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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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

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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

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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

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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

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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

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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

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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
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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

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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

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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

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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

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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
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Applied CCM Motivation PISO SLIM Results
Triangular Cavity
Steady, laminar, 2D lid-driven cavity, ReD = 800
Grid: Hybrid with ∼ 5,500 cells.
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Applied CCM Motivation PISO SLIM Results
Triangular Cavity
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 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

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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
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Applied CCM Motivation PISO SLIM Results
3D Square Cylinder
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 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
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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