Smoother Scheme Oren Peles and Eli Turkel Department of Applied - PowerPoint PPT Presentation

Improvements of Unsteady Simulations for Compressible Navier Stokes Based on a RK/Implicit Smoother Scheme Oren Peles and Eli Turkel Department of Applied Mathematics, Tel-Aviv University In memoriam of Prof. Saul Abarbanel. August 20-24, 2018

Lecture outline • Background and motivation • Governing equations • Two-fluids model • RK/Implicit smoother with source terms • Dual time stepping approach with backward differencing • Dual time stepping approach with DIRK • Adaptive time steps • Lagrangian trajectories of burning droplets • Examples • Conclusions and future work

Motivation - Leidenfrost effect of droplets moving on a ratchet -Leidenfrost maze* The Leid Le idenfrost effect is a physical phenomenon in which a liquid, in near contact with a mass significantly hotter than the liquid's boiling point , produces an insulating vapor layer keeping that liquid from boiling rapidly. Because of this 'repulsive force', a droplet hovers over the surface rather than making physical contact with it. Th is is most commonly seen when cooking : one sprinkles drops of water in a pan to gauge its temperature: if the pan's temperature is at or above the Leidenfrost point, the water skitters across the pan and takes longer to evaporate than in a pan below the temperature of the Leidenfrost point (but still above boiling temperature). *Cheng, C., Guy, M., Narduzzo, A., & Takashina, K. (2015). The Leidenfrost Maze. European Journal of Physics , 36 (3), 035004

• We wish to simulate the maze. • Problem statement – very low Mach number, unsteady, two- phase flow (surrounding air and vapours from the droplets) with moving finite size droplets.

Background • We wish to accelerate the computation of time dependent problems in the framework of a dual-time stepping based code with an explicit scheme for pseudo time and an implicit scheme for the physical time. • Physical models – low Mach compressible flows with perfect gas, two-fluids mixing model, inviscid and viscous flows. • Acceleration methods within the pseudo-time marching: multigrid, local time steps, implicit residual smoothing • For the pseudo time stepping a low-storage RK/Implicit Smoother method is used. • We consider the schemes for the physical time – Backward differencing (BDF) and Diagonally Implicit Runge-Kutta (DIRK). • Adaptive time steps based on DIRK scheme.

Governing equations The Navier-Stokes equations for the conservative variables 𝑅 = 𝜍, 𝜍𝑣, 𝜍𝑤, 𝜍𝑥, 𝜍𝐹 𝑈 in conservation form are: 𝜖𝑅 𝑅 = 𝛼𝐺 𝜖𝑢 + 𝛼𝐺 𝑤 𝑅 + 𝑇 is the inviscid flux 𝐺 = 𝜍𝑣, 𝜍𝑣𝑣 + 𝑞𝑦 𝑈 𝐺 , 𝜍𝑣𝑣 + 𝑞𝑧 , 𝜍𝑣w + 𝑞𝑨 , 𝜍𝑣 𝐹 + 𝑞 𝑤 is the viscous flux 𝐺 𝑈 𝐺 𝑤 = 0, 𝜐 𝑦 , 𝜐 𝑧 , 𝜐 𝑨 , 𝑣𝜐 𝑦 + v𝜐 𝑧 + 𝑥𝜐 𝑨 + 𝑙𝛼𝑈 𝜍 is the gas density, p is the pressure, u, v and w are the velocity vector components and E is the internal energy 𝛿 − 1 + 1 𝑞 2 𝜍 𝑣 2 + 𝑤 2 + 𝑥 2 𝜍𝐹 = 𝜐 𝑦 , 𝜐 𝑧 , 𝜐 𝑨 are the stress vectors, k is the thermal conductivity and T is the temperature

Two-Fluids model • Simple modelling of steady and unsteady mixing of two perfect gases

Model governing equations • The continuity equation is replaced by two continuity equations for the two gases composing the mixture • The modified Navier-Stokes equations are given by                     u         t Sc                     u         t Sc      u u          ˆ u u p x  x t      v u          ˆ u v p y  y t        w u        ˆ u w p z  z t         E            u H T      t Pr                                               u u u h h              x y z       x y z Sc

 p 1 Energy and Total enthalpy     2     E u H E p   1 2 Heat capacities     y y         C w       p w 1 w 1         y y 1 1       C w       v w 1 w 1       C   p C v y y         1  Total density and mean molecular weight  W   w w   𝑞 = 𝜍𝑆𝑈 𝑋 Equation of state

