numerical methods for the navier-stokes equationsphoenics/site_phoenics/apostilas/cfd-1_u... ·...

Computational Fluid Dynamics IPPPPIIIIWWWW

Numerical Methods forthe Navier-Stokes

Equations

Instructor: Hong G. ImUniversity of Michigan

Fall 2001


• Summary of solution methods- Incompressible Navier-Stokes equations- Compressible Navier-Stokes equations

• High accuracy methods- Spatial accuracy improvement - Time integration methods

Outline

What will be covered

What will not be covered• Non-finite difference approaches such as

- Finite element methods (unstructured grid)- Spectral methods


Incompressible Navier-Stokes Equations

Computational Fluid Dynamics IPPPPIIIIWWWWIncompressible Navier-Stokes Equations

wvu

=u

0=⋅∇ u

ρα p

t∇−∇+∇⋅−=

∂∂ uuuu 2

The (hydrodynamic) pressure is decoupled from the rest of the solution variables. Physically, it is the pressurethat drives the flow, but in practice pressure is solved suchthat the incompressibility condition is satisfied.

The system of ordinary differential equations (ODE’s) are changed to a system of differential-algebraic equations (DAE’s), where algebraic equations acts like a constraint.

Computational Fluid Dynamics IPPPPIIIIWWWWVorticity-stream function formulation

Advantages:- Pressure does not appear explicitly (can be obtained later)- Incompressibility is automatically satisfied (by definition of stream function)

Drawbacks:- Limited to 2-D applications(Revised 3-D approaches are available)

Computational Fluid Dynamics IPPPPIIIIWWWWSolution Methods for Incompressible N-S Equations inPrimitive Formulation:

• Artificial compressibility (Chorin, 1967) – mostly steady• Pressure correction approach – time-accurate

- MAC (Harlow and Welch, 1965)- Projection method (Chorin and Temam, 1968)- Fractional step method (Kim and Moin, 1975)- SIMPLE, SIMPLER (Patankar, 1981)

Computational Fluid Dynamics IPPPPIIIIWWWWBack to a system of ODE by

• With properly-chosen , solve until• Originally developed for steady problems• The term “artificial compressibility” is coined from equation of state

• Possible numerical difficulties for large

Artificial Compressibility - 1

02 =⋅∇+∂∂ uc

tp

ρα p

t∇−∇+∇⋅−=

∂∂ uuuu 2

2c : arbitrary constant

2c ( ))(0 xpptp =→

∂∂

ρ2cp =2c

Computational Fluid Dynamics IPPPPIIIIWWWWThe concept can be applied to a time-accurate methodby using “pseudo-time stepping” at every sub-steps.

Artificial Compressibility - 2

02 =⋅∇+∂∂ ucpτ

ρα

τβ p

t∇−∇+∇⋅−=

∂∂+

∂∂ uuuuu 2

At every “real” time step, take “pseudo-time stepping”using explicit time integration until

0,0 →∂∂→

∂∂

ττpu

Since the pseudo time scale is not physical, we canaccelerate the integration however we want.

Computational Fluid Dynamics IPPPPIIIIWWWWMarker-and-Cell (MAC) Method – Harlow and Welch (1965)

• Originally derived for free surface flows with staggered grid

Pressure Correction Method - 1

0=⋅∇ uρα p

t∇−∇+∇⋅−=

∂∂ uuuu 2

)()( 221

nhh

hnh

nh

nh

nh p

tuuuuu ⋅∇∇+∇−=∇⋅⋅∇+

∆⋅∇−⋅∇ +

αρ

)(2 nh

nhh p uu ∇⋅⋅∇−=∇ ρ

Taking divergence of momentum equation,

Poisson equation

ρα p

thn

hn

hn

nn ∇−∇+∇⋅−=∆−+

uuuuu 21

and

Explicit integration

Computational Fluid Dynamics IPPPPIIIIWWWWProjection Method – Chorin (1968), Temam (1969)

• Originally derived on a colocated grid • Identical to MAC except for the Poisson equation


ptpt h

tnh

tn

∇∆−=⇒∇−=∆− +

+

ρρuuuu 1

1 1

01 =⋅∇ +nh u

pthh

th

nh ∇⋅∇∆−⋅∇=⋅∇ +

ρuu 1

thh t

p u⋅∇∆

=∇ ρ2

0

Computational Fluid Dynamics IPPPPIIIIWWWWMAC vs. Projection

1. Integration without pressure


pth

tn ∇∆−=+

ρuu 1

( )nnnt t DAuu +−∆+=

thh t

p u⋅∇∆

=∇ ρ2

2. Poisson equation

3. Projection into incompressible field

)(2 nh

nhh p uu ∇⋅⋅∇−=∇ ρ

nn uu =

( ) ptt hnnnn ∇∆−+−∆+=+

ρDAuu 1

Computational Fluid Dynamics IPPPPIIIIWWWWSIMPLE Algorithm – Patankar (1981)

