Numerical Methods for the Navier-Stokes Equations

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Computational Fluid Dynamics I

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Numerical Methods for

the Navier-Stokes

Equations

Instructor: Hong G. Im University of Michigan

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• 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

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Incompressible

Navier-Stokes Equations

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Incompressible Navier-Stokes Equations













w

v

u

=

u

0 =

⋅

∇

u

ρ

α

p

t

∇

−

∇

+

∇

⋅

−

=

∂

u

₂

The (hydrodynamic) pressure is decoupled from the

rest of the solution variables. Physically, it is the pressure that drives the flow, but in practice pressure is solved such that 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.

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Vorticity-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

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Solution Methods for Incompressible N-S Equations in Primitive 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)

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Back 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

0

2

∇

⋅

=

+

∂

u

c

t

p

ρ

α

p

t

∇

−

∇

+

∇

⋅

−

=

∂

u

₂ 2

c

: arbitrary constant 2

c

0 (

p

(

x

)

t

p

=

→

∂

ρ

2

c

p

=

2

c

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The concept can be applied to a time-accurate method by using “pseudo-time stepping” at every sub-steps.

Artificial Compressibility - 2

0

2

∇

⋅

=

+

∂

u

c

p

τ

ρ

α

τ

β

p

t

∇

−

∇

+

∇

⋅

−

=

∂

+

∂

u

₂

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

0 ,

0 →

∂

→

∂

τ

p

u

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

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Marker-and-Cell (MAC) Method – Harlow and Welch (1965) • Originally derived for free surface flows with staggered grid

Pressure Correction Method - 1

0 =

⋅

∇

u

ρ

α

p

t

∇

−

∇

+

∇

⋅

−

=

∂

u

₂

)

(

)

(

2 2 1 n h h h n h n h n h n h

p

t

u

⋅

∇

+

∇

−

=

∇

⋅

∇

+

∆

⋅

∇

−

⋅

∇

+

α

ρ

)

(

2 n h n h h

p

=

−

∇

⋅

u

⋅

∇

u

∇

ρ

Taking divergence of momentum equation,

Poisson equation

ρ

α

p

t

h n h n h n n n

∇

−

∇

+

∇

⋅

−

=

∆

−

+

u

1 ₂ and Explicit integration

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Projection Method – Chorin (1968), Temam (1969) • Originally derived on a colocated grid

• Identical to MAC except for the Poisson equation Pressure Correction Method - 2

p t p t h t n h t n ∇ ∆ − = ⇒ ∇ − = ∆ − + +

ρ

u u u u 1 1 ₁ 0 1 = ⋅ ∇ n+ h u p t h h t h n h ∇ ⋅∇ ∆ − ⋅ ∇ = ⋅ ∇ +

ρ

u u 1 t h h

t

p

∇

⋅

u

∆

=

∇

2

ρ

0

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MAC vs. Projection

1. Integration without pressure

p

t

h t n+

=

−

∆

∇

ρ

u

1

(

n n

)

n t

t

A

D

u

=

+

∆

−

+

t h h

t

p

∇

⋅

u

∆

=

∇

2

ρ

2. Poisson equation

3. Projection into incompressible field

)

(

2 n h n h h

p

=

−

∇

⋅

u

⋅

∇

u

∇

ρ

n n

u

=

(

)

t

p

t

n n _h n n+

=

+

∆

−

+

−

∆

∇

ρ

D

A

u

1

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SIMPLE Algorithm – Patankar (1981)

(Semi-Implicit Method for Pressure Linked Equations) - Iterative procedure with pressure correction

p p

p = ₀ + ′

1. Guess the pressure field

2. Solve the momentum equation (implicitly)

3. Solve the pressure correction equation

0 p

ρ

α

0 0 2 0 0 0

p

t

n

∇

−

∇

+

∇

⋅

−

=

∆

−

u

(

0

)

2

u

⋅

∇

∆

=

′

∇

t

p

ρ

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_{Pressure Correction Method - 5}

4. Correct the pressure and velocity

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

p p p = ₀ + ′

p

t

_′

∇

∆

−

=

ρ

0

u

Notes:

- Originally developed for the staggered grid system.