Transformation Jacobian for primitive to conservative variables    0 0 0 u u      0 0 0 v v    0 0 0 w w    W        1 p w 1 1 1 p w 1 1 1             2 2 u v w c U c U U               v v     1 w 1 2 w 1 2   a a b b 0 0 0 0 1 0       0 0 0 0 0 1 Conservative to primitive variables transformation Jacobian        1 / 0 0 0 u / u /          0 1 / 0 0 v / v /        0 0 1 / 0 w / w /    U                p w 1 1   1   p w 1 1   1                             2 2 W u 1 v 1 w 1 1 1 c 1 U 1 c 1 U             p p     w 1 2 w 1 2   a a b b 0 0 0 0 1 0       0 0 0 0 0 1

RK/Implicit smoother with source terms We solve the system in conservation form:  Q      F S Q  t Applying the Gauss theorem for an arbitrary control volume and a discretization in time yields the basic scheme:    Q 1 1         S F n d s S F n ds  t V V all faces S

The low-storage Runge-Kutta time marching scheme is: 𝑅 (0) = 𝑅 𝑜 𝑅 (𝑙+1) = 𝑅 (0) − 𝛽 𝑙+1 Δ𝜐𝑆 𝑙 𝑙 = 0 … 𝑞 -1 𝛽 𝑙 are the RK coefficients 𝑅 n+1 = 𝑅 (𝑞) The residual of the k-th step is: 𝑆 k = 1 ⋅ 𝑜 ⅆ𝑡 − 𝑇 𝑙 𝐺 𝑊 all faces For accelerating the calculations using large CFL numbers the residual is replaced by a spatially smoothed residual Δ𝑅

Following Rossow 2006, Swanson et al. 2007, we start with the spatially discretized equation:   Q 1      F n ds S 0  t V all faces Linearizing F and S in time we obtain:      t S          k I A n ds t Q R     V Q  all faces  F  A  - the flux Jacobian Q  S - the source Jacobian  Q

Transforming the equations to primitive variables, the flux Jacobian is written as:     A A A where   1    A A A 2 Finally, the implicit smoothing scheme is given by: 𝐽 + 𝜁 Δ𝜐 − Δ𝜐 𝜖𝑇 𝜖𝑅 Δ𝑅 𝑚𝑝𝑑𝑏𝑚 = 𝑆 𝑙 − 𝜁 Δ𝜐 𝐵 + ⋅ 𝑜ⅆ𝑇 𝐵 − ⋅ 𝑜 ⋅ Δ𝑅 𝑂𝐶 ⅆ𝑇 𝑊 𝑊 𝑏𝑚𝑚 𝑔𝑏𝑑𝑓𝑡 𝑏𝑚𝑚 𝑔𝑏𝑑𝑓𝑡 ε is a relaxation factor. The implicit smoothing scheme is solved iteratively, using symmetric Gauss- Seidel method or red-black iterations (RB is useful for parallel computing).

Dual time stepping approach with backward differencing The system we solve in the pseudo time 𝜐 is 𝜖𝑅 𝜖𝜐 + 𝜖𝑅 𝑅 = 𝛼𝐺 𝜖𝑢 + 𝛼𝐺 𝑤 𝑅 For each physical time step, we solve the NS equation for a steady state . The derivative with respect to the physical time is a source term and approximated via backward differencing scheme: 𝜖𝑢 ≈ − 3𝑅 𝑜+1 − 4𝑅 𝑜⋅ + 𝑅 𝑜−1 𝑇 = − 𝜖𝑅 2Δ𝑢

Since 𝑅 𝑜+1 is unknown, we replace it by the best approximation 𝑅 (𝑙+1) which is the solution in the next pseudo-time RK stage. 𝑇 ≈ − 3𝑅 (𝑙+1) − 4𝑅 𝑜⋅ + 𝑅 𝑜−1 2Δ𝑢 We have 𝑆 k 𝑅 (𝑙+1) = 𝑅 (0) − 𝛽 𝑙+1 Δ𝜐 1 + 𝛽 𝑙+1 3Δ𝜐 2Δ𝑢 −1 3Δ𝜐 Identify 1 + 𝛽 𝑙+1 as a point-implicit smoothing operator and inspired by this 2Δ𝑢 formulation, the modified RK/implicit smoother operator for time dependent problem is: