Group-theoretical model of developed turbulence and renormalization of the Navier-Stokes equation V. L. Saveliev1 and M. A. Gorokhovski2 1. Institute of Ionosphere, Almaty 480020, Kazakhstan 2. Laboratoire de Mécanique des Fluides et AcoustiqueEcole Centrale de Lyon, France PHYSICAL REVIEV E 72, 016302 (2005) Introduction In relation to turbulence, we can roughly express Kadanoff’s idea of “block picture” for the Ising spin field as follows: If, instead of turbulent field v (r ), we consider a field v s r ) that is averaged on ( the scale s , then the last one will “resemble” the original turbulent field v (r ). The exact sense of “resemble” must be defined by the group of transformations for both fields and equations for these fields. In this paper, solely on the basis of the Navier-Stokes equation, we propose a simple model of stationary developed turbulence. This model allows us: 1. To explicitly derive the group of renormalization transformations. 2. We average the Navier–Stokes equation over some small length scale s 0 with the help of our averaging formula. 3. Then we transform this averaged equation by a renormalization group transformation to the equation for the velocity field v s (r ,t ), averaged over any scale s . II. Renormalized averaging formula for <vu> Consider two functions v (r ) and u (r ) depending on radius vector r . The average of v (r v (r ) = ò v (r ¢)Y (r - r ¢)dr ¢, ) is (1) When the weight function Y (r ) is Gaussian, Ys (r ) = 1 3 ( 4ps ) r2 e 4s (2) , the average is defined by Gauss transform (filtering): v (r ) = ò v (r ¢)Ys (r - r ¢)dr ¢ = Gˆ s v . (3) The Gaussian distribution verifies the diffusion equation, ¶ Ys (r ) = Ñ 2 Ys (r ), ¶s where Ñ = 2 ¶ ¶r 2 ( ). (4) Thereby the Gauss transform operator (3) can be represented in the exponential form: 2 Gˆ s = e s Ñ , Gˆ s 1Gˆ s 2 = Gˆ s 1 + s 2 . (5) Operator for averaging the field product Consider the averaged product of two fields: vu 2 r ) = Gˆ s v (r )u (r ) = e s Ñ v (r )u (r ). (6) ( The differentiation of the product of two multipliers can be done by the Leibnitz rule. To this end, we decompose the differentiation operator Ñ into two parts, Ñ = Ñ1 + Ñ 2 , (7) with differentiation operators Ñ 1 and Ñ 2 acting on the first v (r and the second u (r ) ) multipliers respectively. Setting Eq.(7) in Eq. (6) gives e s Ñ v (r )u (r ) = e 2 s (Ñ 1 + Ñ 2 )2 v (r )u (r ) = e 2s Ñ 1 ×Ñ 2 (e s Ñ v (r 2 )(e s Ñ u 2 ) r ( ). ) (8) It then follows that the average of the product of two fields can be written as vu = e 2s Ñ1×Ñ 2 v u . (9) Simplest formula for averaging Instead of using the Taylor-series expansion with an infinite number of terms for the averaging operator e 2s Ñ1 ×Ñ 2 , we use the Taylor series with a residual term, e 2s Ñ 1 ×Ñ 2 = 1 + 2s Ñ 1 ×Ñ 2 + ... + + 1 (2s Ñ 1 ×Ñ 2 )n - 1 + (n - 1 )! (10) 1 1 (2s Ñ 1 ×Ñ 2 )n n! ò d a qn ( a ) e 2a s Ñ 1 ×Ñ 2 , 0 where n is an arbitrary natural number ( n = 1, 2, 3,.... ), and 1 qn (a ) = n (1 - a )n - 1 ; ò d a qn ( a ) = 1. (11) 0 For the case n = 1 we have: s vu s = v s u s + 2ò d s ¢ Ñ v s ¢ ×Ñ u s ¢ s - s ¢. (12) 0 This averaging expression demonstrates that the integral part includes the contribution of all scales less than s in an exact manner. Example of averaging Consider the velocity field in the framework of the following model. Assume that in a region of a typical size L , the velocity field can be represented as vi (r ) = wi + aij rj + virnd (r ), (13) where wi and a ij are constants and virnd (r ) is an isotropic random field. This implies that the considered region is rectilinearly moving as a whole, rotating and stretching. The liquid particles inside this region have random isotropic velocities. For this field we have: Ñ l Ñ m vi = Ñ l Ñ m virnd . (14) Using the averaging formula (9), (10) with n = 2 for the velocity field (13) yields vi vk s = vi s vk s + 2s Ñ l vi s Ñ l vk s + s + 2ò d s ¢2s ¢ Ñ l1 Ñ l2 virnd 0 where l1, l2 = 1, 2, 3 . s¢ Ñ l1 Ñ l2 vkrnd s¢ s- s , ¢ (15) Deviatoric part of stress tensor Accounting for isotropy of the random field, the integral term is proportional to the Kronecker delta dik , vi vk = vi s 2 + dik 3 s vk s + 2s Ñ l1 vi s Ñ l1 vk s s ò d s ¢2s ¢ Ñ l1 Ñ l2 vlrnd 3 s¢ Ñ l1 Ñ l2 vlrnd . 