The Equation for Prandtls Mixing Length

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Frontiers in Aerospace Engineering Volume 3 Issue 2, May 2014 doi: 10.14355/fae.2014.0302.03

The Equation for Prandtls Mixing Length K.O. Sabdenov1, Maira Erzada2 L.N. Gumilyov Eurasian National University Mirzoyana Str., 2, Astana, Republic of Kazakhstan, 010008 sabdenovko@yandex.kz ; 2mayira76@yahoo.co.jp

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Received 5 December, 2013; Revised 10 February, 2014; Accepted 20 February, 2014; Published 28 May, 2014 © 2014 Science and Engineering Publishing Company

Abstract Thev turbulent boundary layer on the solid surface is considered. Semi-empirical equation for mixing length of Prandtl is constructed on the basis of common physical principles. A possibility of turbulent boundary layer description on the basis of two universal dimensionless constants, one of them is Karman constant, is shown.

Keywords Turbulence; Boundary Layer; Viscous Sub Layer; Mixing Length; Karman Constant

Introduction In the existing to date theory of turbulent boundary layer special place takes theory of Prandtl, based on presentation of cross velocity v′ and longitudinal velocity w′ in the form of (Loizianskii, L.G., 1987; Frost, W. and Moulden, T.H., 1977)

dw (1) , dx where is w – longitudinal component of stream average velocity, which is directed lengthways the streamlined solid surface; s′ – pulse coupled mixing length; x coordinate is directed normally to the streamlined surface. Such presentation allows to close Raynolds equation, which figures v′ ~ w′ ~ s ′

2

 dw  − w′v′ = s2   ,  dx  where is s – Prandtls mixing length.

(2)

During many years Prandtls theory was and is still an object of a focused attention of many research workers.

Kutateladze S.S. (1989) shows singular importance of ideas based in this theory. Modern vision of turbulent flame fractal structure (Sabdenov K.O., 1995, 2005; Sabdenov K.O. and Min'kov L.L., 1998), eventually lead to relations, which mathematically equivalent to Eq. (2).

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Prandtls theory is based on integral characteristics, such as mixing length, stream average velocity and its derivatives. In theory these characteristics appear as the result of averaging over all implementations of irregular velocity of turbulent stream and its derivatives. In fact theory of Prandtl is based on the sufficiency of principle of specified integral quantity for description of turbulent boundary layer. Having scales of length, time and mass, it is possible to build accurate of these any force and energy characteristic of stream, using undetermined (structural) constant. This approach does not determine detailed structure of turbulent stream in an explicit form. And in this context Prandtls theory as semi-empirical theory appears incomplete. In this case our lack of knowledge of detailed turbulence structure and mechanism of random motion is contained in undetermined (empirical) constants. Even more late models like Jones – Launder’s (Frost, W. and Moulden, T.H., 1977) and their different modifications are created on the same principles as the Prandtls theory.

In investigation of transition incipient combustion into detonation of gases is necessary to know detailed distribution of average velocity w and mixing length s at each pipe section area and root mean square value of turbulent fluctuation v′, w′ velocity. Last values associate with coefficient of turbulent transport of heat λt and material λd:

dw . dx They play a key role in mechanism of detonation appearance.

λt ~ λd ~ s 2

Requirement for knowledge of w, s, λt, λd also appears during investigation of positive and negative erosive effects in solid propellant rocket engines. Display of that effects associated with characteristics of chemical reactions progress in narrow boundary layer near a

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solid surface of fuel (Sabdenov, K.O., 2008; Sabdenov, K.O. and Erzada M., 2013). Greatest interest represents progress in transition area; it is located between viscous sublayer and logarithmic layer. At the same time to carry out the investigation by analytical methods it is preferably to have simple methods for calculation of the velocity w, mixing length s and coefficients λt, λd.

