Exact and approximate projection methods for transient incompressible and low-Mach flows

Mark ChristonComputational Physics R&DSandia National Laboratories

Collaborators: Phil Gresho, Steve Chan,

Tom Voth, Wing Kam Liu

May 2002

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy under contract DE-AC04-94AL85000.

Overview

Background & driving applicationsThe incompressible Navier-Stokes equationsThe projection methodPressure stabilization and approximate projections The A-conjugate projection conjugate gradient methodAdvective treatment & low-Mach number extensionsLES, two-scale filter & realizable turbulent stressesSummary & Conclusions

Incompressible/low-Mach flows span a diverse range of applications

Vehicle aerodynamicsInitial focus for the incompressible algorithms was submarine flows

Mold filling – casting and resin-transfer moldingEncapsulation of electronic componentsChemically reacting flows – burners, CVD reactorsDispersal of chemical and biological agentsHVAC – environmentally controlled roomsVehicle NVH (noise-vibration-heat)

Flow-induced vibration and noise

Inlet

Outlet

The well-posed incompressible flow problem satisfies “solvability”

BC’s:

IC’s:Constraints:

2Pt

ν∂+ ⋅∇ = −∇ + ∇ +

∂u u u u f

0∇⋅ =u

( ,0) ( )0 x=u x u

0 ˆ 1on= Γi in u n u

0 0 in∇ = Ωiu for a well-posed problem

12 ˆ, 0if

ΓΓ =∅ =∫ in u

1Γ 2Γ

Ω

ˆt t on= Γ1u(x, ) u(x, )n

nuP Fn

ν ∂− + =∂u Fττν

τ∂

=∂

2on Γ

The pressure-Poisson equation provides an alternate formulation

PPE:

With BC’s:

If all solvability conditions are met, momentum + PPE will deliver the same solution as momentum + div constraint!

2

0

Pt

with implies

ν∂+ ⋅∇ = −∇ + ∇ +

∂∇ =i

u u u u f

u2 ( )P in∇ =∇ − ∇ Ωi if u u

21( )P on

n tν∂ ∂

= ∇ + − − ∇ Γ∂ ∂

i iun u f u u

2n

nuP F onn

ν ∂= − Γ∂

( , )Pu

Spatial discretization produces DAE’s not ODE’s

DAE’s are ODE’s with algebraic constraints

Index-2 DAE:

Index-1 DAE:

TC g=u( )M A K CP+ + + =u u u u F Fully-coupled

( )M A K CP+ + + =u u u u F 1 1 ( )T TC M C P C M K A g− − = − − − F u u u

Segregated

2 0,i P dt

φ νΩ

∂+ ⋅∇ ∇ − ∇ − Ω =

∂∫u u u+ u f 0i dψ

Ω∇⋅ Ω =∫ u

jij iM dφ φΩ

= Ω∫ ( )ij i jA dφ φΩ

= ⋅∇ Ω∫u u

ij i jK dφ φΩ

= ∇ ∇ Ω∫ ν i i dφΩ

= Ω∫F fij i jC dψ φΩ

= ∇ Ω∫

,j j j ju u P Pφ ψ= =∑ ∑iil iM dφ

Ω= Ω∫

Some DAE properties

DAE’s with index 2 are “higher index”Higher-index DAE’s are harder to solveHigher-index systems require derivatives of order “index-1” resulting in ill-conditioned systemsHigher-index DAE’s can have hidden constraintsLower-index DAE’s can have more solutions thanoriginal DAE’sDAE’s require consistent IC’sThe “correct” solution satisfies the original DAE’s, i.e., the higher-index system

The indexThe index--1 1 DAE’sDAE’s are hard to solve “correctly” are hard to solve “correctly” -- and and the indexthe index--2 (fully2 (fully--coupled) equations are even harder!coupled) equations are even harder!

≥

Projection methods are one of the most popular schemes for solving the Navier-Stokes

Methods:Chorin (1968) -- Chorin’s original first-order methodVan Kan (1986) Karniadakis, et al. (1991)Gresho, Chan and Christon(1990 - present)Almgren, Bell, Colella, Howell, Lai, Welcome, et al. (1989 - 2000)Guermond & Quartapelle (1997)Guermond & Lu (2000)Kothe, et al. (1995 - present)Rider (1994, 1995)

Analysis:Chorin (1969)Shen

1992: High-order1993: P-3 Scheme1996: 2nd-order

E & Liu (1995, 1996)Jacobs (1995)Guermond & Quartapelle

1996: Convergence1998: Error estimates1999: Incremental method

Wetton (1998)Almgren, Bell & Crutchfield (1999)Armfield & Street (1999)

Projection methods perform betterthan formal analysis suggests!!!