3 s¢ s- s¢ (16) 0 Thus, for the model (13), the deviatoric part of the stress tensor can be expressed through the averaged velocity field in closed form, vi vk s ( - 1 dik v 2 3 + 2s Ñ l1 vi s s = vi Ñ l1 vk s s vk s - 1 d v 3 ik l1 1 - dik Ñ l1 vl2 3 s s Ñ l1 vl2 vl1 s ). s (17) Important remark Starting from Boussinesq, in papers devoted to deriving equations for averaged turbulent fields, it is usual to use the averaging formula on the basis of the gradient-diffusion hypothesis, vi vk s - 1 dik v 2 3 s - vi s vk s - 1 d v 3 ik = - ntur (Ñ i vk 2 s s s + Ñ k vi s ). (18) However, if the velocity field vi (r ) in Eq.(18) is replaced by v%i (r ) = - vi (r ), we obtain v%% i vk s - 1 dik v%2 3 s - v%i s v%k s - 1 dik v%2s 3 = + ntur (Ñ i v%k s s + Ñ k v%i s ). (19) Inevitably, from Eq. (18) and (19), one of two fields vi (r , t ) or v% i (r , t ) has negative turbulent viscosity. Thus, through averaging, it is impossible to derive the universal gradient-type formula of form (18). In other words, one cannot obtain the turbulent viscosity in equations for turbulence by simple averaging of the nonlinear term. The following sections show how the turbulent viscosity can be introduced in a natural way. III. Group-theoretical model of developed turbulence Hereafter, we present a schematic description of the turbulent model, which is solely based on the Euler equation for incompressible flow, ¶v = - Ñ ×vv - Ñ p, ¶t Ñ ×v = 0 r = 1. (20) We will consider continuous symmetry transformations of Eq. (20) where the time variable is not transforming: scaling r ® e t b r , v ® e t b v ; rotation r ® e t W´ r , v ® e t W´ v ; and translation r ® r + t v . These transformations constitute a group and induce the group of the velocity field transformations. The generators of its one-parameter subgroups are the operators that are the linear combination of scaling, translation, and rotation generators, qˆ (b , W, v ) = b (1 - r ×Ñ ) - v ×Ñ - W×r ´ Ñ + (W´ ). (21) Subgroup elements read as gˆ t = e t qˆ , where t is a subgroup transformation parameter, gˆ t 1gˆ t 2 = gˆ t 1 + t 2 . (22) Symmetry parameters The constants b , v , W have the following physical meaning: b is the rate (relative to parameter t ) of the homogeneous scaling, v is the velocity of translation, and W is the angular velocity of rotation. An action of these operators on velocity field v (r ) is as follows: e t b (1- r ×Ñ )v (r ) = e t b v (e - t b r ), e t ((W´ )- W×r ´ Ñ ) ( v r ) = e t W´ v (e - t W´ r ), . (23) e - R ×Ñ v (r ) = v (r - R ) The symmetry of the Euler equation is then expressed by e t qˆ Ñ ×vv = Ñ ×(e t qˆ v )(e t qˆ v ), e t qˆ Ñ p = Ñ p ¢, Ñ ×(e t qˆ v ) = 0 . (24) Self-similar solutions We propose to associate the phenomena of stationary turbulence with self-similar solutions of the Euler equation (20) in relevance to symmetry subgroups with generators (21). The selfsimilar solution implies its dependence on time through the parameter of the space symmetry transformation only, i.e., v (r , t ) = e (t - t 0 )qˆ v (r , t 0 ). (25) The evolution in time of a such self-similar solution v (r ,t ) is governed by the simple equation: ¶v = q̂v ¶t (26) and the spatial configuration, at each time moment, is defined by the equation: qˆv = - Ñ ×vv - Ñ p, Ñ ×v = 0 . (27) Decomposition of turbulent field Requiring solution (25) to be dynamically stationary, the velocity field v (r ,t ) must be represented by a superposition of time harmonics ei wt with real w . Then it follows from selfsimilarity (26) that expansion coefficients v w (r ) are eigenfields of the generator q̂ corresponding to the pure imaginary eigenvalues, v (r , t ) = å e i wt v w (r ). (28) w qˆv w (r ) = i wv w (r ), - ¥ < w< ¥ . (29) Equation (28) indicates that only phases of eigenfields v w (r ) change during the time evolution of turbulent velocity field v (r ,t ). The phases are stochastic and, in principle, unknown for turbulent state. Therefore, only symmetry characteristics (rate of the homogeneous scaling, b ; velocity of translation, v ; angular velocity of rotation, W), and the power spectrum v wv w* (r characterize the turbulent state in our model. ) Invariance of turbulent state The turbulent field with parameters b , W, v is invariant under transformation e t qˆ (b ,W, v ) (accurate to the change of phases of v w (r ), which are immaterial for the turbulent state), Av e t qˆ (b ,W, v )v ( b , W, v ) = v ( b , W, v ). (30) Indeed, applying the operator e t qˆ (b ,W, v ) to the velocity field (28) leads only to the phase change of