Statistical Theory of Turbulence
by the Late Lamented Dr. Shunichi Tsugé
Case Study on Flow through a Grid in Wind Tunnel
Takeo Nakagawa, Hiroyuki Iida
School of Information Science
Japan Advanced Institute of Science and Technology
Nomi, Japan,jp;
AbstractThis paper is concerned with statistical theory of turbulence by the late lamented Dr. Shunichi Tsugª. The theory has
been applied to the primary flow through a grid fixed vertically with respect to the horizontal axis of the wind tunnel. The first
analytical solution has been obtained and explained the well-known “the inverse-linear decay law” of the turbulent intensity. It is
believed that the present result is the first exact solution in the theory of turbulence.
Keywords-grid-produced turbulence;exact solution;turbulent intensity;statistical theory of turbulence;applied mathematics
1. Introduction
Contrary to the firm kinetic theoretical basis of the Navier-
Stokes equation for laminar flows, which verification dates
back to Chapman[1], Enskog[2] and Grad[3], the history of its
turbulent counterpart starts in late 1960s: It is known that two
pioneering workers attempt to two-particle version of the
Euler equation(Zhigulev[4]) and of the Navier-Stokes
equatio n(T sugª=?), which are governing turbulent
correlations for inviscid and viscid gases, respectively. It is
notable that these two papers have proposed a rather striking
thesis, completely contradicting to the conventional belief that
“the kinetic theory is useless for turbulence, because it is
merely concerned with molecular fluctuations having order of
n(the mean number density), and thus are negligibly small
compared with macroscopic turbulent fluctuations of order of
n2”. In fact, human sensors are unable to perceive molecular
fluctuations, consisting of thermal agitation such as molecular
stress and heat-flux fluctuations, as discussed by Landau &
Lifshitz[6], together with fluctuations due to real-gas effects.
An independent innovative hypothesis is proposed by
Tsugª[7] and Grad[8] has made us possible to incorporate
macroscopic turbulent fluctuations into the regime of the
kinetic theory. That is, the two-particle molecular chaos due
to Boltzman is replaced with a less stringent tertiary molecular
chaos. This milder hypothesis leads us to a new finding that in
a shear flow, turbulent correlations are survived over thousand
mean free paths, or a macroscopic fluid-dynamic length, being
detectable with any flow device used currently.
In 1974, it is shown by Tsugª [7]that the equations
governing two-point correlations in an incompressible shear
flow are separable into two Orr-Sommerfeld type equations at
the respective points. It is, however, realized that physical
meaning s of the variables in the equations are much different
from those in the Orr-Sommerfeld equation. In the same
paper, Tsugª=? has proved that the fluid moments obtained
from the one-particle kinetic equation are equivalent with the
Navier-Stokes equation(Nakagawa [9]), and the two-particle
version, the equations governing two-point correlations,
reduces to the KÂrman-Howarth equation.
The main purpose of the present paper is to obtain an exact
solution for the flow through a grid in the wind tunnel based
on the statistical theory of turbulence by Tsugª=?.
2. Equations Governing Boltzmann
Function F and Double Correlation
Function G
In order for Boltzmann function f and double correlation
function g to be identified by using variables in the BBGKY-
hierarchy theory, after Bogoliubov, Born, Green, Kirkwood
and Yvon, the following condition is required, for the
averaging time Ǽ must be longer than the time Ǽg for
satisfying the ergodicity;
Ǽ Ҝ Ǽg (1)
With the assumption (1), the dependent variables (f,g) may
be described by the general framework of the hierarchy
equations(Grad [10]):
У/ Уt+uкУ/Уy)e=J(aфÂ)[e Ô+g(a, Â)], (2)
(У/Уt +uкУ/Уy + ĬкУ/УˠICÂ
,Cф£=GI£Â«ICÂ?,Âф£= Ôg( «, e)+«IÔG?
where tertiary molecular chaos,
h(e, Ô, «)҂қَeَÔَ«Ҝ=0, (4)
has been adopted to truncate the hierarchy system. It may be
worth noting here that if one put g(a, Â)=қَeَÔҜ=0, binary
Open Journal of Applied Sciences
Supplement2012 world Congress on Engineering and Technology
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molecular chaos, the above hierarchy system reduces to the
Boltzman equation:
У/ Уt+uкУ/Уy)e=J(aфÂ)[e Ô].
3. Flow Through a Grid in Wind Tunnel
It may be evident that in the flow through a grid fixed
normal to the main flow direction , turbulence is generated and
then it decays with increasing the distance from the grid, by
experiencing the diffusion as well as viscous dissipation
mainly. This turbulence is the topic to obtain the exact
analytical solution.
