Ecoulements turbulents des eaux peu profondes Turbulent flows in shallow water

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1 Ecoulements turbulents des eaux peu profondes Turbulent flows in shallow water SERGEY GAVRILYUK a, HENRI GOUIN b a. Université d Aix-Marseille & C.N.R.S. U.M.R. 6595, IUSTI, Project SMASH, 5 rue E. Fermi, Marseille Cedex 13 France b. Université d Aix-Marseille & MP, C.N.R.S. U.M.R. 6181, Case 3, Av. Escadrille Normandie-Niemen, Marseille Cedex 0 France Résumé : Les équations de Saint-Venant des eaux peu profondes sans effet de tourbillon sont analogues aux équations de la dynamique des gaz avec une loi d état polytropique. Lorsque le tourbillon est pris en compte, on en déduit des équations qui sont analogues aux écoulements turbulents des gaz avec une structure simplifiée des équations du mouvement. Cette structure correspond aux équations avec un tenseur des contraintes de Reynolds sans termes sources. Les équations d évolution prennent en compte seulement les effets géométriques. Une structure mathématique simple de ces équations nous permet ainsi l intégration des équations de Reynolds. Des lois de conservation non-usuelles nécessaires pour l étude des sauts hydrauliques turbulents sont aussi déterminées. Abstract : The shallow water flows without turbulence effects are similar to the equations of gas dynamics with a polytropic state law. When the vorticity is taken into account, we obtain equations analogous to the ones of turbulent flows of gases with a simplified structure of the equations for the Reynolds stresses. The corresponding mathematical structure allows the integration of Reynolds equations. Non-conventional conservation laws necessary for the study of turbulent bores are also determined. Mots clefs : Tenseur des contraintes de Reynolds ; Ecoulements turbulents ; Equation gyroscopique Reynolds stress tensor ; Turbulent flows ; Gyroscopic equation. 1 Introduction The governing equations of barotropic turbulent compressible fluids are see, for example [1], [], [3] t + U i, i = 0, U i t + U i U j + p δ ij + ρ u i u j, j = 0, ρ u i u j t + ρ u i u j U k, k + ρ u i u j U j,k + ρ u i u j U i,k = S ij. Here brackets mean the averaging, coma means the derivation with respect to the Eulerian coordinates x = {x i }, i {1,, 3}, index t means the partial derivative with respect to time, ρ is the fluid density, U = {U i }, i {1,, 3} is the mass average velocity, p is the pressure, u = {u i }, i {1,, 3} is the velocity fluctuation verifying ρu = 0. Repeated indices mean summation. The term S = {S ij } represents turbulent sources. We introduce the Reynolds stress tensor R = ρ u u, R ij = ρ u i u j. The system 1 can also be rewritten in the tensorial form + div U = 0, t du + p + div RT = 0, 1 dr T + R divu + R + R = S, 1

2 where d/ means the material derivative with respect to the mean motion d = t + UT. The superscript T means the transposition. By using the mass conservation law, the equation for the volumic Reynolds tensor R can be rewritten as the equation for the specific or per unit mass Reynolds tensor where dp + T P + P = S, 3 P = R. The aim of this note is to understand the structure of homogeneous Reynolds equation 3 corresponding to S = 0. The reason is twofold : - first, in the numerical study of compressible turbulent flows, this is a natural step in applying the splitting-up technique see for example [4], - second, the system also appears as an exact asymptotic model of weakly shearing flows of long waves over a flat bottom [5] h t h du + gh dr + div h U = 0, + div R T = 0, T + R divu + R + R = 0. In Eq. 4, h is the fluid depth, the average pressure is given by p = gh /, g is the gravity acceleration, and R = h 0 h Ũ U Ũ Udz, hu = where Ũ is an instantaneous velocity. Equations are written for three-dimensional long waves and the production term is zero in the limit of weakly shearing flows. Also, Eqs. 4 are hyperbolic. Finally, we will focus on the equation of the Reynolds tensor per unit mass 0 Ũdz, dp + T P + P = 0. 5 The particular case rot U = 0 was investigated in [6]. In such a case / T = / and Eq. 5 corresponds to a two-covariant tensor convected by the mean flow. This means that P has a zero Lie derivative d L with respect to the velocity field U and the tensor P 0, image of P in Lagrange coordinates t, X, only depends on X = {X i }, i {1,, 3} d L P dp + P + P = 0, P = F T 1 P0 X F 1, where F = / X is the deformation gradient of the mean motion [7]. The aim of the paper is the study of the homogeneous Reynolds tensor equations structure 5 in the case rot U 0. Geometrical properties of the Reynolds tensor evolution The Reynolds stress tensor P is symmetric, semi-positive definite and consequently can be rewritten in a local basis of orthonormal eigenvectors in the form P = λ e e λ e e T. 4

