± (κi , κj ) are the super- and sub-harmonic nonlinear energy transfer functions for two waves with κi and κj for the Boussinesq-type equations (“app”) and for the Stokes theory. The value of κ�max to compute ε� must be that value of κ where nonlinear effects 29


30

An improved Boussinesq-type model for wave propagation

are indeed negligible. This will depend, of course, on the wave amplitude. If a is the deep water wave amplitude, T its period and we consider that � = 0.02 is the threshold for the nonlinearity to be negligible, recalling the definition of κ in equation (2.11) κ�max =

(2π/T) 2 (a/0.02) 200π 2 a = , g gT2

which yields, for instance, for a = 0.5 m and T = 10 s, a value κ�max ≈ 1.0. We will here consider κ�max = 2 to obtain the coefficients, but other values of

κ�max could be considered in particular problems. Tables 2.3 to 2.5 include, for the given values of α, δ, γ, δσ and γσ , the corresponding values for δ� and γ� and the errors ε� (κ�max = 2) . We note that the corresponding errors for Wei et al. (1995) and Madsen & Schaffer(1998) equations are, respectively, 59% Stokes − 1 for the A set with κ and 53%. The relative errors Gapp max = 5 are ± /G±

plot in Figure 2.7.

No stability analysis for the nonlinear equations has been performed, but the obtained coefficients have not shown any numerical stability problems.

2.5

Concluding remarks

A new set of fully nonlinear Boussinesq-type equations for the wave propagation problem with an improved range of applicability from deep water to shallow water has been presented in this Chapter. The model equations are modifications and improvements of those by Madsen & Schaffer (1998). An optimization procedure is used to minimize the errors compared to those for Airy and Stokes waves, and a linear stability analysis of the resulting equations is presented for uneven sinusoidal bathymetries. The relative errors for each scenarios are also provided. For example, if it is desirable to model very short waves (deep water) with k � h� = 20, the model has a maximum relative error equal to 1% in linear dispersion. 30


2.5 Concluding remarks

31

2 0.1

0.02

!0.02

!0.05

!0.1

G+ !0.1

G!

!j

0.05

!0.05 !0.02

1

0.02

0.02 !0.02

0.02

0 0

1 !

0.05 0.1

2

i

Stokes Figure 2.7: Weakly nonlinear propagation: Gapp − 1 for set A and κmax = 5. ± /G±

31


32

An improved Boussinesq-type model for wave propagation

32


Chapter 3

An explicit numerical scheme for the new model In this Chapter an explicit numerical scheme is implemented to solve the model equations (2.7) presented in Chapter 2. A 4th order Runge-Kutta scheme is employed to step in time, while the spatial derivatives are differenced to an � � accuracy of O ∆x4 . An internal source function is derived for internal wave generation and a CFL-type condition is obtained for the time step. The new

numerical scheme and model equations are validated using some experimental and analytical tests. Along this Chapter all variables are dimensional and primes are neglecting for convenience. The results of this Chapter are under review in Journal of Waterway, Port, Coastal, and Ocean Engineering.

3.1

Source function for wave generation and sponge layers

Two practical problems need to be resolved before a new numerical scheme is developed. We need to be able to generate waves within the computational domain avoiding problems derived from the treatment of waves as boundary conditions. Secondly, we need to radiate the waves generated locally out of 33


34

An explicit numerical scheme for the new model

the computational domain if not reflective boundary conditions are required. Other phenomenon such as wave breaking, runup or wave dissipation due to boundary layer are not considered in the model equations. Nevertheless, they can be included following, e.g.., the same treatment described by Kennedy et al. (2000) and Liu & Orfila (2004).

3.1.1

Source function

It is well known that the generation of waves into the computational domain through the boundary conditions is difficult in wave propagation models. Following Wei et al. (1999) we consider the inclusion of a source function, s, in the continuity equation so as to generate the waves within the computational domain. The function is built to generate the desired linear waves in the far field. We consider the linearized version of the equations (2.7) over a flat bed [Wei et al. 1999] ηt + h∇·u + h2 {mdu h∇2 ∇·u + mdη ∇2 ηt } = s,

ut + g∇η + h2 {mcu ∇∇·ut + gmcη ∇∇2 η} = 0,

(3.1a) (3.1b)

where mdu ≡ d1α + d2α + δ,

mdη ≡ δ,

(3.2a)

mcu ≡ c1α + c2α + γ,

mcη ≡ γ.

(3.2b)

For flat bed there exists a potential function φ such that ∇φ = u. Therefore, taking spatial integration and temporal differentiation of the momentum equation(3.1b), the above equations can be written for the scalar variables η and φ as ηt + h∇2 φ + h2 {mdu h∇2 ∇2 φ + mdη ∇2 ηt } = s,

φtt + gηt + h2 {mcu ∇2 φtt + gmcη ∇2 ηt } = 0. 34

(3.3a) (3.3b)


3.1 Source function for wave generation and sponge layers

35

The solution of the homogeneous version of equations (3.3) is

where i ≡

η = η0 exp (i (kx x + ky y − ωt) ) ,

(3.4a)

φ = φ0 exp (i (kx x + ky y − ωt) ) ,

(3.4b)

−1 is the imaginary constant and k 2 ≡ kx2 + ky2 must satisfy the

frequency dispersion relation derived in Chapter 2 given by (1 − mdu ξ 2 ) (1 − mcη ξ 2 ) ω2 = , gk 2 h (1 − mcu ξ 2 ) (1 − mdη ξ 2 )

(3.5)

where ξ = kh. Besides, η0 and φ0 in equation (3.4) must also satisfy η0 = iΠφ0 , with

k 2 h (1 − mdu ξ 2 ) Π≡ ω (1 − mdη ξ 2 )

(3.6)

ω (1 − mcu ξ 2 ) ≡ g (1 − mcη ξ 2 )

.

(3.7)

The dispersion equation (3.5) can be written as a cubic equation for k 2 3

2

1

0

c3 (k 2 ) + c2 (k 2 ) + c1 (k 2 ) + c0 (k 2 ) = 0,

(3.8)

with c3 ≡ −gmdu mcη h5 ,

c2 ≡ ω 2 mdη mcu h4 + g (mdu + mcη ) h3 , c1 ≡ −ω 2 (mcu + mdη ) h2 − gh, c0 ≡ ω 2 .

The equations by Wei et al. (1995) have mdu = mcη = 0 so that the equation (3.8) reduces to a second order equation for k 2 , which is the case analyzed in Wei et al. (1999). Here we will focus on the extended cubic case. However, for completeness, the solution for mdu = mcη = 0 is included at the end of this Section.

35


36

An explicit numerical scheme for the new model

The equation (3.8) has, for any of the combinations of {mdu , mdη , mcu , mcη }

obtained in the previous Chapter, three real roots for k 2 : one positive (denoted here r12 , with r1 > 0) corresponding to the progressive waves, and two negatives (i.e., −r22 and −r32 , with r2 , r3 > 0) corresponding to evanescent modes. Therefore, the six solutions for the wave number k are k1 ≡ +r1 ,

k2 ≡ +ir2 ,

k3 ≡ +ir3 ,

(3.9)

and their opposite (−r1 , −ir2 and −ir3 ). Following Wei et al. (1999) we assume that, being ψ any of η, φ or s, for the inhomogeneous case we can write ψ (x, y, t) = ψˆ (x) exp (i (ky y − ωt) ) , with |ky | < k1 , so that equations (3.3) read Aφ,1 φˆ���� + Bφ,1 φˆ�� + Cφ,1 φˆ + Bη,1 ηˆ�� + Cη,1 ηˆ = sˆ,

(3.10a)

+Bφ,2 φˆ�� + Cφ,2 φˆ + Bη,2 ηˆ�� + Cη,2 ηˆ = 0,

(3.10b)

where the primes stand for x derivatives, and Aφ,1 ≡ h3 mdu , � � Bφ,1 ≡ h 1 − 2mdu ky2 h2 , � � Cφ,1 ≡ −hky2 1 − mdu ky2 h2 ,

Bφ,2 ≡ −ω 2 mcu h2 , � � Cφ,2 ≡ −ω 2 1 − mcu ky2 h2 ,

Bη,1 ≡ −iωmdη h2 , � � Cη,1 ≡ −iω 1 − mdη ky2 h2 ,

Bη,2 ≡ −iωgmcη h2 , � � Cη,2 ≡ −iωg 1 − mcη ky2 h2 .

To find a particular solution for the problem in equations (3.10) we first consider the Green functions Gη (x, �) and Gφ (x, �) , solutions for a pulse at x = �, i.e., solutions of �� �� Aφ,1 G���� φ + Bφ,1 Gφ + Cφ,1 Gφ + Bη,1 Gη + Cη,1 Gη = δ (x − �) ,

Bφ,2 G��φ + Cφ,2 Gφ + Bη,2 G��η + Cη,2 Gη = 0. 36

(3.11a) (3.11b)


3.1 Source function for wave generation and sponge layers

37

In order to automatically satisfy the above equations at every point except at x = �, the solutions Gη and Gφ of the above problem can be written Gη =

� � 3

exp (ikx,j (x − �) ) , if x > �;

(3.12a)

Gφ =

� � 3

exp (ikx,j (x − �) ) , if x > �;

(3.12b)

and

j=1 aη,j

�3

j=1 aη,j exp (ikx,j (� − x) ) , if x < �,

j=1 aφ,j

�3

j=1 aφ,j

exp (ikx,j (� − x) ) , if x < �,

2 = k 2 − k 2 , with k given in equation (3.9). where kx,j must be so that kx,j j y j

More specifically kx,1 = +

� r12 − ky2 ,

kx,2 = +i

� r22 + ky2 ,

kx,3 = +i

� r32 + ky2 ,

so that the evanescent modes cancel at x → ±∞, where, according to equations (3.12), we get waves travelling rightwards at x → ∞ and leftwards at x → −∞.

Further, in a result consistent with equation (3.6), to satisfy the homogeneous equation (3.11), the coefficients aη,j and aφ,j in equations (3.12) must satisfy aη,j = iΠj aφ,j , where Πj ≡

kj2 h (1 − mdu kj2 h2 ) ω (1 − mdη kj2 h2 )

(3.13)

.

(3.14)

The above Green functions Gη and Gφ satisfy the equation (3.11) at every point except x = �, i.e., satisfy the homogeneous equations. We still require to take into account the point x = �. Integrating equations (3.11) in x from �− to �+ � �+ � Aφ,1 G��� φ �−

+

� �+ Bφ,1 G�φ ��− �

+

� +Bφ,2 G�φ ��−

+ Cφ,1 + Cφ,2

� �

�+ �−

Gφ dx +

�+ �−

Gφ dx +

37

� �+ Bη,1 G�η ��− �

+

� Bη,2 G�η ��−

+ Cη,1 + Cη,2

� �

�+ �−

Gη dx = 1,

�+ �−

Gη dx = 0.


38

An explicit numerical scheme for the new model

Equations (3.12) satisfy

� �+ �−

Gφ dx =

� �+ � �+ G�η ��− = G�φ ��− = 0,

� �+ �−

Gη dx = 0, so that we further impose � �+ � Aφ,1 G��� φ �− = 1,

and

i.e., recalling expressions (3.12),

(3.16)

kx,1 aη,1 + kx,2 aη,2 + kx,3 aη,3 = 0, kx,1 aφ,1 + kx,2 aφ,2 + kx,3 aφ,3 = 0, 3 3 3 kx,1 aφ,1 + kx,2 aφ,2 + kx,3 aφ,3 = i/ (2Aφ,1 ) ,

which, taking into account the expression (3.13), allows to solve for aφ,1 , aφ,2 and aφ,3 . We anticipate that we will be only interested in aφ,1 , which is aφ,1 =

2 (Π kx,1 2

− Π3 ) +

Π2 − Π3 2 kx,2 (Π3 − Π1 )

+

2 (Π kx,3 1

i . − Π2 ) 2kx,1 Aφ,1

Once Gφ in equation (3.12b) has been determined, the solution of φˆ in equation (3.10) is given by the convolution integral � +∞ φˆ (x) = Gφ (x, �) sˆ (�) d�,

(3.17)

−∞

where, following Wei et al. (1999), we consider sˆ (�) = D exp (−β�2 ) ,

(3.18)

where D is still to be determined. The value of the parameter β can be chosen following the recommendations in Wei et al. (1999), i.e., β = 15 k12 for monochromatic wave generation. Substituting the equations (3.12) and (3.18) into (3.17) we get that, for x → +∞, in the far field, the terms corresponding to aφ,2 and aφ,3 , i.e., the evanescent modes, drop off and lim φˆ (x) = DIaφ,1 exp (ikx,1 x) ,

x→+∞

where I≡

� � 2 kx,1 π exp − . β 4β 38

(3.19)


3.1 Source function for wave generation and sponge layers

39

Since we want φˆ to behave as φ0 exp (ikx,1 x) at x → +∞, from the expression (3.19) we get

D=

φ0 η0 = , Iaφ,1 iΠ1 Iaφ,1

(3.20)

and the source function can be written simply as

s (x, y, t) = |D| exp (−βx2 ) sin (ky y − ωt) .

(3.21)

When mcη = mdη = 0, as for the case of Wei et al. (1995), the above expressions (3.20) and (3.21) hold, but now aφ,1 = with kx,1

i 2 − k2 ) , 2kx,1 Aφ,1 (kx,1 x,2

� = + r12 − ky2 ,

kx,2

Π1 ≡

k12 h (1 − mdu k12 h2 ) , ω

� = +i r22 + ky2 ,

k1 = +r1 .

Here, similar to the general case, r1 , r2 > 0 are so that r12 and −r22 are, respectively, the positive and negative roots of the parabola for k 2 2

1

0

gmdu h3 (k 2 ) − (gh + mcu h2 ω 2 ) (k 2 ) + ω 2 (k 2 ) = 0, which is the expression (3.42) in this case.

3.1.2

Sponge layers

Following Wei & Kirby (1995), an artificial damping term d will be added to the momentum equation in a so-called sponge layer so as to allow the radiating waves propagating outside the domain and to avoid possible reflection effects within the domain due to the boundaries. This additional term is given by d = −w1 u + w2 ∇∇·u + w3

g/h η,

(3.22)

i.e., the addition of three different damping forms: Newtonian cooling, viscous damping and sponge filter respectively [Israeli & Orszag 1981].

39


40

An explicit numerical scheme for the new model

In the equation (3.22), wi are space dependent functions that can be written in terms of incident wave frequency, ω, and a smooth monotonically increasing function, f (x, y) which varies from 0 to 1 within the sponge layer [Wei & Kirby 1995]. For instance, in a problem where the wave propagates in the x-direction, for a sponge layer from xa to xb we consider wi = ci ωf (x) ,

f (x) =

exp ( (x − xa ) / (xb − xa ) ) 2 − 1 , exp (1) − 1

(3.23)

so that f (xa ) = 0 and f (xb ) = 1. The empirical coefficients ci govern the magnitude of all three different damping mechanisms. We have obtained satisfactory results using c1 = 7.5, c2 = c3 = 0, and a sponge layer length, xb − xa , equal to two characteristic wave lengths.

3.2

Numerical model

Boussinesq-type models often employ implicit predictor-corrector iterative numerical schemes in the time marching process [Wei & Kirby 1995], where a 3rd -order Adams-Bashford method is used in the prediction and a 4th -order Adams-Moulton method in the corrections until convergence is achieved. Although the implicitness, numerical instabilities have been reported and the use of filters is frequent. In this Section, an explicit 4th −order Runge-Kutta

scheme to solve the fully nonlinear equations for wave propagation presented in previous Chapter is developed. Taking into consideration the source function, s, for wave generation and the damping function, d, for the sponge layers, and being x = {x, y} , the equations (2.7) can be written as

Υ0 (ηt ) + Υ� (η, ηt ) = E0 (u) + E� (η, u) + s (t, x)

(3.24a)

U0 (ut ) + U� (η, ut ) = F0 (η) + F� (η, u) + d (η, u)

(3.24b)

40


3.2 Numerical model

41

where the linear terms are Υ0 ≡ η + (δ − δσ ) ∇· (h2 ∇η) + δσ ∇2 (h2 η) ,

(3.25a)

U0 ≡ u + (c1α + γσ ) h∇X + (c2α + γ − γσ ) h2 ∇Y,

(3.25b)

F0 ≡ −g∇η − g { (γ − γσ ) h2 ∇∇2 η + γσ h∇∇· (h∇η) } ,

(3.25d)

E0 ≡ −X − ∇· { (d1α + δ − δσ ) h2 ∇X + d2α h3 ∇Y} − δσ ∇2 (h2 X) , (3.25c) s ≡ |D| exp (−βx2 ) sin (ky y − ωt) , � d ≡ −w1 u + w2 ∇Y + w3 g/h η,

(3.25e)

Υ� ≡ δ� ∇· {hη∇ηt } ,

(3.26a)

(3.25f)

and the nonlinear terms read

U� ≡ −∇ { (1 + γ� ) η∇· (hut ) + η 2 ∇·ut /2} ,

(3.26b)

E� ≡ −M − ∇· { [ (c1α + δ� ) hη − η 2 /2] ∇X + (c2α h2 η − η 3 /6) ∇Y} − − { (δ − δσ ) ∇· (h2 ∇M) + δσ ∇2 (h2 M) + δ� ∇· (hη∇M) } ,

(3.26c)

F� ≡ −N + γ� ∇ {η∇· [h (N + g∇η) ] } −

− ∇ { (c1α h − η) u·∇X + (c2α h2 − η 2 /2) u·∇Y + (X + ηY) 2 /2} − − { (γ − γσ ) h2 ∇∇·N + γσ h∇∇· (hN) } ,

(3.26d)

with 1 N ≡ ∇ (u·u) . 2

M ≡ ∇· (ηu) ,

Assuming the solution to be smooth, we solve the equations (3.24) using finite differences schemes in space.