Some additional projection literature …

Stewart (1981)Issa, et al. (1985, 1986)Kovacs & Kawahara (1991)Simo, et al. (1994, 1995)Brown & Minion (1995 - 97)Sheu and Lee (1996)Thomadakis & Leschziner (1996)Timmermans, et al. (1996)Nonino, Comini & Croce (1997)

Knio, Najm & Wyckoff (1999)Cummins & Rudman (1999)Minev

1998: BHC correction1999: Stabilization

Codina, et al.1998: predictor-corrector2000: stabilization ideas

Pozrikidis (2001)

The projection method for Navier-Stokes relies on a Helmholtz decomposition

Given: and

A Helmholtz decomposition of yields div-free and curl-free components

The decomposition exists and is unique for a well-posed “incompressible” flow problem

Given is projected onto a div-free and curl-free subspace to obtain

Exact projections are idempotent:

The eigenvalues of are either 0 or 1, so that the projections are stable and norm reducing

2( ) ( )P wheret

ν∂+∇ = = + ∇ − ⋅∇

∂F Fu u u f u u u

( )uF

0, 0and Pt

∂∇ ⋅ = ∇×∇ =

∂u

, ( )Fu u( ( )), ( ( ))and P where

t∂

= ∇ =∂

P F Fu

u uQ2 1 2 1( ) , ( )I and− −= −∇ ∇ ∇⋅ = ∇ ∇ ∇⋅P Q

2 2, and= =P P Q Q,P Q

0∇⋅ =u

The projection algorithm is “realized” by a velocity decomposition

Solve:

Use:

The div-free velocity is obtained with a projection step

Attempts to legitimately decouple the velocity and pressurePreserves a discretely div-free velocity field at each time step

( ) 0,P wheret

∂+∇ = ∇ ⋅ ≠

∂Fu u u

( )uQ

( ) div free−uP

u

1 1[ ] T T nlC M C Cλ− += u

11 1 1, ( )ln n n ntM C P Pλ λ γ

−+ + +∆= − = −u u

0 0I

λ−∇

= −∇ ⋅

u u 1

00

M C n Ml lTC λ

+ =

uu

λ−∇ =u u−∇ =iu 0

Projection methods admit both explicit and semi-implicit time integration

Semi-implicit time integration for the optimal projection algorithm (P2)

1.

2.

3.

4.

The “exact” projection method uses a consistent PPE based on the discrete divergence and gradient operatorsThe “approximate” projection method uses a modified, i.e., a “stabilized” PPESelective mass lumping preserves 4th-order advective phase accuracyAdvection treatment is explicit with a monotonicity preserving predictor-corrector scheme

[ ] [ (1 ) ] nM t K M t Kθ θ+ ∆ = −∆ − +u u

1 1nlM Cλ+ −= −u u

1n nP Pt

λγ+ = +∆

1[ ]T TlC M C Cλ− = u 1( [ ] )T T T n

lor C M C S C Cλ− + = −u u

1 1 (1 ) ( ) n n n nLt A MM CPθ θ+ −∆ + − − −F F u u

The PPE solution can dominate the time step

PPE Properties:is symmetric regardless of the Reynolds number

Can be singular, e.g., contains a hydrostatic modeFor Q1Q0 element can have additional singular modes

Checkerboard mode in 2-DMultiple CB-like modes in 3-D

Right-hand-side, , is noisy, i.e., has a large spectral content and changes rapidly from step-to-step

Source data can control the convergence rate of the solution method -- see Kim and Ro (1995)

Solution of the PPE can consume 80 - 90% of the CPU time per time step.

