Presents a new approach to the Navier-Stokes turbulence. With the Gaussian soft constraint on the Navier-Stokes equation, the author derives the energy spectrum to be E(k) approximately k^-3 and k^-2 in two and three spatial dimensions respectively. The possible future developments are also pointed out

Liao, Wuwell

English

Caltech Authors - Main

J. Phys. A: Math. Gen. 22 (1989) L737-L741. Printed in the UK 
LEmER TO THE EDITOR 
Statistical properties of solutions to the Navier-Stokes 
equation in the limit of vanishing viscosity 
Wuwell Liao 
Condensed Matter Physics, California Institute of Technology, Pasadena, CA 91 125, USA 
Received 8 May 1989 
Abstract. In this letter, we present a new approach to the Navier-Stokes turbulence. With 
the Gaussian soft constraint on the Navier-Stokes equation, we derive the energy spectrum 
to be E ( k )  - k - 3  and k-* in two and three spatial dimensions respectively. We also point 
out the possible future developments. 
This is the first of a series of articles investigating the Navier-Stokes turbulence. The 
objective is to find out what is the relationship, if any, between properties of the fluid 
turbulence and statistical dynamics of the Navier-Stokes equation. The major contribu- 
tion of this letter is to present a new viewpoint and hence a new approach to the 
problem of fully developed turbulence. 
Our starting point is the incompressible Navier-Stokes equation. We take as granted 
that the incompressible Navier-Stokes equation is the underlying evolution equation 
appropriate for the fully developed turbulence. Even if this is true, there is still the 
problem of the existence and uniqueness of solutions to the Navier-Stokes equation 
in three dimensions for large time in the limit of vanishing viscosity. People worry 
about the possibility of a finite-time singularity in solutions of Euler’s equation in 
three dimensionst. However, there is no definitive proof of this scenario. It might be 
possible that the vortex line zigzags around as it stretches so as to avoid a finite-time 
singularity. For the purpose of this calculation, we have to assume that solutions of 
the Navier-Stokes equation exist and are unique for all time in three dimensions when 
the viscosity becomes small. 
The Navier-Stokes equation is first order in time. If we assume the existence and 
uniqueness of the solution, then when the boundary condition and the initial velocity 
profile are specified exactly, we can follow the subsequent evolution of the solution 
and predict its future behaviour exactly. However, in the experiment of the fluid 
turbulence, we cannot control the initial condition precisely. When we repeat the 
experiment, our initial velocity profile is almost the same as before. But there is certain 
probability of deviating from the previous set-up. Nevertheless, we observe similar 
turbulent behaviour. Hence, our measurement is in a sense the average of solutions 
to the Navier-Stokes equation, due to our inability in, setting up the precise initial 
condition. 
I believe that the main task of the theoretical investigation into the fluid turbulence 
is to find a way to translate the experimental situation into a suitable mathematical 
t For two dimensions, one has the existence and uniqueness of solutions for all time, see [l]. 
0305-4470/89/150137 +05f02.50 @ 1989 IOP Publishing Ltd L737 
L738 Letter to the Editor 
language. Experimentally, we can measure the simultaneous n-point correlation of 
velocity fluctuations. In particular, the most often quoted quantity is the velocity- 
velocity correlation in Fourier space. One typical experiment is to set up a mesh wire 
at the upstream of the wind tunnel and then measure the n-point velocity correlations 
at various places downstream. As we continue the experiment, our upstream initial 
velocity profile will have a certain probability distribution, because we cannot control 
the initial condition exactly. Hence, if we fix our probes at certain places downstream, 
the measurement we make is an average over solutions of the Navier-Stokes equation. 
Now, it is found that even though there are complications due to the presence of the 
boundary wall, the region far away from the boundary shows remarkable universal 
statistical behaviours. 
It is the major challenge for theorists to understand these universal behaviours. 
Thus, as a first step towards the understanding of the fully developed turbulence, we 