- The corrected velocity field satisfies the continuity equation

even if the pressure correction is only approximate. - Sometimes tends to be overestimatedp′

) 8 . 0 ( 0 + ′ ≈ = p

ω

p

ω

p underrelaxation

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SIMPLER (SIMPLE Revised)

- Incorporating the projection method (fractional step) Pressure Correction Method - 6

1. Guess the velocity field

2. Solve momentum equation (implicitly) without pressure

3. Solve the pressure Poisson equation

0 u

u

ˆ

₀ ₂

∇

+

∇

⋅

−

=

∆

−

_α

t

(

u

ˆ

)

* 2

∇

⋅

∆

=

∇

t

p

ρ

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_{Pressure 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

=

−

⋅

∇

+

∇

−

∇

∆

−

ρ

α

u

( )

* 2

u

⋅

∇

∆

=

′

∇

t

p

ρ

*

p

t

_′

∇

∆

−

=

ρ

*

u

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_{Stability Consideration}

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

(

)

α

4 and 4 2 2 h t v u t ∆ < + < ∆ t ∆ α 4 2 h t = ∆

(

)

α / 1 4 u v 2 t = + ∆ Re ~ / 1

α

0 → ∆t at both limits! High-Re flow: advection-controlled

Low-Re flow: diffusion-controlled Use implicit schemes for

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_{Accuracy Improvement – Spatial 1}

Spatial Accuracy

• Explicit differencing - use larger stencils ) ( 2 2 1 1 h O h f f f _j′ = j+ − j− + ) ( 12 8 8 ₁ ₁ ₂ ₄ 2 h O h f f f f f _j′ = j− − j− + j+ − j+ +

• Tridiagonal - Padé (compact) schemes

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)

( ) 3 4 ₁ ₁ ₁ 4 1 f f O h h f f f _j′₋ + _j′ + _j′₊ = _j₊ − _j₋ + • Pentadiagonal L + + + + + = ′ + ′ + ′ + ′ + ′₋₂ _j₋₁ _j _j₊₁ _j₊₂ _j₋₂ _j₋₁ _j _j₊₁ _j₊₂ j f f f f af bf cf df ef f

β

γ

δ

ε

α

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_{Accuracy Improvement – Spatial 2}

Ref: Kennedy, C. A. and Carpenter, M. H.,

Applied Numerical Mathematics, 14, pp. 397-433 (1994) .

k∆ k' ∆ 0 1 2 3 0 1 2 3 Exact 2E 4E 6E 6T 8E 8T

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_{Accuracy Improvement – Temporal 1}

Temporal Accuracy • Implicit – Crank-Nicolson

(

)

[

1 2 2 1

]

1/2 1 ) ( ) ( ) ( 2 + + + + ₋ n ₌ ∆ ₋ n ₊ n ₊ _∇ n ₊ _∇ n ₋ ∆ _∇ n n p t t

ρ

α

u u u A u A u u

p

t

=

−

+

−

∇

∂

ρ

1 )

(

)

(

u

D

u

A

u

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Linearization of Advection Terms • For example, a 2-D equation

can be linearized as 0 = ∂ ∂ + ∂ ∂ + ∂ ∂ y x t F E U           = v u ρ ρ ρ U           − − + = xy xx uv p u u τ ρ τ ρ ρ 2 E             − + − = yy xy p v uv v τ ρ τ ρ ρ 2 F

[ ]

= 0 ∂ ∂ + ∂ ∂ + ∂ ∂ y B x A t U U U

[ ]

U F U E ∂ ∂ = ∂ ∂ = B A ,

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Fractional Step Method – Kim & Moin (1985)

• Projection method extended to higher accuracy

[

]

( ) Re 2 1 ) ( ) ( 3 2 1 _n _n ₁ ₂ _t _n n t t A u A u u u u u + ∇ + − − = ∆ − −

Adams-Bashforth (AB2) Crank-Nicolson

t n t n t t u u u u ⋅ ∇ ∆ = ∇     = ⋅ ∇ −∇ = ∆ − + + 1 0 2 1 1

φ

Note that is different from the original pressure

φ

2 Re 2 ∇ ∆ − = t p

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Treatment of implicit viscous terms