3.2.1

Spatial discretization: matrix notation

Following the so-called method of the lines, we first semidiscretize the equations in space considering a uniform unstaggered grid with n = nx ·ny nodes and using finite differences.

41


42

An explicit numerical scheme for the new model

We consider η, ux , uy and h being the column vectors (i.e., n × 1) containing the n nodal values of η, the x and y components of horizontal velocity and h respectively. We define here the column vector u as u≡

ux uy

.

Let further Dx and Dy be the n × n matrices for the finite differences approximations of the first derivatives in x and y, i.e., so that, for instance, Dx ·η

yields the vector containing the nodal values of the approximations of ∂η/∂x. The construction of these matrices comes easily from using either � � � ∂ψ �� ψn+1 − ψn−1 = + O ∆x2 , � ∂x n 2∆x or

� � � ∂ψ �� −ψn+2 + 8ψn+1 − 8ψn−1 + ψn−2 = + O ∆x4 , � ∂x n 12∆x

(3.27a)

(3.27b)

i.e., second or fourth order finite difference approximations.

The divergence, gradient and Laplacian matrices are defined, respectively, as dv ≡ (Dx Dy ) ,

gr ≡

Dx Dy

,

lp ≡ dv·gr {≡ D2x + D2y } . (3.28)

We will further consider Dxx ≡ D2x ,

Dxy ≡ Dx ·Dy {≡ Dy ·Dx } ,

Dyy ≡ D2y ,

(3.29)

as the matrices for the second derivatives, having the same order of accuracy than those for first order derivatives. We denote (·)∗ as a diagonal matrix constructed with the elements of the vector (·) in its diagonal. For later use, we define H ≡ h∗ , Θ ≡ η ∗ and U ≡ (u∗x u∗y ) .

42


3.2 Numerical model

43

Finally, we introduce the (·) operator operating on a squared matrix as

(·) =

(·)

0

0

(·)

,

(3.30)

where 0 denotes the null matrix with the same size of (·). Using the above notation, equations (3.24) can be written as �

(L0 + L� (η, u)) ·

ηt ut

= R0 ·

η u

+ r� (η, u) + χ (t, η, u) .

(3.31)

In the above equation matrices L0 and R0 , which contain linear terms, are from equation (3.25),

L0 ≡

L0,11

0n×2n

02n×n

L0,22

R0 ≡

,

0n×n

R0,12

R0,21 02n×2n

,

(3.32)

where 2

L0,11 = In + (δ − δσ ) dv·H ·gr + δσ lp·H2 ,

2

L0,22 = I2n + (c1α + γσ ) H·gr·dv·H + (c2α + γ − γσ ) H ·gr·dv, 2

3

R0,12 = −dv·H − dv· {(d1α + δ − δσ ) H ·gr·dv·H + d2α H ·gr·dv} − − δσ lp·H2 ·dv·H, 2

R0,21 = −ggr − g { (γ − γσ ) H ·gr·lp + γσ H·gr·dv·H·gr} . The nonlinear matrix L� and nonlinear vector r� in equation (3.31) are

L� ≡

L�,11

0n×2n

02n×n

L�,22

43

,

r� ≡

r�,1 r�,2

,

(3.33)


44

An explicit numerical scheme for the new model

where L�,11 = δ� dv·H·Θ·gr, L�,22 = −gr· { (1 + γ� ) Θ·dv·H + Θ2 ·dv/2} , 2

r�,1 = −m − dv· { [ (c1α + δ� ) H·Θ − Θ /2] gr·x} − 2

3

− dv· { (c2α H ·Θ − Θ /6) gr·y} − 2

− { (δ − δσ ) dv·H ·gr + δσ lp·H2 + δ� dv·H·Θ·gr} m, r�,2 = −N + γ� gr·Θ·dv·H· (N + ggr·η) −

− gr { (c1α H − Θ) U·gr·x + (c2α H2 − Θ2 /2) U·gr·y} − ◦

− gr· ( (x + Θ·y) · (x + Θ·y) T ) /2− 2

− { (γ − γσ ) H ·gr·dv + γσ H·gr·dv·H} n, and m ≡ dv·Θ·u,

x ≡ dv·H·u,

y ≡ dv·u,

� �◦ n = gr· U·UT /2,

with the operator (·) ◦ giving a column vector with the diagonal elements of matrix (·). Finally the vector χ, standing for source function, s, and damping terms, d, is χ=

χs 0n×1

+

0n×n 0n×2n χd,21

χd,22

� � ·

η u

,

(3.34)

where the matrices χs , χd,21 and χd,22 are easily obtained taking into consideration equations (3.21) and (3.22).

3.2.2

Time integration: fourth-order Runge-Kutta scheme

The expression (3.31) represents a nonlinear system of Ordinary Differential Equations (ODE’s) for the nodal values of our unknowns. By defining � � η f≡ , (3.35) u 44


3.2 Numerical model

45

equation (3.31) can be rewritten as ft = (L0 + L� (f ) ) −1 ·r (t, f ) ,

(3.36)

with r (t, f ) ≡ R0 ·f + r� (f ) + χ (t, f ) . Due to the reasons to be argued in Section 3.3, fourth order Runge-Kutta (hereinafter RK4) or third order Adams–Bashforth (AB3) methods are recommended. Note that both schemes are explicit in time. Here we focus on RK4 method. According to the usual (i.e., Kutta) version of this method with f n being the solution at time tn = t0 + n∆t, we can write f n+1 = f n + ∆t

k1 + 2k2 + 2k3 + k4 , 6

(3.37)

where k1 = (L0 + L� (f n ) ) −1 ·r (tn , f n ) ,

k2 = (L0 + L� (f n + ∆tk1 /2) ) −1 ·r (tn + ∆t/2, f n + ∆tk1 /2) ,

k3 = (L0 + L� (f n + ∆tk2 /2) ) −1 ·r (tn + ∆t/2, f n + ∆tk2 /2) , k4 = (L0 + L� (f n + ∆tk3 ) ) −1 ·r (tn + ∆t, f n + ∆tk3 ) ,

so that, in each time step we are required to solve four linear systems or the form (L0 + L� ) ·k = r,

(3.39)

with the matrices being block-diagonal and sparse. To solve the above systems, we consider two options: Gaussian elimination for sparse systems, which requires O (n3/2 ) operations, or the Jacobi iterative

method, with numerical complexity O (nit n) , where nit represents the number of iterations for an acceptable convergence, generally less than 50 for all tests 45


46

An explicit numerical scheme for the new model

carried out. Above, n is the size of the matrix system (twice the number of nodes in the 1D case and three times in the 2D case). For n � 2500 Jacobi method is preferred.

3.3

Linear stability

Before we present, in the following Section, some numerical examples, we introduce here a linear stability analysis of the numerical scheme. Ignoring the source terms, equation (3.36) for the semidiscretized equations reads, in the linear case, ft = A·f ,

(3.40)

with A ≡ L−1 0 ·R0 , where L0 and R0 are defined in equation (3.32). Most

importantly, we first note that, using sinusoidal bathymetries and the coefficients and slope limitations defined in Tables from 2.3 to 2.5, the eigenvalues of the matrix A have shown to be pure imaginary values, so that the system is intrinsically stable. If νj are the eigenvalues of the matrix A and ∆t is the numerical time step, Figure 3.1 shows the regions where the values νj ∆t must fall so that three different ODE solvers will be stable. The regions for the third order Adams– Bashforth (AB3, which is explicit), fourth order Adams–Moulton (AM4, implicit) and fourth order Runge-Kutta (RK4, explicit) are shown, together with a qualitative illustration of the values νj ∆t. We note that stability region for AM4 does not include the imaginary axis, and it is therefore discarded. On the contrary, AB3 and RK4 do include parts of the imaginary axis. As depicted from the Figure, defining νmax ≡ max |νj |, the AB3 method gives linearly stable schemes as long as

νmax ∆t � 0.7236, 46


3.3 Linear stability

47

3

RK4

2 Im(! " t)

AM4

1 AB3

0 !1 !2 !3 !4

!3

!2

!1 Re(! " t)

0

1

2

Figure 3.1: Stability regions for third order Adams–Bashforth (AB3), fourth order Adams-Moulton (AM4) and fourth order Runge-Kutta (RK4) time integration schemes.

and the RK4 method if νmax ∆t � 2.8278.

(3.41)

Because RK4 gives a wider stability region than AB3 does, and also a higher accuracy, the RK4 is the time integration scheme of choice hereinafter. In the one dimensional flat bed case, the value of νmax depends on the chosen coefficients α, δ and γ, on g, h and ∆x, on the order of accuracy used to compute the spatial derivatives (“o”), and, also, on the number of nodes n. Applying dimensional analysis � � νmax ∆x ∆x √ = fx α, δ, γ, Πx ≡ , o, n . h gh

(3.42)

In Figure 3.2 we present the behaviour of function fx using two different sets of coefficients presented in Table 2.3: κmax = 5 (big symbols) and κmax = 10 (small symbols). In all cases n = 50, since the influence of n has shown to be negligible for high values (n � 20). Also, in Figure 3.2 we consider second 47


48

An explicit numerical scheme for the new model

1.5 th

4

1 fx

2nd

0.5

0

!2

0

10

!x

10

10

2

Figure 3.2: General behaviour of function fx using coefficient from Table 2.3 for κmax = 5 (big symbols) and κmax = 10 (small symbols). Spatial derivatives of second order (triangles) and of fourth order (circles).

and fourth order accuracies in space derivatives, i.e., o = 2 and o = 4. From the Figure, for high values of the group Πx , i.e., in “shallow waters” (recall that ∆x ∝ k −1 so that Πx ∝ (kh) −1 ), fx not only becomes independent on the coefficients, which was expected since the coefficients affect the dispersive

performance, but it is a function of the order of accuracy only. The limits are  1.3722, for 4th order in space, lim fx = (3.43) 1.0000, for 2nd order in space, Πx →∞ further indicating that

gh seems to be the proper velocity to construct the

dimensionless group in the LHS of equation (3.42). On the other hand, as depicted in Figure 3.2, after a transition zone (10−1 � Πx � 100 ), whenever Πx � 1, in deep water, the function fx decreases and 48


3.3 Linear stability

49

again becomes Πx independent. Now, however, the limits, which have shown to be always smaller than those in expression (3.43) for Πx � 1, will depend

on the coefficients used in the equations (the higher κmax , the lower the limits of fx in deep waters). This comes as no surprise, for these coefficients affect the deep water performance of the equations. According to expressions (3.41) and (3.42), using a RK4 time integration, ∆t must be chosen so that

gh ∆t 2.8278 � , ∆x fx

and, recalling the shallow water upper bounds in equation (3.43)  √ th gh ∆t 2.061, for 4 order in space, � 2.828, for 2nd order in space. ∆x

(3.44)

(3.45)

which is consistent with the results by Baldauf (2008) for non dispersive equations. The linear stability analysis for the 2D case can be performed similarly to the 1D. Encouraged by the 1D results over uneven beds, we consider here the flat bed case only. The maximum absolute value of the eigenvalues (which are always pure imaginaries), satisfies, for n sufficiently high, � � νmax ∆s ∆s ∆x √ = f s Πs ≡ , Πxy ≡ ,o , h ∆y gh

(3.46)

where ∆s2 ≡ ∆x2 + ∆y 2 . Further, it has been empirically checked that the function fs is related to fx in (3.42) through the simple expression � � Πs −1 fs (Πs , Πxy , o) = (Πxy + Πxy ) fx ,o , Πxy + Π−1 xy so that, recalling expression (3.43), if, e.g., ∆x = ∆y, we get  2.7444, for 4th order in space, lim fs = 2.0000, for 2nd order in space, Πs →∞ 49

(3.47)

(3.48)


50

An explicit numerical scheme for the new model

which yields the corresponding CFL-type condition for RK4 √

 1.030,

gh ∆t 2.8278 � = 1.414, ∆s fs

for 4th order in space, for 2nd order in space.

(3.49)

which is valid for the case ∆x = ∆y. If ∆x �= ∆y, different stability conditions can be readily obtained from the above results.

3.4

Numerical results

In this Section some numerical results are introduced to show the capabilities of i) the equations (2.7) with the coefficients in Table 2.3 and ii) the numerical scheme presented above1 . Linear cases over flat beds are considered first, so as to compare the numerical results with the analytical solutions, which are readily available for simple contours. Secondly, 1- and 2-dimensional nonlinear numerical results are compared to experimental data available in the literature. In all cases we consider 4th -order accuracy for the spatial derivatives.

3.4.1

Linear tests over flat bed

In order to perform preliminary verifications, we consider the comparison of the numerical results with the exact solutions, either for the model equations or for the Airy theory, obtained for linear cases. 1D case: order of convergence A first 1D example is meant to check the order of convergence of the proposed numerical scheme. The convergence rate is expected to be four since the spatial derivatives considered are fourth order accurate and we use a 4th order Runge-Kutta scheme for time integration. We consider the comparison of the numerical results to the exact solution of the linearized model equations 1

source code can be downloaded at http://erkwave.pbworks.com.

50


3.4 Numerical results

51

obtained for a simple case where the domain is defined by −5m ≤ x ≤ 5m. The initial velocity field is null and the initial free surface is � � x2 η (x, t = 0) = η0 exp − 2 , 1m so that η � 0 at the boundaries initially. The exact solution of the linearized Boussinesq-type equations, without sources, can be easily obtained using Fourier analysis. A 1-meter deep (h = 1 m) basin with η0 = 0.05 m is considered. Both for the numerical and the analytical solution, the coefficients in Table 2.3 for κmax = 5 are considered. Five numerical tests with decreasing grid size ∆x were run (Table 3.1). Recalling the 1D stability condition, the time step must satisfy ∆t �

2.061∆x √ ≈ 0.658 m−1 s ∆x, gh

so that in each case we simply set ∆t = 0.5 m−1 s ∆x. Denoting ηn and ηa , respectively, as the numerical and analytical solutions at x = 0 and t = 10 s, the error

is also shown in Table 3.1.

� � � ηn − ηa � � �, ε≡� ηa �

Figure 3.3 shows, in a log-log plot, the error ε against the grid size ∆x, showing that the slope is nearly 4, as expected. 1D case: wave generation and dispersion performance In this second 1D example, the purpose is to show that i) the source function allows to reproduce the desired wave train and, ii) the good dispersive properties of the equations. Monochromatic wave trains propagating over a flat bottom have been simulated and compared with Airy’s solution.

51


52

An explicit numerical scheme for the new model

# of cells

∆x (m)

ε

50

0.2000

3.22·10−2

100

0.1000

2.00·10−3

200

0.0500

1.25·10−4

400

0.0250

7.78·10−6

800

0.0125

4.86·10−7

Table 3.1: Analysis of the order of convergence for a 1D linear case.

10 10

!2

!4

"

10

0

10 10 10

!6

!8

!10

10

!3

!2

10

!1

10 ! x (m)

0

10

Figure 3.3: Analysis of the order of convergence for a 1D linear case.

52


3.4 Numerical results

53

Taking T = 4.5s and g = 9.81m/s2 , three tests with different depths h (m) = {25, 50, 100}, i.e., κ = {4.97, 9.94, 19.87} are considered. They all correspond to deep water conditions (κ � 3). In fact, the exact (or Airy’s) wavelength L

is 31.61m and the celerity is c = 7.026m/s in all cases. We want to generate wave trains with amplitude η0 = 0.05m and we always consider ∆x = 1.25m. In the first case κ � 5 and we consider the coefficients corresponding to κmax = 5 in Table 2.3. The celerity for the Boussinesq model is in this case 7.026 m/s, i.e., indistinguishable from Airy’s result to this accuracy (error in celerity is in this case bounded to 0.008%). Figure 3.4 shows three snapshots of the numerical and Airy solutions. The model represents well the dispersion (Figure 3.4, bottom), and the source function is yielding the desired wave amplitude. Figure 3.5 (top panel) shows, for the same test, the numerical results obtained using the coefficients by Wei et al. (1995), denoted hereinafter as W95, and for coefficients by Madsen & Schaffer (1998), denoted as M98. The results are consistent with the fact that the linear dispersion errors for W95 and M98 are, respectively, 9.2% and 0.73% at κ = 5.

Figure 3.5 does also show the numerical and Airy’s results for the test with h = 50m (κ � 10) and h = 100m (κ � 20) at t = 10T = 45s. This results have been obtained using the coefficients in Table 2.3 corresponding to κmax = 10 (for the case h = 50m) and κmax = 20 (for h = 100m). For the test with h = 50m the celerity of the new coefficients corresponding to κmax = 10 is 7.035m/s (0.13% error relative to Airy’s) and the comparison is again very good, as shown in the middle panel of Figure 3.5. The errors using W95 and M98 are now, respectively, 36% and 8.7%. Finally, for the test with h = 100m (bottom panel in Figure 3.5) the celerity obtained with κmax = 20 coefficients is 7.085m/s (0.84% error) and a small lag is noticed relative to Airy’s solution. For this very deep case the coefficients by W95 and M98 have 53


54

An explicit numerical scheme for the new model

!/!0

1 0 !1

t=T

!/!0

1 0 !1

t = 3T

!/!0

Airy

1 0 !1 0

t = 10T

1

New model equations

2

x/L

3

4

5

Figure 3.4: Monochromatic linear 1D propagation for h = 25 m and T = 4.5 s. Numerical results (circles) and Airyâ&#x20AC;&#x2122;s analytical solution (line).