3 The eigenvalues λ, {1,, 3} are non-negative ; the case λ > 0 is a generic one. For the two-dimensional case, λ 3 0. Let us denote a = λ e, λ > 0 {1,, 3}. Then, From Eq. 6, we deduce By using Eq. 7, Eq. 5 can be written P = dp = a a da a a T. 6 T a T da + a. 7 da + [ a a T da + + T a a] T = 0. 8 The vector da / + / a can be developped in the local basis {a }, {1,, 3} of eigenvectors ; one obtains da + a = A β a β, {1,, 3} 9 β=1 where A β,, β {1,, 3} are the scalar components to be determined. By using Eq. 9, Eq. 8 leads to A g + β=1 A β + A β g β = 0, where g = a a T and g β = g β = a a T β + a β a T,, β {1,, 3}, are six independent symmetric tensors. Consequently, A = 0 and A β + A β = 0,, β {1,, 3}. Equation 8 is equivalent to da + a = Λ i π e with π =A 3 e 1 + A 13 e + A 1 e 3, {1,, 3}, 10 where diagonal matrix Λ and antisymmetric matrix i π are determined in the basis e, {1,, 3} as λ A1 A 13 Λ = 0 λ 0, i π = A 1 0 A λ 3 A 13 A 3 0 The vectors a, {1,, 3} are orthogonal, a T a β = 0, β. If we assume the orthogonality at initial instant, this is equivalent to a T da β + da at β = So, Eqs yield β {1,, 3}, a T Λ i π e β a β + a T β Λ i π e a = 0, or λ λ β e T D e β = a T Λ i π e β + a T β Λ i π e λ e T i π e β + λ β et β i π e 1 3

4 where D = 1 + T is the rate of deformation tensor corresponding to the mean flow. We denote the mixed product of three vectors {a, b, c} as a, b, c a T b c. Hence, Eq. 1 is e λ λ β e T D e β = λ λ β, π, e β = λ β λ π, e, e β = λ β λ π T e γ, where {, β, γ} is a cyclic permutation of the triplet {1,, 3}. Finally, we get Equation 10 can be written Since π T e γ = λ λ β e T D e β λ β. 13 λ λ de + dλ e + λ e Λ i π e = 0, {1,, 3}. 14 e T de by multiplying the left side of Eq. 14 with e T, we get = 0, e T Λ i π e = λ e T i π e = 0, dλ + e T D e λ = 0. By multiplying the left side of Eq. 14 with the projector I e e T, we get Due to the fact there exists a vector Π such that vector Π verifies the condition By multiplying Eq. 15 with e T β By replacing Rel. 13 into Eq. 16 we get de λ + λ I e e T e Λ i π e = 0. de = Π e, λ i Π e + λ I e e T e Λ i π e = Π T e γ = where β,we get λ Π T e γ = λ β π T e γ λ e T β λ β et + = e T β λ β λ λ β T e β 4 T + λ λ β λ e. 16 e T β e e.

