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Exact statistical theory of
1isotropic turbulence
Ran zheng
Shanghai Institute of Applied Mathematics and Mechanics,
Shanghai University, Shanghai 200072,P.R.China
Abstract
Some physically significant consequences of recent advances in the theory of
self-preserved homogenous statistical solutions of the Navier-Stokes equations are presented.
Keywords: isotropic turbulence, Karman-Howarth equation, exact solution
1 Introduction
Homogeneous isotropic turbulence is an idealized concept of turbulence assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity), rotation or reflection (isotropic) of the coordinate system. This is an idealization of real turbulent motions, which are observed in nature or produced in a laboratory have much more complicated structures. This idealization was first introduced by Taylor (1935) to the theory of turbulence and used to reduce the formidable complexity of statistical expression of turbulence and thus make the subject feasible for theoretical treatment. Up to the present, a large amount of theoretical work has been devoted to this rather restricted kind of turbulence. Remarkable progress has been achieved so far in discovering the various nature of turbulence, but nevertheless our understanding of the fundamental mechanics of turbulence is still partial and unsatisfactory (Tatsumi, 1980)
The assumption of self-preservation permits an analytical determination of the energy decay in isotropic turbulence. This idea of similarity and self-preservation has played an important the development of turbulence theory for more than a half-century. The traditional approach to search for similarity solutions in turbulence has been assume the existence of a single length and velocity scale, then ask whether and under what conditions the turbulent motions admit to such solutions. Excellent contributions had been given to this direction. Von Karman and Howarth (1938) firstly deduced the basic equation, also presented a particular set of solutions of Karman-Howarth equation for the final decaying turbulence. Later on, two Russian authors, Loitsiansky (1939) and Millionshtchikov (1941), have each discussed the solutions of Karman-Howarth equation which are obtained when the term describing the effect of the triple 1 National Natural Science Foundation of China (10272018, 10572083)
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velocity correlation is ignored. Their work is an extension of the “small Reynolds number” solution first put forward by von Karman and Howarth. H.Dryden (1943) gave a useful review on this subject. Great deep research on the solutions of Karman-Howarth equation was conducted by Sedov (1944). He showed that one could use the separability constraint to obtain the analytical solution of Karman-Howarth equation. Sedov’s solution could be expressed in terms of the confluent hypergeometric function. Batchelor (1948), who was also the first to carry out a detailed study of this problem, based his analysis on the assumption that the Loitsianskii integral is a dynamic invariant- a widely accepted hypothesis that was later discovered to be invalid. Batchelor conclude that the only complete self-preserving solution that was internally consistent existed at low turbulence Reynolds numbers where the turbulent kinetic energy consistent with the final period of turbulent decay. Batchelor (1948) also found a self-preserving solution to the Karman-Howarth equation in the limit of infinite Reynolds numbers for which Loitsiansky’s integral was an invariant. Objections were later raised to the use of Loitsiansky’s integral as a dynamic invariant: at high Reynolds numbers this integral can be shown to be a weak function of time (see Proudman & Reid 1954 and Batchelor & Proudman 1956). Saffman (1967) proposed an alternative dynamic invariant which yields another power law decay in the infinite Reynolds numbers limit (see Hinze 1975). While the results of Batchelor and Saffman formally constitute complete self-preserving solutions to the inviscid Karman-Howarth equation, it must be kept in mind that they only exhibit partial self-preservation with respect to the full viscous equation. Recently, George (1992) revived this issue concerning the existence of complete self-preserving solutions in isotropic turbulence. In an interesting paper he claimed to find a complete self-preserving solution, valid for all Reynolds numbers. George’s (1992) analysis is based on the dynamic equation for the energy spectrum rather than on the Karman-Howarth equation. Strictly speaking, the solution presented by George was an anlternative self-preserving solution to that of Karman-Howarth and Batchelor since he relaxed the constraint that the triple longitudinal velocity correlation be self-similarity in the classical sense. Speziale and Bernard (1992) reexamined this issue form a basic theoretical and computational standpoint. Several interesting conclusions had been drawn from their analysis.
