Elliptic relaxation for near wall turbulence models

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1 Elliptic relaxation for near wall turbulence models J.C. Uribe University of Manchester School of Mechanical, Aerospace & Civil Engineering Elliptic relaxation for near wall turbulence models p. 1/22

2 Outline Introduction The elliptic relaxation approach The v 2 f model The ϕ f model Test cases and applications Conclusion Elliptic relaxation for near wall turbulence models p. 2/22

3 Introduction Why modelling the near-wall region? In the near-wall region, viscosity and nohomogeneties are dominant. High shear and large rates of turbulence production are present. Here is where the skin friction and heat transfer are controlled, therefore, of vital importance for engineering applications that require these quantities. The wall normal fluctuations are reduced therefore reducing mixing. Elliptic relaxation for near wall turbulence models p. 3/22

4 Introduction The wall effects: No-slip: The boundary condition on the mean velocities creates large gradients where the turbulent production originates. Elliptic relaxation for near wall turbulence models p. 4/22

5 Introduction The wall effects: No-slip: The boundary condition on the mean velocities creates large gradients where the turbulent production originates k -uv.2 P k y y+ Elliptic relaxation for near wall turbulence models p. 4/22

6 Introduction The wall effects: No-slip: The boundary condition on the mean velocities creates large gradients where the turbulent production originates k -uv.2 P k y y+ Low Reynolds number effects: Interaction between energetic and dissipative scales. Elliptic relaxation for near wall turbulence models p. 4/22

7 Introduction The wall effects: Blocking effect: The impermeability condition affects the flow by adjusting the pressure field to ensure the incompressibility condition. Wall echo: Image term in Green s function at the other side of the wall produces an increase in the pressure. Elliptic relaxation for near wall turbulence models p. 5/22

8 Introduction The wall effects: Blocking effect: The impermeability condition affects the flow by adjusting the pressure field to ensure the incompressibility condition. Wall echo: Image term in Green s function at the other side of the wall produces an increase in the pressure. x x x Elliptic relaxation for near wall turbulence models p. 5/22

9 Introduction The wall effects: Blocking effect: The impermeability condition affects the flow by adjusting the pressure field to ensure the incompressibility condition. Wall echo: Image term in Green s function at the other side of the wall produces an increase in the pressure. x x x p i (x, t) = 1 4π S(x, t) ( 1 r + 1 r ) dx Elliptic relaxation for near wall turbulence models p. 5/22

10 Elliptic relaxation approach Starting from the Reynolds-stress equation: Du i u i Dt + ε u iu j k = P ij + ij + x k ( ν Tkl u i u j x l ) + ν 2 u i u j x 2 k with ij = ( ε ) Π ij + ε ij u i u j k To solve ij a non-homogeneous elliptic equation is derived. By integrating the pressure equation and assuming that the two point correlation Ψ ij can be approximated as Ψ ij (x, x ) = Ψ ij (x, x ) exp( r/l) (Durbin,1993) Elliptic relaxation for near wall turbulence models p. 6/22

11 Elliptic relaxation approach The solution is the modified Helmotz-type equation: ij L 2 2 ij = h ij Where any quasi homogenous model can be used for h ij. In fact the model solves for f ij = ij /k in order to enforce correct behaviour at the wall. The length scale is prescribed as: L = C L max ( k 3/2 ε, C η ( ν 3 ) 1/4 ) ε Elliptic relaxation for near wall turbulence models p. 7/22

12 Elliptic relaxation approach The model accurately represents the asymptotic behaviour of the stresses even in the near wall layer. There is no need to prescribe the distance from the wall, so no ill defined in complex geometries. No use of empiricall damping functions. Good representation of the stresses in channel flows and boundary layers. Requires 13 transport equations ( 6 for u i u j, 6 for f ij and one for ε). Numerical difficulties with wall boundary conditions for f 22, f 12, f 23. Equations required to be solved coupled. Elliptic relaxation for near wall turbulence models p. 8/22

13 The v 2 f model In order the simplify the RSM, the elliptic relaxation is introduced to the eddy viscosity approximation (Durbin,1995). Use of correct velocity scale near the wall, ν t = C µ v 2 T 4 3 ν t 2 1 ν t = -uv/du/dy νt = C µ k 2 /ε νt = C µ k υ 2 /ε y+ Elliptic relaxation for near wall turbulence models p. 9/22

14 The v 2 f model Solve transport equations for k, ε and v 2, and elliptic equation for f 22 [ ( Dv 2 Dt = kf ε v2 k + ν + ν ) ] t v 2 x j σ k x j [ ] L 2 2 f x 2 f = 1 j T (C v 1 1) 2 k 2 P C k 2 3 k v 2 has no tensorial meaning, is now a scalar. So is f. Wall boundary condition f = 2νv2 εy 4 Elliptic relaxation for near wall turbulence models p. 1/22