The grid-produced turbulence is neither homogeneous nor
isotropic, but an isotropic, for there exists a specific vector of
the main flow direction(Fig.1).
Fig.1 Cross section of two-dimensional plane waves in
(x2,x3)-plane. x1: primary flow direction.
Let us assume the two-point correlation is separable in the
RǩǪ(y,ˠ,t)=ě[ьo ǩ(y,t:ȁ)ǀě(ˠ,t:ȁ)dȁ], (5)
ǀ=o*(*;conjugate complex), (6)
where ȁis the constant separating the variables, ě[ ]denotes
taking real part.
Then, equations governing o ǩ reduces to a set of integro-
differential equations in the separated 3-dimensional space as
Уor/Уyr=0, (7)
(-iȁ+У/Уt+urкУ/Уyr -ǵкУ2/Уyr2)oj+Уuj /Уyrкor+1/
ǹкУo4/Уyj+У/Уyrьп-пor(Q)oj(ǡ-Q)dQ=0, (j,r=1,2,3).
 These (7) and (8) have been solved for the grid-produced
turbulence in the wind tunnel flow u=(U,0,0) with initial
fluctuations given at the plane;x1=0.
Instead of solving the complete boundary value problem, the
first analytical solution associated with the present theory is
sought to explain the existing experimental finding, viz., ̌the
inverse-linear decay law̍ of the turbulent intensity.
A. Formulation of the Problem
Let the grid-produced turbulence be composed of a plane
non-dispersive wave in the form,
Ǫ2=kкcosǰ, Ǫ3=kкsinǰ,
where ǰ is the azimuth angle of the oblique wave plane
normal to the mean flow direction(Fig.2).
Fig.2 Definition sketch of angles ǰandǾ
Let, then, Qǩmake a Fourier transform into Fǩ in order to
eliminate the nonlinear convolution integral of (8),
Qǩ(x1,ǡ)=1/(2Ǹ) ьп-пFǩ(x1,s)exp(-iǡs))ds. (10)
Note inverse Fourier transform Fǩ(x1,s) is defined as
Fǩ(x1,s)= ьп-пQǩ(x1,ǡ)exp(isǡ))dǡ,
where Qǩ(x1,ǡ) is an infinitely differentiable function of
bounded support.
Substituting (9), together with (10) into (7) and (8), we have
УF1/Уx1+ кУF2/Уs+Ǫ3кУF3/Уs=0, (11)
L(F1 ) + 1 /ǹкУF4/Уx1+NL(F1)=0, (12)
L(F2 ) + 1 /ǹкǪ2УF4/Уs+NL(F2)=0, (13)
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L(F3 ) + 1 /ǹкǪ3УF4/Уs+NL(F3)=0, (14)
L=UУ/Уx1 - ǵЧ, (15)
Ч=У2/Уx12+(Ǫ22+Ǫ32)к У2/Уs2, (16)
NL= F1кУ/Уx1+( Ǫ2F2+Ǫ3F3)к У/Уs. (17)
Then, the following non-dimensional expressions are
introduced into (11)-14),
Ƕ= x1/M, ǯ=s/M, f1= F1/U, f2= F2/U, f3= F3/U, f4= F4/(ǹ
U2), Re=UM/ǵ, (18)
we obtain
Уf1/УǶ+Уf/Уǯ=0, (19)
Ȏ(f1)+ Уf4/УǶ+Ȣ(f1)=0, (20)
Ȏ(f)+ Уf4/Уǯ+Ȣ(f)=0, (21)
J(g)+H(g)=0, (22)
f=Ǫ2f2+Ǫ3f3, (23)
g=Ǫ3f2-Ǫ2f3, (24)
or, inversely
f2=( fкcosǰ+gкsinǰ)/k, (25)
f3=( fкsi nǰ-gкcosǰ)/k, (26)
J=У/УǶ-1/R eкЧ, (27)
Ч=У2/УǶ2+k2к У2/Уǯ2, (28)
H= f1кУ/УǶ+fкУ/Уǯ. (29)
The turbulent correlations defined by (5) may be also
expressed in terms of the Fourier transformed dependent
variables as follows,
U2/(2Ǹ)2ь02Ǹާь-ппfjfi d ǯިdǰ,(j,I;1,2,3). (30)
It may be justified that the grid-produced turbulence is axis-
symmetric with respect to the mean flow direction, namely,
homogeneous in any plane normal to its direction. Such a
turbulent flow is, therefore, described by superimposing the
plane waves considered here, and by averaging over the angle
ǰ within (x2,x3);
, ǯ+krкcos(ǰ-Ǿ)ިdǯުdǰ,
^ 2-x2)2+(x
where ǰ and Ǿ are angles defined in Fig.2
B. The Exact Solution of Grid-produced Turbulence
It may be straight forward that (19)-(22) are two-
dimensional, so that it may be possible to introduce a stream
function Ǡ in the form,
f1=УǠ/Уǯ, f=-УǠ/УǶ.
