Emergence of Multi-Scaling in a Random Force-Stirred Fluid
Victor Yakhot Diego Donzis










 


 
 
 
 
 
 
 
 



 Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843,USA
 

























 


 We consider transition to strong  turbulence in an infinite fluid stirred  by a gaussian random force.  The   transition is defined  as a   first appearance of anomalous scaling of  normalized    moments  of velocity derivatives (dissipation rates) 
 emerging from  the low-Reynolds-number   Gaussian background.  
 It is shown  that due to multi-scaling,  
 strongly intermittent  rare events     
  can be quantitatively described  in terms  of  an infinite number of different “Reynolds numbers”  reflecting multitude of anomalous scaling exponents.   The theoretically predicted  transition disappears at   R_λ≤  3.  The developed theory, is in a quantitative agreement with the outcome of large-scale numerical simulations.


 Emergence of Multi-Scaling in a Random-Force Stirred Fluid. 
     Diego Donzis 
    December 30, 2023
=============================================================

PACS numbers 47.27


Introduction.  
If an infinite  fluid is stirred by a gaussian random force supported in a narrow interval of the wave-numbers k≈ 2π/L, then a very weak forcing  leads to generation of  a random, close-to-gaussian,  velocity field.  In this flow the  mean velocity u=0 and one can introduce the large-scale Reynolds number Re=u_rmsL/ν where the root-mean-square velocity  u_rms=√(u^2).
Increasing   the forcing  amplitude or decrease of viscosity result in a strongly  non-gaussian random flow with moments of velocity derivatives obeying the so-called anomalous scaling.  This means that the moments (∂_xu_x)^2n/(∂_xu_x)^2^n∝ Re^ρ_2n where the exponents ρ_n are,  on the first glance,  unrelated  “strange” numbers. 
In this paper we investigate  the  transition between these two different random/chaotic  flow regimes. First, we discuss some general aspects of the  traditional problem of hydrodynamic stability and transition to turbulence.

  
 Fluid flow can be described by the Navier-Stokes equations subject to boundary and initial conditions (the density is taken ρ=1 without loss of generality): 


    ∂_t u+ u·∇ u=-∇ p +ν∇^2 u +  f


and   ∇· u=0. The characteristic  velocity and length scales u and L,  used for making  the Navier-Stokes equations  dimensionless, are somewhat arbitrary.  In the problem of  a flow past cylinder it is natural to choose f=0, u=U, and L=D where U and D are free-stream velocity and cylinder diameter, respectively.  In a pipe/channel flow u=U=1/H∫_0^Hu(y)dy∝ u_centerline is the mean velocity averaged over  cross-section and L=H is a half-width of the channel. 
In a fully turbulent flow in an infinite fluid one typically   takes 
u=v_rms=√(v^2) and L equal to the integral scale of turbulence. Some other definitions will be discussed below.

Depending on  a setup,  a flow can be generated by pressure/temperature gradients,  gravity,  rotation, electro-magnetic fields etc represented  as forcing functions on the right side of  (1).
If viscosity ν≥ν_tr and the corresponding Reynolds number Re=uL/ν≤ Re_tr=uL/ν_tr, the solution to (1) driven by the regular (not random) forcing f is laminar and regular. As examples, we may recall parabolic velocity profile 
u(y) in pipe/channel flows with prescribed pressure difference between  inlet and outlet. 
In this case the no-slip boundary conditions are responsible for generation of the rate-of-strain S_ij=(∂_iu_j+∂_ju_i)/2. Another important example is  the so called Kolmogorov flow in an infinite fluid driven by the forcing function f=U(0,0,cos kx).  In Benard convection the relevant regular patterns are rolls appearing as  a result of instability of  solution to the conductivity equation. 
Thus, the remarkably successful science of transition to turbulence deals mainly with various aspects of  non-equilibrium order-disorder  or laminar-to-turbulent transition. 


 
In this paper we consider a completely different class of flows. In general,  the  unforced NS equations, being a very important and interesting object,  do not fully describe the  physical reality which includes   Brownian motion,  light scattering, random wall roughness, uncertain inlet conditions, stirring by “random swimmers” in biofluids   etc.   For example, a fluid  in thermodynamic equilibrium satisfies  the   fluctuation-dissipation theorem stating that there  exist an exact relation between viscosity ν  in (1)  and a random noise f which is a Gaussian force  defined by the correlation function [1]:

 
    f_i(  k,ω)f_j( k',ω')= (2π)^d+1D_0d(k)P_ij( k)δ(ω+ω')δ( k+k')


where the projection operator is: P_ij( k)=δ_ij-k_ik_j/k^2.   In  an equilibrium fluid    thermal fluctuations, responsible for Brownian motion   are generated by the forcing (2) with D_0d(k)= k_BT ν/ρk^2≡ D_0k^2.  It is clear that, in general,  the function d(k) in (2) depends on  the physics of  a  flow. 
 