(Semi-Implicit Method for Pressure Linked Equations)

- Iterative procedure with pressure correction


ppp ′+= 0

1. Guess the pressure field2. Solve the momentum equation (implicitly)

3. Solve the pressure correction equation

0p

ρα 0

02

000 p

t

n ∇−∇+∇⋅−=∆− uuuuu

( )02 u⋅∇

∆=′∇

tp ρ

Computational Fluid Dynamics IPPPPIIIIWWWWPressure Correction Method - 5

4. Correct the pressure and velocity

5. Go to 2. Repeat the process until the solution converges.

ppp ′+= 0

pt ′∇∆−=ρ0uu

Notes:- Originally developed for the staggered grid system. - The corrected velocity field satisfies the continuity equationeven if the pressure correction is only approximate.

- Sometimes tends to be overestimatedp′

)8.0(0 ≈′+= ωωppp underrelaxation

Computational Fluid Dynamics IPPPPIIIIWWWWSIMPLER (SIMPLE Revised)

- Incorporating the projection method (fractional step)


1. Guess the velocity field2. Solve momentum equation (implicitly) without pressure

3. Solve the pressure Poisson equation

0u

uuuuu ˆˆˆˆ 20 ∇+∇⋅−=∆− α

t

( )u*2 ⋅∇∆

=∇t

p ρ

Computational Fluid Dynamics IPPPPIIIIWWWWPressure Correction Method - 7

5. Solve the momentum equation with

6. Pressure correction equation

7. Correct the velocity, but not the pressure

8. Go to 2. Repeat the process until solution is converged.

**2**0* 1 p

t∇−∇+∇⋅−=

∆−

ρα uuuuu

( )*2 u⋅∇∆

=′∇t

p ρ

*p

pt ′∇∆−=ρ

*uu

Computational Fluid Dynamics IPPPPIIIIWWWWStability Consideration

Explicit time integration in 2-D requires the stability condition:

( ) αα

4and4 2

2ht

vut <∆

+<∆

t∆α4

2ht =∆

( )α/1

4 2vut

+=∆

Re~/1 α0→∆t at both limits!

High-Re flow: advection-controlledLow-Re flow: diffusion-controlled

Use implicit schemes for appropriate terms!

Computational Fluid Dynamics IPPPPIIIIWWWWAccuracy Improvement – Spatial 1

Spatial Accuracy

• Explicit differencing - use larger stencils

)(2

211 hOh

fff jj

j +−

=′ −+

)(12

88 42112 hOh

fffff jjjj

j +−+−

=′ ++−−

• Tridiagonal - Padé (compact) schemes

( ) )(34 41111 hOff

hfff jjjjj +−=′+′+′ −++−

• Pentadiagonal

L+++++=′+′+′+′+′ ++−−++−− 21122112 jjjjjjjjjj efdfcfbfaffffff εδγβα

Computational Fluid Dynamics IPPPPIIIIWWWWAccuracy Improvement – Spatial 2

Ref: Kennedy, C. A. and Carpenter, M. H., Applied Numerical Mathematics, 14, pp. 397-433 (1994) .

k∆

k'∆

0 1 2 30

1

2

3Exact2E4E6E6T8E8T

Computational Fluid Dynamics IPPPPIIIIWWWWAccuracy Improvement – Temporal 1

Temporal Accuracy

• Implicit – Crank-Nicolson

( )[ ] 2/112211 )()()(2

++++ ∇∆−∇+∇++−∆=− nnnnnnn pttρ

α uuuAuAuu

pt

∇−+−=∂∂

ρ1)()( uDuAu

Nonlinear advection term requires iteration.


Linearization of Advection Terms

• For example, a 2-D equation

can be linearized as

0=∂∂+

∂∂+

∂∂

yxtFEU

=

vu

ρρρ

U

−−+=

xy

xx

uvpu

u

τρτρ

ρ2E

−+

−=

yy

xy

pv

uvv

τρ

τρρ

2

F

[ ] [ ] 0=∂∂+

∂∂+

∂∂

yB

xA

tUUU

[ ] [ ]UF

UE

∂∂=

∂∂= BA ,where Jacobian matrix


Fractional Step Method – Kim & Moin (1985)