54


3.4 Numerical results

55

!/!0

1 0 !1

!/!0

1 0 !1

!/!0

Airy

1 0 !1 0

New model equations

M98

W95

h = 25m

h = 50m

h = 100m

1

2 x/L

3

4

Figure 3.5: Monochromatic linear 1D propagation for T = 4.5 s. Numerical results for the new equations (circles), Madsen & SchaďŹ&#x20AC;er (1998) (squares), Wei et al. (1995) (crosses) and Airyâ&#x20AC;&#x2122;s analytical solution (line) at time t = 10T.

55


56

An explicit numerical scheme for the new model

errors of 81% and 33% respectively, and are clearly unable to represent the linear dispersion.

Besides, three new experiments have been carried out in the DeFrees Hydraulics Laboratory at Cornell University which are also considered for the 1D linear case. The experimental setup, which is shown in Figure 3.6, consists of a 9-meter long basin filled to a constant depth of 0.50 m, and with a dissipative 1:10 (H:V) beach at its end to avoid the reflection. Three different wave periods were generated, T = {0.45 s, 0.55 s, 0.65 s} , so that κ = {9.93, 6.65, 4.76}

respectively. Contrary to the above examples, in this case κ is modified by changing the period instead of the water depth. Wave amplitude in all the cases was η0 = 0.002m, so that nonlinear effects were negligible. The free surface elevation η was measured at two gages (Figure 3.6). The first one (#A) was located at x = 3m, to measure the incident generated wave, and the second (#B), was located 4.9 m from the first gage, to measure the propagated wave.

For the numerical computations, the coefficients in 2.3 corresponding to κmax = 10 were used in the first two experiments, while those for κmax = 5 were considered for the third experiment. Further

∆x = 0.02 m

and

� � 2.061∆x ∆t = 0.01 s < √ , gh

so that the stability condition is satisfied. The comparisons between experimental data and numerical results are shown in Figure 3.6. The agreement in the phase is very good in all three experiments. The small modulation in the experimental results, in the order of 0.1 mm, can be explained as a seiche of the flume. Similar to what it is seen in Figure 3.5, the results of new equations substantially improve those by W95 and M98. 56


3.4 Numerical results

0

x (m) 7.9 9 #B

3 #A

0

z (m)

57

1:10 !0.5 #A

#B

!/!0

T = 0.45s

!/!0

Experimental data

2 1 0 !1 !2

2 1 0 !1 !2

2 1 0 !1 !2

T = 0.55s

!/!0

New model equations

2 1 0 !1 !2

2 1 0 !1 !2 0

2 1 0 !1 !2 0

T = 0.65s

1

2 t/T

3

4

1

2 t/T

3

4

Figure 3.6: Experimental set-up (top panel) and free surface time histories at #A (left) and #B (right) for periods T = 0.45s (top), T = 0.55s (middle) and T = 0.55s (bottom). Experimental data (stars) and numerical results (line).

57


58

An explicit numerical scheme for the new model

2D case: 2D dispersion performance The last linear example is aimed to check the 2D performance of the equations and numerical scheme. We consider a 1-m deep squared basin defined by −5m ≤ x ≤ 5 m and −5m ≤ y ≤ 5 m. The initial velocity field is null and the initial free surface elevation is given by

x2 + y 2 η (x, y, t = 0) = η0 exp − 1m2

with η0 = 0.05m. Figure 3.7 shows the time history of the free surface elevation η at corner and at the center of the basin, for the numerical scheme solving the linearized Boussinesq equations (using here the coefficients for κmax = 5 and ∆x = ∆y = 1m/15 and ∆t = 0.02s) and for the exact solution of the Airy equations, which can be easily found for this simple case using Fourier analysis. The comparison is very good and allows to check, for the 2-dimensional multiharmonic case which includes a whole range of κ� s, the good dispersive performance of the model equations as well as the proposed numerical scheme. Indistinguishable results are obtained using the coefficients for κmax = 10.

3.4.2

Nonlinear tests

Two well-known experiments related to nonlinear waves are considered herein to test the performance of the model equations and the numerical scheme. The first is the 1-dimensional Dingemans’ bar [Dingemans 1994], and the second is the 2-dimensional Vincent and Briggs’ shoal [Vincent & Briggs 1989]. 1D: the Dingemans’ bar The bathymetry of Dingemans’ experiments is a bar (see, e.g., Figures 3.8 and 3.9) situated in a flume where the reference depth is 0.86 m. The most significant aspects of the geometry are i) a minimum depth of 0.20 m at the 58


!/!0

3.4 Numerical results

1.0 0.5

59

Airy

New model equations

corner

0.0 !0.5

!/!0

!1.0 1.0 0.5

center

0.0 !0.5 !1.0 0

5

t(s)

10

15

Figure 3.7: Free surface histories at the corner (top) and center (bottom) in a squared basin. Analytical solution for Airyâ&#x20AC;&#x2122;s theory (line) and numerical solution for the new Boussinesq-type equations (circles).

59


60

An explicit numerical scheme for the new model Case

η0 (m)

A

0.020

C

0.041

T (s) √ 2.02 2 √ 1.01 2

κ

∆x (m)

∆t (s)

0.43

0.05

0.025

1.70

0.05

0.020

Table 3.2: Dingemans’ experiments setup. top of the bar where, depending on the input wave, nonlinear effects are expected to be important, and ii) slopes up to 1:10, so that mild slope conditions are violated. Therefore, if short waves are used as inputs, all three relevant aspects in wave propagation models (dispersion, nonlinearity and bed slope) are to be properly accounted for in the model. This fact makes this series of experiments a useful benchmark problem. Two out of the three experiments reported by Dingemans are analyzed here. They are cases A and C; case B represents a wave breaking case and is, therefore, avoided. Table 3.2 summarizes the experimental and numerical conditions for cases A and C. In Table 3.2, η0 is the incident wave amplitude and κ is computed using the initial depth h = 0.86 m. Nonlinear wave decomposition is expected over the bar, and the values of κ associated to the decomposed waves will be higher than the original. However, the coefficients corresponding to κmax = 5 in Table 2.3 have been considered in both cases. The results using the coefficients for κmax = 10 would be indistinguishable in a figure. Figures 3.8 and 3.9 show the time history comparison between numerical results and experimental data at different reported gages. Section #1 has been used as control section, allowing to synchronize model and experimental time. From Figures 3.8 and 3.9, the comparison between numerical and experimental results is fair, and particularly good for case A (where dispersive effects are smaller than in case C).

60


z (m)

3.4 Numerical results

0 0.5 0 !0.5

5

61

10 #1

15

x (m) 25 30 35 40 #3 #4 #5 #6 #7 #8 1:10

20 #2

1:20

!/!0

#1

#2

!/!0

2.0 0.0 !2.0

#3

#4

2.0 0.0 !2.0

#5

#6

!/!0

2.0 0.0 !2.0

2.0 0.0 !2.0 #7 0 1

50

4

5

Experimental data

!/!0

New model equations

45

#8 2

t/T

3

4

5

0

1

2

t/T

3

Figure 3.8: Dingemansâ&#x20AC;&#x2122; experiments. Case A. Numerical results (lines) and experimental data (stars) for free surface elevation.

61


An explicit numerical scheme for the new model

z (m)

62

0 0.5 0 !0.5

5

10 #1

15

x (m) 25 30 35 40 #3 #4 #5 #6 #7 #8 1:10

20 #2

1:20

!/!0 !/!0

2.0 0.0 !2.0

!/!0

2.0 0.0 !2.0

!/!0

New model equations

2.0 0.0 !2.0

50

4

5

Experimental data

#1

#2

#3

#4

#5

#6

2.0 #7 0.0 !2.0 0 1

45

#8 2

t/T

3

4

5

0

1

2

t/T

3

Figure 3.9: Dingemansâ&#x20AC;&#x2122; experiments. Case C. Numerical results (lines) and experimental data (stars) for free surface elevation.

62


3.4 Numerical results

63

Experimental data

#A7

2.0 !/!0

!/!0

2.0

New model equations

0.0 !2.0 0

t/T

1

M98

W95

#A8

0.0 !2.0 0

t/T

1

Figure 3.10: Dingemans’ experiments. Case A. Comparison at sections #7 and #8 using proposed coefficients (circles), Madsen & Schaffer (1998) (squares) and Wei et al. (1995) (crosses) with experimental data (line).

Figure 3.10 shows comparisons between experimental data and numerical results at gages #7 and #8 for case A using the proposed coefficients and also the coefficients corresponding to M98 and W95. The new model equations produce better agreement with experimental data than previous models.

2D: Vincent and Briggs’ shoal The last test case corresponds to the Vincent and Briggs’ “M2” experiment [Vincent & Briggs 1989], where nonlinear wave propagation, diffraction and refraction effects over a submerged shoal are analyzed. The experiment was carried out in a 35 m-wide, 29 m-long wave basin. The shoal, 30.48 cm high, is located over a flat bottom with a depth of 45.72cm (see the plan view in Figure 3.11, top panel). The location of the gages where the significant wave height Hs was measured are also shown in Figure 3.11 (top panel). We shall only focus in test case “M2”, where a monochromatic wave train 63


64

An explicit numerical scheme for the new model

with a period T = 1.3 s and amplitude η0 = 4.80 cm is propagated along the x-direction. Since κ ≈ 1.0 for the incident wave, the coefficients employed in the numerical results are, again those for κmax = 5 in Table 2.3. Further, in the numerical simulations ∆x = ∆y = 0.10 m and ∆t = 0.025 s (satisfying the CFL-type condition). Figure 3.11 shows the numerical results for Hs at the different transects. In general, good agreement between experimental and numerical results are obtained. For illustrative purposes, Figure 3.12 shows a snap shot at t = 20 s, where the top and minimum value of vertical displacement are located just before the shoal and diffraction and refraction effects due to bathymetry can be observed.

3.5

Concluding remarks

A fourth order explicit scheme has been presented to solve the modified set of fully non-linear Boussinesq-type equations with improved dispersion performance. Using the so-called method of the lines, the equations are first semidiscretized in space and then integrated in time using a 4th -order RungeKutta scheme. A CFL-type condition is given and a specific source function for wave generation derived. The model has been tested to show the linear dispersion performance against the Airy theory and experimental data as well as the nonlinear behaviour tested with the experiments of Vincent and Briggs and Dingemans bar.

64


3.5 Concluding remarks

0

65

5

x(m) 15 20

10

25

30

y(m)

20 15 10

Hs/!0

Hs/!0

Hs/!0

5 3.0 2.0 1.0 0.0

13.72 y(m)

15.25

y = 10.67 m

y = 13.72 m

3.0 2.0 1.0 0.0

x = 24.25 m

12.20

Experimental data

3.0 2.0 1.0 0.0

x = 21.20 m

3.0 2.0 1.0 0.0 3.0 2.0 1.0 0.0 10.67

New model equations

x = 18.15 m

16.77

3.0 2.0 1.0 0.0 18.15

y = 16.77 m

19.68

21.20 x(m)

22.73

24.25

Figure 3.11: Bathymetry and location of gages in Vincent and Briggsâ&#x20AC;&#x2122; experiment (top) and comparison between experimental (stars) and numerical (line) significant wave heights.

65


66

An explicit numerical scheme for the new model

2.5 25

2 1.5

y(m)

20

1 0.5

15

0 !0.5

10

!1 !1.5

5

!2 0 10

15

x(m)

20

25

!2.5

Figure 3.12: Snapshot for free surface elevation at time t = 20s in Vincent and Briggs’ experiment. Values of η/η0 .

66


Chapter 4

Turbulent bed shear stress In this Chapter, we introduce a new instrument to measure directly the wall shear stress under multiharmonic periodic waves avoiding some of the diďŹ&#x192;culties for shear plates. The instrument is conformed by two concentric cylinders, the gap between them filled with water (see Figure 4.1). The inner cylinder is fixed and the outer one, where the boundary layer is to be developed, rotates with a given velocity time history. A torque transducer attached to the rotation axle allows to measure the stress induced by the water boundary layer at the outer cylinder. Due to the sensitivity of the torque meter, only turbulent boundary layers are analyzed. The experimental results for monochromatic waves are compared to those in the literature for checking purposes. Experimental results for both monochromatic and sawtooth waves are compared with a simple turbulent boundary layer model by Simarro et al. (2008).

The results of this Chapter are published in Journal of Hydraulic Engineering [Galan et al. 2011a]

4.1

Theory

In this Section we present some theoretical considerations about the structure of the flow between the cylinders for periodic movements of the outer cylinder. 67


68

Turbulent bed shear stress

As it will be justified below, we consider that the flow is essentially 2D (TaylorCouette flow).

Figure 4.1: General overview of the new instrument designed to measure directly bed shear stress.

4.1.1

Laminar flow

Since the laminar case has an analytical solution and gives a good insight of the flow within the cylinders, we will consider it firstly. In polar coordinates, {r, θ} , the momentum equation in the θ direction is, being ∂/∂θ = ur = 0 68


4.1 Theory

69

since the flow is axisymmetric, ∂u ∂τ ρν ρ = + ∂t ∂r r

∂u u − ∂r r

,

(4.1)

where ρ and ν are the fluid density and viscosity, respectively, u is the tangential velocity (usually denoted uθ ) and τ is the shear stress, given by τ = ρν

∂u . ∂r

(4.2)

Taking into account equation (4.2), the equation (4.1) is a PDE for velocity u = u (r, t) . The boundary conditions are the velocity at the inner side of the outer cylinder (i.e., ub at r = b = 196 mm) and at the outer side of the inner cylinder (ua at r = a = 100 mm). Since the inner cylinder is fixed, ua = 0. Above we have anticipated some dimensions of the device. Considering hereafter 2π/ω periodic motions in time, the Fourier expansion for the velocity reads ∞

u0 � 1� un exp (inωt) = u= + � {un exp (inωt) } , 2 −∞ 2

(4.3)

n>0

where the Fourier components un = un (r) are un ≡ ω/π

� π/ω

−π/ω

u exp (−inωτ ) dτ .

In this laminar case, since ν and ρ are constants, the problem is linear and equation (4.1) reads, for un , ∂ inωun = ν ∂r

∂un ∂r

ν + r

∂un un − ∂r r

,

(4.4)

and the boundary conditions are un (r = a) = 0,

and

un (r = b) = ub,n ,

(4.5)

where ub,n is known from the given function ub (t) . The two-point boundary value problem defined by equations (4.4) and (4.5) can be solved analytically for un and τn . For n = 0 (mean current) the solution, u0 and τ0 , is � � � � ub,0 b ρνub,0 b a2 ∂u0 a2 u0 = 2 r− , τ0 = ρν = 2 1+ 2 . b − a2 r ∂r b − a2 r 69

(4.6)


70

Turbulent bed shear stress

For n �= 0 (oscillatory components), the solution to equations (4.4) and (4.5) is given by

� � r√ � −i + C2 K1 −i −i , (4.7) δ δ where J1 and K1 are Bessel functions and Ci can be found from the boundary un = C1 J1

�r √

conditions in equation (4.5). In the above expression � ν δ≡ , nω

(4.8)

which is a well-known result of the characteristic thickness of the boundary layer. Figure 4.2 shows the velocity profile for the steady case (n = 0) and for a monochromatic one (n = 1 with ω = π/5 s−1 ). In the second case, four significant snapshots are shown. From the Figure it is clear that, for waves, a thin boundary layer develops at the outer cylinder; the thickness of the boundary layer is in the order of the centimeter, consistently with equation (4.8) taking into account that the viscosity is chosen to be ν = 10−5 m2 /s. For the steady case the velocity gradients are small and the stresses at both cylinders are very small. However, for the oscillatory case, huge velocity gradients appear at the outer cylinder, indicating the existence of significant stress, τb , at r = b. The time histories of ub and τb are shown in Figure 4.3. As we can observe, there is a lag between velocity and stress of nearly T/8. The above laminar analysis allows us to introduce, in an analytical manner, the problem under consideration. Before introducing a turbulent boundary layer model in the next Section, it is convenient to make some remarks. First of all, in the present problem the boundary is moving, while in the real world the sea bed is fixed, i.e., we are solving the dual problem. The two problems are, however, perfectly analogous, since here we are computing the shear stress transmitted “from the wall to the water” while in the sea one is interested in 70


4.1 Theory

71

1

ub(t)=1 m/s

u (m/s)

0.5 0 ub(t)=cos(! t) m/s !0.5 !1 0.1

0.15 r (m)

0.2

Figure 4.2: Velocity profile in the laminar case for a steady (ub (m/s) = 1) and unsteady (ub (m/s) = cos (ωt) with ω = π/5 s−1 ) cases. In the latter case, snapshots correspond to t = 0 (—), t = T/8 (− −), t = T/4 (−·−) and t = T/2 (· · · ), where T = 2π/ω = 10 s is the period.

the shear transmitted “from the water to the wall”. Secondly, the thickness of the layer of fluid, which is 96 mm, is orders of magnitude smaller than usual water depths. If the flow is oscillatory, as it will be the case, this aspect is not crucial as long as the boundary layer thickness � δ ∼ ν/ω does not reach the inner cylinder. Obviously, this is the case shown

in Figure 4.2. The distance between the cylinders, b − a, has been chosen so that (b − a) � δ for all experimental conditions.

