FFrom Burgers to Navier-Stokes turbulence
K.P. Zybin ∗ , It is shown that the origin of the Kolmogorov’s law of the fully developed turbulence is the resultof the joint stochastic dynamics of pair points separated by the shock. The result obtained in 1-dcase generalized on 3-d turbulence. A novel procedure of determination of correlation functions in3-d turbulence is proposed.PACS numbers: 47.10.ad, 47.26.E-, 05.40.-a
Burgers equation was scrutinized (see [1]) because ofmany physical applications. I will restrict my considera-tion to the problems of fluid turbulence only. OriginallyBurgers equation was introduced as a simplification ofthe Navier-Stokes (NS) equation with hope of sheddingsome light on issues such as turbulence. Besides, there isstill an idea that time evolution of NS equation leads tosingularity (in the limit of zero viscosity) which is respon-sible for Kolmogorov scaling [2],[3]. It is whell known thatwithout viscosity the time evolution of Burgers equationresults in singularity v ∝ x / but it exists for a momentonly and it is not connected with the velocity statistics.Stochastic Burgers equation under the action of large-scale force takes a form v t + vv x = νv xx + g ( x, t ) (1)Large scale forcing g ( x, t ) implies the existence of the in-ertial interval η ( ν ) (cid:28) l (cid:28) L analogous to NS equation,where η ( ν ) is a dissipative scale and L is the scale of theforce. Actually, the structure functions of velocity in-crements obtained from numerical simulation of Burgersequation exhibit scaling law analogous to the real turbu-lence [1]: S n = (cid:104) [ v ( x + l, t ) − v ( x, t )] n (cid:105) ∝ l ξ n (2)when ξ n = n/ n < ξ n = 1for n ≥
3. The saturation of the scaling exponents couldbe naturally understood in the framework of correlationsof the step functions. As is well known the long timeevolution of the equation (1) results in the appearanceof shocks (in the limit ν → S n ( shock ) = (cid:104) [ sign ( x + l ) − sign ( x )] n (cid:105) ∝ l As for structure functions exponents law ξ n = n/
3, thereis dimensional analysis [4], and exact result for the 3-dorder structure function, plus Polyakov’s idea of spon-taneous breaking of Galilean invariance and dissipativeanomaly [5], but there is no analytical calculation ofthem. Polyakov’s approach was based on introduction of the characteristic function of n -point velocity distri-bution Z n ( λ j , x j , t ) = (cid:68) exp (cid:110)(cid:88) λ j v j ( x j ) (cid:111)(cid:69) (3)Structure functions could be obtained from (3) by differ-entiating on λ i and then putting the lambdas zero : (cid:104) v i ....v i k (cid:105) = (cid:0) ∂ k Z n (cid:1) / ( ∂λ i ...∂λ i k ) (cid:12)(cid:12) λ i =0 This approach was generalized in case of generating func-tional for velocities and velocity derivations [6]. But thisquantum field theory consideration allows to find someinstanton solution when λ is a big parameter [5]-[6]. As aresult the obtained solution does not allow to get struc-ture functions by differentiating this relation on λ andputting λ = 0 afterwords. Thus the statistical descrip-tion of Burgers turbulence does not exists.The aim of my paper to develop statistical theory forBurgers equation and determine the structure functionsexponents in the case of NS equation.For our purpose it is more convenient to use anotherfunction F N which is Laplas transformation of (3): F N = (cid:104) θ ( u − v ( x )) θ ( u − v ( x )) ... θ ( u n − v n ( x n )) (cid:105) Probability distribution function could be easily deter-mined from this relation by the formula: P n = ( ∂ n F N ) / ( ∂u ...∂u n )One can get for F N an equation [5]: ∂ t F N + (cid:88) u k ∂ x k F N − (cid:88) κ ( x i − x j ) ∂ u i u j F N = (4) D N = (cid:88) ∂ u i Γ iN here Γ iN takes the form:Γ iN = ν∂ yy (cid:90) u du ∂ u F N +1 ( u , x i + y ; u , x ; ...u N , x N )The equation (4) is obtained directly from (1) in a propo-sition of Gaussiatity of the large scale force g ( x, t ): (cid:104) g ( x , t ) g ( x , t ) (cid:105) = κ ( x − x ) δ ( t − t ) a r X i v : . [ phy s i c s . f l u - dyn ] D ec Let us consider equation for F function, the term D could be presented in the form: D = lim x → x ( ∂ u d + ∂ u d )here d = ν∂ x x (cid:90) u du ∂ u F ( u , x ; u , x ; u , x )The term d could be written analogously. It is necessaryto remember, that function F here is F = (cid:104) θ ( u − u ( x )) θ ( u − u ( x )) θ ( u − u ( x )) (cid:105) hence ∂ x (cid:90) u du ∂ u F = (cid:104) ∂ x u θ ( u − u ( x )) θ ( u − u ( x )) (cid:105) To calculate derivation on x it is necessary to differen-tiate on u before. So we have: d = ν (cid:104) ∂ x u ∂ x u θ ( u − u ( x )) θ ( u − u ( x )) (cid:105) (5)To calculate it in the limit ν → g ( x, t ) let us solve the equation (1) by standardmethods of expansion of variables and matching obtainedasymptotic solutions [7]. Introducing a new variable y = x/ν one can get from (1): ∂ yy v − v∂ y v = ν ( ∂ t v − g ( νy, t )) (6)Now let us fix the variable y and put ν = 0. The mainasymptotic takes a form2 ∂ y v = v − v After integration of this well-known equation we will get v = v (cid:16) − e v x/ν (cid:17) / (cid:16) e v x/ν (cid:17) + U (7)We choose v = U if y = 0. As we see it is enough to takeinto account antisymmetric part of the velocity only, thesymmetric part is smooth and does not give any inputinto dissipation.To match this solution to an external one let us con-sider F function. κ (0) ∂ u u F + D = 0Multiplying this equation on u and integrating over du we find: κ (0) = ν (cid:104) ∂ x u ∂ x u (cid:105)| x → x Now let us suppose an ergodicity of the obtained solu-tions. It is a usual proposition in experimental study of the developed turbulence. On the basis of this proposi-tion and taking into account (7) we get: (cid:68) ν ( ∂ x v ) (cid:69) = v / (3 L ) (8)This relation should be fulfilled in the limit L → ∞ .Matching it with the previous one we get an idea of “in-frared anomaly”: lim L →∞ (cid:2) v ( L ) / (3 L ) (cid:3) = κ (0) (9)Now let us return to expression for d (5). In accordancewith ergodic hypothesis we define average as: < A ( x ) θ [ x ] θ [ x ] > = 12 L (cid:90) + L − L A ( x + y ) θ [ x + y ] θ [ x + y ] dy Here θ [ x ] is a symbolic name of theta function from F N definition. According to this definition the average (5)could be rewritten as a space integral.As it was mentioned above, our distribution func-tion F contains either fast or slow variables and solu-tion (because of linearity) could be presented as a sum f ( f ast ) + f ( slow ). It is very easy to find f ( f ast ). Infact combining it from solution (7) one can get: f ( f ast ) = 12 L (cid:90) + L − L θ ( u − v ( x − x + y )) θ ( u − v ( y )) dy Where v , ( x ) are smooth-step-like solution (7). Analo-gously one can introduce f n ( f ast ) for arbitrary n . Obvi-ously it will be a solution for generating function (if weneglect an interaction between steps) since (7) is a solu-tion of the Burgers equation and diffusion on velocitieddoes not give any input into function f ( u − u , x − x ).In the limit ν → < ( δv ) n > ∝ l for n >
1. Besides, in this limit the probability of thecoincidence of two shocks in one point is equal zero.Now let us separate the slow part of the distributionfunction. If theta functions changes slowly on the scale1 /ν the production ∂ u∂ u could be presented (in thelimit ν → lim ν → (cid:104) ∂ x u ∂ x u (cid:105) = (cid:2) v ( L ) / (3 L ) (cid:3) δ ( x − x )Thus d = v / (3 L ) × (cid:90) dyδ ( x − x − y ) ∂ u θ ( u − u ( x + y )) θ ( u − u ( x + y )) ≈ [ v / (3 L )] ∂ u f ( u , x ; u , x )Taking into account (9), we will find: u ∂ x f + u ∂ x f = (cid:88) i,j =1 , κ ( x i − x j ) ∂ u i u j f + κ (0) (cid:2) ∂ u u f + ∂ u u f (cid:3) Due to uniformity condition the distribution function de-pends on l = x − x only. Introducing v = u − u and U = u + u , after integration over dU we have got: v∂ l f = [2 κ (0) − κ ( l ) − κ ( − l ) + κ (0)] ∂ vv f (10)Taking into account that l << L , the equation takes asimple form: v∂ l f = κ (0) ∂ vv f It is important to note that stationary solution of thisequation exists only if ( vl ) >
0. That is why structurefunctions < | v | n > are not well defined. In numericalmodeling < | v | > ∝ l . but is not coincide with exactanalytical result l [8]. Possibly the absence of station-arity for ( vl ) < k → ∞ , which corresponds to l →
0. In this case one can find WKB solution: f ∝ exp (cid:110) − √ kv / / (cid:111) , k ≈ [ l κ (0)] − It is easy to get Kolmogorov’s law by integrating thisdistribution function: (cid:104) v n (cid:105) = (cid:90) v n f dv ∝ l n/ Thus we have combination of the solutions: (cid:104) v n (cid:105) = C l + C l n/ It means that for 0 < n < l → n/ n > ν . First let’s put the typical scale of the externalforce equal to infinity and g (+ ∞ ) = g + correspondingly g ( −∞ ) = g − .The condition of the existence of the ν expansion isthe absence of secular term. Integrating (6) over dy from − R → −∞ to R → + ∞ on can get: (cid:90) R − R [2 ∂ t v − ( g + + g − ) − ( g + − g − ) sign y ] dy < ∞ this relation and (7) gives:2 ∂ t U = ( g + + g − ) = φ ( t ) , ∂ t v = ( g + − g − ) = ψ ( t )It is necessary to note also that we should choose v = 0if ψ = 0 and in this case there is no shock and as theresult there is no dissipation on it.Now let us take a look at the dynamics of two points x ( t ) , x ( t ). If these points are both located on the leftor on the right side of the shock the velocity difference v = v ( t ) − v ( t ) and distance l = x ( t ) − x ( t ) does notchange with time. But if we consider a case when, x > x < l andvelocity difference v changes with time: ∂ t l = v , ∂ t v = ψ One can introduce probability distribution function P ( v, l, t ) = (cid:104) δ ( l − l ( t )) δ ( v − v ( t )) (cid:105) Considering ψ ( t ) as Gaussian stochastic process it is easyto get equation ∂ t P + v∂ l P = D∂ vv P which in stationary conditions coincides with (10). Sowe see from our consideration that Kolmogorov’s inputinto structure function is connected with joint stochasticdynamics of the points separated by the shock. It isworth noting that this equation is equivalent to infinitechain for the structure functions S n = < v n > . Actuallymultiplying (10) by v n one can get: ∂ l S n +1 = DS n − (11)It is easy to see that S n ∝ l n/ is the solution of theequation (11).Thus we see that according to discussed theory thestructure function exponents should be bi-fractal if youcalculate them in the region v i l i > t → ∞ . In the case of NS equa-tion a singular-like solution was discussed in [9],[10]. Itwas obtained in inertial interval by introducing large-scale velocity U i ( x, t ) instead force. The external forcewas determined by equation: ∂ t U i + U j ∂x j U i = F i ( x, t )It is necessary to emphasize that U i is given functionand this equation is the definition of the force F i inthe inertial interval. But inside inertial interval force F i does not work. All the energy input is defined bylarge-scale forcing. In this case one can choose U i asuniform and isotropic Gauss stochastic process (or arbi-trary symmetric in time process). Actually in this casewe have < U i F i > = 0. Because of large scale natura ofthe field U i it is possible to expand it into Taylor series.The first term is the most important if we restrict ourself by relation l << L .It was shown that under the action of large-scale veloc-ity gradient tensor A ij ( t ) in the solution of NS equationappears (in the limit t → ∞ ) singular-like vorticity ω [9],[10]. Such kind of solution arises with unit probability.The solution looks like a ribbon, where ω is directed along FIG. 1: ribbon-like solution, the vorticity concentrated in thecentral layer (ribbon), circus with arrows show velocity the ribbon (see Fig.1). The vorticity is concentrated ex-ponentially in time in a sheet which has stochastic rota-tions. In a local frame we have the strongest expansionalong ω and contraction on perpendicular direction. Anaccount of viscosity gives quasi stationary solution. Dueto quasi one-dimension character of the solution (in gen-eral case we have contraction along one direction) theviscosity is important along this direction only. The sta-tionary solution in the rest frame takes the form [9]: λ x (cid:48) ∂ x (cid:48) ω − λ ω = ν∂ x (cid:48) x (cid:48) ω (12)Here parameters λ and λ are Lapunov’s exponents re-sponsible for the discussed above local vorticity dynam-ics. To define them it is necessary to solve stochasticmatrix equation: ∂ t Q ij = Q ik A kj ( t )According to a set of theorems (see for the details [9],[11])the matrix Q in the limit t → ∞ has (with unit proba-bility) solution: Q = Z ∞ e Dt R ( t ) , D ( t ) = diag { λ , λ , λ } here Z ∞ is a number matrix and R ( t ) is a rotation. Dueto incompressibility λ + λ + λ = 0 where λ was chosento be a maximal one. For any symmetrical in time process the value λ = 0[12]. As is known time asymmetry in NS equation isconnected with energy dissipation which is determinedby large-scale stochastic force but not large-scale velocitygradient tensor. Thus λ ≈