• Projection method extended to higher accuracy

[ ] )(Re21)()(3

21 21 ntnn

nt

tuuuAuAuu +∇+−−=

∆− −

Adams-Bashforth (AB2) Crank-Nicolson

t

n

tn

tt u

u

uu⋅∇

∆=∇

=⋅∇

−∇=∆−

+

+

1

0

2

1

1

φφ

Note that is different from the original pressureφ

φφ 2

Re2∇∆−= tp


Treatment of implicit viscous terms

[ ] )(Re21)()(3

21 21 ntnn

nt

tuuuAuAuu +∇+−−=

∆− −

Factorizing,

( )=−

∆−∆−∆−⇒ nt

zzyyxxttt uuδδδ

Re2Re2Re21

[ ] ( ) nzzyyxx

nn tt uuAuA δδδ ++∆+−∆− −

Re)()(3

21

( )=−

∆−

∆−

∆− nt

zzyyxxttt uuδδδ

Re21

Re21

Re21

[ ] ( ) nzzyyxx

nn tt uuAuA δδδ ++∆+−∆− −

Re)()(3

21

TDMA in three directions


Notes on Fractional Step Method

• Originally implemented into a staggered grid system• Later improved with 3rd-order Runge-Kutta method

Ref: Le & Moin, J. Comp. Phys., 92:369 (1991)• The method can be applied to a variable-density problem(e.g. subsonic combustion, two-phase flow) wherePoisson equation becomes

Ref: Rutland, Ph. D. Thesis, Stanford University (1989)Bell, Collela and Glaz, JCP, 85:257 (1989)

( )

∂

∂+⋅∇∆

=∇tt

ttt ρρφ u12 1=Tρ Eq. of State

Computational Fluid Dynamics IPPPPIIIIWWWWBoundary Conditions

Boundary Conditions for Incompressible Flows

• In general, boundary condition treatment is easierthan for the compressible flow formulation due to theabsence of acoustics

• Typical boundary conditions:- Periodic: etc. - Inflow conditions: - Outflow conditions: convective outflow condition

211, ffff NN == +

),,()0( tzyFxf ==

Lxx

Ut ==∂∂+ at0uu

= ∫udA

AU 1


Compressible Navier-Stokes Equations


0=∂∂+

∂∂+

∂∂+

∂∂

zyxtGFEU

=

Ewvu

ρρρρρ

U

++−

−−+

=

x

xz

xy

xx

upEuwuv

puu

ψρτρτρ

τρρ

)(

2

E

++

−

−+

−

=

y

yz

yy

xy

vpEvw

pv

uvv

ψρτρ

τρ

τρρ

)(

2F

++−+

−−

=

z

zz

yz

xz

wpEpvw

vwuww

ψρτρ

τρτρ

ρ

)(

2

G

TchTceRTp pv === ,,ρ

ReTepRcv

)1(,)1(,1

−=−=−

= γργγ

orConstitutive relations

zzzyzxzz

yyzyyxyy

xxzxyxxx

qwvuqwvuqwvu

+−−−=

+−−−=

+−−−=

τττψτττψτττψ

where

Computational Fluid Dynamics IPPPPIIIIWWWWSolution methods for compressible N-S equations

follows the same techniques used for hyperbolic equations

zyxt ∂∂+

∂∂+

∂∂=

∂∂ GFEU

For smooth solutions with viscous terms, central differencing usually works. ⇒ No need to worry about upwind method, flux-splitting,

TVD, FCT (flux-corrected transport), etc.In general, upwind-like methods introduces numericaldissipation, hence provides stability, but accuracybecomes a concern.


- MacCormack method- Leap frog/DuFort-Frankel method- Lax-Wendroff method- Runge-Kutta method

Explicit Methods

Implicit Methods- Beam-Warming scheme- Runge-Kutta method

Most methods are 2nd order. The Runge-Kutta method can be easily tailored to higherorder method (both explicit and implicit).

Computational Fluid Dynamics IPPPPIIIIWWWWMost of the time, an implicit integration method involvesnonlinear advection terms

which are linearized as

zyxt ∂∂+

∂∂+

∂∂=

∂∂ GFEU

[ ] [ ] [ ]z

Cy

Bx

At

nnnnn

∂∂+

∂∂+

∂∂=

∆− ++++ 1111 UUUUU

[ ] [ ] [ ]nnn

CBA

∂∂=

∂∂=

∂∂=

UG

UF

UE ,,

+ ADI, factorization, etc.

Computational Fluid Dynamics IPPPPIIIIWWWWUltimately, compressible Navier-Stokes equations can be written as a system of ODE’s

zyxt ∂∂+

∂∂+

∂∂=

∂∂ GFEU

( )

00 )(

)(,

UU

UU

==

=⇒

tt

ttFdtd

Initial condition

Solution techniques for a system of ODE applies.- Explicit vs. Implicit (Nonstiff vs. Stiff)- Multi-stage vs. Multi-step

Computational Fluid Dynamics IPPPPIIIIWWWWBoundary Conditions

Boundary Conditions for Compressible Flows

• In general, boundary condition for the compressible flow is trickier because all the acoustic waves must beproperly taken care of at the boundaries.

• Typical boundary conditions:- Periodic: still easy to implement - Both inflow and outflow conditions require treatment ofcharacteristic waves(hard-wall, nonreflecting, sponge, etc).

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