From the long history stories, we know that: Decaying homogeneous and isotropic turbulence is one of the most important and extensively explored problems in fluid dynamics and a central piece in the study of turbulence. Despite all the effort, a general theory describing the decay of turbulence based on the first principles has not yet been developed. (Skrbek and Steven R.Stalp, 2000). It thus appears that the theory of self-preservation in homogeneous turbulence has many interesting features that have not yet been fully understood and are worth of further study. (see Speziale and Bernard ,1992,pp.665).
This paper offers a unified investigation of isotropic turbulence, based on the exact solutions of Karman-Howarth equation. Firstly, we will point out that new complete solution set may be exist if we adopt the Sedov (1944) method. Based on the above finds, new results could be obtained for the analysis on turbulence
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features, such as the scaling behaviour, the spectrum, and also the large scale dynamics.
2 Self-preservation solution under Sedov’s separability constraint
For complete self-preserving isotropic turbulence, the Karman-Howarth equation will have a solution if Reynolds number based on the Taylor microscale is constant as first noticed by Dryden (1943). However, this equation also has solutions where Reynolds number based on the Taylor microscale is time dependents when separability is invoked. The separability condition implies that each side of the equation is equal to zero individually, yielding differential equations from which explicit solution for the correlation functions may be determined depending on the choice of parameters. These solutions were first discovered by Sedov (1944) and later compared with experimental data by Korneyev & Sedov (1976). Here, we will discuss the possible new complete solutions under Sedov’s separability constraint.
The two-point double longitudinal velocity correlations read as ( named Sedov equation)
d 2f ⎛4a 1⎞df a 2++⎜+ξ⎟f =0 (1) 2⎜⎟d ξ⎝ξ2⎠d ξ2
with boundary conditionsf (0)=1, f (∞)=0.
In the following analysis, we introduce alternative two parameters denoted by a 1, σ， here
σ=a 2 (2) 2a 1
The complete new set of the solution of the equation (1) with the boundary condition could be given as following：
The first kind of solution：if σ=5，thus 2
−a 124f (ξ)=e (3)
The secondary kind of solution：if κ=σ−5，thus 4
a −1ξ25a ⎛5⎞f (ξ)=e 4F ⎜−σ, , 1ξ2⎟ (4) 24⎝2⎠
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5−σ，thus 4
a −1ξ2⎛5a ⎞f (ξ)=e 4F ⎜σ, , 1ξ2⎟ (5) ⎝24⎠
5The forth kind of solution：if σ=，thus 4
a −12⎛33a ⎞f (ξ)=e 4F ⎜, , 1ξ2⎟ (6) ⎝424⎠
where F (α, γ, z )is the confluent hypergeometric function and the definition of The third kind of solution：if κ=
will be given appendix. From the asymptotic the existing parameterκ
expansions and the limiting forms of the confluent hypergeometric function, we could deduce the existence conditions of these solutions：
For all four kind of solutions：a 1>0,
For the secondary kind of soultions：σ>0；
For the third kind of solution：0
The details could be seen in appendix .A simple comparison shows that the special solution found by Sedov (1944) is belonging to the secondary kind of our new set of solution.
3 Some theoretical results based on the exact solutions
A unified investigation of isotropic turbulence, based on the above exact solutions of Karman-Howarth equation could be given. New results could be obtained for the analysis on turbulence features, such as the scaling behaviour, the spectrum, and also the large scale dynamics, some results could be seen in the following references [19,20,21,22] .
References
[1]Bareenblatt,G.J.& Garilov,A.A. 1974. Sov. Phys. J. Exp. Theor. Phys. 38,399-402.
[2]Batchelor,G.K. 1948.Q. Appl. Maths. 6,97-116.
[3]Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence Turbulence. Cambrige University Press.
[4]Batchelor,G.K. & Proudman,I. 1956. Phil. Trans. R. Soc. Lond. A.248,369-405. [5]Dryden,J.L. 1943. Q. Appl. Maths. 1,7-42.
http://www.paper.edu.cn
[6]George,W.K. 1992. Phys. Fluids A 4,1492-1509.
[6]Hinze,J.O. 1975 Turbulence. McGraw-Hill.
[7]Karman,T.Von & Howarth,L. 1938. Proc. R. Soc. Lond. A164,192-215.
[8]Korneyev & Sedov, L.I. 1976. Fluid Mechanics-Soviet Research 5,37-48.
[9]Lesieur,M. 1990 Turbulence in Fluids, 2nd Edn. Martinus Nijhoff.
[10]Lin,C.C. 1948. Proc. Natl. Acad. Sci. 34,540-543.
[11]Millionshtchikov, M. 1941. Dokl. Akad. Nauk SSSR 32,615-618.
[12]Monin,A.S. & Yaglom,A.M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol.2, MIT Press.
[13]Proundman, I. & Reid, W. H. 1954. Philos. Trans. R. Soc. London, A 247,163-189.
[14]Saffman,P.G. 1967. J. Fluid Mech. 27,581-594.
[15]Sedov, L.I. 1944. Dokl.Akad.Nauk SSSR 42,116-119.
[16]Speziale,C.G. & Bernard, P.S. 1992. J. Fluid Mech. 241,645-667.
[17]Skbek,L. & Steven, R.S. 2000. Phys. Fluids 12,1997-2019.
[18]Tatsumi, T. 1980. Advances in Applied Mechanics,39-133.
[19]Ran zheng, Exact solutions of Karman-Howarth equation. [20]Ran zheng, Scales and their interaction in isotropic turbulence. .
[21]Ran zheng, Dynamic of large scales in isotropic turbulence. .
[22]Ran zheng, On von Karman’s decaying turbulence theory. .
Appendix: Solutions of the correlation coefficients
The ideas of similarity and self-preservation were firstly introduced by von Karman (1938). Following the methods adopted by Sedov (1944,1951), the two point double longitudinal velocity correlations satisfied
d 2f ⎛4a 1⎞df a 2++⎜+ξ⎟f =0 （1） ⎟2⎜d ξ⎝ξ2⎠d ξ2
with the boundary condition
f (0)=1
f (∞)=0
The complete solution are given in this paper, these are：
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−125When σ=，f (ξ)=e 4 2
a −125a 2⎞5⎛5When κ=σ−，f (ξ)=e 4F ⎜−σ, , 1ξ⎟ 424⎝2⎠
−125⎛5a 2⎞When κ=−σ，f (ξ)=e 4F ⎜σ, , 1ξ⎟ 4⎝24⎠
−125⎛33a 2⎞When σ=，f (ξ)=e 4F ⎜, , 1ξ⎟ 4⎝424⎠a a a
The detailed calculation is given as following:
A lot of useful partial differential equations can be reduced to confluent hypergeometric equations. LetP k , m (ς)is the solution of Whittaker equation as that defined by Whittaker and Waston
d 2W ⎡1k 4−m 2⎤+⎢−++⎥W =0 （2） 22d ςς⎦⎣4ς
where
y (z )=z βe f (z )P κ, m (h (z )) （3）
After some reduction, the equation of y (z ) reads
d 2y ⎡h ′′2β⎤dy ′()−++2+g ⋅y (z )=0 （4） f z 2⎢⎥′z dz ⎣h ⎦dz
where
2g =(f ′)−f ′′+2βf ′β(β+1)h ′′⎛β⎞′++f +⎟+g 1 ⎜2′z h ⎝z z ⎠
2h 2⎞⎛h ′⎞⎛12−m +κh −⎟ g 1=⎜⎟⎜⎜⎟4⎠⎝h ⎠⎝4
f (z )=az λ （5）
h (z )=Az λ （6） The solutions of above equation could be deduced in terms of Whittaker function. We discussed this equation in following special case:
The equation under this condition reads as
d 2y ⎡1−λ−2βλ−1⎤dy +−2λα+qy (z )=0 （7） z 2⎢⎥z dz ⎣⎦dz
where
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⎛2A 2⎞2λ−2β(β+λ)+λ24−m 2λ−2q =λ⎜+λ(2αβ+A κλ)z +⎜α−4⎟⎟z z 2⎝⎠2()
The solution of this equation is
λy (z )=z βe αz P κ, m Az λ （8）
For isotropic turbulence, the corresponding parameters satisfied
1−λ−2β=4 （9） ()
λ−1=1 （10）
−2λα=
2a 1 （11） 2⎛2A 2⎞λ⎜⎜α−4⎟⎟=0 （12） ⎝⎠
⎛1⎞β(β+λ)+λ2⎜−m 2⎟=0 （13） ⎝4⎠
a λ(2αβ+A κλ)=2 （14） 2
Hence, we have
λ=2 （15）
a 1 （16） 8
5β=− （17） 2
3m =± （18） 4