 
The random-force-driven NS equation can be written in the Fourier space:


    u_l( k,ω)=G^0f_l( k,ω)-i/2G^0 P_lmn∫ u_m(q,Ω)u_n(k-q,ω-Ω)d kdΩ


where G^0=(-iω+ν k^2)^-1,  P_lmn( k)=k_nP_lm( k)+k_mP_ln( k)  and,  introducing the zero-order solution u_0=G^0 f∝√(D_0), so that u=G^0 f+ v,  one derives the equation for perturbation v:



    v_l(k̂)=-i/2G^0(k̂) P_lmn( k)∫ v_m(q̂)v_n(k̂-q̂)dq̂
    
    -i/2G^0(k̂) P_lmn( k)∫  [v_m(q̂)G^0(k̂-q̂)f_n(k̂-q̂)+ G^0(q̂)f_m(q̂)v_n(k̂-q̂)]dq̂
    
    -i/2G^0(k̂) P_lmn( k)∫ G^0(q̂f_m(q̂)G^0(k̂-q̂)f_n(k̂-q̂)dq̂


 
where the 4-vector k̂=( k,ω).  If the equation (3)-(4) is driven  by a regular force or boundary and/or  initial conditions, then at low-Reynolds number  (Re)  it typically describes  a  regular (laminar) flow field u_0  with  v=0.  With increase of the  Reynolds number Re≥ Re_inst,  this zero-order solution can become unstable, meaning that initially-introduced small perturbations  v grow in time.  Further increase of Re leads  first to  weak interactions between the modes describing the “gas” of these perturbations and, eventually,  when Re-Re_inst/Re_inst≫ 1 mode coupling described by equation (4) becomes very strong.  This regime we call “fully developed” or strong turbulence. The problem of hydrodynamic stability is notoriously difficult and we know very little  about structure of solution for perturbations in the non-universal range Re≈  Re_inst.  


Here we are interested in a simplified   problem of a flow generated by a  gaussian random force (2) with  a well -understood zero-order solution u_0=G^0 f which is not  a result of an instability of a regular  laminar flow but  is prescribed by a choice  of a random force (2).  
The advantages of this formulation are clear from (4)  describing  the dynamics of perturbation 
v driven by  an induced forcing given by the  O(f^2)∝ D_0 last term in (4).  It is easy to see [1],[9] that dimensionless expansion parameter, related to a Reynolds number (see below),  is Γ^2_0=D_0L^4/ν^3Δ where Δ=∫ d(k)d k.  and, since we keep L=O(1),  ν=O(1) and Δ=O(1), the variable  forcing amplitude D_0 can be treated as a dimensionless expansion parameter. 
Thus,  as D_0→ 0,  all  contributions to the right side of (4) can be neglected  and,  if   f stands for  the gaussian random function, then the lowest-order solution u_0  is a gaussian field.. 
 However,   there always exist low-probability  rare events with | v|≥ | u_0| responsible for the 
 strongly non-gaussian tails of the PDF.  Thus, in this flow gaussian velocity fluctuations coexist with the low-probability powerful events where substantial fraction of kinetic energy is dissipated. 
 At  even higher  Reynolds numbers (see below) the non-linearity in (4)  dominates the entire field.  This  complicated dynamics has been observed in experiments on a channel flow with rough (“noisy”) walls [2].

  
This regime is characterized by  the generation of velocity fluctuations v( k,t) in the wave-number  range   k>2π/L  where the “bare” forcing f( k)=0,    which  is  the hallmark   of turbulence.      The above example shows that at least in some range of the Reynolds number    
 low and high-order moments may describe  very different  physical phenomena.  The transition between these two chaotic/random  states of a fluid  is a topic we are interested in this paper.   


Two  cases  are of  a special interest. 
 In  the  low Reynolds number regime  (below transition),  when R_λ=√(5/3 Eν)u_rms^2<R^tr_λ,    the integral (L), dissipation (η)  and Taylor (λ) length scales are of the same order.    Therefore,   (∂_x u_x)_rms=  (u(x+η)-u(x))_rms/η≈ (u(x+L)-u(x))_rms/L  and, since we are interested in  instability of a gaussian flow,  the moments  
 
 
    M_n^<=(∂_xv_x )^2n/(∂_xv_x)^2^n=(2n-1)!!
 


independent on the Reynolds number.  In this case, since the 2n^th-order   moment  can be expressed in powers of the variance,  this means that (∂_xv_x)_rms is a single parameter  (derivative scale) representing statistical properties the flow   in this regime.
This is not always the case. The rms velocity derivative in high Reynolds number turbulent flows,   (∂_xv_x)_rms=√((∂_xv_x)^2)  is   only one of an infinite number of  independent parameters needed to describe the field and in the vicinity of transition Re≥ Re^tr :


    M^>_n= (∂_xv_x )^2n/(∂_xv_x)^2^n= (2n-1)!!C_nRe^ρ_n≈ (2n-1)!!R_λ^ρ_n



where R_λ>R^tr_λ and  the proportionality coefficients  C_n=O(1)  [3], [4].  

