Analysis of trust coefficient and propulsive efficiency of a single heaving airfoil

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 1

INTRODUCTION

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 1 1.

INTRODUCTION The oscillation of an airfoil-like appendage – ‘flapping’ – is a common

form of propulsion for many animals. Flying animals (birds, bats, and insects) generally use flapping to generate both thrust and lift, while many swimming species (fish, aquatic mammals) use flapping primarily to generate thrust.

Recently, much effort has gone into the design and construction of devices propelled by flapping, including robotic fish, ornithopters, and micro -aerial vehicles. One of the primary design challenges in these efforts has been the understanding of the unsteady fluid mechanics of flapping. The generation of thrust by an oscillating airfoil has a notable effect on the qualitative structure of the wake. Behind a slowly oscillating airfoil, the wake consists of a Karman vortex street similar to that behind a bluff body; the sense of rotation of the vortices is a result of the momentum deficit behind the airfoil, corresponding to net drag. Conversely, by oscillating the airfoil more energetically, the wake is ‘inverted’ and the vortices have the opposite sense of rotation to that behind a bluff body, indicative of a jet of fluid behind the airfoil and the development of thrust.

Thrust production by an oscillating airfoil was first described in the early part of the twentieth century. During 1900s scientists independently noted that the vertical motion of an airfoil produces an effective angle of attack so that the resulting normal force vector has a component in the forwa rd direction.

Experimentally, Freymuth (1988) demonstrated that an airfoil undergoing either pure pitching or pure heaving is capable of producing thrust. Thrust production due to pitching was also demonstrated by Koochesfahani (1989). Lai & Platzer (1999) and Jones, Dohring & Platzer (1996) used a water tunnel and laser Doppler velocimetry to obtain high-resolution velocity measurements in the wake of a purely heaving airfoil.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Their experimental results confirmed the calculations of Triantafyllou, Triantafyllou & Grosenbaugh (1993), which identified the Strouhal number, as an important parameter for thrust generation.

St = Strouhal number A = Heave amplitude

In our project we have used non-dimensional velocity instead of Strouhal number. Non dimensional velocity is defined as

Here “k” is reduced frequency and “h” is non-dimensional amplitude.

Lai & Platzer (1999) noted that the structure of the wake (drag or thrust) can be classified according to the non-dimensional velocity, with those motions below a threshold of kh≈0.02 producing drag and those above producing thrust.

Additionally, at large kh (kh>0.2), they observed that the wake b ecomes asymmetric, with the vortex street deflected from the mean position.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Fig.1.1. Drag producing wake for a stationary NACA0012 airfoil

Fig.1.2. Thrust producing wake for a stationary NACA0012 airfoil

1.1. Flapping wing aerodynamics Several experimental and numerical investigations have been performed on flow over flapping wings to understand the unsteady mechanisms of aerodynamic forces generation and also to determine the effects of varying different flapping parameters such as flow Reynolds Number (Re), reduced frequency, plunge amplitude, mode of motion, and phase difference between pitching and plunging motion. We have gathered here some of the recent computational and experimental studies of 2D pure plunging, pitching and 2D combined pitching and plunging motion. Firstly, the studies of pure plunging and pitching motion of 2D airfoil sections are discussed, followed by the discussion of combined pitching and plunging studies. The interest here is to determine the values of flapping frequen cy and flapping amplitude best suited to generate maximum thrust with reasonable efficiency or maximum efficiency with a reasonable thrust.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

1.1.1.

Pure heaving The case of 2D plunging motion has been considered by many researchers to

provide insight into thrust generation and propulsive efficiency (inversion of the vortex street, leading edge separation).

In 2004, Young and Lai numerically simulated the flow over a NACA0012 airfoil undergoing plunging motion at Re = 20,000. They studied the effect of leading edge flow separation on the force generation from a plunging airfoil at different k and h with kh = 0.6. They concluded that for thrust generation, constant kh cannot be a single controlling parameter, rather k and kh must be treated separately for k < 8.

Lewin and Hariri examined numerically the flow over a plunging airfoil with k = 2 to 10, maximum heave velocity kh = 0.8 to 1.5 at Re = 500 using Navier Stokes solver. For the range of parameters covered in their study, they reported aperiodic and asymmetric solutions but with negligible effect on the propulsive efficiency. They examined in detail the effect of k on the thrust generation and propulsive efficiency and found that the leading edge vortex has significant effects on force generation and efficiency of airfoil. This included negative interference between the leading and trailing edge vortices corresponding to low efficiency (Ρ < 0.7) at k = 3.333 for all kh values studied and also the separation of the leading edge vortex corresponding to a sudden drop in output power. They reported a rather low overall efficiency of 0.11 at k = 5.333 and kh = 1.2. The low efficiency (Ρ < 0.1) at high frequencies (k > 5.333) for all kh was due to interaction between the airfoil and the shed vortices inducing drag on the airfoil. At higher frequencies this effect becomes more prominent because of greater proximity of the vortices.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

1.1.2.

Pure pitching

Garrick’s analysis showed that pure pitching motion is not very attractive for thrust generation as it produces positive thrust only at relatively high frequencies (approximately k > 4) and Koochesfahani experimentally verified this result. More recently, Sarkar and Venkatraman investigated numerically the flow over an airfoil undergoing pure pitching motion using a 2D discrete vortex method. They showed that flapping parameters such as mean angle of attack and pitch axis location influence thrust force generation. However, the values of thrust force produced for pure pitching were as low as C Tmean < 0.1 for k > 12 and θo = 5o and were not useful for practical propulsive vehicles.

1.1.3.

Combined pitching and heaving The combined pitching and plunging motion is closer to the real mode of

flapping present in flying and swimming animals in nature, so it is necessary to analyse this mode of motion separately. Garrick calculated the forces not only for pure plunge and pure pitching cases, but also for combined pitching and plunging motion.

Anderson et al. performed experiments to investigate the effects of Strouhal number on the thrust generation and propulsive efficiency of a rigid NACA0012 airfoil undergoing combined plunging and pitching motion at Re = 1100 and 40000.

They reported CTmean ≈ 1 with η as high as 0.87 at kh = 0.56 (St = 0.36), h = 0.75, maximum angle of attack αmax =0.2o and φ = 75o. At low angle of attack (α = 5o), η was less than 0.1. The phase angle between pitching and plunging φ was identified as a critical parameter affecting the propulsive efficiency. Guglielmini and Blondeaux investigated the effect of different flapping parameters on the propulsive efficiency of a 2D elliptic foil undergoing combined pitching and heaving motion by means of numerical solution of the vorticity equation at Re = 1100. They found highest efficiency η = 0.45 for high pitching amplitudes between 30o to 40o and for kh = 0.94 -1.25 (St = 0.3 - 0.4). They also investigated the effect of varying φ on the propulsive efficiency and found that maximum efficiency is obtained at φ = 80o.

They showed an increase in the peak of efficiency around 20% from Re = 1100 to Re = 3300. The pivot position of the pitching axis was also varied from -0.5c to 2c Department of Aerospace Engineering, IIAEM, Jain University 6

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil to find its effect on propulsive efficiency. The location for maximum efficiency was reported to be 1/3c from the leading edge.

