FMIA

ISBN 978-3-319-16873-9

Fluid Mechanics and Its ApplicationsFluid Mechanics and Its ApplicationsSeries Editor: A. Thess

F. MoukalledL. ManganiM. Darwish

The Finite Volume Method in Computational Fluid DynamicsAn Advanced Introduction with OpenFOAM® and Matlab®

The Finite Volume Method in Computational Fluid Dynamics

Moukalled · Mangani · Darwish

113

F. Moukalled · L. Mangani · M. Darwish The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM® and Matlab ®

This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used in the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab®. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems.

With over 220 figures, numerous examples and more than one hundred exercises on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on CFD algorithms, and as a reference for CFD programmers and researchers.

Engineering

9 783319 168739

Solving the Navier-Stokes Equations

Chapter 15

Introduction

• For the solution of the scalar equation we have assumed that the velocity field is known

• How do we actually compute the velocity field is the subject of this lecture

!m = ρv ⋅S

Velocity Pressure Coupling

Navier-Stokes Equations∂ ρφ( )∂t

transient   term!"#

+ ∇⋅ ρvφ( )convection   term!"# $#

= ∇⋅ Γ∇φ( )diffusion   term! "# $# + Q

source   term!

Navier-Stokes equations are a special case of the general scalar equation with Φ = 1, u or v and Γ=0 or μ and appropriate Q

∂ ρv( )∂t

+∇ ⋅ ρvv( ) = ∇ ⋅τ − ∇p + B

∂ρ∂t

+∇⋅ ρv( ) = 0

τ = −µ ∇v +∇vT( )+ 23µI∇⋅v

NS equations are non-linear‣ Not by itself a problem

‣ Can do Picard iteration

Momentum Equations (x,y,z)‣ not a problem

‣ Can solve each sequentially

Source terms?‣ stress tensor sources can be computed

Main Issue is that pressure is not known‣ should find a way to compute it

‣ what PDE to use?

‣ Special case of incompressible flow

Options

• Option 1‣ Vorticity-Stream function formulation

• Option 2‣ Density Based formulation

• Option 3‣ Use the Momentum equations to compute the velocity field‣ Use the Continuity equation to form a pressure equation to

compute the Pressure field‣ This is known as the Primitive Variables Formulation

Option I• Used in 70’s-80’s‣ Derive one PDE for stream function Ψ and one PDE for the

single component vorticity in 2D‣ Eliminates pressure as variable‣ Only 2 PDE;s as opposed to 3 for primitive variable formulation

(u,v,p)

• However no stream function definition in 3D flows

• Vorticity-vector potential formulation in 3D‣ 6 components (3 for vorticity, 3 vector potential)‣ Fewer components for primitive variable formulation

• Difficult to derive bc’s

Option 2: Density Methods

• Popular in compressible flow community

• (u,v,w,ρ,E) used as unknowns

• Artificial compressibility/preconditioning schemes

• Effectively add some artificial compressibility to enable use of density-based schemes for incompressible flows

• Widely used in aeronautics industry

Option 3: Pressure Methods

• Will focus on pressure-based methods since they are the most widely used in the incompressible flow community.

• At convergence, the solution satisfies the discrete continuity and momentum equations

• Choice of pressure-based/density-based solution technique only governs‣ whether we get a solution‣ how fast/cheaply we can get it ‣ how much memory/storage we need

SIMPLE Algorithm

• Semi-Implicit Method for Pressure-Linked Equations

• Proposed by Patankar and Spalding (1972)

• Idea is to start with discrete continuity equation

• Substitute into it the discrete u and v momentum equations

• Discrete momentum equations contain pressure differences

• Hence get an equation for the discrete pressures

• SIMPLE actually solves for a related quantity called the pressure correction

Flow in PipesA portion of a water-supply system is shown in the figure below. The flow rate in a pipe is given by