Then, combining (20), and (21) in order to eliminate f4, we
(J+ǠǯкУ/УǶ-ǠǶкУ/Уǯ)ЧǠ=0. (31)
This suggests that any harmonic function for Ǡ, namely,
solution of the Laplace differential equation, ЧǠ=0, turns out
to be an exact solution of the above full-nonlinear equation
(31). A particular solution, which is no more than a version
of the general solutions, the relevant integral constants being
specified by the boundary conditions at ǯ=дп, and Ƕ=п,
and whose components(ǠǶ, Ǡǯ) exhibiting the decay law,
may be expressed by
Ǡ=ǁarctan(ǯ/kǶ). (32)
It is easy to verify that by substituting (32) into the Laplace
differential equation
Ǡ=(У2/УǶ2+k2к У2/Уǯ2)Ǡ=0,
Ǡ is the solution. Moreover, substitution of (32) into (31)
results in the following relations,
f1=Ǡǯ=ǁ/( kǶ)ާ1+(ǯ/ kǶ)2ި-1, (33)
f=-ǠǶ= ǁǯ/( kǶ2)ާ1+(ǯ/ kǶ)2ި-1. (34)
The turbulent intensity in the Ƕ-direction, which is the non-
dimensional longitudinal coordinate of x1/M, can be
calculated by substituting f1 in (33) into (30), and integrating
it with respect to ǯ and ǰ, and results in
қ(Чu1)2 Ҝ/U2=ǁ2/(4kǶ), (35),
U2/қ(Чu1)2 Ҝ=4 kǶ/ǁ2. (36)
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4. Conclusion
The present result (36) shows that the inverse of the mean
squared fluctuation of the turbulent velocity component in the
x1 mean flow direction is proportional to the normalized
coordinate of Ƕ.
In Fig.3 are compared the predicted inverse decay rate of the
turbulent velocity in the mean flow direction with the classical
data taken by Batchelor & Townsend[11], who have
confirmed experimentally the turbulent energy decay
maintains similarity irrespective of the difference in the
Reynolds number Re=UM/ǵ.
It is believed that (36) is the first exact solution in statistical
theory of turbulence, so it has a permanent value.
(ig.3 Similarity of energy decay at different
Reynolds numbers
(after Batchelor & Townsend[11]).
X:M=0.635, ٨:M=1.27cm, +:M=2.54cm, ڈ:M=5.08cm
г : present theory,
U=longitudinal velocity=1286cm/s,
u1= longitudinal velocity fluctuation,
x=longitudinal coordinate,
M=grid mesh size.
[1] S. Chapman, ̌On the kinetic theory of a gas Τ,̍Phil.
Trans. Roy. Soc. London, vol. A217, pp.115-197, 1917.
2] D. Enskog, “Kinetische Theorie der Vorgˆnge in mˆssig
verdünten Gasen,” Dissertation, Uppsala, Sweden 1917.
[3] H. Grad, “On the kinetic theory of rarefied gases,”
Comm. Pure appl. Math. vol. 2, pp.331-407, 1949.
[4] V. N. Zhigulev, “Equations for turbulent motion of a gas,”
Soviet Physics Doklady, vol. 10, pp.1003-1005, 1966.
[5] S. Tsugé, “On the divergent growth of molecular
fluctuations in classical shear flow,” Phys. Letters, vol. 33A,
pp.145-146, 1970.
[6] L. D. Landau, and E. M. Lifshitz, Fluid Mechanics,
Addison-Wesley, Mass. USA, 1959
[7] S. Tsugé, “ Approach to origin of turbulence on the basis
of two-point kinetic theory,” Phys. Fluids, vol.17, pp.22-
[8] H. Grad, “Singular limits of solutions of Boltzmann’s
equation,” in Rarefied Gas Dynamics (ed. K. Karamcheti),
New York, Academic Press, pp.37-53, 1974.
[9] T. Nakagawa, “A theory of decay of grid-produced
turbulence,” ZAMM, vol.59, pp.648-651, 1979.
[10] H. Grad, “Principles of the kinetic theory of gases,”
Handbuch Der Physik, 12. Auflage, pp.205-294, 1958.
[12] G. K. Batchelor, and A. A. Townsend, “Decay of
isotropic turbulence in the initial period of turbulence,” Proc.
Roy. Soc. vol. London A193, pp.539-558, 1948.
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