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Validation of a Fast Transient Solver based on the Projection Method

This paper presents a fast transient solver suitable for the simulation of incompressible flows. The main characteristic of the solver is that it is based on the projection method and requires only one pressure and momentum solve per time step. Furthermore, advantage of using the projection method in the formulation is the particularly efficient form of the pressure equation that has the Laplacian term depending only on geometric quantities. This form is highly suitable for the high
performance computing that utilises the Algebraic Multi-grid Method (AMG) as the coarse levels produced by the algebraic multi-grid can be stored if the grid is not changing. Fractional step error near the boundaries is removed by utilising the incremental version of the algorithm. The solver is implemented using version
5.04 of the open source library, Caelus. Accuracy of the solver was investigated through several validation cases.The results indicate the solver is accurate and has good computational efficiency.

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Validation of a Fast Transient Solver based on the Projection Method

  1. 1. logo.png Applied CCM Motivation PISO SLIM Results Validation of a Fast Transient Solver based on the Projection Method Darrin Stephens, Chris Sideroff and Aleksandar Jemcov 17 July 2015 Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  2. 2. logo.png Applied CCM Motivation PISO SLIM Results Applied CCM Specialise in the application, development and support of OpenFOAM® - based software Creators and maintainers of Locations: Australia, Canada, USA Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  3. 3. logo.png Applied CCM Motivation PISO SLIM Results Motivation Why develop another transient solver? DES and LES attractive because RANS tends to be problem specific Low cost hardware + open-source software ⇒ DES and LES feasible Traditional transient, incompressible algorithms (PISO and SIMPLE) do not scale well for large HPC, GPU and Many Integrated Core (MIC) environments Let’s review PISO algorithm Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  4. 4. logo.png Applied CCM Motivation PISO SLIM Results PISO Overview Pressure Implicit with Splitting of Operators (PISO)1 method: 1. Solve momentum equation (predictor step) 2. Calculate intermediate velocity, u∗ (pressure dissipation added) 3. Calculate mass flux 4. Solve pressure equation: · ( 1 Ap p) = · u∗ 5. Correct mass flux 6. Correct velocity (corrector step) Repeat steps 2 – 6 for PISO (1 – 6 for transient SIMPLE) 1Isaa, R.A. 1985, “Solution of the implicitly discretised fluid flow equations by operator splitting” J. Comp. Phys., 61, 40. Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  5. 5. logo.png Applied CCM Motivation PISO SLIM Results Fractional Step Error Step 2 main issue with PISO Predicted velocity used only to update matrix coefficients: u∗ = 1 ap Σ anb unb − ( p − p) Pseudo-velocity, u∗, is used on the RHS of pressure equation Therefore requires at least two corrections to make velocity and pressure consistent Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  6. 6. logo.png Applied CCM Motivation PISO SLIM Results Pressure Matrix Non-constant coefficients ( 1 ap ) in pressure matrix affects multi-grid solver performance Multi-grid agglomeration levels cached first time pressure matrix assembled Coefficients ( 1 ap ) only valid for the first time step Turning off caching of agglomeration too expensive Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  7. 7. logo.png Applied CCM Motivation PISO SLIM Results SLIM Overview Semi Linear Implicit Method (SLIM), based on projection method1: decompose velocity into vortical and irrotational components. 1. Solve momentum equation (vortical velocity) 2. Calculate mass flux (pressure dissipation added) 3. Solve pressure equation (irrotational velocity): ∆t 2(p) = · u 4. Correct mass flux 5. Correct velocity (solenoidal) Use incremental pressure approach to recover correct boundary pressure 1Chorin, A.J. 1968, “Numerical Solution of the Navier-Stokes Equations”,Mathematics of Computation 22: 745-762 Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  8. 8. logo.png Applied CCM Motivation PISO SLIM Results Fractional Step Error Velocity split into vortical and potential components - much smaller fractional step error Pressure and velocity maintain stronger coupling Continuity satisfied within one pressure solve because predicted velocity used directly in pressure equation Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  9. 9. logo.png Applied CCM Motivation PISO SLIM Results Pressure Matrix Pressure matrix coefficients purely geometric Multi-grid agglomeration levels assembled during first step now consistent for all time steps Significantly improves parallel scalability for multi-grid solver Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  10. 10. logo.png Applied CCM Motivation PISO SLIM Results Laminar Flat Plate Steady, laminar, 2D flow over a flat plate, Rex = 200, 000 Comparison with Blasius analytical solution cf ≈ 0.644√ Rex Based on NASA NPARC Alliance case Grid: ∼ 220,000 hex cells Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  11. 11. logo.png Applied CCM Motivation PISO SLIM Results Laminar Flat Plate Skin friction distribution compared to Blasius analytical solution Non-dimensional velocity profile at plate exit compared to the Blasius solution Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  12. 12. logo.png Applied CCM Motivation PISO SLIM Results Tee Junction Steady, laminar, 2D tee junction flow, Rew = 300 Grid: ∼ 2,000 hex cells Experimental (Hayes et al.,1989) SLIM Diff Flow split 0.887 0.886 0.112 Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  13. 13. logo.png Applied CCM Motivation PISO SLIM Results Triangular Cavity Steady, laminar, 2D lid-driven cavity, ReD = 800 Grid: Hybrid with ∼ 5,500 cells. Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  14. 14. logo.png Applied CCM Motivation PISO SLIM Results Triangular Cavity Cavity centreline x-velocity distribution compared with experimental data Jyotsna and Vanka (1995) Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  15. 15. logo.png Applied CCM Motivation PISO SLIM Results 2D Circular Cylinder Transient, laminar, incompressible flow past circular cylinder, ReD = 100 Grid: Hybrid with ∼ 9,200 cells. Frequency (Hz) Strouhal Number Experimental 0.0835 0.167 SLIM 0.0888 0.177 Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  16. 16. logo.png Applied CCM Motivation PISO SLIM Results 3D Square Cylinder Transient, turbulent, 3D flow over a square cylinder, ReD = 21, 400 Grid: ∼ 700,000 hex cells; LES model: Smagorinsky Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  17. 17. logo.png Applied CCM Motivation PISO SLIM Results 3D Square Cylinder Comparison with experimental data of Lyn et al. (1995) and numerical results from Voke (1997) Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  18. 18. logo.png Applied CCM Motivation PISO SLIM Results 3D Square Cylinder Set lr St CD Lyn et al. (1995) 1.38 0.132 2.1 SLIM 1.41 0.131 2.44 Other CFD (max) 1.44 0.15 2.79 Other CFD (min) 1.20 0.130 2.03 Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
  19. 19. logo.png Applied CCM Motivation PISO SLIM Results Summary SLIM algorithm was introduced and described Exact velocity splitting improves both convergence and accuracy Geometric pressure matrix coefficients advantageous for parallel efficiency, particularly for multi-grid solvers Accuracy tested through many validation cases (some shown) comprising steady, transient, laminar and turbulent flows Applied CCM © 2012-2015 ICCM2015, Auckland July 2015

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