0. Taking this fact intoaccount one can solve the equation (12) for vorticity andvelocity can be determined by integrating. As a resultin the limit ν → v y (cid:48) ( x (cid:48) ) = v sign x (cid:48) The main difference of this solution from the solutionof Burgers equation is incompressibility. We have gotshear solution. It significant however, that it is a kindof “force free” solution U y ( x ) where pressure has a large-scale gradient along axis y, z but not important alongvelocity gradient x .This step-like solution is obtained in rotation refer-ence frame. Returning to the fix frame we find: v x = − v y (cid:48) sin φ , v y = v y (cid:48) cos φ x (cid:48) = x cos φ + y sin φ Now let us construct a perpendicular velocity incre-ment δv ⊥ = v ⊥ ( r ) − v ⊥ ( r − l ). Here v ⊥ = v y cos φ − v x sin φ and r = xcos φ + ysin φ . Averaging on rotation φ and onspace, on can get perpendicular structure function: (cid:104) δv n ⊥ (cid:105) = 1 L (cid:90) L [ sign r − sign ( r − l )] n rdr ∝ l This solution is analogous to Burgers step solution (2)and connected with fast part of PDF or generating func-tion. An equation for generating function for the caseof NS equation was obtained in [13] and it is quite anal-ogous to equation (4). For our further purpose we willconsider equivalent presentation of generating function –infinite chain of structure function equations (for detailssee [14]): ∂ t D N + ∇ r D N +1 = − T N + 2 ν (cid:2) ∇ r D N − E N (cid:3) here D n = < v j ...v i > , E N = < v k ....v l e ij > and T N = < v j ....v l ( ∂ x i p − ∂ x (cid:48) i p (cid:48) ) > While averaging < > the sum of all terms of a giventype that produce symmetry under interchange of eachpair of indices were taking into account. And [ N ] withinsquare brackets denote the number of indices.As in the case of Burgers equation there is a very im-portant value e ij connected with energy flux: e ij = ∂ x n u i ∂ x n u j + ∂ x (cid:48) n u (cid:48) i ∂ x (cid:48) n u (cid:48) j Actually, (see [14]) 2 νE = 2 νE = 4 ε/ ε isthe energy flux. But this value changes greatly on thedissipation scale. Following ideas of solution (10) let usrestrict our consideration to a slow-changing part of thedistribution function. It means that in calculation of E N all the values except e ij changes slowly (just like in thecase of Burgers equation). Thus E N = < { v k ....v l e ij } > = C εD N − (13)Here C is a constant. Actually, this term couldbe rewritten in the terms of distribution function F N ( v , ..v N ; x , ..x N ) as: E N = (cid:90) e ij ( r, t ) v k ...v l F N Dv and if the distribution function changes slowly in spaceand time we have the expression (13).We are going to find the solution of the chain in thelimit l → ν introducing ε and can put ν = 0 now). But according to our ribbon- like solution in the limit l → ν → ∇ r D N +1 = C εD N − (14)One can see that there is Kolmogorov’s scaling solutionto this chain: D N ∝ r N/ Now let us discuss the results obtained. We see thatour solution of the NS equation is very close to the so-lution of the Burgers equation (2). Actually, structurefunctions S (cid:107)⊥ n obey Kolmogorov’s law for 0 < n < S ⊥ n ∝ l but forperpendicular structure functions only; ribbon-like solu-tion does not give input into main asymptotic of longitu-dinal structure functions. According to modern numeri-cal simulations [15],[16] the value of perpendicular struc-ture function exponents is less than longitudinal one andmore close to Kolmogorov’s law. This fact agrees withsingular-like structures discussed above.The obtained equation (14) is analogous to (11). Theyseem to have close physical interpretation. In case of NSequation the dynamics of pair points, separated by step, is responsible for Kolmogorov’s law too. But in case ofNS equation we have step of tangential velocity.The similarity between NS and Burgers stochastic so-lution could be seen also in equations for PDF. In thecase of NS, the stationary equation for PDF is a diffu-sion type too, and the left side of the equation is v i ∂ l i f .This term should be positive.Thus, in both case – Burgers and NS turbulence theorypredict bifractal behavior of structure functions if condi-tion for velocity difference v i l i > ∗ Electronic address: [email protected][1] J.Bec, K.Khanin, Phys. Rep. 447, 1-66, (2007).[2] U.Frisch, Turbulence. The Legacy of A.N. Kolmogorov(Cambridge: Cambridge Univ. Press, 1995)[3] D.S.Agafontsev, E.A.Kuznetsov, A.A.Mailybaev, Phys.Fluids 27, 085102 (2015)[4] A.Chekhlov, V.Yakhot, Phys. Rev. E, 51 R2739, (1995)[5] A. M. Polyakov Phys. Rev. E , 6183 (1995)[6] G.Falkovich, I.Kolokolov, V.Lebedev and A.Migdal,Phys. Rev. E 54(5), 4896 (1996)[7] Nayfeh A.H. Perturbation methods (Wiley,1973)[8] D.Mitra, J.Bec, R.Pandit, and U.Frisch, Phys. Rev. Lett.94