5 We can now formulate the following result : Theorem 1. The Reynolds tensor can be written in the form R = λ e e. The eigenvectors e and the eigenvalues λ verify the equations : de = Π e, d Ln λ = µ, {1,, 3} 17 where µ = e T D e, Π T e γ = e T β λ β T + λ e λ β. λ The triplet {, β, γ} corresponds to a cyclic permutation of the triplet {1,, 3}. Equation 17 1 is similar to the equations of a rigid body [8]. The vectors e form a natural moving frame G = {e } 3 depending on the mean deformation. The eigenvalues λ of the Reynolds tensor are determined by the evolution equation 17. Note that if λ are initially positive, they will be positive at any time. Hence, the tensor P is always positive definite. Due to the mass conservation law 1 and Eq. 17, we obtain d 3 λ = 0. Consequently, system 17 admits a scalar invariant along the trajectories of mean flow. This invariant was earlier obtained in [6] in a different form. Let us introduce the turbulent specific energy e T = 1 tr P = 1 λ. In the incompressible isochoric case, we have d/ = 0 ; the turbulent energy is minimal in the isotropic case when the three eigenvalues λ are equal λ 1 = λ = λ 3 = λ. In this case, the orthonormal eigenvectors e, {1,, 3} of the Reynolds stress tensor P are also the orthonormal eigenvectors of the mean rate of deformation tensor D. The eigenvalues of D are µ, {1,, 3}. In the compressible isotropic case e T = /3 κ, κ = 3λ / /3, and κ is a classical invariant of isotropic turbulence. In presence of shock waves the quantity 3 λ is not conserved through shocks ; it increases like the classical entropy in compressible fluid dynamics. The estimation of the jump of turbulence entropy in isotropic case was given in [9]. The governing equations admit the energy conservation law t 1 U + e i + e T + div where the internal specific energy e i is defined by U de i = p d 5 1 U + e i + e T 1 + p I + R U = 0,

6 and the mean pressure p is supposed to be a given function of. Indeed, using 17 we immediately obtain 1 1 t U + e i + e T + div U U + e i + e T + p I + R U = 3 Conclusion de T + tr R D = d λ + tr λ µ = 0. The equations of fluid turbulent motions take three equations into account : the equation of the mass balance 1 1, the balance equation of the average momentum 1, and the Reynolds stress tensor equation of evolution 1 3 ; this last equation has been the object of our study. If the turbulent sources are neglected, the turbulent fluid motion is a superposition of the mean motion and turbulent fluctuations. The eigenvectors of the Reynolds tensor carry the fluctuations associated with the mean flow deformation. The amplitude of turbulent deformations is defined by the eigenvalues of the Reynolds stress tensor. Locally, the equations for the directions of turbulent fluctuations describe a small solid whose rotation is given by a gyroscopic type equation Equation The amplitude evolution of turbulent deformations is determined by the diagonal values µ of the mean deformation tensor D expressed in the eigenvector basis of the Reynolds stress tensor. The turbulence increases with the time when µ < 0, and decreases when µ > 0. In the particular case of incompressible fluid motions we have tr D = 0, and hence there always exists a direction in which the turbulence is increasing while in other directions it is decreasing. These mathematical deductions are confirmed by experiments [10]. Acknowledgement : This article is dedicated to the memory of Professor Vladimir M. Teshukov who drew our attention to his model of weakly sheared flows during his visit in Marseille in the fall of year 007. Références [1] S. B. Pope, Turbulent flows, Cambridge University Press, 005. [] D. Wilcox, Turbulence Modeling for CFD, DCW Industries, [3] B. Mohammadi, O. Pironneau, Analysis of the K-epsilon turbulence model, Research in Applied Mathematics, John Wiley & Sons, New York, [4] C. Berthon, F. Coquel, J.M. Hérard, M. Uhlmann, An approximate solution of the Riemann problem for a realisable second-moment turbulent closure, Shock Waves, 11, [5] V.M. Teshukov, Gas dynamic analogy for vortex free-boundary flows, Journal of Applied Mechanics and Technical Physics, 48, [6] J.-F. Debiève, H. Gouin, J. Gaviglio, Evolution of the Reynolds stress tensor in a shock wave-turbulence interaction, Indian Journal of Technology 0 March [7] J. Serrin, Mathematical principles of classical fluid mechanics, in : Encyclopedia of Physics VIII/1, Springer, Berlin, 1959, pp [8] J. E. Marsden, T.S. Ratiu, Introduction to mechanics and symmetry, Series in Applied Mathematics, 17, Springer, Berlin, [9] S. Gavrilyuk, R. Saurel, Estimation of the turbulence energy production across a shock wave, The Journal of Fluid Mechanics, 549, [10] J.-F. Debiève, H. Gouin, J. Gaviglio, Momemtum and temperature fluxes in a shock wave-turbulence interaction, in : Proceedings of the ICHMT/IUTAM Symposium on the Structure of Turbulence and Heat and Mass Transfer, Z. Zarić, Ed., p , Hemisphere Publishing Corporation, London