Similarity Solutions of Velocity Distributions in
Homogeneous Isotropic Turbulence
Zheng Ran
Shanghai Institute of Applied Mathematics and Mechanics,Shanghai University, Shanghai 200072,
P.R.China Abstract
The starting point for this paper lies in the results obtained by Tatsumi (2004) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the one-point velocity distribution function equation obtained by Tatsumi (2004) leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This paper revisits this interesting problem from a new point of view. Firstly, a new complete set of solutions are obtained. Based on these exact solutions, some physically significant consequences of recent advances in the theory of homogenous statistical solution of the Navier-Stokes equations are presented. The comparison with former theory was also made. The origin of non-gaussian character could be deduced from the above exact solutions.
Keywords: isotropic turbulence, PDF, similarity solution
The complete statistical information of turbulence is provided by the probability distribution functional of the velocity field in space and time. The evolution equation for this distribution functional was given by Hopf (1952) in terms of the characteristic functional equation. So far, however, no mathematical method has been known for dealing with such a functional equation generally, and only a few particular solutions given by Hopf himself have been available. A statistically equivalent formalism to distribution functional is given by an infinite set of the joint-probability distribution of the velocities at arbitrary number of points in space-time. The system of equations governing such joint-probability distributions was given by Lundgren (1967) and Monin (1967) independently. The set of equations, however, is indeterminate since each equation involves a higher-order distribution as new unknown. Such indeterminacy is known as the common difficulty of nonlinear physics and the closure of the set of equations is the central problem of turbulence. Recently, the hypothesis of cross-independence of two turbulent velocities has been proposed by Tatsumi (2001) for closing the equations for the velocity distributions given by Lundgren (1967) and Monin (1967). Using this hypothesis, the equations for one-and two-point velocity distributions in homogeneous isotropic turbulence are derived in closed form, and the velocity distributions are obtained from these equations, One-point velocity distribution is obtained as an inertial normal distribution (N1) including the energy dissipation rate as only parameter. The inertial similarity of homogeneous isotropic turbulence associated with the normal one-and two-point velocity distributions seems to give good prospects for the extension of the further approach to inhomogeneous turbulent flows and more complex turbulent phenomena. The start point lives in the work of Tatsumi (2001) with the hypothesis of cross-independence of two turbulent velocities. New similarity exact solutions could be found. This would be change subtlety the results presented by Tatsumi (2001).
Let us consider turbulence in an incompressible viscous fluid, in particular, homogeneous isotropic turbulence whose probability distributions are uniform and isotropic in space. Take the
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coordinates x=(x1,x2,x3) , the time t and denote the fluid velocity at point (x,t) by
u(x,t) and the pressure by p(x,t). We denote the velocities at the two space points x1,x2
and at time t by u(x1,t),u(x2,t), the one-and two-point velocity distributions are defined, respectively, as
f(v1,x1,t)=δ(u(x1,t)−v1) (1)
f2(v1,v2;x1,x2;t)=δ(u(x1,t)−v1)δ(u(x2,t)−v2) (2)
where vi,i=1,2 denotes the probability variable corresponding to the velocities u(xi,t),
δthe three-dimensional delta function, and
initial distribution.
the probability mean with respect to a certain
Tatisumi[5,6] introduced the sum and the difference of two velocities u1,u2 as
1
(u1+u2) (3) 21
u−=(u2−u1) (4)
2u+=
and call them the cross-velocities of two velocities u1,u2. The distributions of the cross-velocities are defined in accordance with the former.
g±(v±,x1,t)=δ(u±(x1,t)−v±) (5)
g2(v±,v±;x1,x2;t)=δ(u±(x1,t)−v1)δ(u(x2,t)−v
where
) (6)
v+=
1
(v1+v2) (7) 21
v−=(v2−v1) (8)
2
Tatsumi (2001) introduced a new independence relation between the cross-velocities by
g2(v+,v−;r,t)=g+(v+,r,t)g−(v−,r,t) (9)
that may be called the cross-independence in contrast to the ordinary independence. Under this consideration, one can obtain the closed form equation for one-point velocity distribution (Ref.6, Eq.50)
⎡∂∂
()t+α⎢
∂v⎢⎣∂t
2
⎤
⎥⋅f(v,t)=0 (10) ⎥⎦
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where
α(t)=ε(t) (11)
13
ε represents the mean energy dissipation rate.
