1

5. Pressure and Velocity

Part 1. Pressure-Velocity CouplingPart 2. Pressure-Correction Methods

Scalar-Transport Equation For Momentum

concentration, velocity (u,v,w)

diffusivity, viscosity,

source, S non-viscous forces

Each component satisfies its own scalar-transport equation:

sourcediffusionadvectionchangeofrate

SAn

fluxmassmasst faces

)Γ()(

d

d

2

Special Features of the Momentum Equations

The momentum equations are:

– non-linear

– coupled

– required also to be mass-consistent

As a result they must be solved:

– iteratively

– together

– in conjunction with the continuity equation

And we also need to specify pressure …

(mass flux) velocity → (ρuA)u

(ρuA)v

uv

mass flux

uA

Solving Mass and Momentum Equations

DO WHILE (not_converged)

CALL SCALAR_TRANSPORT( u )

CALL SCALAR_TRANSPORT( v )

CALL SCALAR_TRANSPORT( w )

CALL MASS_CONSERVATION

END DO

Mass Equation

Scalar-Transport Equation

0)()(d

d

faces

fluxmassmasst

SAn

fluxmassmasst faces

)Γ()(

d

d

For incompressible flow, the mass equation could be regarded

as a special case of the scalar-transport equation ... but only if:

1 !!!

Γ 0

S 0

For compressible flow, the mass equation is a transport

equation for density

V

A

u

un

3

How is Pressure Determined?

Compressible flow:

– mass conservation transport equation for density,

– transport equation for energy temperature, T

– equation of state (e.g. ideal gas law p = RT) pressure, p

Incompressible flow:

– the momentum equations link velocity and pressure

– substitution in the mass equation yields an equation for pressure

A pressure equation arises from the requirement that

solutions of the momentum equation be mass-consistent.

Solving Mass and Momentum Equations

DO WHILE (not_converged)

CALL SCALAR_TRANSPORT( u )

CALL SCALAR_TRANSPORT( v )

CALL SCALAR_TRANSPORT( w )

CALL MASS_CONSERVATION( p )

END DO

Pressure-Velocity Coupling

Q1. How are velocity and pressure linked?

Q2. How does a pressure equation arise?

Q3. Should velocity and pressure be stored

at the same positions?

4

Q1. How are Pressure and Velocity Linked?

Momentum equation:

forcesotherppAuaua

forcepressure

ew

fluxnet

FFPP )(

Velocity-pressure linkage:

)( ewPP ppduP

pa

Ad

A1. (a) The force terms in the momentum equation provide a link between velocity

and pressure.

(b) Velocity depends on the pressure gradient or, when discretised, on the

difference between pressure values ½ cell either side.

pdu Δ

uPp p

w e

area A

Q2. How Does a Pressure Equation Arise?

pdu Δ

The momentum equation links velocity and pressure:

Substituting in the mass equation gives an equation for pressure:

EEppWW

PWwEPe

we

papapa

ppAdppAd

uAuA

)()ρ()()ρ(

)ρ()ρ(0

A2. A pressure equation arises from the requirement that solutions of the

momentum equation be mass-consistent.

EW

N

S

P

n

e

s

w

Q3. Should velocity and pressure be stored at

the same positions?

5

Co-located Pressure and Velocity(i) Effect in the Momentum Equation

)(2

)( 11 iiew ppA

ppANet pressure force: No pi !!!

)( 121

iiw ppp )( 121

iie ppp

w eii-1 i+1

Co-located Pressure and Velocity(ii) Effect in the Continuity Equation

Continuity:

)}()({ρ

}{ρ

)}()({ρ)ρ()ρ(0

2121

2121

21

1121

121

121

iiiiii

ii

iiiiwe

ppdppdA

uuA

uuuuAAuAu

)( 121

iiw uuu )( 12

1 iie uuu

Involves alternate p values only

w eii-1 i+1i-2 i+2

Pressure/Velocity Coupling

Momentum & continuity pressure equation

Remedies:

(1) staggered grid, or

(2) Rhie-Chow interpolation for advective velocities

co-located u, p and

linear interpolation for advective velocities

but the combination of

odd-even decoupling

(“checkerboard” effect)}

x

p

1 2 3 4 5 6 7 8 9 10

6

Staggered Grid(Harlow and Welch, 1965)

u p

v

u p

v

p p

p

pressure/scalarcontrol volume

u control volume

v control volume

Staggered Grid: Advantages

On a Cartesian mesh:

Pressure is stored at the points required to compute pressure forces

)( 1 iiii ppdu

Velocity is stored at the points required to compute mass fluxes

1111

1111

)(

)()(0

iiiiiii

iiiiiiii

pdpddpd

ppdppduu

up pi-1 i i

pp pi-1 i i+1ui ui+1

Staggered Grid: Disadvantages

Added geometric complexity

Rationale fails on non-Cartesian grids

Impossible to implement in unstructured meshes

7

Non-Staggered/Collocated Grid(Rhie and Chow, 1983)

Momentum equation:

pduu Δˆ

Rhie-Chow interpolation: apply this symbolic relation at both nodes and faces

pduu facefaceface Δˆ

)( weFFPP ppAuaua

pduu Δˆ Work out pseudovelocity at nodes

Then interpolate to faces

)()Δ( PEeee ppdpduu

)( weP

P

FF

P ppda

uau

w eW P E

Symbolically:

Mass Conservation → Pressure Equation

pduu Δˆ

EEPPWWwe

PWwEPewe

we

papapauAuA

ppAdppAduAuA

AuAu

)ˆρ()ˆρ(

)()ρ()()ρ()ˆρ()ˆρ(

)ρ()ρ(0

In practice, one solves for a pressure correction:

EEPPWWwe papapaAuAu *)ρ(*)ρ(0

outflowmasscurrentpapapa EEPPWW

w eW P E

Example

For the uniform Cartesian mesh shown below the momentum equation gives a

velocity/pressure relationship

For the u and p values given, calculate the advective velocity on the cell face f :

(a) by linear interpolation;

(b) by Rhie-Chow interpolation if pA = 0.6;

(c) by Rhie-Chow interpolation if pA = 0.9.

pu Δ4

u =

p =

f

A

1 2 3 3

p 0.8 0.7 0.6

8

Looking Ahead ...

Pressure/velocity coupling is the dominant feature in

incompressible flow

Mass and momentum equations must be satisfied

simultaneously

The most popular type of solution algorithm is called a

pressure-correction method

Part 1. Pressure-Velocity Coupling

Part 2. Pressure-Correction Methods

Correcting Pressure to Enforce Mass Conservation

Net mass flux in;

increase cell pressure

Net mass flux out;

decrease cell pressure

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Pressure-Correction Methods

Iterative numerical schemes for pressure-linked equations

Seek to produce velocity and pressure fields satisfying both

mass and momentum equations

Consist of alternating updates of velocity and pressure:

– solve momentum equation with current pressure

– use the velocity-pressure link to rephrase continuity as a pressure-correction

equation

– solve for pressure corrections to “nudge” velocity towards mass conservation

Popular methods:

– SIMPLE

– PISO

Velocity and Pressure Corrections

The momentum equation links velocity and pressure: )( 2/12/1 ppdu

Must simultaneously correct pressure to

retain a solution of the momentum equation: )( 2/12/1 ppdu

Must correct velocity to satisfy continuity: uuu *

)( 2/12/1

*

ppduuVelocity correction formula:

Example

In the steady, one-dimensional, constant-density situation shown, the pressure p

is stored at locations 1, 2 and 3, whilst velocity u is stored at the staggered

locations A, B and C. The velocity-correction formula is

uuu *

where pressure nodes i–1 and i lie on either side of the location for u. The value

of d is 2 everywhere. The boundary condition is uA=10. If, at a given stage in the

iteration process, the momentum equations give u*B = 8 and u*C = 11, calculate

the values of p1, p2, p3 and the resulting corrected velocities.