Below, this anomalous state of a  fluid we call strong turbulence as opposed to the close-to-gaussian low Reynolds number flow field, considered above.  
In a  transitional,  low Reynolds number,  flow  we are interested in here, the forcing, Taylor and dissipation scales are of the same order L≈η≈λ. 
The   Reynolds number  based on the Taylor length-scale is thus:


    R_λ≡  R_λ,1= √(5/ 3 Eν)v_rms^2≈√(5L^4/3 Eν)(∂_xv_x)^2


The physical meaning of this parameter can be seen readily: multiply  and divide (5) by ν and by the dissipation scale η^2. This gives 


    R_λ∝L^2/η^2×η^2√( E/ν^3)≈L^2/η^2



where η^4 E/ν^3=O(1).  The effective Reynolds number O(L^2/η^2), which is the measure of  the spread   of the inertial range in k-space, is a coupling constant, familiar from dynamic renormalization group applications to randomly stirred fluids. 
To describe strong turbulence, one must introduce  an infinite number of   “Reynolds” numbers 

    R_λ,n=  √(5L^4/ 3 Eν)(∂_xv_x))^2n^1/n∝ R_λ^ρ_2n/n∝L^2/η^2 E^n^1/n/ E


where close to transition points  where η≈ L we set R_λ≡ R_λ,1≈ Re. The expressions for exponents ρ_2n 


    ρ_2n=2n+ξ_4n/ξ_4n-ξ_4n+1-1; ξ_n=0.383n/1+n/20




derived in the “mean-field approximation”  in  [4]-[5], 
agree extremely  well with all available experimental and numerical data (see  Refs.[5]-[8]).   Theoretical predictions of anomalous exponents in a random-force-stirred fluid are compared with the results of numerical simulations  [6] on a top panel of Figure 1.  Note that normalized moments of dissipation rate M_n( E) are simply M_2n in the present formulation. 
The same  exponents have been observed in a channel  flow [7] and Benard convection [8], indicating universality of small-scale features in turbulent flows.  



 







 
 
 
 
 

 
 Transition between gaussian and  anomalous flows.   
  In this paper  transition to turbulence is identified with   first appearance of non-gaussian anomalous  fluctuations of velocity derivatives.  
  The concept is illustrated on the bottom  panel of   Fig.1,  where  moments of velocity derivatives from well resolved numerical simulations (described below)  are plotted  against Reynolds numbers R_λ≡ R_λ,1≥ 2..  We can see that transition points of different moments, expressed in terms of R_λ≡ R_λ,1,  are different and below we denote them R^tr_λ,1(n).   
 It is important that transition point for  the lowest  order  moment M_n with n≈ 1  has been found at R_λ≡ R_λ,1≈ 9 first  discovered  in Ref.[5]   and analytically derived in [9]-[10].  This result can be explained as  follows.

  
In accord with the widely accepted methodology, consider the R_λ,1≡ R_λ-dependence of  the normalized n^th  derivative moment  M_n in a flow driven by   a relatively  weak force  f and large viscosity ν.  Then,   gradually decreasing viscosity,  one reaches the critical   magnitude ν=ν_tr  corresponding to R_λ^tr(n)=R^-_λ(n)  which is the upper limit  for gaussianity of the n^th moment.
Then,  consider  the same flow but at a    very large Reynolds number  (small viscosity). In this,  strongly turbulent case,   the large-scale low-order moment,   M_4 for example,  are dominated by a  huge  turbulent viscosity ν_T∝ E^1/3L^4/3, the largest effective viscosity , accounting for velocity fluctuations at the scales r<L  [1].  
 The effective Reynolds number, corresponding to the integral scale L,   is   R_λ^+∝√(L^4/( Eν_T(L)))(∂_xu)_rms^2.  
 This way one reaches the smallest possible Reynolds number R_λ≈ 9 of strongly turbulent (anomalous) flow (see Fig.1).
 If, in accord with experimental and numerical data, we assume   that transition is smooth and at a transition point the Reynolds number is a continuous function meaning that  R_λ^-=R_λ^+, where R_λ^± stand  for the magnitudes just above and below transition, 
   we can write:
 

    R_λ^tr(4)=√(5/3 Eν_tr) v_rms^2=√(5/3 Eν_T(L)) v_rms^2
  

where effective viscosity  of turbulence at the largest (integral) scale calculated in 
 Refs. [9]-[11], is given by 
 
 
    ν_T≡ν(L)≈ 0.084 K^2/ E;



 where K=v_rms^2/2 stands for kinetic energy of velocity fluctuations.  Substituting this into the previous relation gives:


    R_λ^tr(4)=√(5/3 Eν) v_rms^2=√(20/(3× 0.084)) =8.98≈ 9.