assume that we have a fluid occupying the whole space and let the velocity be zero at 
infinity. From experiments, we know that all fluids show the same turbulent behaviour 
when the Reynold’s number is large. Hence it is reasonable to assume that in the limit 
of vanishing viscosity, almost any random initial condition will lead to the same 
turbulent behaviour. In a more realistic calculation, this assumption has to be verified. 
That will be the second phase of the investigation. For the moment, let us assume 
that this is true. Then the problem is to find the scaling behaviour of an n-point 
velocity correlation function, averaged over all solutions of the Navier-Stokes equation. 
We hope that the correct Kolmogorov scaling region will be recovered in the limit of 
vanishing viscosity. 
We start with the incompressible Navier-Stokes equation (with p = 1): 
au 
- + ( u s  V ) u  = -VP+ v v 2 u  
a t  
v . v = o  and lim u(x, t )  = 0. 
l4-= 
The generating functional of correlation functions of the velocity fields, averaged 
over all solutions to the Navier-Stokes equation with certain initial distribution of 
velocity profile, is: 
Z [ J ]  = %(x, t )  dp(A(x))  exp ( I a  ddx Iom dtJ(x,  t )  . U(X, t )  I 
d v + ( u * V ) u + V P - v V 2 ~  ( 3 )  
where p(A(x))  is the probability distribution function of the initial velocity profile 
A(x). We have put a lattice cutoff of order a in the spatial integral. We need an 
ultraviolet cutoff because the Navier-Stokes equation is a hydrodynamic equation. 
From various experiments on the fluid turbulence, the data seem to suggest that 
in the fully developed isotropic and homogeneous turbulence, the two-point velocity 
correlation in Fourier space displays the scaling behaviour with an exponent indepen- 
dent of initial conditions and energy inputs. If this is true, then in the limit of vanishing 
viscosity, we can weight all possible initial conditions equally if we are interested only 
in the derivation of the scaling exponent. This is what we are going to do. The 
generating functional of n-point velocity correlation functions in equation (3 )  is hard 
to compute when one has to deal with the initial velocity profile. 
Letter to the Editor L739 
Next, we reason that all solutions of the Navier-Stokes equation for t 2 O  evolve 
from almost all possible initial conditions at t = 0. In order for this reasoning to work, 
we must demand that solutions to the Navier-Stokes equation as initial value problems 
are unique for all time. We also have to assume that solutions never blow up in finite 
time. Since eventually we are only interested in the limit of vanishing viscosity, we 
do not have to worry about the possibility that all solutions will be damped out by 
the viscosity. 
Therefore we replace the averaging over all possible initial conditions by the 
averaging over all solutions of the Navier-Stokes equation with equal weight: 
Z [ J ]  = 9u(x, t )  exp (la ddx lom d t J (x ,  t )  - U(X, t )  I 
1 x exp { la ddx lom dt [ i+a (: ( u  * V)u, +aaP - vV2ua 
However, there are certain technical difficulties with the above formulation. First 
of all, the usual power counting in determining the dimensions of the fields U and t,b 
is not possible here. The dimensions of the fields are indeterminate. Secondly, we do 
not know how to deal with the free case. In the free case, the Lagrangian is linear in 
both U and t,b. I have not found a way to handle the linear Lagrangian. 
We have to emphasise that the reason we arrive at the linear Lagrangian as the 
free part is because that we must restrict the field configurations to those satisfying 
the Navier-Stokes equation. The reason is obvious: if we believe that the Navier-Stokes 
equation is the underlying master equation for the fluid, then any realisable velocity 
field configuration must evolve through the Navier-Stokes equation. 
In order to make further progress with the calculation, let us try the following trick. 
Since we are interested in the limit of viscosity, we can soften the 8-function constraint 
by the following procedure: 
Next we perform the Fourier transform in space and the cosine transform in time 
on the velocity fields. We can do this because the domain of iterest of u(x, t )  is t > 0 
and hence we are free to extend the definition of u(x, t )  to the range t < 0 in such a 
way that u(x, t )  = u(x, - r ) .  
L740 Letter to the Editor 
After the following transformation: 
u,(x, t )  = 1' * lom g dw cos( wt )  exp(ik * x ) v a (  k, w )  
