validation of a fast transient solver based on the projection method

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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 CCMI Specialise in the application, development and support of

OpenFOAM® - based softwareI Creators and maintainers ofI Locations: Australia, Canada, USA


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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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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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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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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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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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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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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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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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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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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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Triangular CavityI Steady, laminar, 2D lid-driven cavity, ReD = 800I Grid: Hybrid with ∼ 5,500 cells.


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Triangular CavityI Cavity centreline x-velocity distribution compared with

experimental data Jyotsna and Vanka (1995)


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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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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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3D Square CylinderI Comparison with experimental data of Lyn et al. (1995)

and numerical results from Voke (1997)


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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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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


validation of a fast transient solver based on the projection method

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