15 The v 2 f model Advantages: Reproduces the correct behaviour of the turbulent viscosity near the wall No need to include the distance from the wall. Can be used in any geometry Improves predictions on separating flows, as well as heat transfer and skin friction. Drawbacks: One transport and one elliptic equations more than the standard k ε Stiffness of the boundary condition makes it necessary to solve v 2 f coupled. Elliptic relaxation for near wall turbulence models p. 11/22

16 The ϕ f model Instead of solving v 2, a transport equation is solved for the ratio ϕ = v 2 /k (Laurence et al., 24): D(υ 2 /k) Dt = f (υ2 /k) k P + x k [( ν + ν t σ (υ2 /k) Where X is the "cross diffusion" term from the transformation: ( ) X = 2 ν + ν t (υ 2 /k) k k x k x k σ (υ2 /k) ) (υ 2 /k) x k ] + X Elliptic relaxation for near wall turbulence models p. 12/22

17 The ϕ f model Advantages of the substitution are: The term εv 2 /k is not present, leaving stable viscous diffusion as a dominant mechanism near the wall. The wall boundary condition for f is reduced to: f w = lim y 2ν(υ 2 /k) y 2 (1) which is more convenient and easier to reproduce since it has y 2 instead of y 4 of the original model. The LDM modification (Lien and Durbin,1996) proposed as a method to uncouple the equations, does not ensure the correct behaviour in the region far from the wall. Elliptic relaxation for near wall turbulence models p. 13/22

18 The ϕ f model A changes of variable is applied to the original model: f = f + 2ν( ϕ k) k + ν 2 ϕ (2) Leading to: Dϕ Dt = f P ϕ k + 2 k L 2 2 f f = 1 T (C 1 1) ν t ϕ k σ k x j [ ϕ x j ] x j C 2 P k 2ν k [ νt σ k ϕ x j ] (3) ϕ k ν 2 ϕ x j x j Far from the wall the last two terms are negligible, ensuring the correct behaviour. (4) Elliptic relaxation for near wall turbulence models p. 14/22

19 The ϕ f model The boundary condition of f is zero at the wall, improving robustness. The equations can be solved totally uncouple. Useful for unstructured codes (Implemented in the industrial Code_Saturne of EDF) Modification ensures correct behaviour far from the wall (see Laurence et al., 24) ƒ ƒ hom L 2 ƒ term neglected in LDM term neglected in ϕ model y+ y+ a) b) Elliptic relaxation for near wall turbulence models p. 15/22

20 Test cases Flat plate: Channel Flow: (Exp: Weighardt and Tillman) (Kim et al., 1987).5.45 Exp LDM ϕ - f.4 Cf e+6 4e+6 6e+6 8e+6 Re x Elliptic relaxation for near wall turbulence models p. 16/22

21 Test cases Natural convection: Tall cavity (Exp: Betts and Bokhari, 1995). V(m/s) V (m/s) V(m/s) Exp y/h =.5 ϕ - f LDM k-ε L-S Exp y/h =.7 Exp y/h =.95 a) x (m) V (m/s) V (m/s) V (m/s) Exp y/h =.5 ϕ - f LDM k-ε L-S Exp y/h =.7 Exp y/h = x(m) a) Ra =.86x1 6 b) Ra = 1.43x1 6 b) Elliptic relaxation for near wall turbulence models p. 17/22

22 Test cases Separated flows: Asymmetric plane diffuser (Exp: Buice and Eaton, 1997) 4 Profiles at x/h = ϕ model k-ε k-ω y/h 2 y/h Cp.4.2 Exp k-ε k - ω ϕ model U/Ub uu/ub 4 3 y/h 2 y/h x/h vv/ub uv/ub Elliptic relaxation for near wall turbulence models p. 18/22

23 Test cases Flow over periodic hills (Fine LES, Temmerman and Leschziner, 21) LES ϕ model k - ε Y/h a) Y/h U+x/h b) Streamlines: LES, k ε, ϕ Elliptic relaxation for near wall turbulence models p. 19/22

24 Test cases Three dimensional symmetric bump: Work in progress. (Simpson, 22) Elliptic relaxation for near wall turbulence models p. 2/22

25 Test cases Multiple impinging jets. Work in progress. (Geers et. al, 23) Top: Exp, Middle: k ε, Bottom: ϕ f Elliptic relaxation for near wall turbulence models p. 21/22

26 Conclusions The elliptic relaxation approach reproduces the near-wall effects. No need for damping functions or distance from the wall. With full Reynolds stress model too expensive (13 equations) v 2 f with EVM, cheaper but stiff due to boundary conditions. ϕ f better. Same effects but allows for uncoupling of equations, can be used in industrial codes. Tested in separated, impinging and buoyant flows. It is still more expensive than two-equation models. Elliptic relaxation for near wall turbulence models p. 22/22

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