)( 1 ii ppdu where

A 1 B 2 C 3

10

Comments

The continuity equation doesn’t explicitly contain

pressure ... but constraints imposed by the

momentum equation lead to a pressure equation

Matrix equations for pressure are similar to those for

the scalar-transport equation

The source term is minus the current mass outflow

(In incompressible flow) pressure is fixed only up to a

constant

ExampleIn the 2-d staggered-grid arrangement shown below, u and v (the x and y components of velocity),

are stored at nodes indicated by arrows, whilst pressure p is stored at the intermediate nodes A-D.

The grid spacing is uniform and the same in both directions. Velocity is fixed on the boundaries as

shown. The velocity components at the interior nodes (uB, uD, vC and vD) are to be found.

At an intermediate stage of calculation the internal velocity values are found to be

uB = 11, uD = 14, vC = 8, vD = 5

whilst correction formulae derived from the momentum equation are

with geographical (w,e,s,n) notation indicating the relative location of pressure nodes.

Show that applying mass conservation to control volumes centred on pressure nodes leads to

simultaneous equations for the pressure corrections. Solve for the pressure corrections and use

them to generate a mass-consistent flow field.

)(3,)(2 nsew ppvppu

10 20

5 5

5 10

15 10

A B

D

vC vD

uD

uB

x

y

C

SIMPLESemi-Implicit Method for Pressure-Linked Equations

solve pressure-correction equation

correct velocityand pressure

converged?

START

END

yes

no

solve momentumequations

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SIMPLE

1. Solve momentum equation with

the current pressure.

2. Formulate pressure-correction equation:

(i) relate changes in u and p; )( ewP

P

FF

P ppda

uau

(iv) rewrite in terms of p.*)]()ρ()()ρ( mppAdppAd PWwEPe

3. Solve pressure-correction equation:*mpapa FFPP

4. Correct velocity and pressure:

)(*

*

ewPPP

PPP

ppduu

ppp

(iii) apply mass conservation; 0)]*(ρ[)]*(ρ[ we uuAuuA

(ii) make SIMPLE approximation;

)(Σ **

ewP

P

FFp ppd

a

uau

*)ρ()ρ( muAuA we

SIMPLE (continued)

The source of the pressure-correction equation is minus

the current mass imbalance

(Substantial) pressure under-relaxation is usually required:

Iterative process – no need to solve equations exactly at

each stage. Either:– do m iterations of velocity, n iterations of pressure; or

– do enough iterations to reduce the error by a set amount

ppp p α*

Variants of SIMPLE

SIMPLE )( ewP

P

FF

P ppda

uau

SIMPLEC: alternative correction formula:

)(/1

ew

PF

PP pp

aa

du

SIMPLER: precede momentum update with exact pressure equation:

SIMPLEX: solve equations for correction coefficients δp:

)(δ ewPP ppu

)ˆ( uofoutflownetpapa FFPP

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PISO(Pressure-Implicit with Splitting of Operators)

Time-dependent pressure-correction method

Each timestep told tnew is a non-iterative sequence:

1. solve time-dependent momentum eqns with told pressure

2. formulate and solve a pressure-correction equation and update pressure and

velocity

3. second mass-corrector step with time-advanced pressure

More efficient(?) than SIMPLE for time-dependent problems

Summary (1)

Each momentum component satisfies its own scalar-transport

equation

The momentum equations require special treatment because

they are:

– non-linear

– coupled

– required also to be mass-consistent

In incompressible flow, continuity (mass conservation) leads to

a pressure equation

Odd-even decoupling of pressure can be addressed by either:

– staggered velocity grid

– non-staggered grid, but Rhie-Chow interpolation for advective velocities

Summary (2)

Pressure-correction methods are iterative schemes for

solving mass and momentum equations simultaneously

They consist of alternating solutions of:

– the momentum equation (with pressure fixed)

– a pressure-correction equation to nudge the velocity field towards mass

conservation

Widely used pressure-correction methods are:

– SIMPLE

– PISO