Third, the problem is in this case cylindrical. This difference can be shown to be rather small (below 1%), since the solution of the equation (4.1) for our radii is very similar to the equation for the plane case (r → ∞) ρ

∂u ∂τ = . ∂t ∂r

A simple scaling analysis of the equations shows that the curvature is of 71


72

Turbulent bed shear stress

4 !b (N/m2)

2 0

ub (m/s)

!2 !4 0

2

4

t (s)

6

8

10

Figure 4.3: Near bed velocity and bed shear stress time series in the laminar case for a periodic motion ub (m/s) = cos (ωt) with ω = π/5 s−1 .

minor influence if the characteristic radius of the instrument, rc , satisfies rc � ν/ωδ ∼ δ, as it is in our case. Finally, we can point out that the experimental facility has a finite height (1 meter), and the bottom part of it rotates with the outer cylinder, therefore generating a boundary layer at the bottom. As long as no mean velocity (i.e., current) is present, the thickness of this boundary layer will be very small compared to the height, and will not significantly affect the results. We will restrict to the case of pure oscillatory flows, i.e., no wave-current experiments will be carried out, and we will work in 2D.

4.1.2

Turbulent flow

We anticipate that the flow in all the experiments carried out is, as in most engineering problems, turbulent. The laminar case has only been presented in order to gain, in an analytical way, a better understanding of the boundary layer developed in the facility. The results obtained in the experiments will 72


4.1 Theory

73

be compared to an existing turbulent wave-current boundary layer model presented in Simarro et al. (2008), which was originally developed for a flat bed boundary layer. Although the differences between the boundary layer over a flat bed and that in the instrument are small, it is easy, and it seems convenient, to transpose the original model to our instrument geometry. Assuming the eddy viscosity concept to model turbulent shear [Wilcox 2004], the momentum equation reads now � � � � ∂u ∂ ∂u ρνt ∂u u ρ = ρνt + − . ∂t ∂r ∂r r ∂r r

(4.9)

Using the mixing length closure type of model, we can write [Grant & Madsen 1979, Orfila et al. 2007, Simarro et al. 2008] νt = κu∗c ξ,

(4.10)

where κ ≈ 0.4 is the von Karman universal constant, u∗c is a constant characteristic friction velocity and ξ is the distance to the wall. Since the boundary layer develops in the outer cylinder, both u∗c and ξ refer to the outer cylinder. Further, since νt is considered constant in time, we can write again the above expression (4.9) in Fourier components as � � � � ∂ ∂un νt ∂un un inωun = νt + − , ∂r ∂r r ∂r r

(4.11)

with boundary conditions un (r = a) = 0,

un (r = b − ξ0 ) = ub,n ,

and

where ξ0 ≡

ks ν + , 30 9.2u∗c

(4.12)

(4.13)

is the (small) distance to the outer cylinder where the velocity of the fluid is equal to that of the cylinder. In the equation (4.13), ks stands for the equivalent roughness and ν is the molecular kinematic viscosity of the water. 73


74

Turbulent bed shear stress

The equation (4.11) with boundary conditions in equation (4.12) does not have an analytical solution, and it must be solved numerically. Further, the Fourier components of the wall shear stress at the outer cylinder are given by � ∂un �� τb,n = ρνt , (4.14) ∂r �r=b−ξ0 � and τb can be computed using equation (4.3), τb = 12 ∞ −∞ τb,n exp (inωt) .

Finally, the model is closed by an extra relation between the characteristic friction velocity, u∗c , and the stress time history. Three different models are considered following Simarro et al. (2008) |τb,max | , ρ � |τb |� = , ρ � τb u b � = , ρ � u∗ ub �

u2∗c =

(4.15a)

u2∗c

(4.15b)

u∗c

(4.15c)

where τb,max is the maximum value for τb within a period, � · � stands for the

average over a period and u∗ = u∗ (t) is so that τb = ρu∗ |u∗ |. The last closure

equation corresponds to the energetic model by Kajiura (1964). The above options are denoted hereafter, as models I, II and III respectively. The above models allow for the computation of boundary layers with several harmonics (namely, sawtooth shapes). For pure monochromatic waves, i.e., ub = umax sin (ωt) , both the original model for flat bed and the presented here for cylindrical geometry allow to find the friction coefficient, fw , defined as [Jonsson 1966] fw ≡

2τb,max . ρu2max

Being ab ≡ umax ω −1 the near bed particle semi-excursion and ks the equivalent

roughness, dimensional analysis shows that friction coefficient is a function of two groups, namely umax ab Re ≡ ν

u2 = max νω

,

and 74

ab ε≡ ks

� � umax = . ks ω

(4.16)


4.1 Theory

75 0

10

! = 100

!1

10 fw

! = 101 ! = 102

!2

10

! = 103 ! = 104 !3

10

3

10

10

4

5

10 Re

6

10

10

7

Figure 4.4: Function fw (Re, ε) according to the original model II in Simarro et al. (2008)

.

Figure 4.4 shows fw (Re, ε) , meaningful only for monochromatic waves, obtained from the original model II. Models I and III yield slightly higher values (up to 20% higher than model II). In fact, recalling expression (4.10) for the eddy viscosity, νt , and the closure expressions in equations (4.15), it is intuitive that model I will provide larger values for the shear stress than model II. From Figure 4.4, and as a general trend, for high Reynolds numbers, i.e., in the rough case, fw (Re, ε) → fw (ε) , while for small Reynolds, in the smooth

case, fw (Re, ε) → fw (Re) . Simarro et al. (2008) have shown that in the rough case the model works well for ε � 30, while for ε � 30, which in the rough case is equivalent to fw � 0.05, all three models tend to underpredict the friction coefficient (relative to available experimental data).

Anticipating some results from later Sections, we focus only in the turbulent 75


76

Turbulent bed shear stress !1

10 computed f

w

model I !2

10

!3

10

10

!3

!2

10

!1

10

!1

10 computed f

w

model II !2

10

!3

10

10

!3

!2

10

!1

10

!1

10 computed f

w

model III !2

10

!3

10

10

!3

!2

!1

10 10 experimental fw

Figure 4.5: Comparison between the turbulent boundary layer models and experimental data in Kamphuis (1975) (+), Sleath (1988) (�), Jensen (1989) (◦) and Simons et al. (1992) (�). Root Mean Square (RMS) error takes values of 0.276, 0.185 and 0.215 respectively.

76


4.2 Experimental facility and setup

77

region. For turbulent flows the model II is the one with the best agreement with experimental data in the literature, as shown in Figure 4.5. According to the expression (4.16), both Re and ε depend on wave conditions. For instance, a monochromatic wave train with period T = 12 s and wave height H = 0.4 m travelling over a depth h = 3m has a near bed semiexcursion ab ≈ 0.67 m, and therefore Re ≈ 2.4·105 . If the bed has an equivalent roughness of one millimeter, then ε ≈ 6.7·102 .

Due to the scarcity of experimental data, the above model has not been checked for the case of sawtooth shapes. The aim of this Chapter is to perform this comparison. Therefore, experiments will be focused on monochromatic waves (to check the new facility) and on sawtooth shaped waves.

4.2

Experimental facility and setup

The device is conformed by two 4-mm thick concentric cylinders with external diameters of, respectively, 200 mm and 400 mm. The space between the two cylinders, 96 mm thick, can be filled up with water or simply left empty. Besides, the inner side of the outer cylinder can be roughened gluing sand of a given size. Both cylinders are 1m-high (Figure 4.1). The outer cylinder rotates by means of an electronically commutated DC motor following a desired velocity time history ub (t) . At any given time, the desired velocity ub is transmitted by a positioning controller, which compares the actual velocity to the desired one so as to adjust the motor driver for the next time step. The time step for the signal control is fixed at ∆t = 0.125 s and the maximum velocity and acceleration are, respectively, 2.0 m/s and 3.4 m/s−2 for the motor used. The obtained velocity was checked to be indistinguishable to the targeted one in all cases. 77


78

Turbulent bed shear stress

To measure the shear stress at the wall, a torque meter with a capacity of ±20Nm and a sensitivity of ±0.05Nm was coupled to the rotary axle. Ana-

log output data are acquired by the torque transducer each time step, and transmitted by USB connection to the computer, where the torque (in volts) and time are stored for later analysis. The measured torque will depend both on the instrument itself, i.e., the mass of the cylinder and the friction of the system, and on the shear stress produced by the water boundary layer. The component of the torque that does not depend on the water boundary layer can be obtained reproducing the experiment with the gap between the cylinders left empty. The wall shear stress time history for a given velocity time history is therefore the difference between the torque measurements with and without water. In summary, the general procedure for each experiment is this: the gap between the two cylinders is filled to the top with water, and the velocity signal is sent to the motor controller during 70 periods to obtain, at least, 50 valid wave periods after a start-up transient [Sleath 1987]. The torque signal is post-processed summarizing the data from the 50 valid periods in a single period that considers, at each point (or phase within the period), the “most probable value” of the signal obtained through a histogram which considers ten groups between the minimum and maximum signal. The same process is then repeated without water, so as to measure the torque required to move the system. The difference is the torque corresponding to the water boundary layer, which allows the direct computation of the shear stress at the wall of the outer cylinder. As mentioned in previous Section, two different sets of experiments are presented, corresponding to monochromatic and sawtooth waves. For monochromatic waves the near bed velocity ub can be expressed as ub (t) = umax sin (ωt) , 78

(4.17)


4.2 Experimental facility and setup

79

where umax is the maximum velocity and ω ≡ 2π/T is the wave angular velocity.

For sawtooth waves, the velocity ub is given here by 20

ub (t) = umax

3� � 4 i=1

� exp (i(n − 1/2)π) exp (inωt) , 2n−1

(4.18)

where it can be checked that umax is indeed the maximum velocity (Figure 4.6).

1

monochromatic sawtooth

ub/umax

0.5 0 !0.5 !1 0

T/4

T/2

3T/4

T

Figure 4.6: Monochromatic and sawtooth time histories.

In the laminar case, the signal is the same order of magnitude of torque meter sensitivity (0.05Nm). According to Nielsen (1992), in the laminar flow case √ (Re � 104 ), the maximum stress is given by τmax = ρ νω umax . Further, for monochromatic waves the wave angular velocity is ω = amax /umax being amax the maximum acceleration. Taking into account device’s limitations for velocity and acceleration, and imposing Re � 104 , the maximum expected torque for the laminar case can be shown to be � 0.10Nm, i.e., which is in the order

of the device’s sensitivity. Therefore, we will focus only on turbulent boundary layers, which are the most important for engineering purposes. 79


80

Turbulent bed shear stress test

T (s)

umax (m/s)

Re

mc T08 u125 k00

8

1.25

1.99·106

mc T13 u135 k00

13

1.35

3.77·106

st T08 u125 k00

8

1.25

st T13 u135 k00

13

1.35

Table 4.1: Smooth test cases for sinusoidal and sawtooth waves.

Two different roughnesses were employed in the experiments. The first one corresponds to the material of the cylinders (methacrylate, ks ≈ 10−2 mm), yielding smooth boundary layers. This is not an interesting condition for

engineering purposes and, therefore, only few tests were carried out for this condition (Table 4.1). In Table 4.1, for example, test “mc T08 u125 k00” corresponds to a monochromatic (“mc”, while “st” stands for sawtooth) wave with T = 8 s, umax = 1.25 m and a roughness ks = 0 mm (for the roughness is negligible). The rest of the tests were performed gluing a very uniform (σg < 1.1) 1 mmdiameter sand to the inner side of the outer cylinder. Table 4.2 shows the conditions tested considering that ks ≈ 2.5 d50 (proposed by Nielsen, 1992) with d50 = 1.0 mm.

4.3

Results

Figure 4.7 shows the velocity ub and the bed shear stress τb time histories for monochromatic and sawtooth waves for smooth test cases (Table 4.1). The experimental results for τb (crosses) compare well, in general, with the proposed model II. The above is in spite of the fact that in smooth boundary layer the stress transmitted to the water is small, the noise of the system becoming more important. 80


4.3 Results

test

81

T (s)

umax (m/s)

Re

ε

mc T04 u04 k25

4

0.4

1.02·105

1.02·102

mc T08 u04 k25

8

0.4

2.04·105

2.04·102

mc T12 u04 k25

12

0.4

3.06·105

3.06·102

mc T16 u04 k25

16

0.4

4.07·105

4.07·102

mc T04 u08 k25

4

0.8

4.07·105

2.04·102

mc T08 u08 k25

8

0.8

8.14·105

4.07·102

mc T12 u08 k25

12

0.8

1.22·106

6.11·102

mc T16 u08 k25

16

0.8

1.63·106

8.15·102

mc T04 u12 k25

4

1.2

9.17·105

3.06·102

mc T08 u12 k25

8

1.2

1.83·106

6.11·102

mc T12 u12 k25

12

1.2

2.75·106

9.17·102

mc T16 u12 k25

16

1.2

3.67·106

1.22·103

mc T08 u16 k25

8

1.6

3.26·106

8.15·102

mc T12 u16 k25

12

1.6

4.89·106

1.22·103

mc T16 u16 k25

16

1.6

6.52·106

1.63·103

st T08 u04 k25

8

0.4

st T08 u06 k25

8

0.6

st T08 u08 k25

8

0.8

st T08 u10 k25

8

1.0

Table 4.2: Rough test cases for sinusoidal and sawtooth shaped waves.

81


82

Turbulent bed shear stress

5

T = 8s, umax = 1.25 m/s

T = 13s, umax = 1.35 m/s

ub, !b

2.5 0 !2.5 !5 5 ub, !b

2.5 0 !2.5 !5 0

2

4 t(s)

6

8

0

3.25

6.5 t(s)

9.75

13

Figure 4.7: Velocity ub (m/s) (dashed line) and bed shear stress τb (N/m2 ) (×, experimental; full line, model II) time histories for smooth tests in Table 4.1. Monochromatic waves (top panels) and sawtooth waves (bottom panels).

Figure 4.8 shows the equivalent results for the monochromatic tests in Table 4.2, corresponding to ks = 2.5 mm. Again, there is, in general, a very good agreement between experimental and modelled (model II) results regarding the shape, the phase lag and the absolute values. For the experiments in Figure 4.8, Table 4.3 shows the error �, defined as � model � � experimental � �τ � �τ � − max max �≡ , (4.19) |τ model |max obtained by using the three models. As for the experimental data in the literature, the model II gives, in general, better results (Figure 4.5), and the 82


4.3 Results

83

models I and III tend to overpredict the shear stress. Only in the tests with smaller velocity the model I seems to work better. However, those tests correspond to smaller values of ε, and we already know that all the models tend to underpredict the shear stress for small values of ε. By construction, all three models give shear stress times histories with the same phase. To analyze the errors in the shear stress phase relative to the experimental results, we consider that the error in the phase, �φ (relative to T), is the one minimizing the function � j

2

[τ experimental (t = tj ) − τ model (t = tj + �φ T) ] ,

(4.20)

where tj indicates time values where experimental data are available. Table 4.3 includes the phase errors for the monochromatic tests (|�φ | � 0.078). To further check both the validity of the new instrument and of the proposed models, Figure 4.9 shows, in terms of fw , our experimental results together with those in the literature already presented in Figure 4.5, comparing them with the above models. The new experimental data follow the same trend as those in the literature, confirming the validity of the new experimental approach. Finally, Figure 4.10 shows the results for the sawtooth waves presented in Table 4.2. The corresponding errors appear in Table 4.4. We remark that here |�φ | � 0.062 and that, again, model II is the one yielding better results when compared to the experimental data. The new experimental results show,

similar to the experimental results by Suntoyo et al. (2008) using indirect measuring techniques, that shear stress time history has a “flat” zone which the models are unable to represent. Besides, the experimental results in Figure 4.10 clearly show that the maximum positive stress has a higher absolute value than the negative one. In sawtooth shaped time histories (dashed line in Figure 83


84

Turbulent bed shear stress

error � test

|�φ |

model I

model II

model III

mc T04 u04 k25

0.343

0.133

0.248

0.061

mc T08 u04 k25

0.016

-0.316

-0.131

0.031

mc T12 u04 k25

0.097

-0.214

-0.038

0.031

mc T16 u04 k25

-0.036

-0.387

-0.188

0.031

mc T04 u08 k25

0.383

0.173

0.288

0.061

mc T08 u08 k25

0.279

0.032

0.173

0.046

mc T12 u08 k25

0.194

-0.068

0.081

0.052

mc T16 u08 k25

0.187

-0.067

0.076

0.047

mc T04 u12 k25

0.405

0.198

0.313

0.061

mc T08 u12 k25

0.318

0.095

0.223

0.062

mc T12 u12 k25

0.288

0.066

0.190

0.062

mc T16 u12 k25

0.286

0.062

0.187

0.070

mc T08 u16 k25

0.297

0.076

0.202

0.062

mc T12 u16 k25

0.316

0.101

0.220

0.072

mc T16 u16 k25

0.313

0.085

0.210

0.078

Table 4.3: Monochromatic tests errors with the three models. For each case, the model with minimum � is shown in bold font.

test

error � |�φ |

model I

model II

model III

st T08 u04 k25

0.5536

0.2372

0.3911

0.062

st T08 u06 k25

0.4891

0.1147

0.2997

0.031

st T08 u08 k25

0.4605

0.0605

0.2607

0.031

st T08 u10 k25

0.4572

0.0557

0.2586

0.046

Table 4.4: Sawtooth tests errors with the three models. For each case, the model with minimum � is shown in bold font.