a A =±1 （19） 4
⎧a 5⎫κ=±⎨2−⎬ （20） ⎩2a 14⎭α=−
From above analysis, we can introduce two parameters to classification turbulence, they are: a 1, σ=a 2。 2a 1
According to Whittaker and Waston, if 2m isn’t an integral,
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z
21m 2⎞⎛1F ⎜+m −κ, 1+2m , z ⎟ （21） ⎠⎝2
z 1−−m ⎛1⎞P κ, −m (z )=e 2z 2F ⎜−m −κ, 1−2m , z ⎟ （22） ⎠⎝2
For the caseκ=0，we must use the secondary Kummer formula,
1+m ⎛z 2⎞2P 0, m (z )=z 0F 1⎜⎜1+m ; 16⎟⎟ （23） ⎝⎠P κ, m (z )=e z −
By making use of the boundary condition, we could chose the rational parameters for isotropic turbulence. The solution of equation could be rewritten in
y (z )=A 1+m 2⋅e A ⎞2⎛⎜α−⎟z 2⎠⎝z β+λm +λ
2⎛1⎞⋅F ⎜+m −κ, 1+2m , Az λ⎟ （24） ⎝2⎠
Let A >0，this resulted in the definition of exponent.
If we chose m =−3，in the above solution, the exponent ofz reads as 4
2
5⎛3⎞=−+2×⎜−⎟+1 （25） 2⎝4⎠
=−3
The boundary conditiony (0) would be broken under this condition. So we only chose
m =
Another condition must be satisfied 3 （26） 4
α+
The solution is
5
4A =0 （27） 2
There is an important parameter κin the above solution, the multiple values could be existed： When κ=⎨25⎛5⎞y (z )=A ⋅e −Az ⋅F ⎜−κ, , Az 2⎟ （28） 2⎝4⎠⎧a 25⎫−⎬， ⎩2a 14⎭
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y (z )=A ⋅e
when κ=−⎨54−Az 25⎛5⎞⋅F ⎜−σ, , Az 2⎟ （29） 2⎝2⎠⎧a 25⎫−⎬， ⎩2a 14⎭
5
42⎛5⎞y (z )=A ⋅e −Az ⋅F ⎜σ, , Az 2⎟ （30） ⎝2⎠
We must treat the other special caseκ=0，by using the secondary Kummer formula
P 0, m (z )=z
where 1+m 2⎛z 2⎞0F 1⎜⎜1+m ; 16⎟⎟ （31） ⎝⎠
z −⎛z 2⎞20F 1⎜⎜1+m ; 16⎟⎟=e F (m , 2m , z ) ⎝⎠
For this case, the solution of equation is
y (z )=A ⋅e
Another reduced case for σ=54−Az 2⎛33⎞⋅F ⎜, , Az 2⎟ （32） ⎝42⎠5，the solution is 2
f (ξ)=e −a 124 （33）
At last, we have already obtained a complete set solution of isotropic turbulence , depending on two parameters, these are： −125when σ=，f (ξ)=e 4 2
a −1ξ25a 2⎞5⎛5when κ=σ−，f (ξ)=e 4F ⎜−σ, , 1ξ⎟ 424⎠⎝2
−1ξ25⎛5a 2⎞when κ=−σ，f (ξ)=e 4F ⎜σ, , 1ξ⎟ 4⎠⎝24
−1ξ25⎛33a 2⎞If σ=，f (ξ)=e 4F ⎜, , 1ξ⎟ 4⎝424⎠a a a
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References
[1]Whittaker,E.T. and Waston, G.N., A course of modern analysis. Cambridge University Press, 1935
[2]M.Abramowitz and I.A.Stegun, Handbook of mathematical functions. Dover, New York,1965
[3]Wang, Z.X. and Guo,D. R., Special functions. The series of advanced physics of Peking University. Peking University Press,2000 (In Chinese)
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Exact statistical theory of
1isotropic turbulence
Ran zheng
Shanghai Institute of Applied Mathematics and Mechanics,
Shanghai University, Shanghai 200072,P.R.China
Abstract
Some physically significant consequences of recent advances in the theory of
self-preserved homogenous statistical solutions of the Navier-Stokes equations are presented.