Read et al. and Schouveiler et al. experimentally investigated the flow over a NACA0012 airfoil undergoing combined pitching and plunging motion at Re = 40,000 with the pitch axis location at 1/3c, h = 0.75, φ = 90o. The effect of varying pitch amplitude θ = 10o - 40o and kh = 0.188 – 1.38 was studied. Read et al. reported maximum propulsive efficiency η = 0.715 at αmax= 15o and kh = 0.502 with CTmean = 0.18. More significant results (simultaneously combining high thrust and efficiency) reported were η = 0.556, CTmean = 0.79 at αmax = 20o, kh = 1.25 and η = 0.508, CTmean = 1.08 at αmax = 25o, kh = 1.38.

They found very high thrust coefficient CTmean = 2.43 with η = 0.49 at φ = 100o, kh = 1.88, αmax= 35o when higher-order heave motions were employed. They also reported that adding pitch bias angles to the pitching motion i.e., a mean pitching angle other than zero can produce a very strong side force which can be useful for manoeuvring.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 2

UNSTEADY FLOW

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 2 2.

UNSTEADY FLOW

In steady two processes may play significant roles: (1) The unsteady forces due to the trailing-edge vortex growth, (2) The shedding of the leading-edge vortex as a function of non dimenstional velocity .

Investigating these effects in a single stroke moving with velocity U at an angle of attack α, which serves as a snapshot of a flapping wing, with U the sum of u 0 and u 1 , α = tan -1 [u 1 (t)=u 0 ]. The lift is defined to be orthogonal to U, and its forward component corresponds to the instantaneous thrust in the flapping model.

2.1. Unsteady Forces To illustrate the unsteady behaviour, let’s compute the time -dependent lift on the wing after an impulsive start with an angle of attack α. Examining the lift coefficient C l , as a function of time.

Figure 2.1. shows a case with α = 40 o and Re = 1000.

Fig.2.1. Vorticity contour plot (a) and lift coefficient (b) on an impulsively started ellipse with an angle of attack 40_. Re = 1000. The inset shows the lift coefficient over a longer time period.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil -The force coefficient curve on the right side of fig ure 2.1. shows three different regions. Region A corresponds to the diffusion of the boundary layer vorticity immediately after an impulsive start and region B corresponds to the roll-up and the growth of the vortex sheet near the tips of the airfoil and t he formation of the attached leading-edge vortex. Remark that the transitional point between regions A and B gives a direct measure of the onset of the shedding.

Finally, in region C the force settles into a quasi-steady state as the vortices are convected downstream from the body. At a much later time, the leading -edge vortex interacts with the trailing-edge vortex and triggers the onset of the von Karman wake. The periodicity, however, is on the order of T = 10 as shown in the inset, which is much larger than the time scale of our interest, the flapping period. The relevant time scale for our problem is the characteristic time in region B, which is denoted τ, and is on the order of unity. Although the impulsively started flow imposes a special initial condition, τ depends mostly on Re and the geometry of the wing; thus it is expected to be of the same order of magnitude as in general flows.

A straightforward consequence of the observed unsteady effects is to modify the forces obtained from the quasi-steady analysis. More interestingly, the unsteady effect also sets an intrinsic time scale τ to be compared with the reduced frequency. Roughly speaking, the wing should flap slowly enough, compared to τ, to allow the growth of the vortex.

2.2. Leading Edge Vortex In figure 2.2 the pressure distribution for three typical cases are shown, α = 4:50, α = 42:50, and α= 720 at time t = 2.2. For each case, the corresponding vorticity configuration is shown. In the cases of the smallest and largest α, both leading and trailing-edge vortices are shed, while in the other case, the leading-edge vortex remains bound. The presence of this leading-edge vortex induces a lower pressure region, seen as a dip in the pressure the distribution, which enhances the lift. Although the three-dimensional leading-edge vortex is different, and has a strong component in the axial direction, applied the twodimensional potential theory to flows around a plate with the presence of a leading-edge vortex at an assumed position, and they found a substantial increase of the force.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Fig. 2.2 Vorticity contours (a) and pressure distribution (b) around a wing for different angle of attack Îą = [0;180_] correspond to the upper surface, and Îą = [180_; 360_] the lower surface.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

2.3. Optimal flapping explanation Returning to the flapping mode and interpret the observed optimal flapping. In the flapping motion, we are interested in the average thrust. The lift analysed in the previous section has a component in the forward direction which has to overcome the viscous drag in order to generate a net thrust. In figure 2.3 we show the thrust coefficient of an impulsively started flow at Re = 1000. The thrust exists only at or above a minimum angle of attack, α

min

= 20°, on figure 2.3(a). More interestingly, both lift and drag are

time-dependent, and the thrust only occurs in a time window ts ϵ (0:2; 1:5), or equivalently, t ϵ (0:4; 3). The lower bound can be associated with the time scale governing the growth of the trailing-edge vortex, hence the lift. The upper bound can be attributed to the time scale governing the shedding of the leading-edge vortex which reduces the lift. Each curve, corresponding to a given α, has a maximum |Cx|, whose dependence on α is shown on figure 2.3(b).Now we can imagine that during a flapping motion, the effective angle of attack increases from 00 to a maximum angle of attack over time T/4. In order to achieve a high thrust, there are two conditions. First, as suggested by figure 2.3, the maximum angle of attack should be in the range of 450-600 corresponding to kh between 0.5 and 0.85. Secondly, for each fixed kh, the wing stroke should be just long enough to stay inside the thrust window. Among the four different frequencies, f = 2 Hz falls outside the thrust window, and indeed we find a drag instead of a thrust; f = 0.5 Hz corresponds to a situation where the flap just reaches its max (Cx) at the end of the half-cycle to benefit from the whole thrust window to produce a large thrust. The other two cases correspond to the intermediate situation. To confirm this picture, we repeat similar experiments for different kh. The results are summarized on figure 2.4, where the absolute values of the thrust coefficient and effciency are plotted against kh. Note that for each fixed kh, |Cx| increases with kh as expected. Furthermore the maxima occur at kh 2.2 independent of kh at higher kh. This is consistent with figure 2.4, where we observe that the maximum thrust occurs at a time which is insensitive to kh for sufficiently high kh.

Based on the mechanisms discussed above, we can also speculate on the effect of non-zero mean angle of attack on frequency selection. In particular, in the case of small mean angle of attack and in the limit of linear approximations, the total force is simply rotated from that obtained in the zero mean angle of attack case; thus we expect that the Department of Aerospace Engineering, IIAEM, Jain University 12

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil selected frequency remains the same. On the other hand, in the case of large mean angle of attack, the dynamic stall would occur on a smaller time scale, and we expect that the selected frequency would shift toward a slightly higher value. These predictions will be tested in future studies. In general peaks of efficiency and thrust need not to coincide.

Here the efficiency shows similar behaviour to the thrust. The low values of thrust indicate that heaving motion alone at low Reynolds numbers is not the best way to generate thrust. Indeed heaving motion is mostly employed by fish, which have much larger Reynolds numbers, and insects typically use the combination of heaving and pitching motions.