Derivation of Equation for Mixing Length The mixing length s, contained in (2), is integral scale of turbulence and is not local value (Frost, W. and Moulden, T.H., 1977; Reynolds, A.J., 1974) and it depends of spatial variables. Its functional form is found by consideration of different physical assumtions and depends on character of concerned liquids flux. But alocal character of mixing length means that functional form of s should be determinates as the result of differential equation solution. There are some attempts to find such equation. In the present article is given a derivation of an equation for the mixing length inside the boundary layer on the basis of general principles.

By deriving specified equation we will use the following assumtions: a) mixing length s in the case of boundary layer flow is a solution of an ordinary differential equation of the second order; b) boundary layer can be conveniently divided into two areas: viscous sublayer and outer layer with a thickness δ, in which the liquid kinematic viscosity ν is not determinative dimensional value; c) for specified two zones there are numerical “fundamental” constants, which characterize the turbulent flow in these regions.

These assumtions can be easily proved. The Navier – Stokes equations are the second order equations in regard to spatial variables. Expressions for pulsating velocities and their correlation are the same type equations (Frost, W. and Moulden, T.H., 1977; Monin, А.S. and Jaglom, А.М., 1965). But in a boundary layer all average values can be considered as depending only on one coordinate x. Therefore, taking into equations (1) and (2), we conclude, that first assumption (a) is acceptable.

Second assumption (b) expresses well known fact: existence of viscous sublayer was stated by Prandtl and later experimentally confirmed.

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Third assumption (c) is based on carrying out the analogy with the common structure of the Unified field theory (Okun, L.B., 1985; Spiridonov, O.P., 1991).

In it each type of known interactions (strong, electromagnetic, low, gravitational) is characterized by own dimensionless fundamental constant. The stronger interactions the bigger the constants. For example, constant of electromagnetic interaction approximately equal to 1/137. There is no any evidence against possibility of application of such general conditions of the field theory to the turbulence.

So we are looking for mixing lengths equation in the class of the second order ordinary differential equations. It is known that viscous sub layer is very thin and turbulent viscosity is less then kinematic viscosity. According to Eq. (2) in the viscous sub layer mixing length value is also small. Outside of viscous sub layer mixing length is large, and turbulent viscosity is much higher than kinematical viscosity. So, mixing length as solution of equation should grossly change in the relatively small region of space. From mathematical physics we know that such properties posses differential equations with a small parameter at highest derivative (Nayfeh, A.H., 2004). Therefore under hypothesis that desired equation resolved for highest derivative, we can write

d 2s  ds dw  (3) , s, δ ,ν  , = F , 2 dx dx dx   where F – is an unknown function of the arguments specified in brackets.

ν

In the right part of Eq. (3) the velocity w is not included because of invariance of physical processes along streamlined surface (Monin, А.S. and Jaglom, А.М., 1965; Schlichting, H., 1960). Derivatives from w of higher orders also should be excluded from Eq. (3).

Justification is given below.

Now let’s look on the equation (3) in the viscous sublayer. Here dimensional quantity δ should disappear from Eq. (3) according to theory of similarity and dimensionality, because.

1

dx << 1. dw Left part of Eq. (3) has dimensionality of velocity. The right part of Eq. (3) also has to have the same dimension. In the simplest linear form function F can be presented as

δ

ν

ν  ds  dw , = F p   +γ s s  dx  dx n

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where p, γ, n – are constant numbers. Then Eq. (3) can be written as

d s dw ν  ds  (4) 0. − p   −γ s = 2 s dx dx dx   In this equation valid for viscous sub layer, dw/dx ≈ const.

ν

2

n

Simple analysis shows that physically meaningful solutions of Eq. (4), wiсh remains limited together with first and second derivatives of s, if s → 0, results only with n = 2. It is easy to find out integrating Eq. (4) one time without the third member.

Further we will note that p can not be equal to zero, because the turbulent viscosity νt theoretically follows (Monin, А.S. and Jaglom, А.М., 1965) to

ν t ~ s2 ~ xk , k ≥ 3 .

Such behavior of viscosity νt is possible when p ≠ 0.