eigenfields ( v w ® e i wt v w ) in the turbulent velocity field decomposition. Hereafter we will skip the symbol A v , and writing "invariance of the turbulent velocity field," the sentence "accurate to the change of stochastic phases" will be omitted for the sake of simplicity. Renormalization-group invariance We introduce an averaged velocity field using the Gauss weight function with scale s , v = e s Ñ v (r ) = 2 s 1 ( 4ps )3/ 2 ò d r ¢e - 2 (r - r ¢) 4s v (r ¢). (31) The invariance (30) for turbulent fields provides the corresponding invariance for averaged turbulent fields v s : ˆ 2bs Ñ 2 ) e t (q - v = e s Ñ e t qˆ e - s Ñ v 2 s 2 = e s Ñ e t qˆ v = e s Ñ v = v 2 s 2 s (32) . This equation can be rewritten in an equivalent form: e t s qˆ v s0 = v s , ts = 1 s ln , b s0 (33) The change of the scale of averaging from s 0 to s is equivalent to the composition of scaling, rotation, and translation transformations. Note also that the scaling coefficient e t s b = s / s0 depends only on initial s 0 and final s scales and does not depend on turbulence parameters. We call the property (33) a renormalization-group invariance of averaged turbulent fields. IV. Renormalization of the averaged Navier-Stokes equation Averaging over small scale We will study the case when characteristics of turbulence are changing in space and time slowly. Consider now the Navier-Stokes equation ¶v + Ñ ×vv + Ñ p - nÑ 2v = 0, ¶t Ñ ×v = 0, r = 1. (34) We average this equation with some small scale s 0 . On the scales less than s 0 , s £ s 0 , we assume that the velocity pulsations become purely isotropic (13). Making use of our averaging formula in form (17), we have ¶ v s0 + Ñ ×( v ¶t Ñ × v s 0 = 0. s0 v s0 + 2s 0Ñ l v s0 Ñl v s0 ) + Ñ p ¢ = nÑ 2 v s0 , (35) Inner threshold of turbulence We assume that s 0 is the inner threshold of the turbulence such that, on scales larger than s 0 , s > s 0 , we have Euler turbulence with self-similarity. The exact determination of s 0 is an open problem. Here we propose a simple method of estimating its value. The gradients of the velocity field tend to destroy the continuous structure of flow due to nonlinear steeping while the viscosity processes smooth over the flow. To derive the equation for s 0 , we equate these two factors on the basis of dimensional consideration. The antisymmetric part, which is connected with the rotation as a whole, is excluded from gradient Ñv , n = c éë S ik s0 s0 S ik 1/ 2 ù s0 û , S ik s = 1 (Ñ v 2 i k s + Ñk vi s ) (36) Here c is some unknown constant. s0 = n c éë S ik s0 S ik 1/ 2 ù s0 û . (37) Equation for averaged turbulent fields Excluding molecular viscosity n from the Navier-Stokes equation with the help of the equation for s 0 , we have: ¶ v s0 + Ñ ×( v ¶t v s0 s0 + 2s 0Ñ l v s0 ntur (s 0 ) = cs 0 éë S ik Ñl v s0 s0 S ik ) + Ñ p ¢ = ntur (s 0 )Ñ 2 v s0 , 1/ 2 ù = n. s0 û (38) (39) 1 s , along with renormalization group ln b s0 Applying transformation e t s q̂ with t s = symmetry, we obtain the final equation for the averaged turbulent field for scales s > s 0 , ¶ v ¶t s + Ñ ×( v s v s + 2s Ñ l v s Ñl v s ) + Ñ p ¢ = ntur (s )Ñ 2 v s , (40) ntur (s ) = cs [ S ik s S ik s ]1/ 2 , Ñ×v s = 0. Remarks about viscosity 1. The equation for averaged fields (40) are similar to the known equation of Leonard applied in the LES approach ( the small difference is that the dissipative term with molecular viscosity is, in principle, not present here). 2. The equation for averaged fields, due to absence of the molecular viscosity term, is invariant under the scale transformation, which involves changing the averaging scale s : e ( ln a r ×Ñ + 2s ¶ - 1 ¶s ) v s (r , t ) = 1 v a a 2s (a r , t ) (41) 3. It is important to stress that we have shown that the turbulent viscosity appeared not as a result of averaging of the nonlinear term in the Navier-Stokes equation, but from the molecular viscosity term with the help of renormalization-group transformation. V. Conclusion 1. We have obtained the regularized averaging formula for averaging a two velocity field product. 2. Assuming that on the small length scale s 0 , the turbulent velocity field can be approximated as the sum of a smooth velocity field and a random isotropic field, we averaged the NavierStokes equation over this small scale by making use of our averaging formula. 3. 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