Let us consider homogeneous turbulence without external energy supply and obtain the velocity distribution as the solution of the governing equation (10), and we introduce the self-preserving hypothesis as
⎛v⎞
f(v,t)≡b(t)Y⎜⎟ (12) ⎜l(t)⎟
⎠⎝
ξ=
νl(t) (13)
On substitution from (12), Eq. (10) reduces to the following form
d2Y⎛2a1⎞dYa2
+⎜+ξ⎟+Y=0 (14) 2⎜⎟2dξ⎝ξ2⎠dξY(0)=1,Y(∞)=0. (15)
where
−
1dla
⋅=1 (16)
α(t)l(t)dt2
l2(t)dba2
(17) ⋅=
α(t)b(t)dt2
We call these the scaling equations. Now, let us confine ourselves to the self-similar solution in
time. Under this consideration
ai=constant, (18)
For i=1,2.
The turbulence energy equation reads as
dE=−ε (19) dt
For decaying turbulence, we have
E(t)=E0t−1 (20)
ε(t)=ε0t−2 (21)
Substitution these into the scaling equations, we have
l2(t)=
a1ε0−12
t+l0 (22) 6
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b(t)−σ=(1+ωt) (23) b0
where
6l02
ω=
a1ε0
σ=
a2
a1
The complete solution are given in this paper, these are(the details could be seen in Appendix)： When
σ=3(denotes type SI)，
Y1(ξ)=e
−a12ξ4
(24)
When
κ=σ−(denotes type SI)，
Y2(ξ)=e
−a12ξ4
32
3a⎛9⎞
F⎜−σ,,1ξ2⎟ (25)
24⎝4⎠
When
κ=
3
−σ(denotes type SIII), 2
Y3(ξ)=e
−a12ξ4
33a⎛⎞
F⎜σ−,,1ξ2⎟ (26)
424⎝⎠
When
σ=
3
(denotes SVI) 2
Y4(ξ)=e
−a12ξ4
⎛11a⎞
F⎜,,1ξ2⎟ (27) ⎝424⎠
We could deduce the corresponding asymptotic expansions of the above solutions, and give the
existence conditions for different type of solutions. (1) Existence condition for SI
In order to satisfy the boundary condition at infinity, we must have
a1>0 (28)
This is the only existence condition for SI, and also one of existence conditions for other three solutions.
(2) Existence condition for SII
F2(ξ)∝ξThe boundary condition leads to
⎛3⎞2⎜−σ⎟4⎝⎠
(29)
σ>
3
(30) 4
which is one of the existence conditions for SII.
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(3) Existence condition for SIII
F3(ξ)∝ξSo
9⎞⎛
2⎜σ−⎟
4⎠⎝
(31)
σ<
(4) Existence condition for SIV
9
(32) 4
−12
F4(ξ)∝ξ (33)
So the only existence condition for SIV is a1>0.
We can draw the following conclusion on the existence conditions for the complete set of solutions:
(1)For all kind of solution: a1>0. (2)For the second kind of solution：σ>
33；σ≠; 4293
(3)For the third kind of solution：0<σ<；σ≠;
42
A simple comparison shows that the special solution found by Tatsumi (2001) belongs to one
special kind of our new set of solution. Other differences between Tatsumi’s solution and our new set of solutions would be discussed below.
In order to give a complete description of turbulence features, we must introduce a probability structure based on these solutions. Letpj,j=1,2,3,4 denotes the probability of the j’th class solution，and the set P is called the state vector of this process defined by
P≡(p1,p2,p3,p4). (34)
The state vector is a function of the parameterσ, we could be easy to write as follows (1)If 0<σ≤
3
，the probability vector is 4
P≡(p1,p2,p3,p4)=(0,0,1,0) (35)
(2)If
33
<σ<，we have 42
⎛11⎞
P≡(p1,p2,p3,p4)=⎜0,,,0⎟ (36)
⎝22⎠
(3)If
σ=
3
, we have 2
P≡(p1,p2,p3,p4)=(0,0,0,1) (37)
(5)If
39<σ<, 24
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⎛11⎞
P≡(p1,p2,p3,p4)=⎜0,,,0⎟ (38)
⎝22⎠
(6)If
9
≤σ<3，we have 4
P≡(p1,p2,p3,p4)=(0,1,0,0) (39)
(7)If
σ=3，
⎞⎛11
P≡(p1,p2,p3,p4)=⎜,,0,0⎟ (40)
⎠⎝22
(8)If
σ>3，we have
P≡(p1,p2,p3,p4)=(0,1,0,0) (41)
Based on the above deduced probability structure, the average of turbulence could be given, which
represents the statistical distributions of turbulence. In the following discuss, we will deal with the one-point velocity distribution function, in principle, we could give the average of these physical quantities.