!mA

!mB

!mF = 20

!mE

!mD

!mC

P1 = 275

P6 = 40

P2 = 270

P5

P4 = 0

P3

!m = C∆ PCA = 0.4

CB = CD = CF = 0.2

CC = CE = 0.1

Pressure drop over the length of the pipe

Hydraulic conductance

1 Guess values for the unknowns

2Compute the flow rates using the guessed pressure

3Construct a pressure correction equation, and solve for the pressure correction

Solution

!mk∗ + ! ′mk( )

~k∑ = 0

! ′mk

~k∑ = − !mk

∗

~k∑

1 P3 = 200 P5 = 30

!mB∗ = CB P3

∗ − P2( ) !mA∗ = CA P1 − P3

∗( ) !mC∗ = CC P4 − P3

∗( ) !mD∗ = CD P3

∗ − P5∗( ) !mE∗ = CE P6 − P5

∗( )2

!mk∗ ≠ 0

~k∑

!mk = 0

~k∑

!m∗ = C∆ P∗

! ′m = C∆ ′P( ) !m = C∆ P

3

! ′mA = CA − ′p3( )

! ′mB = CB ′p3( )

! ′mC = CC − ′p3( ) ! ′mD = CD ′p3 − ′p5( )

! ′mE = CE − ′p5( )

!mA

!mB

!mF = 20

!mE

!mD

!mC

P1 = 275

P6 = 40

P2 = 270

P5

P4 = 0

P3

! ′mA − ! ′mB + ! ′mC − ! ′mD = − !mA

∗ − !mB∗ + !mC

∗ − !mD∗( )

! ′mD + ! ′mE = − !mD∗ + !mE

∗ − !mF( )

!mk∗ + ! ′mk( )

k=nb 3( )∑ = 0

CA − ′p3( )−CB ′p3( ) +CC − ′p3( )−CD ′p3 − ′p5( ) = − !mA∗ − !mB

∗ + !mC∗ − !mD

∗( )

!mk∗ + ! ′mk( )

k=nb 5( )∑ = 0

CD ′p3 − ′p5( ) +CE − ′p5( )= − !mD

∗ + !mE∗ − !mF( )

3

! ′mA = CA − ′p3( )

! ′mB = CB ′p3( )

! ′mC = CC − ′p3( ) ! ′mD = CD ′p3 − ′p5( )

! ′mE = CE − ′p5( )

!mA

!mB

!mF = 20

!mE

!mD

!mC

P1 = 275

P6 = 40

P2 = 270

P5

P4 = 0

P3

−0.7 ′p3( ) + 0.2 ′p5( ) = 100.2 ′p3( )− 0.3 ′p5( ) = −15

⇒′p3 = 0′p5 = 50

⎧⎨⎪

⎩⎪

! ′mA = 0

! ′mB = 0

! ′mC = 0

! ′mD = −10

! ′mE = −5

!mB∗ = −14

!mA∗ = 30

!mC∗ = −20

!mD∗ = 34

!mE∗ = 1

!mF = 20P1 = 275

P6 = 40

P2 = 270

P4 = 0

p3∗ = 200

p5∗ = 30

′p3 = 0

′p5 = 50

SIMPLE Based Algorithms

The SIMPLE AlgorithmStaggered Grid

aCvvC

• + aFvvF

•

F=NB(C )∑ = −VC ∇p(n)( )C

∇.(ρvv) = −∇p +∇⋅τ +B

∴v n( ),P n( )

!m•f

f =nb(C )∑ ≠ 0

∇⋅(ρv) = 0

Predictor

∴v•,P n( )

Corrector∴ ′v , ′P

!mf

f =nb(C )∑ = 0

!m•f + ! ′mf( )

f =nb(C )∑ = 0 !m

•ff = ρv•f ⋅Sf

! ′mf = ρ ′v f ⋅Sf

aeuue

• + afuu f

•

f =NB(e)∑ = −Ve ∇p(n)( )e

C EWw e

ρv f• ⋅Sf

f =NB(C )∑ = ρue

•Ae − ρuw•Aw

ue• +

afuu f

•

aeu

f =NB(e)∑ = − Ve

aeu

∂p(n)

∂x⎛⎝⎜

⎞⎠⎟ e

ue• = He u

•( )− de ∂p(n)