1[ ]TlC M C−

TC u

Geometrical singularities canexcite “local” pressure modes

The “bi-harmonic catastrophe” problem shows large amplitude CB mode decays spatiallyPressure stabilization filters the modes and improves PPE conditioning

Un-stabilized

Stabilized (β=0.05)

Diverging channel (Gresho, et al., 1995)

0,5.1 == vu

)1100ln(2.0 += yu

Un-stabilized

Stabilized (β=0.05)

The Q1Q0 approximate projection algorithm relies on pressure stabilization

Replace the consistent PPE by an operator that is “easier” to solve -- at the expense of the divergence

Equal-order Q1Q1 approximate projection --Gresho, Chan, Christon and Hindmarsh, 1995Related work of Rider, Almgren – Bell – Colella

Pressure stabilization for Q1Q0 element yields a regularized saddle-point problem

Silvester & Kechkar, 1990, Silvester, 1995,Norburn & Silvester, 1997

1

IJ

Tl IJ

IJ I JIJ

C M CS dβ ψ ψ

−

Γ= − Γ

Γ

∫0.0 1.0β≤ ≤ Global

LocalI

Jh

Pressure

h

I J

“Project thedifference”

( )1

0

nl

nlT

M C M

C S λ

+ −=

−

u uu

( )1[ ]T T nlC M C S Cλ− + = −u u

Pressure stabilization reduces computational work for PPE solve

minimizeseigenspectraof the PPE

yields inaccuratevelocity fieldsNo. of iterationsrelativelyinsensitive to βProjections areapproximate so∇ ≠iu 0

0.01 0.1β≤ ≤

0.25β ≥

β=0.05 typically

Pressure stabilization also improves the iterative convergence rate

2-D: 20% reduction in iterations and computational work3-D: 7-10X reduction in iterations and computational work“Realistic” convergence criteria yields more modest gains

12 1 1210 , 10 , 0.05n n nb Ax b x x x β− − −− ≤ − ≤ =

Global JumpLocal Jump

No Stabilization

0 500 1000 1500 2000 2500 3000

Iterations

||b-A

x||/|

|b||

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

62720 2116807840

No StabilizationGlobal JumpLocal Jump

0 50 100 150 200 250 300 350 400 450 500

Iterations

||b-A

x||/|

|b||

10-10

10-12

10-14

10-8

10-4

10-6

10-2

100

2800 11200 44800

2-D Vortex Shedding 3-D Post & Plate

The A-conjugate projection-CG methodfurther reduces the PPE solution time

Efforts with algebraic multigrid (AMG) of Ruge and Stuben (1987)Lack of robustness in computing coarse-grid operators -- particularly for complex geometryFactor of 17 slower than direct resolve cost

A-conjugate projection CG uses short and long-wavelength information to, in effect, “deflate” the eigenspectra of the PPEConstruct initial solution that minimizes the residual in the A-norm (Christon, 2002).Given a set of A-conjugate vectors,

1.

2.

3.

4.

5.

( )r b Ax= −

( )T ni i bα φ=

Φ

1N

i iix α φ==∑n nSolve A x b Ax∆ = −

n nx x x= + ∆11 1 11

, lnll l i iil Axψφ ψ α φψ

++ + =+= = −∑

1 additional mat-vecrequired per time-stepto add A-conjugate vector

A-conjugate projection selectively uses information from previous pressure fields

• Solving relies on N-prior solutions that are A-conjugate

• where are A-conjugate• A-conjugate construction is achieved with

one additional mat-vec per time step

( )1[ ]T Tl

nC M C S Cλ− + = −u u

∑+∆= iinn φαλλ iφ

λ∆ λ

11φα

1 1 2 2 3 3n nλ λ α φ α φ α φ= ∆ + + +

A-conjugate projection-CG with SSOR preconditioning approaches direct resolve cost

Grind-time comparison for 1000 time steps relative to resolve time for PVS direct solver (3D Cylinder: Nel=11250, Nnp=11466, Nb=146, ε=1.0e-5)

49.21.602660.42.032966.62.398750

51.71.713162.32.183469.42.6610025

51.11.753859.92.134166.52.5312110

59.72.024769.42.634975.03.201485

74.73.3610083.65.0511087.46.523280

% PPEGrindTimeNIT% PPE

GrindTimeNIT% PPE

GrindTimeNIT

No. of Vectors

ESSOR-PCGSSOR-PCGJPCGβ=0

*

46.01.522357.91.922663.72.207650

49.41.652758.82.002966.52.448825

48.11.683756.81.983663.72.3510810

56.21.874466.22.394372.42.911305

74.03.239682.84.8310586.66.163090

% PPEGrindTimeNIT% PPE

GrindTimeNIT% PPE

GrindTimeNIT

No. of Vectors

ESSOR-PCGSSOR-PCGJPCGβGlob=0.05

*

Coarse-grained parallelism is treated with domain-decomposition message-passing

Explicit message passing via MPI (SPMD model)Non-overlapping sub-domainswith cache-blockingFinite element assembly procedure induces communication