[

]

( ) Re 2 1 ) ( ) ( 3 2 1 _n _n ₁ ₂ _t _n n t t A u A u u u u u + ∇ + − − = ∆ − − Factorizing,

(

−

)

=       ₋ ∆ ₋ ∆ ₋ ∆ ⇒ t n zz yy xx t t t u u

δ

Re 2 Re 2 Re 2 1

[

]

(

)

n zz yy xx n n t t u u A u A − + ∆

δ

+

δ

+

δ

∆ − − Re ) ( ) ( 3 2 1

(

−

)

=       ₋ ∆       ₋ ∆       ₋ ∆ t n zz yy xx t t t u u

δ

Re 2 1 Re 2 1 Re 2 1

[

]

(

)

n zz yy xx n n t t u u A u A − + ∆

δ

+

δ

+

δ

∆ − − Re ) ( ) ( 3 2 1

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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) where

Poisson equation becomes

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

( )

      ∂ ∂ + ⋅ ∇ ∆ = ∇ t t t t t

ρ

φ

1 u 2

ρ

_T =₁ _{Eq. of State}

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_{Boundary Conditions}

Boundary Conditions for Incompressible Flows

• In general, boundary condition treatment is easier than for the compressible flow formulation due to the absence of acoustics

• Typical boundary conditions:

- Periodic: etc. - Inflow conditions:

- Outflow conditions: convective outflow condition

2 1 1, f f f f_N = _N₊ = ) , , ( ) 0 (x F y z t f = = L x x U t = = ∂ ∂ + u 0 at u     ₌

∫

udA A U 1

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Compressible

Navier-Stokes Equations

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0 =

∂

+

∂

+

∂

+

∂

z

y

x

t

G

F

E

U

                = E w v u ρ ρ ρ ρ ρ U                 + + − − − + = x xz xy xx u p E uw uv p u u ψ ρ τ ρ τ ρ τ ρ ρ ) ( 2 E                   + + − − + − = y yz yy xy v p E vw p v uv v ψ ρ τ ρ τ ρ τ ρ ρ ) ( 2 F                   + + − + − − = z zz yz xz w p E p vw vw uw w ψ ρ τ ρ τ ρ τ ρ ρ ) ( 2 G T c h T c e RT p = ρ , = _v , = _p R e T e p R c_v , ( 1) , ( 1) 1 − = − = − = γ ρ γ γ or Constitutive relations z zz yz xz z y yz yy xy y x xz xy xx x q w v u q w v u q w v u + − − − = + − − − = + − − − = τ τ τ ψ τ τ τ ψ τ τ τ ψ where

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Solution methods for compressible N-S equations

follows the same techniques used for hyperbolic equations

z

y

x

t

∂

+

∂

+

∂

=

∂

U

E

F

G

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 numerical dissipation, hence provides stability, but accuracy

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- 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 higher order method (both explicit and implicit).

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Most of the time, an implicit integration method involves nonlinear advection terms

which are linearized as

z

y

x

t

∂

+

∂

+

∂

=

∂

U

E

F

G

[ ]

z C y B x A t n n n n n ∂ ∂ + ∂ ∂ + ∂ ∂ = ∆ − + + + +1 1 1 1 U U U U U

[ ]

A n

[ ]

B n

[ ]

C n      ∂ ∂ =       ∂ ∂ =       ∂ ∂ = U G U F U E , ,

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Ultimately, compressible Navier-Stokes equations can be written as a system of ODE’s

z

y

x

t

∂

+

∂

+

∂

=

∂

U

E

F

G

(

)

0 0

)

(

)

(

,

U

=

⇒

t

F

dt

d

Initial condition Solution techniques for a system of ODE applies.

- Explicit vs. Implicit (Nonstiff vs. Stiff) - Multi-stage vs. Multi-step

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_{Boundary Conditions}

Boundary Conditions for Compressible Flows

• In general, boundary condition for the compressible flow is trickier because all the acoustic waves must be

properly taken care of at the boundaries. • Typical boundary conditions:

- Periodic: still easy to implement

- Both inflow and outflow conditions require treatment of characteristic waves