∂x⎛⎝⎜

⎞⎠⎟ e

uw• = Hw u•( )− dw ∂p(n)

∂x⎛⎝⎜

⎞⎠⎟ w

ue• = He u

•( )− de ∂p(n)

∂x⎛⎝⎜

⎞⎠⎟ e

′ue = He ′u( )− de∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

ue = He u( )− de∂p∂x

⎛⎝⎜

⎞⎠⎟ e

! ′mf

f =nb(C )∑ = − !m•

ff =nb(C )∑

′v f = H f ′v( )− d f ∇ ′p( ) f

Pv

Pressure Equation

ρ ′v f ⋅S f

f =nb(C )∑ = − !m•

ff =nb(C )∑ ′v f = H f ′v( )− d f ∇ ′p( ) fSubstituting

H f ′v( )− d f ∇ ′p( ) f⎡⎣ ⎤⎦ ⋅S f

f =nb(C )∑ = − !m•

ff =nb(C )∑

−d f ∇ ′p( ) f ⋅S f

f =nb(C )∑ = − !m•

ff =nb(C )∑ − H f ′v( ) ⋅S f

f =nb(C )∑

− −affu ′v ff

a fu

ff =NB( f )∑

⎛

⎝⎜⎞

⎠⎟⋅S f

f =nb(C )∑

C EWw e

−d f ∇ ′p( ) f ⋅S ff =nb(C )∑ = −

d f Af

CF′pF − ′pC( )

f =nb(C )∑

= aCp ′pC + aE

p ′pE + aWp ′pW( )

− !m•

ff =nb(C )∑ = − ρue

•Ae − ρuw•Aw( )

⇒ −d f Af

CF′pF − ′pC( )

f =nb(C )∑ = − ρue

•Ae − ρuw•Aw( )

aCp ′pC + aE

p ′pE + aWp ′pW( ) = bCp ⇒ ′v f = −d f ∇ ′p( ) f ⇒ v••f = v

•f − d f ∇ ′p( ) f

SIMPLESIMPLECSIMPLERSIMPLESTSIMPLE-MPISO

The SIMPLE Algorithm

assemble and solve momentum equation for v*

assemble and solve Pressure Correction equation for P'

repeat until convergence

SIMPLE iteration assemble using Rhie-Chow interpolation

m∗

f

correct , v*, ρ(n) and P(n) to get

improved , v**, ρ* and P*

m∗

f

m∗∗

f

start with nth iteration values

, v(n), ρ(n), P(n)m

(n)f

∴v n( ),P n( )

Predictor∴v•,P n( )

Corrector

∴v••, ′P

aCp ′pC + aE

p ′pE + aWp ′pW( ) = bCp

aCvvC

• + aFvvF

•

F=NB(C )∑ = −VC ∇p(n)( )C

v••f = v•f − d f ∇ ′p( ) f

p•C = p(n)C + ′pC

Sequential Solution

• Pressure methods were also developed as sequential methods as opposed to coupled methods where all the Navier-Stokes equation are solved simultaneously

• In sequential methods each of the following equation is solved individually in sequence‣ Solve u-momentum equation for u velocity component, with all

other variables assumed known‣ Solve v-momentum equation for v velocity component‣ Modify the continuity equation to yield a pressure equation and

solve for pressure‣ Repeat process until convergence

Collocated Grids

Collocated Grids

C EW

w e

C EWw e

u P

Pu

!mf

uw ueu f

??u f

C EW

w e

C EWw e

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟f

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟C

ρ ′v f ⋅S f

f =nb(C )∑ = − !m•

ff =nb(C )∑

′!mf

f =nb(C )∑ = − !m•

ff =nb(C )∑

′v f = 12 ′vC + ′vF( )

′ue = He ′u( )+ De∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

′uw = Hw ′u( ) + Dw∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

′ue − ′uw = He ′u( )− Hw ′u( )+ De∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

− Dw∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

≈ D∆ x

12

′pEE − 2 ′pC + ′pWW[ ]