Fig. 2.3(a) Thrust window for different angles of attack. Thrust is the projection of the total force in the mean flow direction.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Fig. 2.3(b) Thrust window for different angles of attack. Thrust is the projection of the total force in the mean flow direction.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 3

COMPUTATIONAL FLUID DYNAMICS

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 3 3.

COMPUTATIONAL FLUID D YNAMICS CFD or computational fluid dynamics is predicting what will happen,

quantitatively, when fluids flow, often with the complications of simultaneous flow of heat, mass transfer (eg perspiration, dissolution), phase change (eg melting, freezing, boiling), chemical reaction (eg combustion, rusting), mechanical movement (eg of pistons, fans, rudders), stresses in and displacement of immersed or surrounding solids. Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that use numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the millions of calculations required to simulate the interaction of fluids and gases with the complex surfaces used in engineering. Even with simplified equations and high -speed supercomputers, only approximate solutions can be achieved in many cases. Ongoing research, however, may yield software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is often performed using a wind tunnel with the final validation coming in flight test.

The most fundamental consideration in CFD is how one treats a continuous fluid in a discretized fashion on a computer. One method is to discretize the spatial domain into small cells to form a volume mesh or grid, and then apply a suitable algorithm to solve the equations of motion (Euler equations for inviscid and Navier Stokes equations for viscous flow). In addition, such a mesh can be either irregular (for instance consisting of triangles in 2D, or pyramidal solids in 3D) or regular; the distinguishing characteristic of the former is that each cell must be stored separately in memory. Where shocks or discontinuities are present, high resolution schemes such as Total Variation Diminishing (TVD), Flux Corrected Transport (FCT), Essentially Non Oscillatory (ENO), or MUSCL schemes are needed to avoid spurious oscillations (Gibbs phenomenon) in the solution. If one chooses not to proceed with a mesh-based method, a number of alternatives exist, notably Smoothed particle hydrodynamics (SPH), a Lagrangian method of solving fluid Department of Aerospace Engineering, IIAEM, Jain University 16

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil problems, Spectral methods, a technique where the equations are projected onto basis functions like the spherical harmonics and Chebyshev polynomials, Lattice Boltzmann methods (LBM), which simulate an equivalent mesoscopic system on a Cartesian grid, instead of solving the macroscopic system (or the real microscopic physics). It is possible to directly solve the Navier- Stokes equations for laminar flows and for turbulent flows when all of the relevant length scales can be resolved by the grid (a direct numerical simulation). In general however, the range of length scales appropriate to the problem is larger than even today's massively parallel computers can model. In these cases, turbulent flow simulations require the introduction of a turbulence model. Large eddy simulations (LES) and the Reynolds averaged Navier-Stokes equations (RANS) formulation, with the k-Îľ model or the Reynolds stress model, are two techniques for dealing with these scales. In many instances, other equations are solved simultaneously with the Navier -Stokes equations.

These other equations can include those describing species concentration (mass transfer), chemical reactions, heat transfer, etc. More advanced co des allow the simulation of more complex cases involving multi-phase flows (e.g. liquid/gas, solid/gas, liquid/solid), non-Newtonian fluids (such as blood), or chemically reacting flows (such as combustion).

3.1 Discretization Methods In CFD The stability of the chosen discretization is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretization handles discontinuous solutions gracefully.

The Euler equations and Navier-Stokes equations both admit shocks, and contact surfaces.

Some of the discretization methods being used are: â&#x20AC;˘ Finite volume method (FVM). This is the "classical" or standard approach used most often in commercial software and research codes. The governing equations are solved on discrete control volumes. FVM recasts the PDE's (Partial Differential Department of Aerospace Engineering, IIAEM, Jain University 17

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil Equations) of the N-S equation in the conservative form and then discretize this equation. This guarantees the conservation of fluxes through a particular control volume. Though the overall solution will be conservative in nature there is no guarantee that it is the actual solution. Moreover this method is sensitive to distorted elements which can prevent convergence if such elements are in critical flow regions. This integration approach yields a method that is inherently conservative (i.e. quantities such as density remain physically meaningful • Finite element method (FEM). This method is popular for structural analysis of solids, but is also applicable to fluids. The FEM formulation requires, however, special care to ensure a conservative solution. The FEM formulation has been adapted for use with the Navier-Stokes equations. Although in FEM conservation has to be taken care of, it is much more stable than the FVM approach. Subsequently it is the new direction in which CFD is moving. Generally stability/robustness of the solution is better in FEM though for some cases it might take more memory than FVM methods.

• Finite difference method. This method has historical importance and is simple to program. It is currently only used in few specialized codes. Modern finite difference codes make use of an embedded boundary for handling complex geometries making these codes highly efficient and accurate. Other way s to handle geometries are using overlapping-grids, where the solution is interpolated across each grid.

• Boundary element method. The boundary occupied by the fluid is divided into surface mesh.

• High-resolution schemes are used where shocks or discontinuities are present.

To capture sharp changes in the solution requires the use of second or higher order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of flux limiters to ensure that the solution is total variation diminishing.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

3.2 How Is The Working Done In CFD Working in CFD is done by writing down the CFD codes. CFD codes are structured around the numerical algorithms that can be tackle fluid problems. In order to provide easy access to their solving power all commercial CFD packages include sophisticated user interfaces input problem parameters and to examine the results. Hence all codes contain three main elements:

1. Pre-processing.

2. Solver 3. Post - processing.

3.2.1 PRE-PROCESSING Preprocessor consists of input of a flow problem by means of an operator friendly interface and subsequent transformation of this input into form of suitable for the use by the solver.

The user activities at the Pre-processing stage involve:

1) Definition of the geometry of the region: The computational domain. Grid generation is the subdivision of the domain into a number of smaller, no overlapping sub domains (or control volumes or elements Selection of physical or chemical phenomena that need to be modelled).

2) Definition of fluid properties: Specification of appropriate boundary conditions at cells, which coincide with or touch the boundary. The solution of a flow problem (velocity, pressure, temperature etc.) is defined at nodes inside each cell. The accuracy of CFD solutions is governed by number of cells in the grid. In general, the larger numbers of cells better the solution accuracy. Both the accuracy of the solution & its cost in terms of necessary computer hardware & calculation time are dependent on the fineness of the grid. Efforts are underway to develop CFD codes

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil with a (self) adaptive meshing capability. Ultimately such programs will automatically refine the grid in areas of rapid variation.

3.2.2 SOLVER These are three distinct streams of numerical solutions techniques: finite difference, finite volume& finite element methods. In outline the numerical methods that form the basis of solver performs the following steps:

1) The approximation of unknown flow variables are by means of simple functio ns

2) Discretization by substitution of the approximation into the governing flow equations & subsequent mathematical manipulations.

3.2.3 POST-PROCESSING As in the pre-processing huge amount of development work has recently has taken place in the post processing field. Owing to increased popularity of engineering work stations, many of which has outstanding graphics capabilities, the leading CFD are now equipped with versatile data visualization tools.

These include:

1) Domain geometry & Grid display 2) Vector plots 3) Line & shaded contour plots 4) 2D & 3D surface plots 5) Particle tracking 6) View manipulation (translation, rotation, scaling etc.)

