Lie symmetry group theory for turbulence modelling and simulation Aziz Hamdouni∗, Dina Razafindralandy∗, Marx Chhay†, and Nazir Al Sayed‡ ∗Laboratoire des Sciences de l’Ingénieur pour l’Environnement – UMR 7356 Université de La Rochelle – France †Laboratoire Optimisation de la Conception et Ingénieurie de l’Environnement – UMR 5271 Université de Savoie – France ‡Laboratoire de Mécanique des Structures Industrielles Durables – UMR EDF/CNRS/CEA 8193 EDF R&D – France Abstract—In this paper, we present some applica- II. Lie symmetry group theory and application tions of the Lie-group theory to fluid mechanics prob- to the non-isothermal Navier-Stokes equations lems. We present how to use the Lie-group theory to de- duce conservation laws of the Navier-Stokes equations Instead of presenting the Lie symmetry group theory and scaling laws of turbulent flows. We also develop on a general equation, we apply it directly to the non- symmetry preserving turbulent models, and show how isothermal Navier-Stokes equations: the symmetry theory can contribute to build robust numerical schemes. ∂u 1 + div(u ⊗ u) + ∇p − ν∆u − βgθ e = 0 ∂t ρ 3 I. Introduction ∂θ (1) + div(θu) − ∆θ = 0 ∂t The literature shows that the Lie-symmetry group the- ory constitutes a powerful modelling tool in engineering div u = 0 science. They allow, for instance, the computation of A continuous symmetry of equations (1) is a transforma- Green function of linear equations. In addition, symme- tion tries are extensively used in literature to compute self- ˆ similar solutions of various equations. In turbulence, vor- T : q = (t, x, u, p, θ) 7−→ qˆ = (t,ˆ xˆ, uˆ, p,ˆ θ), (2) tex solutions of the Navier-Stokes equations was found depending smoothly on the parameter , and which leave as special self-similar solutions. Finally, we mention that the set of solutions of (1) invariant. In other words, T is the symmetries may give an information on the large-time a symmetry of (1) if behaviour of the solution. To some extent, the symmetries traduce the physics of the equations. NS(q) = 0 =⇒ NS(qˆ) = 0 (3) In this communication, we present some applications to where NS designate equations (1). A Lie symmetry group fluid mechanics, and especially to the modelling and sim- of NS is a set of continuous transformations which has a ulation of turbulent flows. In section II, we recall briefly Lie group structure. the theory of Lie-symmetry group and how to compute Definition (3) permits to compute some but not all them. We then list the Lie-symmetry transformations of symmetry groups of NS. In order to be exhaustive, one the Navier-Stokes equations in the non-isothermal case. introduces the tangent vector: Next, we recall the Nœther’s theorem which links the symmetries of an equation to conservation laws in the case ∂ ∂ ∂ ∂ ∂ X = ξ + ξ · + ξ · + ξ + ξ (4) of a Lagrangian-described system. We then show how to t ∂t x ∂x u ∂u p ∂p θ ∂θ extend this theorem to non-Lagrangian systems and how where to find conservation laws of the Navier-Stokes equations. ∂qˆ ξ = . (5) In section IV, we are especially interested in turbulent q ∂ flows. We use the Lie-group theory to find wall and scaling =0 laws of non-isothermal turbulent flows. Next, we develop A tangent vector of a Lie group is also called its generator symmetry preserving turbulence models and show their since, knowing the components ξq of X, one can determine numerical performance. the elements T of the group by solving the equation In section V, we build robust numerical schemes, based dqˆ on symmetry preservation. = ξ(qˆ), qˆ( = 0) = q. (6) d Definition (3) is equivalent to the infinitesimal condition • the group of the first scaling transformations: NS(q) = 0 =⇒ X(2) · NS(q) = 0. (7) (t, x, u, p, θ) 7−→ (e2 t, e x, e− u, e−2 p, e−3 θ) (21) X(2) is a prolongation of X which accounts for the first which shows how u, p and θ change when the spatio- and second order partial derivatives in the equations (see temporal scale is multiplied by (e, e2), [1]–[3]). This condition enables to compute the tangent • and the group of the second scaling transformations: vectors of all the Lie symmetry groups of (1): 2 2 2 ∂ (t, x, u, p, θ, ν, κ) 7→ (t, e x, e u, e p, e θ, e ν, e κ) X = (8) (22) 1 ∂t which shows the consequence of a modification of the ∂ spatial scale. X2 = ζ(t) (9) ∂p Equations (1) own other known