Requires remapping of on-processor nodes, elements, etc.Formation of consistent and lumped mass matricesMatrix-vector products foriterative solversGradient operators for flow solverFormation of right-hand-side

P0 P1

P3P2

P0

P1 P2 P3

Fine-grained parallel speedupsdemonstrate scalability

Communication costs scale with the number of CG iterations required for the pressure/projection solveO(1000) elements per processor yields 12.6% communication overhead on 1024 processors

• Re=100 3-D Post & Plate• Efficiencies for 1000 time steps• ~250 Elements per Processor

• Re=800 Backward Facing Step• Efficiencies for 1000 time steps• ~250 Elements per Processor

Predictor-corrector advection scheme uses operator-limiting to preserve monotonicity

15 mm slot Re=4000, Fr=326

SOUC

C

QUICK

FEMLumped FEM

Preserve 4th-order FEM phase accuracyOperator-limiters are characteristic-based Forward-Euler inviscid predictorTrapezoidal-rule correctorLimiters DO NOT act on smooth data

Re=4000 slot-jet test for the operator limited explicit advection treatment

Temperature Vorticity15 mm slotRe=4000Fr=326Experimental Strouhal: St=0.16 (Yu and Monkewitz, 1990)Computational Strouhal: St=0.18∆t based on a fixed grid-CFL=1.0

High-Grashof number flow test for the monotonicity preserving advection scheme

Insulated

InsulatedGr = 1.32 x 109

Pr = 0.7L = 1 m∆T = 10o C201 x 201 grid∆t based on a fixed grid-CFL=2.0

Th Tc

g

Acoustically filtered equations for low-speed, reacting flow neglect fast (and weak) waves

Majda & Sethian (1985) formulation assumes:Two species - burnt and unburnt gas with same γ-law and molecular weights, one-step Arrhenius kineticsSolution algorithm adapted from Lai (1993), similar to Day & Bell (1999)

2, ( ), , exptotp p Ep p p O M k Ap RT RT

ρ∆ = + ∝ = = −

2pt

ρ µ∂ + ⋅∇ = −∇ + ∇ + ∂ u u u u f

20p

T dpC T T q k Zt dt

ρ κ ρ∂ + ⋅∇ = + ∇ + ∂ u

( )Z Z D Z k Zt

ρ ρ ρ∂ + ⋅∇ = ∇ ∇ − ∂ iu

( )( )20

1 1dp q k Z Tp dt

γ ρ κγ

∇ − + − + ∇

iu =

Low-Re variable density and reacting flowshave similar wake structures

, -- no buoyancy forces100, 100, 1, 1.4Re Pe Sc γ= = = =

Non-Reacting Reacting

Large Eddy Simulation relies on resolving the large-scale fluid dynamics

Energy is added to a turbulent flow at the large scales via stirring, shear, etc., and cascades to smaller scales where dissipation occurs

Length and time scales span a broad spectrum from the integral to the KolmogorovLarge eddy simulation (LES) models the self-similar fine-scale physics than can not be resolved at the grid-scale

Energy Input Cascade Dissipation

Increasing wave number(Decreasing length scale)

LES relies upon filtering the Navier-Stokes Equations

LES Filtering is based upon a finite-domain convolution

• For a filter scale,

The filtered, incompressible Navier-Stokes equations

Postulated SGS stresses:Smagorinsky model:

• Based on a balance between production and dissipation of turbulent kinetic energy

• Does not account for wall bounded or anisotropic flows.

τ

µ ρ

, ( ) ( ) ( , , )a f x f x K a x dξ ξ ξΩ

= −∫

0∇⋅ =u

( ) 2Tp

tµρ ρ∂ + ⋅∇ = −∇ +∇ ⋅ ∇ + ∇ −∇ ⋅ ∂

u u u u u τ

where ij i j i ju u u uτ = −

1, ,2

1 2 ,3ij ij kk t ij ij i j j iS S u uδ τ µ− = − = +

( )t s ij ijC a S S= 2

GILA LES computation captures large-scale vortical structures and mean drag

Ahmed’s body with 30o slant andExperimental drag coefficient:Predicted short-time average drag coefficient:LES crimes:

RANS type graded grid (~500,000 elements)No explicit wall functionsUnder-resolved energy spectrum

Re .= 4 29 106xCw = 0 378.