!m•e − !m

•w = ρA u•

E − u•W( )Similarly

≈ D 12

∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ E

+ ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ C

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ C

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟W

⎡

⎣⎢

⎤

⎦⎥

= 12 HC ′v( )− DC

∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ C

+ HF ′v( )− DF∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ F

⎛⎝⎜

⎞⎠⎟

aCp ′pC + aEE

p ′pEE + aWWp ′pWW = − !mE

• − !mW•( )

Rhie Chow Interpolation

H f v( )= gPHP v( )+ gFHF v( )

vP = HC v( )−DC∇pC

v f = v f +∇p f( )−D f∇pf

Df =Vf

a fv

HP v( )=−aFvvFaCv

F∑

DP =VCaCv

v f = H f v( )−D f∇pf

vF = HF v( )−DF∇pF

!mf• = ρv f

• ⋅Sf

C EW

w e

C EWw e

u P

Pu uw ue

u f

C EW

w e

C EWw e

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟f

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟C

!mf !mf

!me !mw

= gP vP +DP∇pP( )+ gF vF +DF∇pF( )≈ v f +Df∇pf

= v f −D f ∇pf −∇p f( )

!′mf = ρ ′v f ⋅Sf

1

2

3

Pressure Correction

u P

ρ ′v f ⋅S ff =nb(C )∑ = − !m•

ff =nb(C )∑

′!mf

f =nb(C )∑ = − !m•

ff =nb(C )∑

′v f == v f − D f ∇pf −∇p f( )′ue = ′ue − De

∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

⎛

⎝⎜⎞

⎠⎟

′ue − ′uw = ′ue − De∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥− ′uw − Dw

∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

≈ D∆ x

12

′pE − 2 ′pC + ′pW[ ]

!m•e − !m

•w = ρ f ue

•Ae − ρwuw•Aw

Similarly

≈ D 12

∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ e

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

⎡

⎣⎢

⎤

⎦⎥

′uw = ′uw − Dw∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

− ∂ ′p∂x

⎛⎝⎜

⎞⎠⎟ w

⎛

⎝⎜⎞

⎠⎟

C EW

w e

C EWw e

Pu uw ue

u f

C EW

w e

C EWw e

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟f

∂ p∂x⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟C

!mf

!me !mw

aCp ′pC + aE

p ′pE + aWp ′pW = − !mf

•

f =nb C( )∑

The Pressure-Correction Equation

assemble and solve momentum equation for v*

assemble and solve Pressure Correction equation for P'

repeat until convergence

SIMPLE iteration assemble using Rhie-Chow interpolation

m∗

f

correct , v*, ρ(n) and P(n) to get

improved , v**, ρ* and P*

m∗

f

m∗∗

f

start with nth iteration values

, v(n), ρ(n), P(n)m

(n)f

∴v n( ),P n( )

∴v•,P n( )

∴v••, ′PaCp ′pC + aE

p ′pE + aWp ′pW( ) = bCp

aCvvC

• + aFvvF

•

F=NB(C )∑ = −VC ∇p(n)( )C

v••f = v•f − d f ∇ ′p( ) f

p•C = p(n)C + ′pC

! ′mf = −ρ (n) D f∇ ′pf( ) ⋅S f

!mf• = ρ (n)v f

• ⋅S f = ρ (n) v f• −D f ∇ ′pf −∇ ′p f( )⎡⎣ ⎤⎦ ⋅S f

′v f = −D f∇ ′pf

Pressure-Velocity Coupling

�

− ρ f* D f ∇ ′ P ( ) f ⋅S f( )

f∑ = − ρ f

*U f*( )

f∑ − ′ ρ f ′ v f ⋅S f( )

f∑ − ρ f

* H ′ v [ ] f ⋅S f( )f∑

First Predictor

€

vP

* =H[v*]P−D

P∇P

n( )( )P

First Corrector

( ) ( ) ( )( )ρ′+ρ=ρ′+=′+=ρ′′′ n*n****,PPP,,P, vvvv

€

vP

** = H[v**

]P−D

P∇P

*( )P

= H[v* + ′ v ]