Keywords: isotropic turbulence, Karman-Howarth equation, exact solution
1 Introduction
Homogeneous isotropic turbulence is an idealized concept of turbulence assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity), rotation or reflection (isotropic) of the coordinate system. This is an idealization of real turbulent motions, which are observed in nature or produced in a laboratory have much more complicated structures. This idealization was first introduced by Taylor (1935) to the theory of turbulence and used to reduce the formidable complexity of statistical expression of turbulence and thus make the subject feasible for theoretical treatment. Up to the present, a large amount of theoretical work has been devoted to this rather restricted kind of turbulence. Remarkable progress has been achieved so far in discovering the various nature of turbulence, but nevertheless our understanding of the fundamental mechanics of turbulence is still partial and unsatisfactory (Tatsumi, 1980)
The assumption of self-preservation permits an analytical determination of the energy decay in isotropic turbulence. This idea of similarity and self-preservation has played an important the development of turbulence theory for more than a half-century. The traditional approach to search for similarity solutions in turbulence has been assume the existence of a single length and velocity scale, then ask whether and under what conditions the turbulent motions admit to such solutions. Excellent contributions had been given to this direction. Von Karman and Howarth (1938) firstly deduced the basic equation, also presented a particular set of solutions of Karman-Howarth equation for the final decaying turbulence. Later on, two Russian authors, Loitsiansky (1939) and Millionshtchikov (1941), have each discussed the solutions of Karman-Howarth equation which are obtained when the term describing the effect of the triple 1 National Natural Science Foundation of China (10272018, 10572083)
http://www.paper.edu.cn
velocity correlation is ignored. Their work is an extension of the “small Reynolds number” solution first put forward by von Karman and Howarth. H.Dryden (1943) gave a useful review on this subject. Great deep research on the solutions of Karman-Howarth equation was conducted by Sedov (1944). He showed that one could use the separability constraint to obtain the analytical solution of Karman-Howarth equation. Sedov’s solution could be expressed in terms of the confluent hypergeometric function. Batchelor (1948), who was also the first to carry out a detailed study of this problem, based his analysis on the assumption that the Loitsianskii integral is a dynamic invariant- a widely accepted hypothesis that was later discovered to be invalid. Batchelor conclude that the only complete self-preserving solution that was internally consistent existed at low turbulence Reynolds numbers where the turbulent kinetic energy consistent with the final period of turbulent decay. Batchelor (1948) also found a self-preserving solution to the Karman-Howarth equation in the limit of infinite Reynolds numbers for which Loitsiansky’s integral was an invariant. Objections were later raised to the use of Loitsiansky’s integral as a dynamic invariant: at high Reynolds numbers this integral can be shown to be a weak function of time (see Proudman & Reid 1954 and Batchelor & Proudman 1956). Saffman (1967) proposed an alternative dynamic invariant which yields another power law decay in the infinite Reynolds numbers limit (see Hinze 1975). While the results of Batchelor and Saffman formally constitute complete self-preserving solutions to the inviscid Karman-Howarth equation, it must be kept in mind that they only exhibit partial self-preservation with respect to the full viscous equation. Recently, George (1992) revived this issue concerning the existence of complete self-preserving solutions in isotropic turbulence. In an interesting paper he claimed to find a complete self-preserving solution, valid for all Reynolds numbers. George’s (1992) analysis is based on the dynamic equation for the energy spectrum rather than on the Karman-Howarth equation. Strictly speaking, the solution presented by George was an anlternative self-preserving solution to that of Karman-Howarth and Batchelor since he relaxed the constraint that the triple longitudinal velocity correlation be self-similarity in the classical sense. Speziale and Bernard (1992) reexamined this issue form a basic theoretical and computational standpoint. Several interesting conclusions had been drawn from their analysis.