 
 
extremely close to the outcome of numerical simulations. 
The coefficient C_μ=0.084, derived in [9]-[11] is to be compared with C_μ=0.09 widely used in engineering turbulent modeling for  half a century [12].  It follows from the  relations (5)-(6):


    R^tr_λ(n)≡ R^tr_λ,1(n)=(R^tr_λ,n)^n/ρ_2n

 
  
The Reynolds number dependence of normalized moments  of velocity derivative is shown on Fig.1.  The data in the bottom panel  of Fig.1 was generated from  a new set of simulations 
at very low Reynolds numbers.
As in [6],
numerical solutions to Navier-Stokes equations 
are obtained from Fourier pseudo-spectral
calculations with second-order Runge-Kutta integration in time. 
The turbulence is
forced numerically at the large scales,
using a combination of independent Ornstein-Uhlenbeck processes
with Gaussian statistics and finite-time correlation. 
Only low wavenumbers modes
within a sphere of radius k_F≈ 2
in wavenumber space are forced.
In order to obtain different Reynolds numbers, 
viscosity is changed accordingly while the forcing at 
large scales remains constant. 
In this approach, thus, large scales, and thus the energy
flux, remain statistically similar.
Resolution is at least k_maxη≈ 3 at the 
highest Reynolds number which was found to produce converged 
results at the Reynolds numbers investigated here.

Velocity fields are saved at regular time intervals that are 
sufficiently far apart
(of the order of an eddy-turnover time) to 
ensure statistical independence between them. For each field
velocity gradients moments are computed and averaged over space. 
Ensemble average is computed across these snapshots in time
and are used to compute confidence intervals also shown in Fig.1. 
 
The intersection points of curves describing gaussian moments  
(horizontal dashed lines) and  those corresponding to the  fully- turbulent anomalous scaling 
  give transitional  R_λ^tr(n)  for each moment.  These are compared  to the theoretical prediction of Eq.(10) with R_λ,n^tr≈ 8.5 in Fig.2.
This result can be understood as follows:  in accord with theoretical predictions the  transitional  Reynolds number R^tr_λ,n≈ const≈ 9  in each statistical realization.
 If R_λ,1<R^tr_λ≈ 9,  the transition is triggered by the low -probability violent velocity fluctuations 
  (∂_xv_x)^n^1/n>(∂_xx_x)_rms coming    from the tails of probability density.   
   
  It is also interesting to evaluate the limiting, smallest,  transitional Reynolds number  following  (10)  in the limit n→∞. The relations (5)-(6),(10)  give R_λ= R^tr_λ,1→ 2.92.  Evaluated on a popular model ξ_n= n/9+2(1-(2/3)^n/3) [13], one readily derives  R^tr_λ,1→ 3.81.  According to both models,  in a   flow with R_λ≤ 3,   no transition to strong turbulence defined by anomalous scaling of moments of velocity derivatives  exist. 

  
  Summary and conclusion. In this paper a    problem of transition between two different random  states has been studied both  analytically and numerically.  It has been shown that  while the gaussian state can be described in terms of the Reynolds number based on the  variance of probability density,  the description of the  intermittent state of strong  turbulence requires  an infinite number of "Reynolds numbers"  R_λ,n reflecting  the multitude of anomalous scaling exponents of different-order moments  (n)  of velocity derivatives.  This   novel  concept   enables one to account  for   both typical  and violent extreme  events  responsible for emergence of anomalous scaling  in the “sub-critical” state when the widely used Reynolds number R_λ,1<R^tr_λ is small.   It has also been demonstrated  that,  in accord with the theory,  the critical  R^tr_λ,n≈ 9 is   independent of n.  The proposed theory is in a good quantitative agreement with the  results of large-scale  direct numerical simulations presented above.   The role of turbulent bursts in  low Reynolds number flows in various physico-chemical processes  and the problem of  universality will be discussed in  future communications.
  
  

     
We are grateful to  H. Chen,  A.Polyakov,  D. Ruelle, J. Schumacher,  I. Staroselsky, Ya.G. Sinai, K.R. Sreenivasan  and M.Vergassola    for many stimulating and informative discussions.  DD acknowledges support from NSF.
          



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