u,(k, w ) =  ddx jOm p d t c o s ( w t )  IT exp(-ik.x)v,(x, t )  
( 2 d d  
the generating functional becomes 
ddk * 
z [ J ( ~ ,  w)]=lim J fi 9v,(k, w) exp( I - dwJ,(-k, w)u,(k, w )  
v - r o  , = I  ( 2 d d  0 
x e x p [ - l l s l o m d w u u ( - k ,  w) 
x [S(0, +a,- w ) +  S(0, +a,+ w)]  (8) 
where we have conveniently set the upper momentum cutoff to be 1. 
Now, it is well known that when the Lagrangian is multiplied by a constant, then 
in the limit of that constant becoming infinite, the only surviving graphs are those 
without loops [2]. Hence, with the softening of the 6-function constraint by the 
procedure described in equation (9, the current theory is on the same level as the 
Ginzburg-Landau theory of phase transitions of the second kind. However, the 
three-point and four-point vertex functions ( I P I  proper functions) diverge as v + 0. I 
suppose this is not very surprising. The limit of vanishing viscosity is a very singular 
limit. 
Since we have to take the limit v + O  first, the only non-zero two-point Green 
function is: 
(u,(k, O ) U a ( - k ,  0))x 1/k4. (9) 
Because we are averaging over all solutions equally and taking the limit v+O, there 
is no timescale in the problem. The correlation function is time invariant: 
Trkd-'(va(k,O)u,(-k,O))= E(k)  d t a E ( k ) .  J 
Thus: 
E ( k ) o c l / k 2  in 3~ 
E ( k ) a  l / k 3  in 2 ~ .  
Letter to the Editor L741 
Perhaps I should compare my approach to previous attempts on the Navier-Stokes 
turbulence. In the literature, the best reference in the subject of applying the field- 
theoretic method to the Navier-Stokes turbulence is the paper by De Dominicis and 
Martin [3]. However, their approach is quite different. De Dominicis and Martin 
considered the Navier-Stokes equation with a Gaussian-generated random force. They 
had to adjust the correlation of the random force properly in order to match the 
Kolmogrov 2 spectrum. As a matter of fact, the majority of the work in applying the 
renormalisation group method to the Navier-Stokes turbulence starts with the 
Gaussian-generated random force. There, the force-force scaling exponent, which is 
a free parameter in the theory, actually determines the scaling exponent of the velocity- 
velocity correlation in k-space. In the current approach, on the contrary, there is 
absolutely no adjustable parameter. 
In summary, I have presented a new approach to the Navier-Stokes turbulence. 
In this approach, one does not have to consider separately the energy cascade in three 
dimensions and the enstrophy cascade in two dimensions in order to derive the right 
energy spectrum. We feel strongly that one should not need further information besides 
the Navier-Stokes equation to derive the correct energy spectrum. Of course, the 
answers in equation (11) are not quite correct. We believe that the problem lies with 
the softening of the 8-function constraint on the Navier-Stokes equation. It is well 
known in the graph-bipartitioning problem that one would arrive at the wrong solution 
if one does not enforce the constraint exactly [4]. We think this is what happens here. 
We are currently investigating equation (4) more rigorously. The result will be published 
soon. We hope then to shed light on the validity of the assumption that solutions of 
the Navier-Stokes equation exist and are unique for all time in three dimensions in 
the limit of vanishing viscosity. 
I would like to thank Professor M Cross, Jan Segert, Vipul Periwal, Michael Grabinski, 
Jonathan Miller, Professor Y Avron, Peter Weichmann and Ken Aoki for interesting 
conversations. Support from the Division Research Fellowship and a US National 
Science Foundation Grant No DMR-8715474 are gratefully acknowledged. 
References 
[ I ]  Kat0 T 1967 Arch Rat. Mech. Anal. 25 188-200 
[2] Amit D J Field theory, the Renormalization Group, and Critical Phenomena, (Singapore: World Scientific) 
[3] De Dominicis C and Martin P C 1979 Phys. Rev. A 9 419-22 
[4] Liao W 1987 Phys. Rev. Lerr. 59 1625-8 
ch 6 


Statistical properties of solutions to the Navier-Stokes equation in the limit of vanishing viscosity

theory, the Renormalization Group, and Critical Phenomena,

theory, the Renormalization Group, and Critical Phenomena, (Singapore: World Scientific)

http://authors.library.caltech.edu/6587/1/LIAjpa89.pdf

Statistical properties of solutions to the Navier-Stokes equation in the limit of vanishing viscosity

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