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 4

IMMERSED BOUNDARY METHOD

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 4 4.

IMMERSED B OUNDARY METHOD The term “immersed boundary method” was first used in reference to a

method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids. These methods have been extended to simulate unsteady viscous flows and thus have capabilities similar to those of IB methods. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all. The interaction equations of the numerical scheme involve a smoothed approximation to the Dirac delta function, constructed according to certain principles which were first discussed by Peskin.

Consider the simulation of flow past a solid body shown in Figure 1a.

The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps.

First, a surface grid covering the boundaries Γb is generated. This is then used as a boundary condition to generate a grid in the volume Ωf occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines. Because the grid conforms to the surface of the body, the Department of Aerospace Engineering, IIAEM, Jain University 22

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil transformed equations can then be discretized in the computational domain with relative ease. If a finite-volume technique is employed, then the integral form of the governing equations is discretized and the geometrical information

regarding the grid is incorporated directly into the discretization.

Figure 4.1 (a) Schematic showing a generic body past which flow is to be simulated. The body occupies the volume Ίb with boundary Î&#x201C;b. The body has a characteristic length scale L, and a boundary layer of thickness δ develops over the body. (b) Schematic of body immersed in a Cartesian grid on which the governing equations are discretized.

If an unstructured grid is employed, then either a finite-volume or a finiteelement methodology can be used. Both approaches incorporate the local cell geometry into the discretization and do not resort to grid transformations.

Now consider employing a nonbody conformal Cartesian grid for this simulation, as shown in Figure 1b. In this approach the IB would still be represented through some means such as a surface grid, but the Cartesian volume grid would be generated with no regard to this surface grid. Thus, the solid boundary would cut through this Cartesian volume grid. Because the grid does not conform to the solid boundary, incorporating the boundary conditions would require modifying the equations in the vicinity of the boundary. However, assuming that such a procedure is available, the governing equations would then be discretized using a finite Department of Aerospace Engineering, IIAEM, Jain University 23

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil difference, finite-volume, or a finite-element technique without resorting to coordinate transformation or complex discretization operators.

At this point it is useful to compare the advantages and disadvantages of these two approaches. Clearly, imposing the boundary conditions is not straightforward in IB methods and, furthermore, the ramifications of the boundary treatment on the accuracy and conservation properties of the numerical scheme are not obvious. In addition, alignment between the grid lines and the body surface in body-conformal grids allows better control of the grid resolution in the vicinity of the body and this has implications for the increase in grid size with increasing Reynolds number. For example, consider the case of the two-dimensional (2D) body of characteristic length L in Figure 1a with a boundary layer of thickness δ wherein it is required to provide an average grid spacing of Δn and Δt in the directions normal and tangential to the body surface. It is now easy to show that for moderately high Reynolds numbers for which δ<<L, the size of a body-conformal grid scales as (L/ Δt )(δ/ Δn ), whereas that of a Cartesian grid scales as (L2/ Δn2). Assuming further that Δn ∝ δ and Δt ∝ L, we find that the ratio of sizes for a Cartesian to a bodyconformal grid will scale as (L/δ) 2. For a laminar boundary layer, (L/δ) ∝ Re0.5, which implies that the grid-size ratio will scale with (Re)1.0 for 2D bodies. For a three-dimensional (3D) body, it can similarly be shown that the grid-size ratio would scale with (Re)1.5. Thus, as the Reynolds number increases, the size of a Cartesian grid increases faster than a corresponding body-conformal grid. However, note that this faster increase in grid size does not necessarily imply a corresponding increase in the computational cost because a substantial fraction of the grid points can be inside the solid body where the fluid flow equations need not be solved. This fraction is proportional to the ratio of the volume of the solid to the volume of the bounding box for the body and therefore depends on the shape and orientation of the body. Furthermore, in comparison with structured curvilinear body-formal grids, using a Cartesian grid can significantly reduce the per-grid-point operation count due to the absence of additional terms associated with grid transformations. When compared with unstructured grid methods, the Cartesian grid–based IB method retains the advantage of being amenable to powerful line-iterative techniques and

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil geometric multigrid methods, which can also lead to a lower per-grid-point operation count.

The primary advantage of the IB method is associated with the fact that the task of grid generation is greatly simplified. Generating body-conformal structured or unstructured grid is usually very cumbersome. The objective is to construct a grid that provides adequate local resolution with the minimum number of total grid points. For anything but the simplest geometries, these conflicting requirements can lead to a deterioration in grid quality, which can negatively impact the accuracy and the convergence properties of the solver (Ferziger & Peric 1996). Thus, even for simple geometries, generating a good-quality body-conformal grid can be an iterative process requiring significant input from the person generating the grid. As the geometry becomes more complicated, the task of generating an acceptable grid becomes increasingly difficult. Within the structured grid approach, complex geometries are often handled by decomposing the volume into subdomains and generating a separate grid in each subdomain (Quarteroni & Valli 1999). Apart from the complexity that is introduced into the solution algorithm due to the presence of multiple subdomains, grid smoothness can deteriorate at the interface between subdomains. The unstructured grid approach is inherently better suited for complex geometries, but here, too, grid quality can deteriorate with increasing complexity in the geometry. In contrast, for a simulation carried out on a non-body conformal Cartesian grid, grid complexity and quality are not significantly affected by the complexity of the geometry.

The advantage of the Cartesian gridâ&#x20AC;&#x201C;based IB method also becomes eminently clear for flows with moving boundaries. Simulating such flows on body conformal grids requires generating a new grid at each time step as well as a procedure to project the solution onto this new grid (Tezduyar 2001). Both of these steps can negatively impact the simplicity, accuracy, robustness, and computational cost of the solution procedure, especially in cases involving large motions. In contrast, including body motion in IB methods is relatively simple due to the use of a stationary, nondeforming Cartesian grid. Thus, despite the significant progress made in simulating flow with moving boundaries on body-conformal grids (Baum et al. 1998, Ramamurti&Sandberg 2001, Tezduyar 2001), due to its inherent Department of Aerospace Engineering, IIAEM, Jain University 25

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil simplicity, the IB method represents an extremely attractive alternative for such flows.

4.1. IMPOSITION OF BOUNDARY CONDITION ON IMMERSED BOUNDARIES Imposition of boundary conditions on the IB is the key factor in developing an IB algorithm. It is also what distinguishes one IB method from another. Consider the simulation of incompressible flow past the body in Figure 1b, which is governed by the following equations:

where

is the fluid velocity, p is the pressure, and ρ and μ the density and viscosity,

respectively. The solid body occupies the domain Ωb, with boundary denoted by Γb , and the surrounding fluid domain denoted by Ωf. The outer boundary of the flow domain is disregarded for the purposes of this discussion. For ease of discussion, the coupled system of momentum and continuity equations can be notionally written as

where U = ( , p) and L is the operator representing the Navier-Stokes equations as in Equation 1. It should be noted that in the context of the incompressible NavierStokes equations, pressure is determined by the continuity constraint and, consequently, the continuity equation is considered an implicit equation for pressure. A number of numerical integration schemes such as fractional-step (Chorin 1968) and SIMPLE (Patankar 1980) also explicitly derive and solve a Poisson Department of Aerospace Engineering, IIAEM, Jain University 26

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil equation for pressure, which, depending on the particular implementation, also requires appropriate boundary conditions.