Cw = 0 386.

Streamwise Vorticity

Flow Direction

U∞ 30o

Artificial non-stationary flows can occur when div(u) is unconstrained

Re=10,000 (Zang, Street & Koseff 1993

Unconstrained div(u)

Constrained div(u)

0.000

0.005

0.010

0.015

0.020

0.025

Kin

etic

Ene

rgy

0 250 500

Time [S]

750 1000 1250 1500

Collect Statistics

0.00

2.00E-6

4.00E-6

6.00E-6

8.00E-6

1.00E-5

1.20E-5

1.40E-5

div(

u)

0 250 500 750

Time [s]1000 1250 1500

Filtering can be used to control and reduce numerical errors for LES

Application of the LES SGS model at 2∆x wavelengths introduces dispersive errorsSetting the filter scale larger than the grid scale promises to reduce the numerical errors over the entire discrete spectrum (Ghosal, 1996)Commutative RKPM filter may be used with a grid-independent filter size for LES

E(k)

η−1∆x−1l−1

EnergyContainingEddies

Intertialrange

SOUC

C

2 x λ∆

QUICKFEM

Lumped FEM

Filter Scale

LES filters are based on the Reproducing Kernel Particle Method (RKPM)

Commutation error, , is determined by filter moments

RKPM consistency is determined by moments as well …

Commutivity error for RKPM is set by the degree of consistency (see Wagner & Liu, Voth & Christon)

RKPM filters permit treatment of arbitrary geometry and unstructured meshes (with an increased computational cost associated with the initial search)

( ) ( ) ( )M x x K a x dk k= −∫ ξ ξ ξ

Ω

, , 0 1, 0, 1, 2,kM M k== = …

( ) ( )M x x xa

dk k= −−

∫ ξ φ

ξξ

Ω

0 11, 0 yields linear consistencyM M= =

df df d fdx dx dx

= −

a( )kO xdf

dx ∆

∝

Dynamic SGS stress model removes the empiricism associated with a model constant

Application of the coarse-scale filter yields the test-scale stresses:

Use of the same postulated form of the test-scale stresses yields the “dynamic constant” in terms of known quantities

A second filter is required at a length scale larger than the original filter scale

T u u u uij i j i j= −

P u P u Q u P u x x u xa a a a a ii

Np

i i= + = −=∑2 2

1

, ( ) ,where andφ ∆ Q u u xa a ai

Np

i i2 21

= −=∑( )φ φ ∆

CL MM M

L Tij ij

ij ijij ij ij= = −where andτ , M a S S a S Sij ij ij ij ij= − −( )2

2 2

aP u 2aP u 2aQ u

Re 395 DNS data provided by William Cabot, CTRτ =

Positivity and consistency yield realizable instantaneous sub-grid scale stresses

Realizability requires positivity in the three principal invariants of the SGS stresses:

( )

1

22

3

0

0

det 0

iii

ii jj iji j

ij

I

I

I

τ

τ τ τ

τ

≠

= ≥

= − ≥

= ≥

∑

∑

ij i j i ju u u uτ = −

DNS data: William Cabot, CTR395Reτ =

Two-scale decomposition using RKPM filter with linear consistency

P ua Q ua2P ua2

Re 395 DNS data provided by William Cabot, CTRτ =

Improving LES reliability on unstructured grids will rely on understanding:

the side effects of a finite divergencethe influence of commutative errors for LES in complex geometry and unstructured gridsfilter construction that yields second-order commutative errorsdispersive and diffusive errors on SGS modelsthe influence of grid anisotropy on SGS modelsthe interaction between explicit filters, filter scales, SGS models and resolved-scale numericsthe interaction between advective schemes and sub-grid scale modelsthe interaction between SGS models and the resolved-scale numericsthe errors introduced by VLES, i.e., under-resolved LES

Summary & Conclusions

Projection methods have evolved dramatically over the past decadeA-conjugate projection-CG is an effective way to treat the PPETheir numerical performance is difficult to predict from analysisSimplicity & flexibility makes them capable of treating a large range of flow regimes and physics

Preserving is a critical part of incompressible LES computations

unconstrained divergence can introduce artificial dissipation independent of advective schemes

RKPM filterscontrol commutative errors (on unstructured meshes)provide multi-scale decomposition for dynamic SGS modelsyield realizable SGS stresses due to enforced consistency

0∇ =iu