f=∑pjfj (42)
j=1
4
(1)If 0<σ≤
3
，the probability vector is 4
f=f3 (43)
(2)If
33
<σ<，we have 42
f=
(3)If
1
(f2+f3) (44) 2
σ=
3
, we have 2
f=f4 (45)
(5)If
39<σ<, 24
f=
(6)If
1
(f2+f3) (46) 2
9
≤σ<3，we have 4
f=f2 (47)
(7)If
σ=3，
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f=
(8)If
1
(f2+f1) (48) 2
σ>3，we have
f=f2 (49)
Actually, in a turbulent viscid fluid, the statistics of the velocity field at a fixed point is non-gaussian. For example, let us set (2)If
σ=1.0, and it belongs to the secondary case:
33
<σ<，we have 42
f=
the part in terms of the Yj(ξ) )
1
(f2+f3) (44) 2
So, we could easily write the expression of the one-point velocity distribution as : (only presented
Y=
a
1
(Y2+Y3) (45a) 2
1−41ξ2⎧⎛53a12⎞⎛13a12⎞⎫
,,ξY=eFF+⎜⎟⎜,,ξ⎟⎬⎨
2424⎠⎝424⎠⎭⎩⎝≈e
a−1ξ24
⎧2a12⎫⎨1+⋅ξ+...⎬⎩34⎭
(45b)
We can see that Yis non-guassian type distribution. The comparison is presented in Figure 1.
10.90.80.70.60.50.40.30.20.1002PresentSolutionGaussianSolutionfr46
Figure.1 Comparison of the one-point velocity distribution solution
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[1]Hopf,E., Statistical hydromechanics and functional calculus. J. Rat. Mech. Anal. 1,87-123.(1952).
[2]Kida, S., Murakami,Y., J.Phys. Soc. Japan, 57,3657-3660(1988). [3]Lundgren,T.S., Phys. Fluids. 10,969-975(1967).
[4]Monin, A.S., PMM J.Appl.Mech.31,1057-1068(1967).
[5]Tatsumi,T., Mathematical physics of turbulence. In: Kambe, T., et al. (Ed.), Geometry and Statistics of Turbulence. Kluwer Academic Publishers, Dordrecht, pp,3-12. [6]Tatsumi, T. & Yoshimura, T., Fluid Dynamics Research, 35,123-158(2004).
Appendix : Solutions of the PDF equation
The ideas of similarity and self-preservation were firstly introduced by von Karman (1938). Following the methods adopted by von Karman (1938), the one-pointl velocity distribution function satisfied
d2f⎛2a1⎞dfa2
+f=0 （1） +⎜+ξ⎟⎟2⎜dξ⎝ξ2⎠dξ2
with the boundary condition
f(0)=1 f(∞)=0
The complete solution are given in this paper, these are： When
a12
ξ4
σ=3，f(ξ)=e
−
a
−1ξ233a2⎞⎛9
When κ=σ−，f(ξ)=e4F⎜−σ,,1ξ⎟
224⎝4⎠−1ξ2333a2⎞⎛
When κ=−σ，f(ξ)=e4F⎜σ−,,1ξ⎟
2424⎝⎠−1ξ23⎛11a2⎞
When σ=，f(ξ)=e4F⎜,,1ξ⎟
2⎝424⎠
a
a
The detailed calculation is given as following:
A lot of useful partial differential equations can be reduced to confluent hypergeometric equations,
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Pk,m(ς)is the solution of Whittaker equation as that defined by Whittaker and Waston
d2W⎡1κ14−m2⎤
+⎢−++⎥W=0 （2） 22
ςdς⎦⎣4ςwhere
y(z)=zβef(z)Pκ,m(h(z)) （3）
After some reduction, the equation of y(z) reads
d2y⎡h′′2βdz2−⎢⎣h′+z+2f′(z)⎤⎥dy⎦dz
+g⋅y(z)=0 where
g=(f′)2
−f′′+2βf′β(β+1)h′′⎛βz+z2+h′⎜⎝z+f′⎞⎟⎠
+g1 g⎛h′⎞
2
⎛1h21=⎜⎝h⎟
⎠
⎜⎞⎜24−m+κh−4⎟⎝⎟ ⎠