Conventional methods proceed by developing a discretization of Equation 3 on a body-conformal grid where the boundary condition (Equation 4) on the immersed boundary Γb is enforced directly. In an IB method, Equation 1 would be discretized on a nonbody conformal Cartesian grid and the boundary condition would be imposed indirectly through modification of Equation 3. In general, the modification takes the form of a source term (or forcing function) in the governing equation that reproduces the effect of boundary. Introduction of a forcing function into the governing equations can be implemented in two different ways and this leads to a fundamental dichotomy in IB methods. In the first implementation, the forcing function, denoted here by f

b

, is included into the continuous governing

Equation 3 leading to the equation L(U) = f domain (Ωf + Ωb ). Note that f b = (

m

b

, f p ) where

, which then applies to the entire m

and f p are the forcing functions

applied to the momentum and pressure, respectively. This equation is subsequently discretized on a Cartesian grid, leading to the following system of di screte equations:

[L]{U}= f

b

(5)

and this system of equations is solved in the entire domain.

In the second approach, the governing equations are first discretized on a Cartesian grid without regard to the immersed boundary, resulting in the set of discretized equations [L] {U} = 0. Following this, the discretization in the cells near the IB is adjusted to account for its presence, resulting in a modified system of equations [L']{U} = {r }, which are then solved on the Cartesian grid. In this equation, [L'] is the modified discrete operator and {r } represents known terms associated with the boundary conditions on the immersed surface. The above system of equations can be rewritten as [L]{U} = { f 'b },

(6)

Department of Aerospace Engineering, IIAEM, Jain University 27

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil where { f 'b } = {r}+[L]{U}−[L']{U}. Comparing Equations 5 and 6 clearly shows the connection between the two approaches. In the first approach, which we term “continuous forcing approach,” the forcing is incorporated into the continuous equations before discretization, whereas in the second approach, which can be termed the “discrete forcing approach,” the forcing is introduced after the equations are discretized. An attractive feature of the continuous forcing approach is that it is formulated independent of the underlying spatial discretization. On the other hand, the discrete forcing approach very much depends on the discretization method.

However, this allows direct control over the numerical accuracy, stability, and discrete conservation properties of the solver.

4.1.1.

CONTINUOUS FORCING APPROACH

Among existing methods in this category, elastic and rigid boundaries require different treatments and we discuss these separately in this section.

4.1.1.1.

Flows with Elastic Boundaries

The original IB method by Peskin (1972, 1981) was developed for the coupled simulation of blood flow and muscle contraction in a beating heart and is generally suitable for flows with immersed elastic boundaries. In this method the fluid flow is governed by the incompressible Navier-Stokes equations and these are solved on a stationary Cartesian grid. The IB is represented by a set of elastic fibers and the location of these fibers is tracked in a Lagrangian fashion by a collection of massless points that move with the local fluid velocity. Thus, the coordinate

k

of

the kth Lagrangian point is governed by the equation

Department of Aerospace Engineering, IIAEM, Jain University 28

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil The stress (denoted by

) and deformation of these elastic fibers is related by a

constitutive law such as the Hookeâ&#x20AC;&#x2122;s law. The effect of the IB on the surrounding fluid is essentially captured by transmitting the fiber stress to the fluid through a localized forcing term in the momentum equations, which is given by

where δ is the Dirac delta function. Because the location of the fibers does not generally coincide with the nodal points of the Cartesian grid, the forcing is distributed over a band of cells around each Lagrangian point (see Figure 2 a) and this distributed force is imposed on the momentum equations of the surrounding nodes. Thus, the sharp delta function is essentially replaced by a smoother distribution function, denoted here by d, which is suitable for use on a discrete mesh. Due to the fibers, the forcing at any grid point xi, j is then given by

Figure 2 (a) Transfer of forcing

k

from Lagrangian boundary point (

k)

to

surrounding fluid nodes. Shaded region signifies the extent of the force disctribution.(b) Distribution functions employed in various studies.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

The fiber velocity in Equation 7 is also obtained through the use of the same smooth function. The choice of the distribution function d is a key ingredient in this method.

4.1.1.2.

Flows with Rigid Boundaries

The previous method is naturally well suited for elastic bodies but its application to flows with rigid bodies poses problems because the constitutive laws used for elastic boundaries are not generally well posed in the rigid limit. This problem could be circumvented by considering the body to be elastic but extremely stiff. A second approach is to consider the structure attached to an equilibrium location (Beyer & Leveque 1992, Lai & Peskin 2000) by a spring with a restoring force

given by

where κ is a positive spring constant and

e

k

is the equilibrium location of the kth

Lagrangian point. Accurately imposing the boundary condition on the rigid IB requires large values of κ. However, this results in a stiff system of equations that is subject to severe stability constraints (Lai & Peskin 2000, Stockie &Wetton 1998).

On the other hand, lower values of κ can lead to spurious elastic effects such as excessive deviation from the equilibrium location as noted in the low Reynolds cylinder wake simulations of Lai & Peskin (2000).

The above approach can be viewed as a specific version of the model developed by Goldstein et al. (1993) to simulate the flow around rigid bodies. In this model, the effect of the rigid body on the surrounding flow is modelled through a forcing term

Department of Aerospace Engineering, IIAEM, Jain University 30

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil where the coefficients α and β are selected to best enforce the boundary condition at the immersed solid boundary. The original intent behind Equation 11 was to provide feedback control of the velocity near the surface (Goldstein et al. 1993), but from a physical point of view, it can also represent a damped oscillator (Iaccarino & Verzicco 2003). This method has been used to simulate start-up flow past a circular cylinder at a moderate Reynolds number and a low Reynolds number turbulent flow in a channel with streamwise grooves (Goldstein et al. 1993). In general, results are promising at low Reynolds numbers but accurately enforcing the boundary conditions requires large values of α and β, which can lead to stability problems, especially for highly unsteady flows.

4.1.1.3.

General Considerations

The continuous forcing approach is very attractive for flows with immersed elastic boundaries. For such flows, the method has a sound physical basi s and is simple to implement. Consequently, many of the applications of these methods are found in biological and multiphase flows where elastic boundaries abound.

However, using this approach for flows with rigid bodies poses some challenges associated with the fact that the forcing terms used are generally not well behaved in the rigid limit. This problem is essentially tackled by employing simplified models that attempt to mimic the effect of the solid boundary on the flow. The parameters introduced in these models, however, have implications for numerical accuracy as well as stability. The smoothing of the forcing function inherent in these approaches also leads to an inability to provide a sharp representation of the IB and this can be especially undesirable for high Reynolds number flows. Finally, all of these methods require the solution of the governing equations inside the immersed body. As noted earlier, with increasing Reynolds numbers, the proportion of grid points inside the IB also increases, and the requirement of solving the equations inside the solid can be a burdensome overhead.

4.1.2.