From the long history stories, we know that: Decaying homogeneous and isotropic turbulence is one of the most important and extensively explored problems in fluid dynamics and a central piece in the study of turbulence. Despite all the effort, a general theory describing the decay of turbulence based on the first principles has not yet been developed. (Skrbek and Steven R.Stalp, 2000). It thus appears that the theory of self-preservation in homogeneous turbulence has many interesting features that have not yet been fully understood and are worth of further study. (see Speziale and Bernard ,1992,pp.665).
This paper offers a unified investigation of isotropic turbulence, based on the exact solutions of Karman-Howarth equation. Firstly, we will point out that new complete solution set may be exist if we adopt the Sedov (1944) method. Based on the above finds, new results could be obtained for the analysis on turbulence
http://www.paper.edu.cn
features, such as the scaling behaviour, the spectrum, and also the large scale dynamics.
2 Self-preservation solution under Sedov’s separability constraint
For complete self-preserving isotropic turbulence, the Karman-Howarth equation will have a solution if Reynolds number based on the Taylor microscale is constant as first noticed by Dryden (1943). However, this equation also has solutions where Reynolds number based on the Taylor microscale is time dependents when separability is invoked. The separability condition implies that each side of the equation is equal to zero individually, yielding differential equations from which explicit solution for the correlation functions may be determined depending on the choice of parameters. These solutions were first discovered by Sedov (1944) and later compared with experimental data by Korneyev & Sedov (1976). Here, we will discuss the possible new complete solutions under Sedov’s separability constraint.
The two-point double longitudinal velocity correlations read as ( named Sedov equation)
d 2f ⎛4a 1⎞df a 2++⎜+ξ⎟f =0 (1) 2⎜⎟d ξ⎝ξ2⎠d ξ2
with boundary conditionsf (0)=1, f (∞)=0.
In the following analysis, we introduce alternative two parameters denoted by a 1, σ， here
σ=a 2 (2) 2a 1
The complete new set of the solution of the equation (1) with the boundary condition could be given as following：
The first kind of solution：if σ=5，thus 2
−a 124f (ξ)=e (3)
The secondary kind of solution：if κ=σ−5，thus 4
a −1ξ25a ⎛5⎞f (ξ)=e 4F ⎜−σ, , 1ξ2⎟ (4) 24⎝2⎠
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5−σ，thus 4
a −1ξ2⎛5a ⎞f (ξ)=e 4F ⎜σ, , 1ξ2⎟ (5) ⎝24⎠
5The forth kind of solution：if σ=，thus 4
a −12⎛33a ⎞f (ξ)=e 4F ⎜, , 1ξ2⎟ (6) ⎝424⎠
where F (α, γ, z )is the confluent hypergeometric function and the definition of The third kind of solution：if κ=
will be given appendix. From the asymptotic the existing parameterκ
expansions and the limiting forms of the confluent hypergeometric function, we could deduce the existence conditions of these solutions：
For all four kind of solutions：a 1>0,
For the secondary kind of soultions：σ>0；
For the third kind of solution：0
The details could be seen in appendix .A simple comparison shows that the special solution found by Sedov (1944) is belonging to the secondary kind of our new set of solution.
3 Some theoretical results based on the exact solutions
A unified investigation of isotropic turbulence, based on the above exact solutions of Karman-Howarth equation could be given. New results could be obtained for the analysis on turbulence features, such as the scaling behaviour, the spectrum, and also the large scale dynamics, some results could be seen in the following references [19,20,21,22] .
References
[1]Bareenblatt,G.J.& Garilov,A.A. 1974. Sov. Phys. J. Exp. Theor. Phys. 38,399-402.
[2]Batchelor,G.K. 1948.Q. Appl. Maths. 6,97-116.
[3]Batchelor, G.K. 1953 The Theory of Homogeneous Turbulence Turbulence. Cambrige University Press.
[4]Batchelor,G.K. & Proudman,I. 1956. Phil. Trans. R. Soc. Lond. A.248,369-405. [5]Dryden,J.L. 1943. Q. Appl. Maths. 1,7-42.
http://www.paper.edu.cn
[6]George,W.K. 1992. Phys. Fluids A 4,1492-1509.
[6]Hinze,J.O. 1975 Turbulence. McGraw-Hill.
[7]Karman,T.Von & Howarth,L. 1938. Proc. R. Soc. Lond. A164,192-215.