The solutions of above equation could be deduced in terms of Whittaker function. We discussed this equation in following special case:
f(z)=azλ h(z)=Azλ The equation under this condition reads as
d2ydz2+⎡⎢1−λ−2β⎣z−2λαzλ−1⎤⎥dy⎦
dz+qy(z)=0 where
q=λ2
⎛2⎜⎜α2−A⎞2λ−2+λ(2αβλ−2
β(β+λ)+λ2(14−m2)⎝
4⎟⎟⎠z+Aκλ)z+z2The solution of this equation is
y(z)=zβe
αzλP(λκ,mAz) For isotropic turbulence, the corresponding parameters satisfied
1−λ−2β=2 λ−1=1 −2λα=
a1
2
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（4） （5） （6）
（7） （8）
（9）
10）
11） （ （http://www.paper.edu.cn
⎛2A2⎞λ⎜⎟=0 （12） ⎜α−4⎟
⎠⎝
2
β(β+λ)+λ2⎜−m2⎟=0 （13）
λ(2αβ+Aκλ)=
Hence, we have
⎛1⎝4⎞⎠
a2
（14） 2
λ=2 （15） α=−
a1
（16） 8
321
m=± （18）
4
β=− （17）
A=±
a1
（19） 4
⎧a23⎫
−⎬ （20） ⎩a12⎭
κ=±⎨
From above analysis, we can introduce two parameters to classification turbulence, they are:
a1,σ=
a2
. a1
According to Whittaker and Waston, if 2m isn’t an integral,
Pk,m(z)=ez
−
−
z21+m2
⎞⎛1
F⎜+m−κ,1+2m,z⎟ （21）
⎠⎝2
⎞⎛1
F⎜−m−κ,1−2m,z⎟ （22）
⎠⎝2
⎛z2⎞
0F1⎜⎟ （23） ⎜1+m;16⎟
⎠⎝
Pk,−m(z)=ez
For the case
z
21−m2
κ=0，we must use the secondary Kummer formula,
P0,m(z)=z
1+m2
By making use of the boundary condition, we could chose the rational parameters for isotropic turbulence. The solution of equation could be rewritten in
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y(z)=zβeαzPκ,mAzλλ()⎞⎛1
F⎜+m−κ,1+2m,Azλ⎟ （24）
⎠⎝2
⎞⎛1
⋅F⎜+m−κ,1+2m,Azλ⎟
⎠⎝2
λ2
=ze=A
βαzλ⋅e
A
−z22
(Az)z
λ2+m
1
1+m2
⋅e
A⎞2⎛
⎜α−⎟z
2⎠⎝
β+λm+
LetA>0，this resulted in the definition of exponent. If we chose m=−
1
，in the above solution, the exponent ofzreads as 4
23⎛1⎞
=−+2×⎜−⎟+1 （25）
2⎝4⎠=−2<0
β+λm+
λThe boundary conditiony(0) would be broken under this condition. So we only chose
m=
Another condition must be satisfied
1
（26） 4
α+
The solution is
A
=0 （27） 2
23⎛3⎞
y(z)=e−Az⋅F⎜−κ,,Az2⎟ （28）
2⎝4⎠
There is an important parameter kin the above solution, the multiple values could be existed： When
κ=⎨
⎧a23⎫
−⎬， ⎩a12⎭
23⎛9⎞
y(z)=e−Az⋅F⎜−σ,,Az2⎟ （29）
2⎝4⎠
when
κ=−⎨
⎧a23⎫
−⎬， ⎩a12⎭
233⎛⎞
y(z)=e−Az⋅F⎜σ−,,Az2⎟ （30）
42⎝⎠
We must treat the other special casek=0，by using the secondary Kummer formula
P0,m(z)=z
1
+m2
⎛z2⎞0F1⎜⎟ （31） ⎜1+m;16⎟
⎠⎝
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where
z−⎛z2⎞2
0F1⎜⎟=eF(m,2m,z) ⎜1+m;16⎟
⎠⎝
For this case, the solution of equation is
y(z)=e
Another reduced case for
−Az2
⎛11⎞
⋅F⎜,,Az2⎟ （32） ⎝42⎠
a12
ξ4
σ=3，the solution is
f(ξ)=e
−
（33）
At last, we have already obtained a complete set solution of isotropic turbulence , depending on two parameters, these are： When
σ=3，f(ξ)=e
−
a12ξ4
a
−1ξ233a2⎞⎛9
When κ=σ−，f(ξ)=e4F⎜−σ,,1ξ⎟
224⎝4⎠−1ξ233a2⎞3⎛
When κ=−σ，f(ξ)=e4F⎜σ−,,1ξ⎟
4242⎝⎠−1ξ23⎛11a2⎞
When σ=，f(ξ)=e4F⎜,,1ξ⎟
2⎝424⎠
a
a
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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