84


8 4 0 !4 !8 0 1 2 3 4

t(s)

T=8s

T = 12 s

10 5 0 !5 !10 0 2 4 6 8 t(s)

0 3 6 9 12 t(s)

T = 16 s

0 4 8 1216 t(s)

umax = 1.2m/s2 umax = 0.8m/s2 umax = 0.4m/s2

ub, !b ub, !b

4 2 0 !2 !4

ub, !b

2 1 0 !1 !2

ub, !b

T=4s

85

umax = 1.6m/s2

4.3 Results

Figure 4.8: Velocity ub (m/s) (dashed line) and bed shear stress Ď&#x201E;b (N/m2 ) (Ă&#x2014;, experimental; full line, model II) time histories for smooth tests in Table 4.2. Monochromatic waves only.

4.6), the maximum positive acceleration, which corresponds to the ascending steep slope of ub (t) , is higher, in absolute value, than the negative one, which corresponds to the descending gentle slope. Because the bed shear stress depends on the local acceleration [Liu & Orfila 2004, Simarro et al. 2008], the above behaviour of the bed shear stress can be explained. In the monochromatic case (full line in Figure 4.6) there is symmetry, i.e., the shape is the same running the time rightwards or leftwards, and therefore, the positive and negative shear stresses have the same absolute values. 85


86

Turbulent bed shear stress !1

computed fw

10

model I model II model III

!2

10

!3

10 !3 10

!2

10 experimental fw

!1

10

Figure 4.9: Experimental and analytical values for fw : new data (black figures) and data available in the literature (white figures).

The asymmetric behaviour of the bed shear stress under sawtooth waves, which is in fact very well captured by the model, is of major importance in sediment transport processes, since the sediment transport rate has a nonlinear dependence on the shear stress.

In order to show in a more clear fashion the influence of velocity time history shape on the bed shear stress, Figure 4.11 shows the numerical and experimental bed shear stress time histories for tests mc T08 u08 k25 (monochromatic) and st T08 u08 k25 (sawtooth), which share the same period, maximum velocity and roughness. From the Figure, the sawtooth time history gives, relative to the monochromatic, slightly (â&#x2030;&#x2C6; 10%) bigger maximum shear stress, while the diďŹ&#x20AC;erences in the minimum values are more significant (â&#x2030;&#x2C6; 30%). 86


4.3 Results

87

ub, !b

3 1.5 0 !1.5 !3

umax=0.4m/s

umax=0.6m/s

umax=0.8m/s

umax=1.0m/s

ub, !b

5 2.5 0 !2.5 !5 0

2

4 t(s)

6

8

0

2

4 t(s)

6

8

Figure 4.10: Velocity ub (m/s) (dashed line) and shear stress Ď&#x201E;b (N/m2 ) (Ă&#x2014;, experimental; full line, model II) time histories for smooth tests in Table 4.2. Sawtooth waves only.

87


88

Turbulent bed shear stress

4 3

ub, !b

2 1 0 !1 !2 !3 0

Figure 4.11:

2

4 t(s)

6

8

Shear stress Ď&#x201E;b (N/m2 ) time histories for tests mc T08 u08 k25

(monochromatic) and st T08 u08 k25 (sawtooth). Numerical results (lines) and experimental data (symbols).

88


4.4 Concluding remarks

4.4

89

Concluding remarks

In this Chapter we have presented an instrument to measure the bottom shear stress under symmetric (monochromatic) and asymmetric (sawtooth shaped) waves in a direct way. The instrument is similar to Taylorâ&#x20AC;&#x2122;s viscosimeter, but introduces a transducer to measure the torque required for the outer cylinder to follow a given velocity time history. The experimental results have been validated against experimental results for monochromatic waves in the literature. Further, the experimental results have been compared to an existing simple multiharmonic turbulent boundary layer model. Experiments have considered both smooth and rough turbulent boundary layers as well as monochromatic and sawtooth waves, and the agreement between experimental and numerical results is fair. The closure option in model II has shown to perform the best.

89


90

Turbulent bed shear stress

90


Chapter 5

General conclusions and future work In this Thesis a new set of low order Boussinesq-type equations, where the maximum order of the spatial derivatives is three, has been derived. The equations have been derived as a modification from the original set by Madsen and Schaffer (1998) by adding some weighting coefficients which improve the dispersion characteristics of the original model. To obtain the values of the coefficients, an optimization method has been applied so as to minimize the errors in linear dispersion and linear shoaling respect to the Airy’s theory (deeper waters) and weakly nonlinear energy transfer between different harmonics respect to the Stokes’ theory (shallower waters). The new set of equations are able to accurately propagate water waves from deep to shallow waters.

After the optimization process, the new set of equations provides errors in wave celerity below 1% for κ ≤ 20 whereas the original equations the errors

in celerity is 33%. Besides, the new equations present errors in linear shoaling bounded by 12% in the region where κ ≤ 10 whereas for the original set the error is 99%. Finally, the derived equations have an error bounded by 15% in nonlinear energy transfer for κ ≤ 2, while the Madsen & Shaffer’s equations 91


92

General conclusions and future work

is 59%.

A linear stability analysis of the linearized Boussinesq-type equations has been performed over uneven beds, which have been characterized by the maximum slope of the bathymetry, smax . Some stability problems have been reported when Madsen & Schaffer’s equations are employed for smax ≥ 2.5. All sets of coefficients presented in this Thesis yield more stable equations than those by Madsen & Schaffer (1998).

To solve the modified Boussinesq-type equations a new numerical scheme has been developed. An explicit 4th order Runge-Kutta method with a source function for wave generation within the domain has been implemented. Numerical tests have been carried out to validate both the new set of equations and the numerical scheme. In all tests, the model provides very good agreement with both analytical solutions and experimental data. A CFL-type stability condition has been derived for the time step taking into account the definition of stability region for the 4th order Runge-Kutta method. This general procedure avoids the use of numerical filtering during the simulations, which to a greater or lesser extent introduces artificial numerical diffusion in the wave propagation.

In this Thesis the effect of boundary layer under different wave conditions has been studied experimentally. A new experimental device similar to Taylor’s viscosimeter has been designed and constructed in order to measure directly the bottom shear stress under monochromatic and non-monochromatic waves. The new measuring device has been tested by some experiments available in the literature providing very good agreement in the comparisons. The device has been also used to validate a turbulent bed shear stress model which will be used to couple the Boussinesq type of equations with a sediment transport module. 92


93

Future work During the development of this Thesis the author identified several scientific issues, not yet properly solved. Some of these points have been already explored by the author and they will constitute some of the future research lines of the candidate in the next years. First of all, and of great importance when dealing with non linear wave models in shallow waters, is the inclusion of the wave breaking. Nowadays, wave breaking models require aprioristic knowledge of breaking location. The author will investigate the inclusion on breaking through depth-integration of a k − � turbulence model, which has been used successfully in Navier-Stokes models [Lin et al. 1999]. Moreover

the dry and wet algorithm to deal with the wave run-up in beaches will be implemented in the numerical code in the next months.

Bottom boundary layer effects which are of major concern for sediment transport models will be coupled with the new set of Boussinesq-type equations. The boundary layer effects will be incorporated by modifying the bottom boundary condition for the core region (potential flow) in order to incorporate the irrotational velocity induced by the boundary layer. This approach has been already incorporated in traditional weakly non linear as well as in highly non linear Boussinesq models by several authors [Liu & Orfila 2004, Liu et al. 2006, Orfila et al. 2007] but the enormous computational cost of their approaches restrict current models to the study of small scale processes.

Once the bottom boundary layer is treated in the new Boussinesq model, the next problem to be solved is the sediment transport module. Available models for sediment transport are usually derived from the quadratic law of the velocity to obtain the bottom shear stress that drives de sediment. This approach leads to innacuracies not only in the sediment transported but also in the phase lag. In fact, one of the main problems in coastal morphodynamics is the 93


94

General conclusions and future work

generation, destruction and movement of submerged sandbars and this problems constitutes today one of the milestones to be solved [Simarro et al. 2008]. This problem will be treated not only under a numerical perspective but also flume and field experiments will be carried out in the next year. Dynamics and morphodynamics in very shallow areas are in many cases the result of the nonlinear interaction of oscillatory flow and mean current. A common and not well resolved example is the generation of rip currents as the result of this interaction. In order to study such coastal processes the candidate pretends to include in the new formulation presented in the Thesis the interaction of the oscillatory flow with the mean current by performing an accurate multiple scale analysis and obtaining a new set of equations.

94


Appendix A

Features of the new equations In this appendix the linear and weakly nonlinear behaviour of the expressions (2.7) are analyzed. Without loss of generality, we consider here the one-dimensional case only. Although some of the results below are already expressed in Madsen & Shcaffer (1998), they will be presented below for completeness and clarity.

A.1

Linear propagation

Assuming mild slope conditions, i.e., O (σ) � 1, the one-dimensional linear versions of equations (2.7) are

ηt + hux + µ2 h2 (mdu huxxx + mdη ηxxt ) + +hx {u + µ2 h (mσdu huxx + mσdη ηxt ) } = O (σ 2 ) ,

(A.1a)

+hx µ2 h {mσcu uxt + gmσcη ηxx } = O (σ 2 ) ,

(A.1b)

ut + gηx + µ2 h2 (mcu uxxt + gmcη ηxxx ) +

where mdu ≡ dα + δ,

mdη ≡ δ,

(A.2a)

mcu ≡ cα + γ,

mcη ≡ γ,

(A.2b)

95


96

Features of the new equations

are the coefficients corresponding to the terms O (σ 0 ) and mσdu ≡ 2d1α + 3 (dα + δ) + 2 (δ + δσ ) ,

mσdη ≡ 2 (δ + δσ ) ,

mσcu ≡ 2c1α + 2γσ ,

mσcη ≡ 2γσ ,

(A.2c) (A.2d)

those in terms O (σ 1 ) . The solutions of equations (A.1) can be expressed as [Dingemans 1997] η = (Aη + iAησ ) exp(iS),

(A.3a)

u = (Au + iAuσ ) exp(iS), (A.3b) � √ where i ≡ −1 is the imaginary constant, S ≡ kdx − ωt is the wave phase angle and Aη , Au , Aησ and Auσ are real valued amplitudes. Both Aησ and Auσ are O (σ) and, besides, k and all four amplitude functions are slowly

varying in space. Introducing the expressions (A.3) into (A.1), we can collect the following two leading problems � � � � � � ωgdη −khgdu Aη 0 · = , −gkgcη ωgcu Au 0 and

where

ωgdη −gkgcη

−khgdu ωgcu

� � ·

Aησ Auσ

gdu ≡ 1 − mdu ξ 2 ,

=

qη qu

(A.4)

,

gdη ≡ 1 − mdη ξ 2 ,

gcu ≡ 1 − mcu ξ 2 ,

gcη ≡ 1 − mcη ξ 2 ,

(A.5)

(A.6a) (A.6b)

with ξ ≡ µkh, and qη ≡ − (1 − 3mdu ξ 2 ) hAu,x + µ2 h2 (3mdu khkx Au − mdη ω (kx Aη + 2kAη,x ) ) − − hx { (1 − mσdu ξ 2 ) Au + µ2 mσdη khωAη } ,

(A.7a)

− hx {µ2 mσcu khωAu − gµ2 mσcη k 2 hAη } .

(A.7b)

qu ≡ −g (1 − 3mcη ξ 2 ) Aη,x + µ2 h2 (3gmcη kkx Aη − mcu ω (kx Au + 2kAu,x ) ) −

96


A.1 Linear propagation

97

For future use, we define gd ≡

A.1.1

gdη , gdu

gc ≡

gcη . gcu

(A.8)

Linear frequency dispersion

To have a non-trivial solution for (A.4), the determinant of the coefficient matrix of these equations must vanish, i.e. � � ω gk Au gd = gc ≡ m = , kh ω Aη

(A.9)

which yields the following frequency dispersion relation for the wave system � 2 � 2 ω c gc = = {≡ gcd } , (A.10) 2 gk h gh gd allowing the computation of k and the celerity c as a function of ω and h. The goodness of the above dispersion relation is usually measured by comparing the result to Airy’s wave celerity given by c2 tanh ξ = . gh ξ

(A.11)

Following the notation in the main text, the results using Airy’s theory are denoted with “Airy”, and hereafter those from the Boussinesq-type equations (2.7) with “app”. Since ξ ≡ µkh depends on the whether we use the approximate expression (A.10) or the exact one, we favour the use of a k-independent variable, κ, defined as [Nwogu 1993] µ2 ω 2 h κ≡ g

� � c2 2 = ξ , gh

(A.12)

which, recalling the expressions (A.10) and (A.11), in the approximate case can be rewritten as κ = gcd ξ 2 and in the exact one as κ = ξ tanh ξ, so that κ ≈ ξ for κ � 3, i.e., in deep water.

97


98

Features of the new equations

Rewriting ξ 2 as κ/gcd in equations (A.6), the expression (A.10) gives −gcd (gcd − mdη κ) (gcd − mcu κ) + (gcd − mdu κ) (gcd − mcη κ) = 0, allowing to solve gcd , i.e. capp , directly as a function of κ.

A.1.2

Linear wave shoaling

Following Chen & Liu (1995), we consider the linear shoaling performance in terms of the propagated wave amplitudes as Aapp η AAiry η

= exp

��

Airy app αA − αA η η

κ

κ∗

0

dκ∗

,

(A.13)

Airy app where αA and αA are the shoaling gradients for both approaches. η η

The “α-operator” [Madsen & Sorensen 1992] operating on a variable “a”, which depends on water depth h, is defined as αa ≡ −ha−1 ∂a/∂h, so that αa measures the percentage change of the variable “a” caused by a unit percentage change in water depth h. It is of interest to note that αab = αa + αb and that, being b = b (a (h) ) , αb = αa ab−1 db/da. Airy The shoaling gradient αA within equation (A.13) is obtained from the Green’s η

law, i.e., A2η cg = ctt, and expressed as Airy αA η

αc G =− g = 2 (1 + G) 2

� G 1 + [1 − cosh (2ξ) ] , 2

(A.14)

with G≡

2ξ . sinh 2ξ

Above, αcg is obtained applying the α-operator to the group celerity cg , given from Airy’s dispersion equation (A.11) as cg ≡ dω/dk = c (1 + G) /2.

98


A.1 Linear propagation

99

app To obtain αA we remark that the solvability of the system (A.5) requires the η

condition qη = −nqu to be satisfied, with n≡

ωgdη gkgcη

=

khgdu ωgcu

,

and qη and qu given in equations (A.7). Recalling that Au = mAη , after some manipulation of the condition qη = −nqu we get app αA = η

zh − zk αkapp − zm αm , zA η

(A.15)

where σ 2 1 − mσdu ξ 2 mdη ξ mσ ξ 2 mσcη ξ 2 zh ≡ + + cu − , gdu gdη gcu gcη

zk ≡ −

3mdu ξ 2 mdη ξ 2 mcu ξ 2 3mcη ξ 2 + + − , gdu gdη gcu gcη

1 − 3mdu ξ 2 2mcu ξ 2 + , gdu gcu 1 − 3mdu ξ 2 2mdη ξ 2 2mcu ξ 2 1 − 3mcη ξ 2 ≡ + + + . gdu gdη gcu gcη

zm ≡ zA η and

Γ , 1+Γ � � 1 1 1 1 1 = + − + − (αkapp − 1) , 2 gdu gdη gcu gcη

αkapp = αm

which have been obtained by applying the α-operator to dispersion equation (A.10) and to the definition of m in equation (A.9). Above �

1 1 1 1 Γ≡1+2 − + + − gdu gdη gcu gcη 99

.


100

A.2

Features of the new equations

Weakly nonlinear propagation

We now consider the nonlinear behaviour of the equation sets (2.7) over flat beds and for the one dimensional case. In this case we get ηt + hux + µ2 h2 (mdu huxxx + mdη ηxxt ) + �pη = O (�2 ) , ut + gηx + µ2 h2 (mcu uxxt + gmcη ηxxx ) + �pu = O (�2 ) ,

(A.17a) (A.17b)

where pη ≡ (ηu) x + µ2 h2 (mdη (ηu) xx + m�cu ηuxx ) x + µ2 h (m�cη ηηxt ) x ,

(A.18a)

pu ≡ uux + µ2 h2 (mcη (uux ) x + cα uuxx + u2x /2) x −

− µ2 h (η (m�u uxt + gm�η ηxx ) ) x , (A.18b)

with m�cu ≡ cα + δ� ,

m�cη ≡ δ� ,

m�u ≡ 1 + γ� ,

m�η ≡ γ� .