Taking into account n = 2, let’s write solution Eq. (4), fulfilling condition s(0) = 0: m

 (1 − p )γ 2 dw  1 , s C= x  , m 1  sh 1− p dx  ν  where C1 – integration constant. If p = 1/2, then this expression for s becomes empirical formula Ranny (Reynolds, A.J., 1974).

In case of arbitrary values of p and within x → 0 for the mixing length the exponential law of decrease s ~ xm will be valid. There are different opinions about m number. A lot of authors consider that m = 2. To this value of m brings widely known formula of Van Driest E.R. (Loizianskii, L.G., 1987; Frost, W. and Moulden, T.H., 1977; Levich, V.G., 1962). Then νt ~ x4, that shows fast damping of turbulence in the viscous sublayer.

Mentioned arguments and comparesion with results of other authors allows to decide on p = 1/2. By this means we get 2

 γ 2 dw  (5) s = C1  sh x  .

2ν dx   Value of constant γ remains ill-defined. First of all, let us state that this parameter is characterizes balance between turbulence generation and molecular viscous dissipation. This number is a dimensionless constant of turbulent viscous sublayer. It is clear that γ > 0.

Otherwise Eq. (5) would not result asymptotical behavior of s → ∞ when x → ∞. This condition is necessary for equations with small parameter at highest derivative, describing the boundary layer

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(Nayfeh, A.H., 2004).

So, in viscous sub layer equation for Prandl mixing length can be written as

d 2 s ν  ds  dw (6) −   − γ s2 = 0. dx dx 2 2  dx  Now we will try to deepen potentials of this equation.

According to the second assumption (b) outside of viscous sub layer the terms in the required equation proportional to viscosity ν, asymptotically become zero. To save equality in Eq. (6) with dw/dx ≠ 0, third summand in (6) should be multiplied by dimensionless group of s, δ and derivative of s not higher than first order: f(s/δ, ds/dx). Here derivative of dw/dx, having dimension of times reciprocal is eliminated. Because a viscosity is not a mean parameter any more, then there is no physical quantity containing dimension of time which could be used for formation of a dimensionless complex together with dw/dx. Becide of this, in viscous sublayer should be f(s/δ, ds/dx) ≈ 1. Also, within x → δ should be performed inequality (Frost, W. and Moulden, T.H., 1977) 2

νs

s ≤ δ.

(7)

Simplest form of function f(s/δ, ds/dx), following all mentioned above features is a linear representation

1  ds s   s ds  , f  , = 1−  + µ  dx λδ   δ dx  where µ, λ – numbers to be determined. Now equation for the mixing length can be written as d 2 s ν  ds  dw  s ds  f  , = 0. −   − γ s2 2 2  dx  dx  δ dx  dx It means that search the expression for the mixing of the viscous sublayer is reduced to the equality f = 0, or, 2

νs

1−

1  ds

µ  dx

+

s 

= 0.

(8)

λδ  Let us consider extreme case δ → ∞. It gives a solution s = µx. It leads to the universal logarithmic low of the velocity distribution w(x) ~ lnx. When compare the obtained result with a known data, we find out, that µ – is Karman constant (µ ≈ 0,4), which is fundamental constant of logarithmic layer. According to the third assumption (c) µ retains its significance even outside logarithmic layer. It means, that number λ is stated through the Karman constant µ: λ(µ). Let’s take the simplest form λ = µ. With a regard to this equality, the solution of equation (8) under the condition s(x = 0) = 0 looks as

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  x  (9) = µ 2 1 − exp  −  . δ  µδ    If in this formula µ ≈ 0,395 is taken, than values of mixing length are consistent with experimental date (Loizianskii, L.G., 1987). Herewith, relative error does not exceed 0,02. On the border of boundary layer x = δ value of mixing length s ≈ 0,14δ, and condition (7) is fulfilled. s

The advantage of formula (9) in comparison with the formulas suggested previously (Loizianskii, L.G., 1987) is clear: it contains only one empirical constant µ, which digit value is known and generally agreed.