P−D

P∇ P

n( ) + ′ P ( )[ ]P

€

′ v P

= H[ ′ v ]P−D

P∇ ′ P ( )

P

′ ρ = Cρ ′ P

Condition

€

ρP

* − ρP

o( )δt

V + Δ ρ*v**.S[ ]P

= 0

€

ρP

n( ) + ′ ρ P− ρ

P

o( )δt

V + Δ ρn( ) + ′ ρ ( ) v

* + H[ ′ v ]−D ∇ ′ P ( )( ).S[ ]P

= 0

€

VCρ

δt′ P P

+ Δ CρU* ′ P [ ]

P

−Δ ρ n( )D ∇ ′ P ( ).S[ ]

P

= −ρ

P

n − ρP

o( )δt

V −Δ ρ n( )U

*[ ]P

−Δ ρn( )

H[ ′ v ].S[ ]P

−Δ ′ ρ ′ v ( ).S[ ]P

Neglect

Approximation

€

Δ ′ ρ ′ v ( ).S[ ]P

€

H[ ′ v ]

€

′ v P

= −DP∇ ′ P ( )

P

Approximation Equation

€

VCρ

δt′ P P

+ Δ CρU* ′ P [ ]

P−Δ ρ

n( )D ∇ ′ P ( ).S[ ]

P

= −ρ

P

n − ρP

o( )δt

V −Δ ρn( )U

*[ ]P

Second Predictor

€

vP

*** =H**[v

**]P−D

P

** ∇P*( )P

Second Corrector

€

′ ′ v , ′ ′ P , ′ ′ ρ ( ) v****

= v***

+ ′ ′ v ,P**

= P*

+ ′ ′ P ,ρ**= ρ*

+ ′ ′ ρ ( )

€

vP

**** =H**[v

****]

P−D

P

** ∇ P* + ′ ′ P ( )[ ]

P

€

vP

*** =H**[v

**]P−D

P

** ∇P*( )P

€

′ ′ v P

= H**

[v**** − v

**]

P−D

P

** ∇ ′ ′ P ( )P

= H**

[v*** − v

** + ′ ′ v ]P−D

P

** ∇ ′ ′ P ( )P

′ ′ ρ = Cρ ′ ′ P

€

ρP

* + ′ ′ ρ P− ρ

P

o( )δt

V + Δ ρ* + ′ ′ ρ ( ) v*** + H

**[v

*** − v** + ′ ′ v ]−D

P

** ∇ ′ ′ P ( )( ).S[ ]P

= 0

Condition

€

ρP

** − ρP

o( )δt

V + Δ ρ**v****.S[ ]P

= 0

€

VCρ

δt′ ′ P P

+ Δ CρU*** ′ ′ P [ ]

P

−Δ ρ*D

P

** ∇ ′ ′ P ( ).S[ ]P

= −ρ

P

* − ρP

o( )δt

V −Δ ρ*U

***[ ]P

−Δ ρ*H

**[v

*** − v** + ′ ′ v ].S[ ]

P−Δ ′ ′ ρ ′ ′ v ( ).S[ ]

P

Neglect

Approximation

€

H**

[v*** − v

** + ′ ′ v ]P

€

Δ ′ ′ ρ ′ ′ v ( ).S[ ]P

€

′ ′ v P

= −DP

**∇ ′ ′ P ( )

P

Approximation Equation

€

VCρ

δt′ ′ P P

+ Δ CρU*** ′ ′ P [ ]

P−Δ ρ*D

P

** ∇ ′ ′ P ( ).S[ ]P

=−ρ

P

* − ρP

o( )δt

V −Δ ρ*U***[ ]P

collocated variable arrangementRhie-Chow Interpolation

treatment leadsto variety of schemes

neglected

SIMPLESIMPLECSIMPLERSIMPLESTSIMPLE-MPISO

Problem 1- Staggered Grid

• Use the SIMPLE procedure to compute p2, uB, and uC from the following data:

• As an initial guess, set

1 2 3uB

uC