[8]Korneyev & Sedov, L.I. 1976. Fluid Mechanics-Soviet Research 5,37-48.
[9]Lesieur,M. 1990 Turbulence in Fluids, 2nd Edn. Martinus Nijhoff.
[10]Lin,C.C. 1948. Proc. Natl. Acad. Sci. 34,540-543.
[11]Millionshtchikov, M. 1941. Dokl. Akad. Nauk SSSR 32,615-618.
[12]Monin,A.S. & Yaglom,A.M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol.2, MIT Press.
[13]Proundman, I. & Reid, W. H. 1954. Philos. Trans. R. Soc. London, A 247,163-189.
[14]Saffman,P.G. 1967. J. Fluid Mech. 27,581-594.
[15]Sedov, L.I. 1944. Dokl.Akad.Nauk SSSR 42,116-119.
[16]Speziale,C.G. & Bernard, P.S. 1992. J. Fluid Mech. 241,645-667.
[17]Skbek,L. & Steven, R.S. 2000. Phys. Fluids 12,1997-2019.
[18]Tatsumi, T. 1980. Advances in Applied Mechanics,39-133.
[19]Ran zheng, Exact solutions of Karman-Howarth equation. [20]Ran zheng, Scales and their interaction in isotropic turbulence. .
[21]Ran zheng, Dynamic of large scales in isotropic turbulence. .
[22]Ran zheng, On von Karman’s decaying turbulence theory. .
Appendix: Solutions of the correlation coefficients
The ideas of similarity and self-preservation were firstly introduced by von Karman (1938). Following the methods adopted by Sedov (1944,1951), the two point double longitudinal velocity correlations satisfied
d 2f ⎛4a 1⎞df a 2++⎜+ξ⎟f =0 （1） ⎟2⎜d ξ⎝ξ2⎠d ξ2
with the boundary condition
f (0)=1
f (∞)=0
The complete solution are given in this paper, these are：
http://www.paper.edu.cn
−125When σ=，f (ξ)=e 4 2
a −125a 2⎞5⎛5When κ=σ−，f (ξ)=e 4F ⎜−σ, , 1ξ⎟ 424⎝2⎠
−125⎛5a 2⎞When κ=−σ，f (ξ)=e 4F ⎜σ, , 1ξ⎟ 4⎝24⎠
−125⎛33a 2⎞When σ=，f (ξ)=e 4F ⎜, , 1ξ⎟ 4⎝424⎠a a a
The detailed calculation is given as following:
A lot of useful partial differential equations can be reduced to confluent hypergeometric equations. LetP k , m (ς)is the solution of Whittaker equation as that defined by Whittaker and Waston
d 2W ⎡1k 4−m 2⎤+⎢−++⎥W =0 （2） 22d ςς⎦⎣4ς
where
y (z )=z βe f (z )P κ, m (h (z )) （3）
After some reduction, the equation of y (z ) reads
d 2y ⎡h ′′2β⎤dy ′()−++2+g ⋅y (z )=0 （4） f z 2⎢⎥′z dz ⎣h ⎦dz
where
2g =(f ′)−f ′′+2βf ′β(β+1)h ′′⎛β⎞′++f +⎟+g 1 ⎜2′z h ⎝z z ⎠
2h 2⎞⎛h ′⎞⎛12−m +κh −⎟ g 1=⎜⎟⎜⎜⎟4⎠⎝h ⎠⎝4
f (z )=az λ （5）
h (z )=Az λ （6） The solutions of above equation could be deduced in terms of Whittaker function. We discussed this equation in following special case:
The equation under this condition reads as
d 2y ⎡1−λ−2βλ−1⎤dy +−2λα+qy (z )=0 （7） z 2⎢⎥z dz ⎣⎦dz
where
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⎛2A 2⎞2λ−2β(β+λ)+λ24−m 2λ−2q =λ⎜+λ(2αβ+A κλ)z +⎜α−4⎟⎟z z 2⎝⎠2()
The solution of this equation is
λy (z )=z βe αz P κ, m Az λ （8）