So equation of the mixing length is as follows

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v∗ v∗ w S s= , z x = , W = , v∗ = ν ν v∗

τs , ρ

where τs − the stress at the wall.

Exact accordance of calculated velocity distribution W(z) and mixture length S(z) (Fig. 1) with experimental data of Nikuradze, contained in book (Loizianskii, L.G., 1987; Schlichting, H., 1960), comes out by γ = 0,15 and q = 4,23⋅10−3. Herewith, the taken value q is approximately 7 times smaller than Deissler’s experiment data shows, but practically it is align with Ranny’s data (Reynolds, A.J., 1974).

However, it is necessary to consider great difficulties when determining q in experiments.

2 d 2 s 1  ds  γ 2 dw  1  ds s  0 . (10) − 1 −  +  =   − s 2 2  dx  ν dx  µ  dx µδ   dx For the flow in a long tube with a radius R it is necessary to state that δ = R. In this form Eq. (10) is correct in all flow space, it is not necessary to consider separately the turbulent layers having big differences in the characteristics.

s

In the Eq. (10) the value of γ is still unknown. To find out γ it is necessary to make additional investigation, which is shown below.

Analysis of Equation (10) by the Example of Flow Velocity Distribution Calculation in a Boundary Layer Reynolds equation together with Eq. (2) after single integration becomes the form (Loizianskii, L.G., 1987; Schlichting, H., 1960)

dw 2  dw  τ , +s   = dx ρ  dx  where τ – shearing stress; ρ – liquid density.

2

ν

(11)

Equation (10), (11) have been solved using numerical methods with δ = ∞, for viscous sublayer and logarithmic layer. It is enough for γ determination. Full statement of problem in dimensionless variables is as following 2

S

d 2 S 1  dS  dW  1 dS  −   −γ S2 = 1− 0, 2 2  dz  dz  µ dz  dz 2

dW  dW  1, + S2   = dz  dz 

d 2S = S (0) 0,= W (0) 0, = (0) q , dz 2

FIG. 1. DISTRIBUTION OF VELOCITY W AND MIXING LENGTH S. v∗ = 5 m/s; ν = 3,5⋅10 –5 m2/s; ● − BY EXPERIMENTAL DATA OF NIKURADZE (BENNETT C.O. AND MYERS J.E., 1962).

The velocity of liquid in a tube can be provided from equation (10), (11). But the scales for length and velocity in a viscous sublayer and in the range of the developed whirl flow strongly differ. It complicates a little the task of calculation of flow in a tube. It is possible to obtain the simple expression for w if instead of the Eq. (10) the simplified Eq. (9) is used.

Therefore we need to know the shearing stress τ(x). In addition, we can determine the shearing stress τ(x) by the next equation (Loizianskii, L.G., 1987; Reynolds, A.J., 1974; Bennett C.O. and Myers J.E., 1962)

x  = τ ρ v*2 1 −  , δ   where δ = R – is tubes radius. From Eq. (11) follows   4 s 2τ  1+ , 1 − 2   ρν   and after integrating the w(x) becomes: dw ν = dx 2 s 2

w( x) =

ν x

 dx 4 s 2τ  2 , 1 − ∫  1+ s 2 0 ρν 2  

(12)

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where for the way of mixture length s we will take expression from a formula (9).

For calculation of velocity the equations (9), (12) must be solved together.

Earlier by Karman was obtained the following equation (Loizianskii, L.G., 1987; Reynolds, A.J., 1974):

wmax − w( x) 1  x x (13) = − ln 1 − 1 −  + 1 −  . v∗ R R  µ   Results of calculations of velocity using formulas (12) and (13) are given in Fig. 2.

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means of γ turned out to be approximately ≈3 times smaller than µ. Same tendency can be observed in case of turbulent boundary layer. Vortex interaction in the outer part of boundary layer is stronger than in viscous sublayer because of low turbulence intensity in it. This affected the interrelation of numbers γ and µ.