Discrete Forcing Approach

In this section, methods are categorized into those that are formulated so as to impose the boundary condition on the immersed boundary through indirect means, and those that directly impose the boundary conditions on the IB. Department of Aerospace Engineering, IIAEM, Jain University 31

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

4.1.2.1.

Indirect BC Imposition

For a simple, analytically integrable, one-dimensional model problem, it is possible to formally derive a forcing term that enforces a specific condition on a boundary inside the computational domain (Beyer&Leveque 1992). The same is not usually feasible for the Navier-Stokes equations because the equations cannot be integrated analytically to determine the forcing function. Consequently, all the approaches in the previous section employ what are essentially simplified models of the required forcing. To avoid this issue, Mohd-Yosuf (1997) and Verzicco et al.

(2000) developed a method that extracts the forcing directly from the numerical solution for which an a priori estimate can be determined. Starting from the discretized Navier-Stokes equations without any modification due to the presence of the IB, and using the same notation introduced in section 2, the system [L] {U*} = 0 is solved at each time step where {U*} represents a prediction of the velocity field.

The forcing { f b} in Equation 6 is then defined as

where {r} = {U Γ} δ(|

k

̶

i,j

|) and [L'] = [L] + ([I] ̶ [L]) δ(|

k

̶

i,j

|), [I] being

the identity matrix. As before, the Dirac delta function is replaced by a smooth distribution function d and the Equation 6 for this method then becomes:

and this formally represents the enforcement of the boundary condition at location k

on the immersed surface.

The major advantage of the discrete forcing concept is the absence of user

specified parameters in the forcing and the elimination of associated stability Department of Aerospace Engineering, IIAEM, Jain University 32

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil constraints. However, the forcing still extends into the fluid region due to the use of a distribution function and the details of the implementation depend strongly on the numerical algorithm used to discretize the governing equations.

4.1.2.2.

Direct BC Imposition

Although the application of IB methods to low and moderate Reynolds number flows has been successful, their extension to higher Reynolds numbers is challenging due to the need to accurately resolve the boundary layers on (immersed) surfaces not aligned with the grid lines. In such cases the local accuracy of the solution assumes greater importance, and the spreading of the effect of the IB introduced by the smooth force distribution function is less desirable. For this reason, other approaches can be considered where the immersed boundary is retained as a “sharp” interface with no spreading and where greater emphasis is put on the local accuracy near the IB. This can usually be accomplished by modifying the computational stencil near the immersed boundary to directly impose the boundary condition on the IB. Here we describe two methods that fit into this category.

4.1.2.2.1.

Ghost-Cell Finite-Difference Approach

The boundary condition on the IB is enforced here through the use of “ghost cells.” Ghost cells are defined as cells in the solid that have at least one neighbour in the fluid. For instance, cell “G” in Figure 2 is a ghost cell. For each ghost cell, an interpolation scheme that implicitly incorporates the boundary condition on the IB is then devised. A number of options are available for constructing the interpolation scheme (Majumdar et al. 2001). One simple option is bilinear (trilinear in 3D) interpolation where a generic flow variable Φ can be expressed with reference to Figure 2 as Φ = C 1 x1 x2 + C 2 x1 + C 3 x 2 + C 4

(14)

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil The four coefficients in the above equation can be evaluated in terms of the values of Φ at fluid nodes F1, F2, and F3, and at the boundary point B2, which is the normal intercept from the ghost node to the IB. Boundary point B1, which is midway between points P1 and P2, can also be used instead of B2. Note that P1 and P2 are the intercepts with the y- and x-lines passing through the ghost point, respectively.

A less accurate, linear interpolation scheme (i.e., C1 ≡ 0 in Equation 14) would not employ the fluid node F3 and therefore would retain a discrete form, which is well suited for line-solution techniques (Ferziger & Peric 1996). Applying a linear reconstruction is acceptable for laminar flows or for high Reynolds number flows when the first grid point is located in the viscous sublayer (Iaccarino & Verzicco 2003). At high Reynolds numbers when the resolution is marginal, linear reconstruction could lead to erroneous predictions. For such cases higher-order interpolation can be used. For instance, one could employ an interpolant that is linear in the tangential direction and quadratic in the normal direction (Majumdar et al. 2001), such as Φ = C1n2 + C2nt + C3 n + C4t + C5,

(15)

where n and t are local coordinates normal and tangent, respectively, to the IB. The five coefficients can be determined by using the four fluid points values F1 to F4 and the boundary condition at point B2 where the selection of point F4 depends on the surface normal. Alternatively, the points F1 to F3 and the two boundary points P1 and P2 could be used without losing generality.

Irrespective of the particular interpolation scheme used, the value of the variable at the ghost-cell node, ΦG, can be expressed as ∑ɷi ɸi = ɸG ,

(16)

where the summation extends over all the points in the stencil, including one or more boundary points, and ωi are known geometry dependent coefficients. The above equation represents the modified discrete Equation 6 for the ghost cell and this can now be solved simultaneously with the discretized Navier-Stokes equations Department of Aerospace Engineering, IIAEM, Jain University 34

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil for fluid nodes. This method has been used for simulating a wide variety of flows including compressible flow past a circular cylinder and an airfoil (Ghias et al.

2004) at Reynolds numbers up to (10 5 ).

4.1.2.2.2.

Cut-Cell Finite-Volume Approach None of the IB methods discussed so far are designed to satisfy, the

underlying conservation laws for the cells in the vicinity of the IB. Strict globa l and local conservation of mass and momentum can only be guaranteed by resorting to a finite-volume approach and this is the primary motivation for the cut-cell methodology. This methodology was first introduced in the context of Cartesian grid methods for inviscid flow computations (Clarke et al. 1986) and was later applied to simulation of viscous flows (Udaykumar et al. 1996, 2001, 2002; Ye et al. 1999).

Figure 3a shows a schematic of a Cartesian grid with an IB that demarcates a solid from a fluid. In this method, cells in the Cartesian grid that are cut by the IB are identified, and the intersection of the boundary with the sides of these cut cells is determined. Next, cells cut by the IB, whose cell center lies in the fluid, are reshaped by discarding the portion of these cells that lies in the solid. Pieces of cut cells whose centers lie in the solid are absorbed by neighbouring cells. This results in the formation of control volumes, which are trapezoidal in shape (Ye et al. 1999), as shown in Figure 4.2.

Figure 4.2 Representation of the points in the vicinity of an immersed boundary used in the ghost-cell approach. Fi are fluid points, G is the ghost point, and Bi and Pi are locations where the boundary condition can be enforced.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Finite-volume discretization of the Navier-Stokes equations requires the estimation of mass, convective and diffusive flux integrals, and pressure gradients on the faces of each cell and the issue is to evaluate these on the cell faces of the trapezoidal cells. The approach proposed by Ye et al. (1999) is to express a given flow variable Φ in terms of a two-dimensional polynomial interpolating function in an appropriate region and evaluate the fluxes f based on this interpolating function.