(A.19a) (A.19b)

the parameters involved in the nonlinear terms. Following, e.g., Schaffer (1996), we consider here the asymptotic expansions η = η1 + �η2 + . . . and u = u1 + �u2 + . . . , and two leading order, i.e., O (�0 ) , waves with different frequencies (ωi and ωj ), so that

η1 = η1i + η1j = Aiη cos Si + Ajη cos Sj

(A.20a)

u1 = η1i + η1j = Aiu cos Si + Aju cos Sj ,

(A.20b)

where Aau = ma Aaη with m’s given in equation (A.9). Also, the phases are written as Sa = ka x − ωa t + θa . Substituting (A.20) into equations (A.18) for 100


A.2 Weakly nonlinear propagation

101

pη and pu we can write pη = ρi+j rηi+j sin Si+j + ρi−j rηi−j sin Si−j + + ρi+i rηi+i sin Si+i + ρj+j rηj+j sin Sj+j , pu = ρi+j rui+j sin Si+j + ρi−j rui−j sin Si−j + + ρi+i rui+i sin Si+i + ρj+j ruj+j sin Sj+j , where Sa±b ≡ Sa ± Sb = ka±b x − ωa±b t + θa±b , being ka±b ≡ ka ± kb , ωa±b ≡ ωa ± ωb and θa±b ≡ θa ± θb . In the above expressions for pη and pu ρa±b ≡ with δab = and, finally

δab ka±b Aaη Abη , 2

 1/2 1

if ωa = ωb , if ωa �= ωb ,

a±b rηa±b = −ma+b gdη + µ2 h (m�cu h {k 2 m} a+b − m�cη {ωk} a+b ) , a±b rua±b = −ma mb gcη +

+ µ2 h (m�u {ωkm} a+b + (cα hma mb − gm�η ) {k 2 } a+b ± hka kb ma mb ) ,

a±b a±b a±b , g a±b follow the where ma+b ≡ ma + mb and so on, while gdu , gdη , gcu cη

definitions in equation (A.6) but using ξa±b ≡ µka±b h instead of ξ.

Recalling the expression (A.17), the second order terms η2 and u2 will necessarily be of the form i−j i+i j+j η2 = Ai+j η� cos Si+j + Aη� cos Si−j + Aη� cos Si+i + Aη� cos Sj+j , i−j i+i j+j u2 = Ai+j u� cos Si+j + Au� cos Si−j + Au� cos Si+i + Au� cos Sj+j ,

and, substituting, it must hold � � � i±j � � i±j � i±j i±j ωi±j gdη −ki±j hgdu Aη� rη · = −ρi±j . i±j i±j i±j −gki±j gcη ωi±j gcu Au� rui±j 101

(A.21)


102

Features of the new equations

Finally, the transfer function is defined as Gapp ± ≡

hAi±j η� Aiη Ajη

= δij h

i±j i±j 2 hg i±j r i±j ωi±j ki±j gcu rη + ki±j du u 2 hg i±j g i±j − 2ω 2 g i±j g i±j 2gki±j i±j dη cu du cη

,

having used the solution of the system (A.21) above. The transfer function for the Stokes theory is [Schaffer 1996] � � Hi±j Stokes G± = δij h µωi±j − Li±j , Di±j

(A.22)

where, following the above notation, � 2 � � � µ ωi ωj gki kj µ2 {ω 3 } i±j g k2 Hi±j ≡ ωi±j ± − + − , g ωi ωj g 2 2 ω i±j 2 Di±j ≡ gki±j tanh (µki±j h) − µωi±j , � � µ2 {ω 2 } i+j µ 2 ωi ωj 1 gki kj Li±j ≡ ∓ − . 2 ωi ωj g g

Since ω defines κ for a given h, GStokes and Gapp ± ± are functions of κi and κj .

Further, they are symmetric, so that G+ and G− can be presented in a single

plot [Madsen et al. 2003]. Figure A.1 shows the shape of GStokes in the range ± 0 � κi , κj � 2. Figure A.2 shows the relative errors of Wei et al. (1995)

equations relative to the Stokes theory. The maximum error is 59% in the considered range (0 � κi , κj � 2). The results for equations in Madsen & Schaffer (1998) are similar. If both leading order waves are equal we get hAapp gc gd g4cu m−1 rη+ + gd g4du m−2 ru+ η� = , A2η 4 gd g4du g4cη − gc g4dη g4cu with rη+ ≡ −2m {1 − (4mdη + m�cu − m�cη gd−1 ) ξ 2 } ,

ru+ ≡ −m2 {1 − (4mcη + 2m�u gd−1 + 2cα − 2m�η gc−1 gd−1 + 1) ξ 2 } , 102

(A.23)


A.2 Weakly nonlinear propagation

2

10 5

103

3

G+ !1

!j

!

G

1 10 5

3 !2

!2

!3

0 0

!3 !5 !10

!5 !10

1 !i

2

Figure A.1: Weakly nonlinear propagation: GStokes . ±

2 0.1

G+ 0.1

G

!

0.05

!j

0.05

1

0.02 0.01 !0.05

0 0

!0.02 !0.01 0.01

1 !i

!0.2 !0.1 !0.05 !0.02 !0.01 0.01 0.02 0.05 0.1

2

Stokes Figure A.2: Weakly nonlinear propagation: Gapp − 1 for W95. ± /G±

103


104

Features of the new equations

while the transfer function for the Stokes theory in (A.22) reduces to hAStokes 1 η� = 2 Aη 2

� ξ 2 (3 coth ξ − 1) coth ξ . 2

104

(A.24)


Appendix B

Stability analysis for sinusoidal bathymetries σσ a+b ≡ ah2 + bh2 , and Defining mσσ c 1 du ≡ d1α + δσ + δ, mcu ≡ c1α + γσ , and h

denoting n + j as n+j , the coefficients for the continuity equation are aη,−2 ≡ h21 k 2 µ2 {mdη n2−2 + mσdη n−2 + 4δσ } , aη,+2 ≡ h21 k 2 µ2 {mdη n2+2 − mσdη n+2 + 4δσ } ,

aη,−1 ≡ h1 hc k 2 µ2 {2mdη n2−1 + mσdη n−1 + 2δσ } , aη,+1 ≡ h1 hc k 2 µ2 {2mdη n2+1 − mσdη n+1 + 2δσ } , aη,0 ≡ − {1 − k 2 µ2 mdη h1+2 n2 } ,

and σσ −bu,−3 ≡ ih31 k 3 µ2 {mdu n3−3 + mσdu n2−3 + (7mσσ du + δσ ) n−3 + 3 (mdu + δσ ) } , σσ −bu,+3 ≡ ih31 k 3 µ2 {mdu n3+3 − mσdu n2+3 + (7mσσ du + δσ ) n+3 − 3 (mdu + δσ ) } , σσ −bu,−2 = ih21 hc k 3 µ2 {3mdu n3−2 + 2mσdu n2−2 + 10mσσ du n−2 + 4mdu } , σσ −bu,+2 = ih21 hc k 3 µ2 {3mdu n3+2 − 2mσdu n2+2 + 10mσσ du n+2 − 4mdu } ,

−bu,−1 ≡ ih1 k 3 µ2 {3mdu h1+1 n3−1 + mσdu h1+1 n2−1 } +

1+0 0+1 + ih1 k 3 µ2 { (3mσσ + 5 (mσσ } n−1 + du − δσ ) h du − δσ ) h

1+0 0+1 + ih1 k 3 µ2 {+ (mσσ + (3mσσ } + ih1 k {−n−1 } , du − δσ ) h du − 5δσ ) h

105


106

Stability analysis for sinusoidal bathymetries

−bu,+1 ≡ ih1 k 3 µ2 {3mdu h1+1 n3+1 − mσdu h1+1 n2+1 } +

0+1 + ih1 k 3 µ2 { (3mdu − δσ ) h1+0 + 5 (mσσ } n+1 + du − δσ ) h

1+0 0+1 + ih1 k 3 µ2 {− (mσσ − (3mσσ } + ih1 k {−n+1 } , du − δσ ) h du − 5δσ ) h

0+1 −bu,0 ≡ +ihc k 3 µ2 {mdu h1+6 n3 + 4 (mσσ n} − ihc kn. du − 2δσ ) h

For the momentum equation, the coefficients are 2 au,−2 ≡ h21 k 2 µ2 {mc12 n2−2 + mσc12 mn−2 + mσσ cu m } ,

au,+2 ≡ h21 k 2 µ2 {mc12 n2+2 − mσc12 n+2 + mσσ cu } ,

au,−1 ≡ h1 hc k 2 µ2 {2mcu n2−1 + mσcu n−1 + mσσ cu } , au,+1 ≡ h1 hc k 2 µ2 {2mcu n2+1 − mσcu n+1 + mσσ cu } , 0+1 au,0 ≡ k 2 µ2 {mc12 h1+2 n2 + 2mσσ } − 1, cu h

and −bη,−2 ≡ ih21 gk 3 µ2 {mcη n3−2 + mσcη n2−2 + γσ n−2 } ,

−bη,+2 ≡ ih21 gk 3 µ2 {mcη n3+2 − mσcη n2+2 + γσ n+2 } ,

−bη,−1 ≡ ih1 ghc k 3 µ2 {2mcη n3−1 + mσcη n2−1 + γσ n−1 } , −bη,+1 ≡ ih1 ghc k 3 µ2 {2mcη n3+1 − mσcη n2+1 + γσ n+1 } , −bη,0 ≡ +igk 3 µ2 {mcη h1+2 n3 + 2γσ h0+1 n} − igkn.

106


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Wave Motion 46 (2009) 489–497

Contents lists available at ScienceDirect

Wave Motion journal homepage: www.elsevier.com/locate/wavemoti

Bottom friction effects on linear wave propagation G. Simarro a,*, A. Orfila b, A. Galán a,b, G.A. Zarruk b,1 a b

E.T.S.I. Caminos, Canales y Puertos, Universidad de Castilla–La Mancha, 13071 Ciudad Real, Spain Institut Mediterrani d’Estudis Avançats (IMEDEA), C. Miquel Marques 21, 07190 Esporles, Spain

a r t i c l e

i n f o

Article history: Received 18 March 2008 Received in revised form 12 January 2009 Accepted 2 June 2009 Available online 7 June 2009 Keywords: Linear theory Boundary layer WKB aproximation

a b s t r a c t Bottom boundary layer effects on the linear wave propagation over mild slope bottoms are analyzed. A modified WKB approximation is presented including boundary layer effects. Within the boundary layer, two cases are considered: laminar (constant viscosity) and turbulent. Boundary layer effects are introduced by coupling the velocity inside the boundary layer to the irrotational velocity in the core region through the bottom boundary condition. This formulation properly accounts for the phase between near bed velocity and bed shear stress. The resulting differential equation for the energy conservation introduces a new term accounting for the energy losses due to the boundary layer effects. ! 2009 Elsevier B.V. All rights reserved.

1. Introduction The bottom boundary layer in water wave propagation is important, at least, in two fundamental aspects related to coastal and environmental engineering. First, it determines the stress that the water transmits to the bottom, which is important in the near shore morphodynamics and ecosystems, since bottom shear stress is responsible for sediment transport [4,9]. Secondly, the energy dissipation in the boundary layer is responsible for wave damping, modifying not only the wave amplitude but also wave celerity and phase. Boundary layer effects can be of first order importance if propagation over long distances is considered. Therefore, in order to obtain accurate models for water waves propagation, the energy dissipation within the boundary layer has to be taken into account in their formulation. This is often done by introducing the bed shear stress in the horizontal momentum equation. The shear stress transmitted to the bottom is usually expressed as the square of the near bottom velocity as [3,13],

s ¼ qC f ujuj; where s is the bottom shear stress, q is the water density, u is the near bed velocity and C f is a dimensionless friction coefficient, which is a function of a relative bed roughness and a Reynolds number [4]. As long as C f is properly chosen, the above frictional model is appropriate if the primary concern is the amount of energy dissipated in a time scale bigger than one wave period. However, this bottom stress model does not correctly describe the phase of the bottom stress relative to the bottom velocity, since it is well known, for instance, that the bottom stress is p=4 out of phase with the bottom velocity for an oscillatory laminar boundary layer [9]. This phase lag is smaller in the turbulent case [4]. A bottom stress such as the above described is not adequate when computing sediment transport rate, unless an empirical phase shift is introduced. For the laminar boundary layer case and linear water wave propagation, some results introducing the proper phase have already been obtained [1,2]. More recently [7] introduced the result of integrating the linearized and laminar boundary layer

* Corresponding author. Tel.: +34 926295300. E-mail address: gonzalo.simarro@uclm.es (G. Simarro). 1 Present address: Institut for Energy Technology, P.O. Box 40, N-2027, Kjeller, Norway. 0165-2125/$ - see front matter ! 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2009.06.009


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G. Simarro et al. / Wave Motion 46 (2009) 489–497

into the bottom boundary condition for the potential in the core, obtaining a set of Boussinesq-type equations including the bottom laminar boundary layer effects. The boundary layer effects were introduced in the continuity equation as a convolution integral. The extension for uneven bottoms was derived by [8] for mild slope conditions. The above derivations were made under the assumption of a laminar or constant viscosity. Sometimes the model of constant eddy viscosity is used to describe the bottom boundary layer and, further, the eddy viscosity is modeled as a linear function of the distance to the bottom [6,5,12]. Following this concept, Orfila et al. [10] extended the formulation by [8] to include the effects of a fully developed turbulent bottom boundary layer in a nonlinear wave propagation model. In this work, we estimate the effects of the bottom friction within the framework of such a model leaving aside the discussion of the accuracy of its performance. In reality the turbulent boundary layer is certainly nonlinear. In many real situations, water wave linear theory is accurate enough for propagating waves from deep to shallow water, so that simulations can be done in a much more efficient way. The aim of this paper is to introduce the above mentioned boundary layer formulations (laminar and turbulent) in linear models. Since the goal is to present the influence of the boundary layer, numerical results will only be presented in one dimensional cases. The paper is structured in the following manner. For completeness, we summarize the governing equations, boundary conditions and the Fourier expansion for periodic waves in Section 2. The equations for the WKB approximation are derived in Section 3. The leading order solution provides the modified equation for the energy conservation where the dissipation effects are included. These set of equations are expressed in terms of the Fourier (harmonic) components of the free surface amplitude. A brief discussion of the results is presented in Section 4. 2. Governing equations Hereinafter, dimensional variables are primed and dimensionless variables are unprimed. We will consider the domain 0 between the sea bed, which is located at z0 ¼ "h ðx0 ; y0 Þ, and the free surface at z0 ¼ g0 ðx0 ; y0 Þ (Fig. 1). The mean water level (MWL) is considered to be at z0 ¼ 0. For convenience, here $0 will stand for ð@=@x0 ; @=@y0 Þ and a0 for ða0x ; a0y Þ. The no flux boundary conditions at the bottom and the free surface are obtained considering that they are material surfaces. These conditions read 0

@h 0 þ u0 & $0 h þ w0 ¼ 0; @t 0 @ g0 þ u0 & $0 g0 " w0 ¼ 0; @t 0

0

z0 ¼ "h ;

ð1aÞ

z0 ¼ g 0 ;

ð1bÞ

where u0 is the horizontal velocity, w0 is the vertical component and t 0 is time. In water wave propagation problems, it is usually assumed that the fluid is inviscid, so that it is allowed to slip over the contours. In that case, the no flux conditions in expressions (1) are the only kinematic conditions to be used. Moreover, the velocity is considered to be irrotational, i.e., there exists a velocity potential U0 so that

u0 ¼ $0 U0 ;

and w0 ¼

@ U0 : @z0

ð2Þ

For incompressible fluids, the continuity equation for the potential reads

r02 U0 þ

@ 2 U0 ¼ 0; @z02

0

"h 6 z0 6 g0 ;

ð3aÞ 0

and the boundary conditions (1) for rigid bed (i.e., @h =@t 0 ¼ 0) are, 0

@U 0 ¼ 0; z0 ¼ "h ; @z0 0 @ g0 @U þ $0 U0 & $0 g0 " 0 ¼ 0; z0 ¼ g0 : @t 0 @z

$0 U0 & $0 h0 þ

Fig. 1. Illustration of the variables and the boundary conditions for the two dimensional wave propagation problem.

ð3bÞ ð3cÞ


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G. Simarro et al. / Wave Motion 46 (2009) 489–497

The dynamic free surface boundary condition can be obtained from Bernoulli’s equation imposing the continuity of the pressure field at the water–air interface. Assuming constant atmospheric pressure, it reads

! " @ U0 1 @ U0 @ U0 0 0 0 0 þ $ U & $ U þ þ g 0 g0 ¼ 0; @t 0 2 @z0 @z0

z0 ¼ g0

ð3dÞ

being g 0 the acceleration of gravity. 2.1. Dimensionless equations The above equations are scaled by considering usual dimensionless variables

1 0 0 0 fz ; h g; h0

fz; hg '

g'

g0 a00

;

0

fx; yg ' k0 fx0 ; y0 g;

and

U'

0

0

k 0 h0 qffiffiffiffiffiffiffiffiffi U0 ; 0 0 a0 g 0 h0

u'

0

0

h q0ffiffiffiffiffiffiffiffiffi u0 ; 0 0 a0 g 0 h0

w'

0

02

k0 h0 qffiffiffiffiffiffiffiffiffi w0 ; 0 0 a0 g 0 h0

0

t ' k0

qffiffiffiffiffiffiffiffiffi 0 g 0 h0 t0 ; 0

with a00 and h0 being characteristic lengths for the wave amplitude and the water depth, respectively, and k0 is the characteristic wave number. The boundary value problem defined in Eq. (3) reads, in dimensionless form

r2 U þ

1 @2U

l2 @z2

$U & $h þ

¼ 0;

1 @U

l2 @z

"h 6 z 6 !g;

ð4aÞ

z ¼ "h;

ð4bÞ

¼ 0;

@g 1 @U þ ! $U & $g " 2 ¼ 0; z ¼ !g; @t l @z ! " @U ! 1 @U @U þ g ¼ 0; $U & $U þ 2 þ @t 2 l @z @z

ð4cÞ z ¼ !g;

ð4dÞ

where

!'

a00 0 ; h0

and

l ' k00 h00 ;

are the dimensionless parameters representing, respectively, the nonlinear and the dispersive effects. Throughout this paper only linear waves are to be analyzed and, hence, hereafter, it will be considered that ! ' 0. The dispersive parameter, l, is small in shallow water conditions, but throughout this work no assumption about its value will be made. 2.2. The boundary layer In real fluids, viscous effects become important within a thin layer attached to the bottom (Fig. 1), where velocity gradients are large. Therefore, the hypothesis of inviscid fluid is no longer valid in this region (boundary layer) and rotational and irrotational velocity components have to be considered, i.e.,

u ¼ $ U þ ur ;

and w ¼

@U þ wr ; @z

ð5Þ

with ur and wr standing, respectively, for the horizontal and vertical components of the rotational velocity. The no flux boundary condition at the bottom in (4b) reads now

ð$U þ ur Þ & $h þ

1 @U

l2 @z

þ

wr

l2

¼ 0;

z ¼ "h:

ð6Þ

Following [8], to solve the rotational velocity at the seabed we will consider a coordinate system locally parallel to the bed ^ r Þ is (hats in Fig. 2). We note that the rotational velocity component normal to the bottom ðw

^ r ¼ wr ð1 þ Oððl$h2 ÞÞÞ þ l2 ur & $h; w where Oð$hÞ ( Oð1Þ since mild slope conditions will be assumed. Therefore, the expression (6) is also

$U & $h þ

1 @U

l2 @z

þ

^r w

l2

¼ 0;

z ¼ "h:

ð7Þ

^ r =l2 which accounts for the boundary layer effects (compare to expression (4b)). Above expression (7) introduces the term w


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G. Simarro et al. / Wave Motion 46 (2009) 489–497

Fig. 2. Schematic view of the coordinate system locally parallel to the bed.