For isotropic turbulence, the corresponding parameters satisfied
1−λ−2β=4 （9） ()
λ−1=1 （10）
−2λα=
2a 1 （11） 2⎛2A 2⎞λ⎜⎜α−4⎟⎟=0 （12） ⎝⎠
⎛1⎞β(β+λ)+λ2⎜−m 2⎟=0 （13） ⎝4⎠
a λ(2αβ+A κλ)=2 （14） 2
Hence, we have
λ=2 （15）
a 1 （16） 8
5β=− （17） 2
3m =± （18） 4
a A =±1 （19） 4
⎧a 5⎫κ=±⎨2−⎬ （20） ⎩2a 14⎭α=−
From above analysis, we can introduce two parameters to classification turbulence, they are: a 1, σ=a 2。 2a 1
According to Whittaker and Waston, if 2m isn’t an integral,
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z
21m 2⎞⎛1F ⎜+m −κ, 1+2m , z ⎟ （21） ⎠⎝2
z 1−−m ⎛1⎞P κ, −m (z )=e 2z 2F ⎜−m −κ, 1−2m , z ⎟ （22） ⎠⎝2
For the caseκ=0，we must use the secondary Kummer formula,
1+m ⎛z 2⎞2P 0, m (z )=z 0F 1⎜⎜1+m ; 16⎟⎟ （23） ⎝⎠P κ, m (z )=e z −
By making use of the boundary condition, we could chose the rational parameters for isotropic turbulence. The solution of equation could be rewritten in
y (z )=A 1+m 2⋅e A ⎞2⎛⎜α−⎟z 2⎠⎝z β+λm +λ
2⎛1⎞⋅F ⎜+m −κ, 1+2m , Az λ⎟ （24） ⎝2⎠
Let A >0，this resulted in the definition of exponent.
If we chose m =−3，in the above solution, the exponent ofz reads as 4
2
5⎛3⎞=−+2×⎜−⎟+1 （25） 2⎝4⎠
=−3
The boundary conditiony (0) would be broken under this condition. So we only chose
m =
Another condition must be satisfied 3 （26） 4
α+
The solution is
5
4A =0 （27） 2
There is an important parameter κin the above solution, the multiple values could be existed： When κ=⎨25⎛5⎞y (z )=A ⋅e −Az ⋅F ⎜−κ, , Az 2⎟ （28） 2⎝4⎠⎧a 25⎫−⎬， ⎩2a 14⎭
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y (z )=A ⋅e
when κ=−⎨54−Az 25⎛5⎞⋅F ⎜−σ, , Az 2⎟ （29） 2⎝2⎠⎧a 25⎫−⎬， ⎩2a 14⎭
5
42⎛5⎞y (z )=A ⋅e −Az ⋅F ⎜σ, , Az 2⎟ （30） ⎝2⎠
We must treat the other special caseκ=0，by using the secondary Kummer formula
P 0, m (z )=z
where 1+m 2⎛z 2⎞0F 1⎜⎜1+m ; 16⎟⎟ （31） ⎝⎠
z −⎛z 2⎞20F 1⎜⎜1+m ; 16⎟⎟=e F (m , 2m , z ) ⎝⎠
For this case, the solution of equation is
y (z )=A ⋅e
Another reduced case for σ=54−Az 2⎛33⎞⋅F ⎜, , Az 2⎟ （32） ⎝42⎠5，the solution is 2
f (ξ)=e −a 124 （33）
At last, we have already obtained a complete set solution of isotropic turbulence , depending on two parameters, these are： −125when σ=，f (ξ)=e 4 2
a −1ξ25a 2⎞5⎛5when κ=σ−，f (ξ)=e 4F ⎜−σ, , 1ξ⎟ 424⎠⎝2
−1ξ25⎛5a 2⎞when κ=−σ，f (ξ)=e 4F ⎜σ, , 1ξ⎟ 4⎠⎝24
−1ξ25⎛33a 2⎞If σ=，f (ξ)=e 4F ⎜, , 1ξ⎟ 4⎝424⎠a a a
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References
[1]Whittaker,E.T. and Waston, G.N., A course of modern analysis. Cambridge University Press, 1935
[2]M.Abramowitz and I.A.Stegun, Handbook of mathematical functions. Dover, New York,1965
[3]Wang, Z.X. and Guo,D. R., Special functions. The series of advanced physics of Peking University. Peking University Press,2000 (In Chinese)