non-Lie symmetries, ∂ 1 ∂ which are: X3 = βg x3 + (10) ∂p ρ ∂θ • the reflections about x1 and x2 axes • and the material indifference in the limit of a 2D ∂ ∂ ∂ ∂ horizontal flow in a simply connected domain [4]: X4 = x2 − x1 + u2 − u1 (11) ∂x1 ∂x2 ∂u1 ∂u2 (t, x, u, p) 7→ (t , R(t)xˆ , R(t)u + R˙ (t)x , pˆ) (23) ∂ ∂ ∂ X4+i = αi(t) +α ˙ i(t) − ρ xiα¨i(t) , i = 1, 2, 3, ∂xi ∂ui ∂p with (12) pˆ = p − 3ωφ + ω2kxk2/2 ∂ ∂ ∂ ∂ ∂ X8 = 2t + x · − u · − 2p − 3θ (13) where R(t) is an horizontal 2D rotation matrix with ∂t ∂x ∂u ∂p ∂θ angle ωt, ω a real parameter, and φ the usual 2D where ζ and the αi’s are arbitrary functions of time. The stream function. dot symbol ( ˙ ) stands for time derivation. In the next section, we recall Nœther’s theorem which We can also consider symmetries (which are sometimes links symmetries to conservation laws. called equivalence transformations) of the form (t, x, u, p, θ, ν, κ) 7−→ (t,ˆ xˆ, uˆ, p,ˆ θ,ˆ ν,ˆ κˆ). (14) III. Symmetry and conservation laws Consider a system described by a Lagrangian action Such symmetries take a solution of (1) into a solution of Z other non-isothermal Navier–Stokes equations with differ- L[w] = L(y, w, w˙ ) (24) ent values of ν and κ. Applying condition (7) leads to the Ω infinitesimal generator and the corresponding Euler-Lagrange equation ∂ ∂ ∂ ∂ ∂ ∂ ∂L ∂L X9 = x· +u· +2p +θ +2ν +2κ . (15) − Div = 0, (25) ∂x ∂u ∂p ∂θ ∂ν ∂κ ∂w ∂w˙ From these generators, the symmetry group of (1) can P ∂f be identified. They are, respectively: where Div f = i for any smooth function f. ∂yi • the group of time translations: Nœther showed that to each Lie group which leaves the action L invariant corresponds a conservation law [5]. More (t, x, u, p, θ) 7−→ (t + , x, u, p, θ), (16) precisely, if • the group of pressure translations: X ·L = 0 (26) (t, x, u, p, θ) 7−→ (t, x, u, p + ζ(t), θ), (17) then Div C = 0. (27) • the group of pressure-temperature translations: The conserved quantity C is defined by (t, x, u, p, θ) 7−→ (t, x, u, p + βg x3, θ + a/ρ), (18) " #T ∂L • the group of horizontal rotations: C = Lξ + (ξ − γξ ) (28) y ∂γ w y (t, x, u, p, θ) 7−→ (t, Rx, Ru, p, θ) (19) where R is a 2D (constant) rotation matrix, j ∂wj where γk = . • the three-parameter group of generalized Galilean ∂yk Unfortunately, Nœther’s theorem does not apply di- transformations: rectly to equations (1) for these later do not have a (t, x, u, p, θ) 7−→ (t, x+α(t), u+α˙ (t), p+ρ xα¨(t), θ), Lagrangian structure. However, one can associate to (1) (20) adjoint equations such that, together, they derive from a “Bilagrangian” function. Indeed, consider, for example, Using the described theory, one can compute the Lie the isothermal case. Then, the Navier-Stokes equations symmetry groups of (33) which, for lack of space, are not ∂u 1 listed here but can be found in [6]. + u · ∇u + ∇p − ν∆u = 0 A scaling law of equations (33) verifies the condition ∂t ρ (29) X · U = 0, X · Θ = 0 (34) 1 div u = 0 where X is the generator of one of the Lie symmetry and their adjoint equations groups of (33). Solving these equations leads to the fol- ∂v 1 lowing scaling laws (among others). − + v · ∇u − u · ∇v − ∇s − ν∆v = 0 ∂t ρ • The linear law which can be found in the middle (30) region of a Couette flow and in the viscous sublayer of a boundary flow: div u = 0 derive from the Bilagrangian U1 = C1x2 + C3, Θ = C2x2 + C4. (35) ! 1 du dv 1 1 • The classical logarithmic wall law and its correspond- L = · v − u · + s − u · v div u 2 dt dt ρ 2 ing temperature law: (31) 1 U = C ln(x +b)+C , Θ = C [x +b]−1+C . (36) − p div v + ν tr(∇uT ∇v). 1 1 2 3 2 2 4 ρ • The exponential law: In these expressions, v and s are adjoint variables. With this extension of the notion of Lagrangian system, U1 = C1 exp(Cx2) + C3, Θ = C2 exp(2Cx2) + C4. Nœther’s theorem can be applied to the Navier-Stokes (37) equations to find conservation laws. • The power law of velocity discovered by Oberlack In the next section, we show some applications of sym- [7] in the buffer region of a boundary layer and its metry analysis to the understanding and simulation of temperature counterpart: turbulence. a 2a−1 U1 = C1(x2 + b) + C3, Θ = C2(x2 + b) + C4. IV. Symmetry and turbulence (38) In the first subsection, we present the derivation of scal- As we now know, symmetries contain fundamental phys- ing laws of turbulent flows from symmetry.