^ r at z ¼ "h is, for the laminar case [7] The result for w

vl2 p

where

^ r ðz ¼ "hÞ ¼ " pffiffiffi w

0

t

r2 Uðz ¼ "h; t ¼ nÞ pffiffiffiffiffiffiffiffiffiffi t"n

dn;

v is a dimensionless parameter accounting for the boundary layer strength and defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1u u m0 0 t qffiffiffiffiffiffiffiffiffi ( 1; h0 k 0 g 0 h0

v' with

Z

0

0

ð8Þ

m0 being the kinematic viscosity, which is constant for the laminar case.

2.2.1. Turbulent boundary layer for periodic waves As mentioned, boundary layer is usually turbulent in real cases. In the case of turbulent boundary layer, the problem is much more complex. The constant eddy viscosity linear model [6,5] has been used often to describe the bottom boundary layer. In this paper we use such model leaving aside the boundary layer nonlinearity issue. Considering the flow periodic in time, any time dependent function n can be expressed as

X 1X n n expð"inxtÞ ¼ 0 þ Rfnn expð"inxtÞg; 2 n n 2 n>1

ð9Þ

^ r can be written in compact form for both with x being the main frequency and nn the Fourier components. The solution for w the laminar and the turbulent cases as [11]

where

1þi ^ r;n ðz ¼ "hÞ ¼ "vl2 r2 Un ðz ¼ "hÞ pffiffiffiffiffiffiffiffiffiffi #n ; w 2nx #n ¼

8 laminar BL; < 1; $ ffiffiffiffiffiffiffiffiffiffi% pffiffiffiffi K1 2pffi"in x^z0 $ % pffiffiffiffiffiffiffiffiffiffiffi ; turbulent BL; : ^z0 K0 2

"inx^z0

ð10Þ

ð11Þ

and with ^z0 defined as

^z0 '

1 ^z00

v h00

;

where ^z00 is the elevation over the seabed where the velocity cancels. The elevation ^z0 depends on the bed roughness and a Reynolds number. Besides, the parameter v is given in Eq. (8): in the case of turbulent boundary layer (which is the usual one), m0 must be replaced by a characteristic kinematic viscosity, which


G. Simarro et al. / Wave Motion 46 (2009) 489–497

493

depends on the own solution. The solution is, therefore, to be obtained in a iterative way [10]. The performance of this boundary layer model in terms of the friction factor can be found in [12]. 2.3. Potential equations for periodic waves Since the above boundary layer results were presented in terms of the Fourier components, the equations for the core region will be hereinafter presented in terms of Fourier components (assuming that the movement is periodic). Recalling expression (9), the above Eq. (4) read now, replacing (4b) by (7) and linearizing (i.e., setting ! ' 0)

r2 Un þ

1 @ 2 Un

l2 @z2

$Un & $h þ

1 @ Un

l2 @z 1 @ Un

" inxgn "

"h 6 z 6 0;

¼ 0;

1þi " vr2 Un pffiffiffiffiffiffiffiffiffiffi #n ¼ 0; 2nx ¼ 0;

ð12aÞ z ¼ "h;

z ¼ 0;

l2 @z " inxUn þ gn ¼ 0; z ¼ 0:

ð12bÞ ð12cÞ ð12dÞ

We remark that the boundary layer effects appear only at the bottom boundary condition (12b). Besides, combining the boundary conditions at the free surface we get

1 @ Un

l2 @z

" n2 x2 Un ¼ 0;

z ¼ 0:

ð13Þ

Since the problem is linear, there is no interaction between different harmonics and, hereafter, only n ¼ 1 will be considered (and n will be omitted). Further, in order to solve the potential, only Eqs. (12a), (12b) and (13) are to be used, i.e.,

r2 U þ

1 @2U

l2 @z2

$U & $h þ 1 @U

l2 @z

"h 6 z 6 0;

¼ 0;

1 @U

l2 @z

" vXr2 U ¼ 0;

" x2 U ¼ 0;

ð14aÞ z ¼ "h;

z ¼ 0;

ð14bÞ ð14cÞ

with

1þi

X ' pffiffiffiffiffiffiffi #:

ð15Þ

g ¼ ixU; z ¼ 0:

ð16Þ

2x

Once the potential is solved, the free surface elevation can be computed from Eq. (12d), i.e.,

The above Eq. (14) are solved in the following section using the WKB approximation. 3. The WKB approximation The WKB is usually employed in water wave propagation to handle with the fact that there are two characteristic length scales: one corresponding to the wave length and another corresponding to the bottom variations. Here, three different horizontal characteristic lengths will be present: the two above mentioned plus the one corresponding to the boundary layer effects on the wave damping. For the sake of clarity, we consider first the case when the bed is flat. As above mentioned, in this case there are two different scales (the wave length, which is order 1 according to the scaling, and the characteristic damping scale, which will be order v"1 ). The WKB approximation can be here performed as usual, except for the boundary layer effects playing the role usually played by the bed being uneven. The governing Eq. (14) for the potential are, in this case

r2 U þ 1 @U

l2 @z

1 @2U ¼ 0; l2 @z2 " vXr2 U ¼ 0;

" x2 U þ

1 @U

l2 @z

¼ 0;

"h 6 z 6 0;

ð17aÞ

z ¼ "h;

ð17bÞ

z ¼ 0:

ð17cÞ


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G. Simarro et al. / Wave Motion 46 (2009) 489–497

As usual in WKB, we first consider the potential Uðx; zÞ written as

Uðx; zÞ ¼ Aðx; zÞ expðiSðx; zÞÞ;

ð18Þ

with A and S real functions. Substituting (18) into (17) we get

r2 A " A$S & $S þ A @A

l2 @z

1

l2

@2A @S @S "A @z2 @z @z

!

¼ 0;

"h 6 z 6 0;

þ vIfXg$ & ðA2 $SÞ " vRfXgðAr2 A " A2 $S & $SÞ ¼ 0;

" x2 A þ

1 @A

l2 @z

¼ 0;

ð19aÞ z ¼ "h;

z ¼ 0;

ð19bÞ ð19cÞ

from the real parts, while from the imaginary parts we get

$ & ðA2 $SÞ þ 2

A @S

l2 @z A @S

l2 @z

& ' 1 @ 2 @S ¼ 0; A @z l2 @z

"h 6 z 6 0;

" vRfXg$ & ðA2 $SÞ " vIfXgðAr2 A " A2 $S & $SÞ ¼ 0; ¼ 0;

ð20aÞ z ¼ "h;

z ¼ 0:

ð20bÞ ð20cÞ

In the above expressions, ‘‘I” and ‘‘R” stand, respectively, for ‘‘imaginary part of” and ‘‘real part of”. The essence of the WKB approximation is to consider two different horizontal scales, usually one corresponds to the wave and the other to the bottom variations. In this case, the latter will correspond to the boundary layer effects (which are order v). Following usual WKB procedure (see, e.g., [3] for details), we consider the asymptotic expansions

A ¼ a0 þ v2 a1 þ v4 a2 þ & & & ; 2

ð21aÞ

4

S ¼ v ðb0 þ v b1 þ v b2 þ & & &Þ: "1

ð21bÞ

^ . We remark that functions ai and b in the above expansions are ^ ¼ vx, so that $ ¼ v$ The slow variable is now defined as x i ^ b Þ ¼ Oð1Þ. ^ ai Þ ¼ Oð$ slowly varying, i.e., Oð$ i Introducing expansions (21) into Eq. (20), the leading order implies the well known result that @b0 =@z ¼ 0, i.e., b0 ¼ b0 ðxÞ. Taking this into account, and substituting (21) into Eq. (19) we get, to the leading order 2 ^b & $ ^ b þ 1 @ a0 ¼ 0; "h 6 z 6 0; " a0 $ 0 0 l2 @z2 a0 @ a0 ¼ 0; z ¼ "h; l2 @z 1 @ a0 ¼ 0; z ¼ 0; " x2 a0 þ 2 l @z

ð22aÞ ð22bÞ ð22cÞ

which, at it is well known, implies

a0 ðx; zÞ ¼ aU ðxÞf ðx; zÞ; f ðx; zÞ '

coshðlkðz þ hÞÞ ; coshðlkhÞ

ð23Þ

2

^b & $ ^ b . Further, the expression (22c) implies where k satisfies the Eikonal equation k ¼ $ 0 0

x2 ¼

k

l

tanhðlkhÞ;

ð24Þ

which is the dispersion relationship (in dimensionless form) allowing to compute k as a function of x and h. In order to know the spatial evolution of a0 ðx; zÞ, i.e., aU ðxÞ, the terms order Oðv2 Þ are to be analyzed. Introducing expansions (21) into Eq. (20) we get, to the following order

^ & ða2 $ ^ $ 0 b0 Þ þ

& ' 1 @ 2 @b1 ¼ 0; a l2 @z 0 @z

"h 6 z 6 0;

a20 @b1 ^b & $ ^ b ¼ 0; z ¼ "h; þ IfXga20 $ 0 0 l2 @z a0

@b1 ¼ 0; @z

z ¼ 0:

ð25aÞ ð25bÞ ð25cÞ


G. Simarro et al. / Wave Motion 46 (2009) 489–497

495

Depth integrating the continuity Eq. (25a), using the boundary conditions at z ¼ "h and z ¼ 0, and recalling expression (23), we get

&

^b $ & a2U $ 0

Z

0

"h

' f 2 dz ¼ "vIfXg

a2U k

2

2

cosh ðlkhÞ

;

ð26Þ

where the boundary layer effects, order v, appear in the right hand side. R0 It is well known (see, e.g., [3]) that "h f 2 dz ¼ ccg , where c and cg are the wave and group celerities, respectively, given by

c'

x k

;

and cg ¼

@x c ¼ ð1 þ GÞ; @k 2

with

G'

0 0

2lkh 2k h : ¼ sinhð2lkhÞ sinhð2k0 h0 Þ

ð27Þ

The analysis for the mild slope case is similar. However, the bottom boundary condition introduces an extra term accounting for the bed slope (see expression (14b)). The leading order yields the same results in Eqs. (23) and (24), and now the expression (25b) becomes

^h þ a20 $^ b0 & $

a20 @b1 ^ b ¼ 0; z ¼ "h; ^b & $ þ IfXga20 $ 0 0 l2 @z

ð28Þ

but, depth integrating, the expression (26) remains valid. 4. Discussion of the boundary layer effects Summarizing the above results, the WKB approximation yield

U ¼ aU

coshðlkðz þ hÞÞ expðiv"1 b0 Þ; coshðlkhÞ

where kðxÞ is given by the dispersion expression (24), b0 ðxÞ satisfies

^b & $ ^ b ¼ k2 i:e: $b & $b ¼ v2 k2 ; $ 0 0 0 0

and the amplitude aU ðxÞ satisfies the Eq. (26). Besides, recalling expression (16), the free surface satisfies

g ¼ iag expðiðv"1 b0 ÞÞ; with ag ' xaU . In order to show the influence of the boundary layer on the wave propagation, let us focus on the behavior of ag in the one dimensional case. According to the expression (26), and recalling Eq. (15),

(

)

^ b ¼ "v $ & a2g ccg $ 0

! " 1þi I pffiffiffiffiffiffiffi # ; 2x cosh ðlkhÞ 2

a2g k 2

^ b ¼ @b =@ ^ so that in the one dimensional case, being $ x ¼ k and ck ¼ x, 0 0

! " 2 a2g k @ ( 2 ) 1þi I pffiffiffiffiffiffiffi # ; ag cg ¼ "v 2 @x 2x xcosh ðlkhÞ

or, alternatively

pffiffiffiffi ! " a2g G x @ ( 2 ) 1þi I pffiffiffi # ; ag cg ¼ "v h @x 2

ð29Þ

! " pffiffiffiffiffiffiffiffiffiffi a02 @ ( 02 0 ) 1þi gG ag cg ¼ " m00 x0 0 I pffiffiffi # ; 0 @x h 2

ð30Þ

with G defined in Eq. (27). In dimensional form, expression (29) reads

where m00 is the already mentioned characteristic kinematic eddy viscosity (or the viscosity if it is considered constant). We remark that the RHS stands for the damping. For the laminar case, # ¼ 1, the above expression is equivalent to the results 0 0 presented in [2]. According to the definition of G, this term rapidly decreases in deep water ðk h ) 1Þ. If viscous effects 0 c ¼ ctt is recovered. are ignored, the well know energy conservation expression a02 g g


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G. Simarro et al. / Wave Motion 46 (2009) 489–497

10

bottom (m) group celerity cg (m/s)

5

0

−5

−10 0

100

200 300 x position (m)

400

500

Fig. 3. Evolution of the group celerity across 500 m propagation. The incident wave is composed only by one component with a period of 8 s.

normalized amplitude (−)

1.4

without boundary layer with boundary layer

1.3

1.2

1.1

1 0

100

200 300 x position (m)

400

500

Fig. 4. Normalized wave amplitude along the distance considering the damping term in Eq. (30) (solid line) and neglecting the viscous effects (dashed line).

Further, potential and free surface Fourier components read, in dimensional form and expressed using the wave amplitude a0g ,

U0 ¼

g0

x

0

& &Z '' 0 0 a0g f exp i k dx þ d ;

ð31aÞ

and

& &Z

g0 ¼ ia0g exp i

0

0

k dx þ d

'' :

ð31bÞ

From the potential, the horizontal velocity is

0 ( ) 1 & &Z '' 0 0 @ a0g f @ U g 0 0 A 0 0 u0 ' 0 ¼ 0 @ þ ik a f exp i k dx þ d : g 0 @x @x x

ð32Þ

Comparing expressions (32) and (31b), velocity and free surface elevation are slightly out of phase due to the term

( ) @ a0g f @x0

;

that stands for both, depth variations and bottom boundary layer effects. To illustrate some of the above results, we shall focus on the constant viscosity case (i.e., laminar or constant eddy viscosity). For constant eddy viscosity, m00 is a constant given value and, according to expression (11), # ¼ 1. We consider a monochromatic wave train with a period of 8 s propagating over a 500 m distance (Fig. 3), the depth varying linearly from 10 to 2 m. Fig. 3 shows also the evolution of c0g , since it is important in shoaling aspects according to expression (30). Note that, because c0g decreases in the propagation direction, shoaling is expected. Fig. 4 compares the results for the wave amplitude obtained with and without boundary layer. A constant eddy viscosity m00 ¼ 10"4 m2 =s has been used. As depicted from the figure, boundary layer reduces the wave amplitude as long as waves propagates: in this case, however, shoaling due to depth variations are stronger than boundary layer effects.


G. Simarro et al. / Wave Motion 46 (2009) 489–497

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5. Concluding remarks A formulation for linear wave propagation with boundary layer effects has been presented. The boundary layer effects are introduced in the wave propagation boundary value problem through a modification of the bottom boundary layer. The model can be applied with either laminar or turbulent eddy viscosity. The boundary layer effects appear as a modification of the energy equation where a new term accounts for the effects of the bottom friction. The results have been here presented in a one dimensional case, but it can be easily implemented to two dimensional linear wave propagation problems in an efficient computational way. Acknowledgement The authors thank financial support from MEC thought Project CTM2006-12072. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

N. Booij, Gravity Waves on Water with Non-Uniform Depth and Current. Ph.D. Thesis, Technical University of Delft, The Netherlands, 1981. R.A. Dalrymple, J.T. Kirby, P.A. Hwang, Wave diffraction due to areas of energy dissipation, J. Waterway Port Coast. Ocean Eng. 110 (1984) 67–79. M.W. Dingemans, Water Wave Propagation over Uneven Bottoms, World Scientific, Singapore, 1997. J. Fredsoe, R. Deigaard, Mechanics of Coastal Sediment Transport, World Scientific, New York, New York, 1992. W.D. Grant, O.S. Madsen, Combined wave and current interaction with a rough bottom, J. Geophys. Res. 84 (C4) (1979) 1797–1808. K. Kajiura, On the bottom friction in an oscillatory current, Bull. Eqrthquake Res. Int. 42 (1964) 147–174. P.L.-F. Liu, A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech. 520 (2004) 83–92. P.L.-F. Liu, G. Simarro, J. Vandever, A. Orfila, Experimental and numerical investigations of viscous effects on solitary wave propagation in a wave tank, Coast. Eng. 520 (2-3) (2006) 181–190. P. Nielsen, Coastal Bottom Boundary Layers and Sediment Transport, World Scientific, Singapore, 1992. A. Orfila, G. Simarro, P.L.-F. Liu, Bottom frictional effects on periodic long wave propagation, Coast. Eng. 54 (11) (2007) 856–864. G. Simarro, A. Orfila, Boundary Layer Effects on the Propagation of Weakly Nonlinear Long Waves. In Nonlinear Wave Dynamics, World Scientific, 2009. G. Simarro, A. Orfila, P.L.-F. Liu, Bed shear stress under wave-current turbulent boundary layer, J. Hydraul. Eng. 134 (2008) 2, 225–23. I.A. Svendsen, Introduction to nearshore hydrodynamics, Advanced Series on Ocean Engineering, World Scientific, Singapore, 2005.


Under review in Geophysical GEOPHYSICAL RESEARCH LETTERS, VOL. ???, XXXX, DOI:10.1029/, Research Letters Mixing rise in coastal areas induced by waves A. Galan,1 A. Orfila,2 G. Simarro,3 C. Lopez4, and I. Hernandez-Carrasco5 We study the horizontal surface mixing and the transport induced by waves using local Lyapunov exponents and high resolution data from numerical simulations of waves and currents. By choosing the proper spatial (temporal) parameters we compute the Finite Size and Finite Time Lyapunov exponents (FSLE and FTLE) focussing in the local stirring and diffusion inferred from the Lagrangian Coherent Structures (LCS). The methodology is tested by deploying a set of eight lagrangian drifters and studying the path followed in front of LCS derived under current field and wave and currents.

Finite Time Lyapunov Exponent (FTLE) for a fluid particle located initially in a position x at time t0 with a finite time integration T is given by, λT (x, t) =

δx(t0 + T ) 1 ln , |T | δx(t0 )

(1)

where δx(t0 + T ) is the distance between the particle and a neighboring particle advected by the flow after time T . Analogous, the Finite Size Lyapunov Exponent (FSLE) for two particles initially separated a distance δ0 is, λδf (x, t, δ0 , δf ) =

1. Introduction

δf 1 ln , |τ | δ0

(2)

being τ the average time it takes the two particles separated by an initial distance δ0 to reach a separation of δf . In FTLE, one has to fix a finite integration time while in FSLE the initial and final distance between two adjacent particles (δ0 , δf ) are fixed. This two magnitudes have to be equivalent if they are measured with the corresponding time T and distance δf . Both exponents can be integrated forward and backwards in time providing information of the barriers, boundaries or lines of strong stretching, by means of LCS. Under a transport perspective these LCS provide information about those areas where trajectories of initially close particles are quickly separated or particles of different origin attracted. In this work we analyze the influence of waves on the modification on the LCS in a coastal area. FSLE are derived from ocean circulation model as well as from wave model and results are compared with available data from drifters.

Transport, dispersion and mixing of coastal waters are of crucial interest due to the ecological and economical importance of these areas. Despite the increasing advance in the scientific description of the physical processes that take place in the coastal ocean a high degree of uncertainty still remains when predicting trajectories of particles advected by the flow in coastal environments. On one hand coastal dynamics is influenced by deep water conditions over a complex topography and driven at the surface by highly variable (spatially and temporally) wind conditions. On the other hand the effects of wind generated waves modify the current field by the excess of momentum flux induced by waves. In summary, flows in coastal areas are the combination of currents with variations of hours or days and wave oscillatory flows with periods of seconds to tens of seconds. This interaction is usually accounted in coastal ocean models by the radiation stress concept [Longuet-Higgins and Stewart, 1964]. Trajectory of water particles have been studied exten¨ okmen, 2000; sively under a lagrangian point of view [Ozg¨ Molcard et al. , 2006]. Moreover, the stretching by advection is usually analyzed by means of the Lyapunov exponents [Haller and Yuan, 2000; Mancho et al., 2008; HernandezCarrasco et al., 2011]. Local finite-time Lyapunov exponents are the exponential rate of separation, averaged over infinite time, of fluid parcels separated infinitesimally [d’Ovideo et al., 2004]. These lagrangian descriptors provide the existence of patterns that are a proxy of the whole flow [Wiggins, 1992]. Mixing properties of passive tracers in time dependent flows, depend on the chaotic nature of the Lagrangian particle trajectories [Lapeyre, 2002]. These structures are known as Lagrangian Coherent Structures (LCS).

2. Data and Methods The study has been performed in a semi enclosed bay located in the southern side of the Island of Mallorca, Western Mediterrenean Sea (Figure 1). Velocity data was obtained from the Regional Ocean Model System (ROMS), a free-surface, hydrostatic, primitive equation ocean model that uses stretched, terrain-following coordinates in the vertical and orthogonal coordinates in the horizontal [Song and Haidvogel , 1994]. Three different domains were implemented in order to obtain high resolution currents in the area of study. The coarser mesh with a resolution of dθ = dλ = 1/74◦ (e.g. dx � dy = 1500m) take boundary conditions from an operational general ocean circulation model (MFSTEP). This domain, is nested to a second domain with a mesh of dx = dy = 300 m and the later to a third domain covering the study area which has a grid resolution of dx = dy = 75 m (Figure 1, right). This area is around 18 km wide with depths at its open boundary around 80 m. The Bay is open to southerly to southwesterly swells. The final grid is 348 × 260 nodes with 10 vertical levels . All domains were forced using wind provided by the PSU/NCAR mesoscale model MM5 [Grell et al., 1995]. The study was done from November 10th to November 24th 2009. During this period two different simulations

1 ETS. Caminos. Universidad de Castilla la Mancha, 13072 Ciudad Real, SPAIN. IMEDEA(CSIC-UIB), 07190 Esporles, SPAIN 2 IMEDEA(CSIC-UIB), 07190 Esporles, SPAIN. Corresponding author 3 Institut de Ciencies del Mar ICM (CSIC), 08003 Barcelona, SPAIN. 4 IFISC (CSIC-UIB). 07122 Palma de Mallorca, SPAIN. 5 IFISC (CSIC-UIB). 07122 Palma de Mallorca, SPAIN.

Copyright 2011 by the American Geophysical Union. 0094-8276/11/$5.00

1


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GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS

34’

#75m 40’

32’

30’

#75m 20’

30

40

10

#300m

10

20

39oN 30.00’

10’ 28’

50

39oN #1500m 2oE

20’

40’

3oE

20’

40’

26’

10

2oE 39.00’

30

50

36’

20

60

70

40

80 33’

42’

45’

Figure 1. Area covering the domains for the wave and current models (left). Fine resolution grid detail (right).

were performed by forcing the ocean model with realistic wind fields (hereinafter set I) and forcing the model with the same wind fields plus the additional gradients of the radiation stress tensor (hereinafter set II). The vertical structure of temperature and salinity was obtained from Levitus database [Locarnini et al., 2006; Antonov et al., 2006]. The radiation stresses were obtained in the same domains of the circulation model by integrating WAM model, a third generation spectral wave model specifically designed for global and shelf sea applications [Komen et al., 1994]. Boundary conditions for the first domain were taken from the operational model for the Western Mediterranean operated by the Spanish Harbor Authority. Although the wave model is not appropriate for very shallow waters since typical physical processes at those areas such as diffraction, triad-wave interactions or depth-induced wave breaking are not considered, the model provides for typical wavelengths the correct wave field for depths higher than 10m.

For set II, the effects of waves are included in the three domains by adding the gradient of the radiation stresses as an additional forcing term acting on the surface. Stresses are updated every three hours since it is a reasonable interval to characterize the wave climate. For both simulations, velocity fields were? stored every 5 minutes. On November 19th eight drifting buoys were deployed in the area of study for three days. The buoys were specifically designed for coastal studies and provide the position through a GPS position via GSM transmission every 5 minutes. Drifters were deployed at the vertex of a square in groups of 2 buoys. Deep water significant wave heights during this period reach 1m from the southwest.

3. Results We have performed simulations with the circulation model for the period considered storing surface velocity fields every 5 minutes. For both set I and set II the wind forcing and waves were updated every 3 hours. A grid of particles is launched every 3 hours for all the period. Particles are advected using a first order Euler algorithm which integrate velocity data from the numerical model in time. The FSLE at each point is defined as the maximum Lyapunov exponent of the 4 neighboring particles. On the other hand, the FTLE is computed using the spatial gradient of the flow at the four closest neighbors once the final position of each particle at the desired T has been reached [Shadden et al., 2005]. LCS are dynamical patters organizing the flow which can not be crossed by particles [Joseph and Legras , 2002]. Besides, LCS obtained from Lyapunov exponents integrated forward in time, characterize repelling material lines (unstable structures) while LCS provided by Lyapunov exponents integrated backward in time, identify lines of attracting material (stable structures) [d’Ovideo et al., 2004]. To

Figure 2. Finite Time Lyapunov Exponents for time integration T = 3h, T = 6h, T = 12hand T = 24h (top panel). Finite Size Lyapunov Exponents for final separation of δf = 150m, δf = 250m,δf = 500m and δf = 750m (bottom panel) (November 23, 2009; 18 : 00h). Positive values of the exponents correspond to forward integration and negative values to backward integration.


GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS show the general behavior of the Lyapunov exponents we performed for a given day the FTLE and FSLE forward and backward in time using the numerical simulations of set I. The FTLE for November 23rd 2009 at 18.00h computed for time integration of T = 3h, T = 6h, T = 12h and T = 24h are are shown in Figure 2 (top panels) where only the LCS defined as the maximum values of the FTLE are displayed. The structures with positive values correspond to FTLE integrated forward in time (hereinafter FTLEf) while structures with negative values to those exponents computed backwards in time (hereinafter FTLEb). As seen when time integration in small, many LCS are present in the flow which are filtered when longer times are used. Intersection of LCS forwards and LCS backwards correspond to hyperbolic points where stable and unstable directions of particles coexist. Analogous, the FSLE computed for final separation of particles of δf = 150m, δf = 250m, δf = 500m and δf = 750m are shown in Figure 2 (bottom panels) starting at November 23rd at 18.00h. Similarly the LCS of the FSLE with positive values correspond to those exponents computed forward in time (hereinafter FSLEf) and those with negative values to exponents computed backward in time (hereinafter FSLEb). As seen, FTLE for time integration T = 12h present a pattern of LCS similar to FSLE integrated for final separation of δf = 250m. Maximum values for the Lyapunov exponent are of the order of 2 days−1 corresponding to mixing times of 12h. It is noticeable that LCS from FSLE are more defined than those obtained by FTLE since for the latter, all particles of the domain will have a finite Lyapunov exponent but only those particles that reach the final δf will have a value of the Lyapunov exponent when FSLE are computed. Since we are interested in the mixing and presence of barriers of transport induced by waves, we will show only the LCS obtained from the FSLEb. The final distance considered is δf = 250m. which is roughly 3 mesh points. The influence of waves in the stirring of the surface layer in the study area is assessed by analyzing the LCS from periods with different wave conditions. Starting on November 15th relatively mild wave conditions were present in the area. For this period, we compute the LCS of the FSLEb for both set I ( Figure 3, left panel) and for set II (Figure 3, right panel). LCS are shown for November 15th to November 18th at 00.00h. Not surprisingly the structure of the LCS display a similar pattern for both set of data in this 4-days analyzed. A barrier parallel to the coast is almost present during all the period separating two areas in the shallow zone. Shallow areas are characterized by stronger mixing as seen by the complex pattern of the LCS. Mean values of the FSLEb for currents are −0.13, −0.17, −0.31 and −0.31 days−1 for November 15th , 16th , 17th and 18th respectively. For the same period these values for waves and currents are −0.18, −0.17, −0.29 and −0.31 days−1 . To show the utility of LCS in the study of the dispersion of surface material such as pollutants as well as the indication of such structures to be zones of attracting flow, we deployed 296 virtual neutral particles at four different locations. For the period comprised between November 15th 18th , the path followed general behaviour of lagrangian particles are similar for wind conditions and wind and waves conditions. It is noticeable that those particles located over LCS remain there and those particles located at different sides of the structures do not cross the LCS tending to the more attractive structures. LCS provide therefore a dynam-

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ical picture of the organization of the flow that is a powerful tool when analyzing areas of strong converge (divergence) of the flow.

Figure 3. Daily snapshots of Finite Size Lyapunov Exponents backward in time for a final separation of δf = 250m computed by currents (left panel) and with waves and currents (right panel) starting on November 15th . Virtual particles at four different areas are marked by circles, diamonds, squares and stars. A different situation is obtained for November 21st to November 24th . During this period, significant wave height (Hs ) measured at deep waters reached 1m. The effect of waves is to increase mixing and therefore to modify the transport at the surface. The LCS for this period are displayed in Figure 4 for set I (left) and for set II (right). LCS are displayed each day at 00.00h. Mean values for the wind stress during this period is 0.1N/m2 which is increased by the effects of waves up to 0.25N/m2 . The barrier that was located near the coast in mild wave conditions is moved onshore appearing new areas of strong mixing in the middle of the Bay.


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GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS

Figure 4. Daily snapshots of FSLEb for a final separation of δf = 250m computed by currents (left panel) and with waves and currents (right panel) starting on November 21th . Virtual particles at four different areas are marked by circles, diamonds, squares and stars. Contrarily to previous situation, the LCS from set I and the LCS from set II show a very different pattern. In general, the effect of waves is to break the LCS generating chaotic tangles that increase the horizontal stirring and mixing. The mean values of the FSLE for set I are −0.27, −0.31, −0.27 and −0.40 days−1 for currents at November 21th , 22th , 23th and 24th respectively, being −0.31, −0.33, −0.30 and −0.42 days−1 for set II. Those data suggest that in this analized case the effects of waves is to increase mixing around a 10% respect the simulations made only with wind forcing. Again, we deployed 296 virtual particles at four different areas selected so as to be at both sides of the LCS of the initial day. On November 22nd particles launched at the flow from set I are located on areas characterized by low values of FSLEb implying small dispersion (Figure 4 left). The initial shape of the particles deployed are maintained after 24 hours. Nevertheless particles within the flow driven by set II are distributed in deep waters over a LCS that is the results of the waves that broke the original shape (Figure 4 right).

Figure 5. 6 hour snapshot of FSLEb for a final separation of δf = 250m computed by currents and trajectories of lagrangian drifters. Moreover two different sets collapsed to this line and finally got mixed (see diamonds and stars on Figure 4). One day after, on November 23rd , differences become more important. The two set of particles originally deployed at the east, have been moved to the center of the Bay since they have been attracted by the LCS which cross the bay in a south-north direction. At evening on November 23th wave energy decrease which can be inferred from the LCS displayed on the 24th . Both LCS fields present the same structure of the manifolds (Figure 4, bottom). However, due to the history of the dynamics, particles are disposed over different lines of attraction being particles from different sets totally mixed in set II (see circles, diamonds, squares and star at Figure 4, bottom left and right). To elucidate the role of the radiation stress from waves waves on the computation of LCS in coastal areas, eight lagrangian drifters were deployed on November 19th at 12.00. Figure 5 and Figure 6 display the LCS every 6 hours, starting the date of the deployment for set I and for set II, respectively, and trajectories followed by the buoys. The first 18


GALAN ET AL.: MIXING BY WAVES IN COASTAL AREAS hours waves were below 30 cm and it is reflected in the pattern of the LCS. Along this period, there is a clear separation between two areas of the Bay being the deeper part more diffusive than the shallow part due to the fact that dynamics within the Bay ia mainly driven by shelf slope dynamics and relatively mild small scale wind driven circulation acts in the Bay. Drifters at the shallow part travel small distances indicating high residence time of waters for this period. These buoys fall in small diffusive areas in both, set I (Figure 5) and set II (Figure 6). Drifters deployed at deeper waters, are attracted by the strong structure that cross the Bay form East to West moving along it. On November 20th at 6.00 Hs increased and differences between the two patterns of LCS became evident. A new line of attraction appeared for set II parallel to the main structure. This line is fully developed on November 20t h at 18:00 not being present when computing the LCS in absence of waves (Figure 5). This new structure continues evolving and moving onshore increasing

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mixing in set II. The path of the lagrangian drifters follow the LCS from FSLEb provided by set II.

4. Conclusions We have shown that either FTLE and FSLE can be used by choosing the proper scale to analyze the dynamical field of coastal areas. The LCS integrated backward in time, provide valuable information when trying to response and mitigate possible spills in these areas. However we have shown that a proper characterization of currents including the effects induced by waves is necessary in order to obtain a correct description of the mixing activity and the coherent structures that control the transport at the scale of interest. Acknowledgments. Authors would like to thank financial support from Spanish MICINN thought project CTM2010-16915 and from Med Project TOSCA (G-MED09-425).

References

Figure 6. 6 hour snapshot of FSLEb for a final separation of δf = 250m computed by currents and waves and trajectories of lagrangian drifters.

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Tesis Alvaro