For instance, to approximate the flux on the southwest face, fsw , Φ (in the shaded trapezoidal region shown in Figure 3b) is expressed in terms of a function that is linear in x1 and quadratic in x2:

where C1 to C6 are six unknown coefficients that can be expressed in terms of values of Φ at the six stencil points shown in Figure 3b and an expression similar to Equation 16 is developed for fsw . Equation 17 represents the most compact function that allows at least a second-order accurate evaluation of φ or its derivative at the sw location. A similar approach can be employed to evaluate the flux on the east -face fe as well as the interface flux fi . This approach results in a discretization scheme that is globally as well as locally second-order accurate and also satisfies conservation of mass and momentum exactly irrespective of the grid resolution.

This method has been used to simulate various flows with stationary and moving boundaries including flow-induced vibrations (Mittal et al. 2003), flapping foils (Mittal et al. 2002b), objects in free fall through a fluid (Mittal et al. 2004), and diaphragm-driven synthetic jets (Utturkar&Mittal 2002). Extending this approach to three dimensions, however, is nontrivial because the cut-cell procedure leads to complex polyhedral cells, and discretization of the full Navier-Stokes equations on such cells is complicated. Extension to three dimensions would likely be based on “cell-trimming” procedures (Berger & Aftosmis 1998) that generate body-fitted grids from a Cartesian grid.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

4.1.2.3.

General Considerations

The methods presented in this section and other related methods not discussed here introduce the boundary condition directly into the discrete equations. The forcing procedure is therefore intimately connected to the details of the discretizatio n approach and practical implementation is not as straightforward as the continuous forcing approach. However, discrete forcing enables a sharp representation of the IB, and this is desirable, especially at higher Reynolds numbers. Furthermore, the discrete forcing approach does not introduce any extra stability constraints in the representation of solid bodies. Finally, this approach decouples the equations for the fluid nodes from those for the nodes in the solid, thereby obviating the solutions of the governing equations for the solid grid nodes. This is highly desirable for high Reynolds number flows. As we discuss in the following section, one disadvantage of the discrete forcing approach is that inclusion of boundary motion can be more difficult. Finally, methods in this category also usually require imposition of a pressure boundary condition on the immersed boundary (see, for instance, Udaykumar et al. 2001), whereas no pressure boundary condition is needed for methods that employ continuous forcing.

4.2. Flows with Moving Boundaries In the context of flows with moving boundaries, most of the methods described here can be viewed as Eulerian-Lagrangian, wherein the Eulerian form of the governing equations (as in Equation 7) is solved on a stationary grid and moving boundaries are tracked in a Lagrangian fashion (as in Equation 7). The use of a stationary, non deforming grid and the associated retention of the Eulerian form of the governing equations greatly simplify the incorporation of moving boundaries into IB methods.

In contrast, Lagrangian methods have to deal with moving/deforming grids (Tezduyar 2001) as well as discretized equations that incorporate time derivatives of cell volumes and other grid-related quantities.

Further distinctions among these methods can be made based on the technique used to track the IB as well as the approach used to represent its effect on Department of Aerospace Engineering, IIAEM, Jain University 37

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil the underlying Eulerian flow-field variables. For instance, in Peskin’s IB method (1981), the boundary is tracked as a distinct and sharp Lagrangian entity while it is treated as diffuse in accounting for its effect on the fluid phase. In contrast, for methods such as cut cell and ghost cell, the IB is tracked as a sharp, Lagrangian entity and also treated as such when incorporating its effect on the fluid phase.

Using this taxonomy, IB methods can also be contrasted with so-called Eulerian methods such as Volume-of-Fluid (Anderson et al. 1998), which retain the diffuse nature of the interface both in tracking as well as representing i ts effect on the flow field.

For sharp-interface methods such as cut-cell and ghost-cell, one additional issue has to be dealt with in order to enable boundary motion. As shown in Figure 4, as the IB moves across the fixed Cartesian grid, “freshly-cleared” cells, i.e., cells in the fluid which were inside the solid at the previous time step, are encountered.

Figure 4.3 Schematics showing the key features of the cut-cell methodology. (a) Trapezoidal finite volume formed near the immersed boundary for which f denotes the face-flux of a generic variable. (b) Region of interpolation and stencil employed for approximating the flux fsw on the southwest face of the trapezoidal finite volume.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Figure 4.4 Schematic showing the creation of “freshly-cleared” cells on a fixed Cartesian grid due to boundary motion from time step (t −Δt) to t. Schematic indicates how the flow variables at one such cell could be obtained by interpolating from neighbouring nodes and from the immersed boundary

In effect, for cases involving boundary motion, the spatial discontinuity associated with the sharp IB leads to a temporal discontinuity for cells near the boundary.

Straightforward temporal discretization of the momentum equation for these cells is not possible since flow variables in these cells do not have a valid time-history. One approach to handle this issue is to merge these cells with adjacent fluid cells for the first time step after a cell emerges from the body. This approach is essentially similar to what is done in body-conformal Lagrangian methods and does not affect the spatial accuracy of the method. Another approach is to determine the flow velocity in this cell for one time step by interpolating from neighbouring cells. The issue of freshly-cleared cells is not encountered in IB methods that employ continuous forcing since the spreading of the effect of the IB over a few grid cells on both sides of the boundary, provides a smooth transition between the fluid and solid phases, and removes the temporal discontinuity for cells emerging into the fluid. Thus, inclusion of boundary motion is quite straightforward in these methods.

+

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

4.3. Conclusions The last decade has seen a tremendous rise in the popularity of IB methods. The primary factor driving this is the relative ease with which this methodology allows researchers to develop computational models of flows with extremely complex geometries and/or moving boundaries. By eliminating the need for complex grids, these methods also significantly reduce the time and effort required to set up and initiate a simulation. The largest numbers of applications of these methods are currently found in biological and multiphase flows. In addition to these, IB methods will see increased application in complex turbulent flows, fluid-structure interaction, and multimaterial and multiphysics simulations.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 5

RESULTS AND ANALYSIS

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 5 5.

RESULTS AND ANALYSIS The results are based upon the output of a solver that implements immersed

boundary method which is based upon a discrete forcing scheme. A NACA 0012 airfoil has been used for the purpose of evaluation of the critical parameters. The airfoil heaves in a fluid flow with Reynolds number of 500. The non-dimensional velocity (kh) is set to 1, keeping that a constant, the reduced frequency and nondimensional amplitude has been varied accordingly. The code has been run for reduced frequency ranging from 2 to 10.

5.1. Contours The contours of the all the reduced frequency ranging from 2 to 10 is obtained. Following are the contours of various reduced frequency and time step

5.1.1.

Reduced frequency k=2 Here in the figure 5.1 we can see that at time equals 0.15 times the time

period of an oscillation, there is formation of the leading and trailing edge vortex. Leading edge vortex is bound to the airfoil whereas trailing edge vortex is about to separate.

Fig 5.1 Vorticity contour at k=2 at 0.15T. Following in figure 5.2 to figure 5.4 are the contours that show the separation of the trailing edge vortex and increase in strength of the leading edge vortex.

Fig.5.2 Vorticity contour at k=2 at 0.0.25T. Department of Aerospace Engineering, IIAEM, Jain University 42

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Fig 5.3 Vorticity contour at k=2 at 0.0.35T.

Fig 5.4 Vorticity contour at k=2 at 0.0.55T Once the airfoil is halfway of the oscillation, the Leading edge vortex starts shedding, this can be seen in following figures 5.5-5.6

Fig 5.5 Vorticity contour at k=2 at 0.0.65T

Fig 5.6 Vorticity contour at k=2 at 0.0.75T When the airfoil is at 3/4 th of one oscillation, the leading edge experiences a new bound vortex and the former leading edge vortex interacts with the newer trailing edge vortex. This is illustrated in following figure 5.7.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Fig 5.7 Vorticity contour at k=2 at 0.0.85T Same type of flow can be seen when the reduced frequency is 3 i.e . k=3. Hence the contours are same as in k=2.

5.1.2.

Reduced frequency k=4

In this section we will talk about the flow over the airfoil heaving with reduced frequency 4. As the frequency has increased, the leading edge vortex doesnâ&#x20AC;&#x2122;t get enough time to clear the airfoil since the airfoil reverts its direction of motion before it could separate. The figure 5.8 shows the leading edge vortex.

Fig. 5.8 Vorticity contour at k=4 at 1.15T

Fig 5.9 Vorticity contour at k=4 at 1.25T

Department of Aerospace Engineering, IIAEM, Jain University 44

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil As we have seen in k=2 that the leading edge vortex interacts with the trailing edge vortex at trailing edge. In this case the leading edge vortex is shedding and hence interacts with trailing edge vortex away from the airfoil.

5.1.3.

Reduced frequency k=5

While k=5 the flow is almost same as the flow during the re duced frequency k=4. At this frequency it has been seen that the leading edge vortex generated in first stroke will interact with trailing edge vortex of the third stroke.

In the figure we can see the first leading edge vortex interacting with third stroke trailing edge. We can notice second and third stroke leading edge vortex.

First stroke leading edge vortex Fig. 5.10 Vorticity contour at k=4 at 1.65T

5.1.4.

Reduced frequency k=6 and above

For the reduced frequency of 6 or more, the frequency become s large enough that the leading edge vortex losses its strength because of its long lasting contact with the airfoil. The leading edge vortex is shown in the following figure 5.12.

First stroke leading edge vortex Fig. 5.11 Vorticity contour at k=4 at 1.95T

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

5.2. Plots We have plotted couple of graphs. First plot is co efficient of thrust and reduced frequency and second plot is efficiency and reduced frequency.

Fig. 5.12 Coefficient of thrust versus Reduced Frequency plot

Fig. 5.13 Efficiency versus Reduced Frequency plot Department of Aerospace Engineering, IIAEM, Jain University 46

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil The substantial jump in efficiency in the intermediate frequencies corresponds directly to the transition from a shed LEV to one that is dissipated. The comparatively low efficiency observed at k =2 for all kh in this study corresponds directly to the negative interference at the trailing edge between the Leading edge Vortex and the Trailing edge vortex of opposite sign. At high frequencies, the efficiency slowly tapers off as k increases for all kh, mimicking the theoretical calculations for inviscid heaving airfoils. Interaction between the airfoil and the shed vortices induces a drag on the airfoil, which has a stronger effect at higher frequencies owing to the greater proximity of the vortices. In addition, for high values of kh, the wakes are highly energized by the deposition of many vortices (LEVs, TEVs, and secondary vortices) into the flow. Much of this energy does not go towards providing useful work (thrust) and is thus wasted.

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 6

CONCLUSION AND FUTURE SCOPE

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Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

Chapter 6 6.

CONCLUSION AND FUTURE SCOPE The immersed boundary method, used to solve the viscous incompressible flow

problem, is based on a discrete forcing scheme that allows for a sharp representation of the immersed boundary. Based on the output of this solver, it is realised that for the given rigid airfoil (heaving symmetrically), NACA-0012, both the mean thrust coefficient and the propulsive efficiency exhibit peak values with respect to reduced frequency. The coefficient of thrust initially increases with increasing k value and peaks around k=8, after which it drops. The propulsive efficiency shows a similar trend but with its peak value measured around k=7. It is found that the thrust generation and propulsive efficiency is significantly affected by the leading edge vortex. These include negative interference between the leading and trailing edge vortices corresponding to lower efficiencies and also that the separation of the leading edge vortex corresponds to a sudden drop in the output power. The low efficiencies at high frequencies can be attributed to the fact that the shed vortices induce drag on the airfoil.

Extension of the current flow problem can be conceived by relaxing the constraint that demands the airfoil to be rigid. Under such case the airfoil is free to deform under the influence of aerodynamic forces impressed on it. This deformation will provide an added influence on the flow, apart from that induced by the heaving of the airfoil. Other extensions can be brought in by restricting heaving and allowing pitching motion only, allowing simultaneous heaving and pitching, bringing in asymmetries in the motion of the airfoil, positioning a second airfoil in the wake of the first airfoil etc. Various other possibilities exist as the combination of the above stated extensions.

The understanding of the unsteady nature of the flow gained from such studies can help us understand the bird and insect flights better. This understanding can then be implemented in building small scale aerial vehicles with robust characteristics.

Predicting the optimal value of the variables in unsteady flight can help us improve flight endurance and other such important parameters drastically.

Department of Aerospace Engineering, IIAEM, Jain University 49

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

APPENDIX 7.

VALIDATION OF CODE

The performance of the model was assessed through several test simulations.

Preliminarily, flow around a stationary circular cylinder was simulated at Re=100 using a 128Ă&#x2014;384 grid. The simulated St was 0.164 and the force coefficients on the cylinder were found to be C D,avg =1.26, C L,rms =0.219 and CL ,max =0.304, all of which agree well with published numerical results(G. C. LEWIN AND H. HAJHARIRI)

Department of Aerospace Engineering, IIAEM, Jain University 50

Analysis of thrust coefficient and propulsive efficiency of a single heaving airfoil

REFERENCES

[1]

Numerical Analysis of Flapping Wing Aerodynamics M.A. Ashraf, J.C.S. Lai and J. Young School of Aerospace, Civil and Mechanical Engineering University of New South Wales at Australian Defence Force Academy, Canberra, ACT 2600 AUSTRALIA

[2]

On the Onset of Transition at Low Reynolds Number Flow Over Airfoils Y. Elimelech * Technion-IIT, Haifa 32000, ISRAEL R. Arieli â&#x20AC; RAFAEL Ltd., Haifa 22500, ISRAEL G. Iosilevskii â&#x20AC;Ą Technion-IIT, Haifa 32000, ISRAEL 4th AIAA Theoretical Fluid Mechanics Meeting 6 - 9 June 2005, Toronto, Ontario Canada

[3]

Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow

By G. C. LEWIN AND H. HAJ-HARIRI Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4146, USA.

[4]

A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries

R. Mittal a,*, H. Dong a, M. Bozkurttas a, F.M. Najjar b, A. Vargas a, A. von Loebbecke a Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC 20052, United States b Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States a

[5]

Vortex shedding and frequency selection in flapping flight

By Z. JANE WANG Courant Institute of Mathematical Sciences, New York University, NY 10012, USA

Department of Aerospace Engineering, IIAEM, Jain University 51