The obtained equation (10), more likely, is the simplest among all possible. It is acceptable for all occasions, when physical parameters in boundary layer are grossly change normally to the solid surface and weakly change in the direction of liquid motion.

Authors thank Johann Dueck for useful discussions of the work and valuable recommendations.

REFERENCES

Bennett C.O. and Myers J.E. Momentum Heat and Mass Transfer., (1962). McGraw-Hill Book Co., N.Y. Frost, W. and Moulden, T.H., Handbook of Turbulence: Fundamentals and application, (1977), Plenum Press, New York.

Kutateladze, S.S., Izbrannye trudy, (1989), Novosibirsk, Nauka.

FIG. 2. DISTRIBUTION OF VELOCITY w ALONG THE TUBE RADIUS; v∗ = 0,0487wmax, wmax = 12,5 m/s; THE LINE 1 CALCULATION FOR A FORMULA (12), THE LINE 2 - FOR A FORMULA (13); ●, ■ − BY EXPERIMENTAL DATA OF NIKURADZE (BENNETT C.O. AND MYERS J.E., 1962).

Here is R = 0,1 m, kinematic viscosity of liquid ν = 1,5⋅10 –5 m2/s. The difference of the velocity calculated by 12 formulas and 13 formulas does not exceed 26%. this difference decreased along with the increase in velosity wmax (Generally the number of Re = wmaxR/ν increases) and v∗/wmax go down with it.

Besides, in limit x → 0 formula (13) has logarithmic divergence, and the formula (12) doesn't possess such defect.

Summary There exist a lot of arguments around problem of fundamentality of Karman’s constant. Afterwards it became to be believed that Karman’s number is a function of coordinate x. But results of this work show the constancy of Karman’s number, having physical meaning of the balance measure between turbulence generation and turbulent dissipation in all region outside viscous sublayer. It means that µ can occur a fundamental constant in wide sense. The same relates to the number γ. Such argument is supported by

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Levich,

V.G.

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Hydrodynamics.,

(1962).

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Nayfeh, A.H., Perturbation Methods, (2004), WILEY-VCH Verlag Gmb & Co. KGaA, Weinheim.

Okun, L.B., αβγ…Z. An elementary introduction to the physics of elementary particles, (1985), Moscow, Nauka.

Reynolds, A.J., Turbulent Flows in Engineering, (1974), John Wiley and Sons, London, New York, Sydney, Toronto.

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Sabdenov, K.O. and E. Maira, Analytical calculation of the negative erosive burning rate // Combustion, Explosion and Shock Waves, Vol. 49, No. 6 (2013), pp. 690-699.

Sabdenov, K.O. and E. Maira, The mechanism of occurrence of negative erosive effect // Combustion, Explosion and Shock Waves, Vol. 49, No. 3 (2013), pp. 273-282.

Sabdenov, K.O., On the threshold nature of erosive burning

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// Combustion, Explosion and Shock Waves, Vol. 44, No. 3 (2008), pp. 300-309.

Sabdenov, K.O., The fractal theory of the slow deflagrationto-detonation transition // Combustion, Explosion and Shock Waves, Vol. 31, No. 6 (1995), pp. 705-710.

Sabdenov, K.O., The theory of spontaneous detonation. Part 1. Formulation of the main provisions // Izvestia Tomskogo politehnicheskogo universiteta (in Russia), Vol. 308, N. 3

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Sabdenov, K.O., The theory of spontaneous detonation. Part 2.

Modeling of explosive processes // Izvestia Tomskogo politehnicheskogo universiteta (in Russia), Vol. 308, No. 4 (2005), pp. 19-25.

Schlichting, H., Boundary Layer Theory, 4th ed., (1960), McGraw-Hill Book Co., N.Y. Spiridonov, O.P., Fundamental physical constants, (1991), Moscow, Vyshaia shkola.

(2005), pp. 16-22.

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