validation of a fast transient solver based on the projection method
TRANSCRIPT
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Applied CCM Motivation PISO SLIM Results
Validation of a Fast Transient Solver basedon the Projection Method
Darrin Stephens,Chris Sideroff and Aleksandar Jemcov
17 July 2015
Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
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Applied CCM Motivation PISO SLIM Results
Applied CCMI Specialise in the application, development and support of
OpenFOAM® - based softwareI Creators and maintainers ofI Locations: Australia, Canada, USA
Applied CCM © 2012-2015 ICCM2015, Auckland July 2015
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Applied CCM Motivation PISO SLIM Results
Motivation
Why develop another transient solver?I DES and LES attractive because RANS tends to be
problem specificI Low cost hardware + open-source software⇒ DES and
LES feasibleI Traditional transient, incompressible algorithms (PISO and
SIMPLE) do not scale well for large HPC, GPU and ManyIntegrated Core (MIC) environments
I Let’s review PISO algorithm
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Applied CCM Motivation PISO SLIM Results
PISO OverviewPressure 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 flux4. Solve pressure equation:∇ · ( 1
Ap∇p) = ∇ · u∗
5. Correct mass flux6. 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.
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Applied CCM Motivation PISO SLIM Results
Fractional Step ErrorI Step 2 main issue with PISOI Predicted velocity used only to update matrix coefficients:
u∗ = 1ap
(Σ anb unb − (∇p−∇p)
)I Pseudo-velocity, u∗, is used on the RHS of pressure
equationI Therefore requires at least two corrections to make velocity
and pressure consistent
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Applied CCM Motivation PISO SLIM Results
Pressure MatrixI Non-constant coefficients ( 1
ap) in pressure matrix affects
multi-grid solver performanceI Multi-grid agglomeration levels cached first time pressure
matrix assembledI Coefficients ( 1
ap) only valid for the first time step
I Turning off caching of agglomeration too expensive
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Applied CCM Motivation PISO SLIM Results
SLIM OverviewSemi Linear Implicit Method (SLIM), based on projectionmethod1: decompose velocity into vortical and irrotationalcomponents.
1. Solve momentum equation (vortical velocity)2. Calculate mass flux (pressure dissipation added)3. Solve pressure equation (irrotational velocity):
∆t∇2(p) = ∇ · u4. Correct mass flux5. Correct velocity (solenoidal)
Use incremental pressure approach to recover correctboundary pressure
1Chorin, A.J. 1968, “Numerical Solution of the Navier-Stokes
Equations”,Mathematics of Computation 22: 745-762
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Applied CCM Motivation PISO SLIM Results
Fractional Step ErrorI Velocity split into vortical and potential components - much
smaller fractional step errorI Pressure and velocity maintain stronger couplingI Continuity satisfied within one pressure solve because
predicted velocity used directly in pressure equation
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Applied CCM Motivation PISO SLIM Results
Pressure MatrixI Pressure matrix coefficients purely geometricI Multi-grid agglomeration levels assembled during first step
now consistent for all time stepsI Significantly improves parallel scalability for multi-grid
solver
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Applied CCM Motivation PISO SLIM Results
Laminar Flat PlateI Steady, laminar, 2D flow over a flat plate, Rex = 200, 000I Comparison with Blasius analytical solution cf ≈ 0.644√
Rex
I Based on NASA NPARC Alliance caseI Grid: ∼ 220,000 hex cells
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Applied CCM Motivation PISO SLIM Results
Laminar Flat PlateI Skin friction distribution compared to Blasius analytical
solution
I Non-dimensional velocity profile at plate exit compared tothe Blasius solution
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Applied CCM Motivation PISO SLIM Results
Tee JunctionI Steady, laminar, 2D tee junction flow, Rew = 300I Grid: ∼ 2,000 hex cells
Experimental (Hayes et al.,1989) SLIM DiffFlow split 0.887 0.886 0.112
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Applied CCM Motivation PISO SLIM Results
Triangular CavityI Steady, laminar, 2D lid-driven cavity, ReD = 800I Grid: Hybrid with ∼ 5,500 cells.
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Applied CCM Motivation PISO SLIM Results
Triangular CavityI Cavity centreline x-velocity distribution compared with
experimental data Jyotsna and Vanka (1995)
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Applied CCM Motivation PISO SLIM Results
2D Circular CylinderI Transient, laminar, incompressible flow past circular
cylinder, ReD = 100I Grid: Hybrid with ∼ 9,200 cells.
Frequency (Hz) Strouhal NumberExperimental 0.0835 0.167
SLIM 0.0888 0.177
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Applied CCM Motivation PISO SLIM Results
3D Square CylinderI Transient, turbulent, 3D flow over a square cylinder,
ReD = 21, 400I Grid: ∼ 700,000 hex cells; LES model: Smagorinsky
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Applied CCM Motivation PISO SLIM Results
3D Square CylinderI Comparison with experimental data of Lyn et al. (1995)
and numerical results from Voke (1997)
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Applied CCM Motivation PISO SLIM Results
3D Square Cylinder
Set lr St CDLyn et al. (1995) 1.38 0.132 2.1
SLIM 1.41 0.131 2.44Other CFD (max) 1.44 0.15 2.79Other CFD (min) 1.20 0.130 2.03
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Applied CCM Motivation PISO SLIM Results
Summary
I SLIM algorithm was introduced and describedI Exact velocity splitting improves both convergence and
accuracyI Geometric pressure matrix coefficients advantageous for
parallel efficiency, particularly for multi-grid solversI Accuracy tested through many validation cases (some
shown) comprising steady, transient, laminar and turbulentflows
Applied CCM © 2012-2015 ICCM2015, Auckland July 2015