2-d conservation of momentum (contd.)

CIS/ME 794Y A Case Study in Computational Science & Engineering

2-D conservation of momentum (contd.)

xx

P

y

uv

x

uu

t

u:directionx

yx

yy

P

y

vv

x

vu

t

v

:directiony

xy

Or, in cartesian tensor notation,

jxij

ixP

jxiu

jutiu

Where repeated subscripts imply Einstein’s summation convention, i.e.,

22

11

jj x

)(u

x

)(u

y

)(v

x

)(u

x

)(u


i

j

j

i

k

kijij x

u

x

u

x

u

3

2

Conservation of momentum (contd.):

The shear stress ij is related to the rate of strain (i.e., spatial derivatives of velocity components) via the following constitutive equation (which holds for Newtonian fluids), where is called the coefficient of dynamic viscosity (a measure of internal friction within a fluid):

Deduction of this constitutive equation is beyond the scope of this class. Substituting for ij in the momentum conservation equations yields:

ix

ju

jxiu

kxku

3

2

ixixP

jxiu

jutiu


Navier-Stokes equations for 2-D, compressible flow

The conservation of mass and momentum equations for a Newtonian fluid are known as the Navier-Stokes equations. In 2-D, they are:

0y

)v(x

)u(t

y

u

x

v

yx

u2

y

v

x

u

3

2

xx

P

y

uv

x

uu

t

u:directionx

x

v

y

u

xy

v2

y

v

x

u

3

2

yy

P

y

vv

x

vu

t

v:directiony


Navier-Stokes equations for 2-D, compressible flow in Conservative Form

The Navier-Stokes equations can be re-written using the chain-rule for differentiation and the conservation of mass equation, as:

0y

)v(

x

)u(

t

y

u

x

v

yx

u2

y

v

x

u

3

2

xx

P

y

uv

x

u

t

u:directionx2

x

v

y

u

xy

v2

y

v

x

u

3

2

yy

P

y

v

x

uv

t

v:directiony

2

(1)

(2)

(3)


Conservation of energy and species

The additional governing equations for conservation of energy and species are:

222

222

2222

x

v

y

u

y

v

x

u2

y

v

x

u

3

2

y

Tk

yx

Tk

xvP

2

v

2

uRT

2

3

y

uP2

v

2

uRT

2

3

x2

v

2

uRT

2

3

t

:EnergyofonConservati

ii

ii

iiii n

y

nD

yx

nD

xvn

yun

xt

n

(4)

(5)


Summary for 2-D compressible flow

UNKNOWNS: , u, v, T, P, ni N+5, for N speciesEQUATIONS:• Navier-Stokes equations (3 equations: conservation

of mass and conservation of momentum in x and y directions)

• Conservation of Energy (1 equation)• Conservation of Species ((N-1) equations for n

species)• Ideal gas equation of state (1 equation)

• Definition of density: (1 equation)

N

1iiinm


Extension of LBI method to 2-D flows

• Non-dimensionalize the 2-D governing equations exactly as we did the quasi 1-D governing equations.

• Take geometry into account. For example,

CenterBody

Outer Body


• Let ri(x) represent the inner boundary, where x is measured along the flow direction.

• Let ro(x) represent the outer boundary, where x is along the flow direction.

ri(x)ro(x)


• The real domain

is then transformed into a rectangular computational domain, using coordinate transformation:

x

y or r


• The coordinate transformation is given by:

• The governing equations are then transformed:

)x(r)x(r

)x(ry,x

io

i

2io

ioi

iio

rr

dx

dr

dx

drry

dx

drrr

xxx


Or,

and

etc.

dx

dr

dx

dr

rrdx

dr

rr

1xxx

io

io

i

io

io rr

1

yyy

2

2

2io

2

2

rr

1

yyy


This will result in a PDE with and as the independent variables; for example,

Recall that for quasi 1-D flow, we had equations of the form

)(D)(Dt

)(Dt x


Applying the LBI method yielded:

or,

2D

t

n1n

x

n1n

n

n1n

xn1n

2D)t(

nx

n1nx D)t(D

2

)t(I


Applying the same procedure to our transformed 2-D problem would yield:

Recall that after linearization of the quasi 1-D problem, the resulting matrix system was:

nn1n DD)t(D2

)t(D

2

)t(I

1Nx

nN

n3

n2

n1

1x)2N(

1N

N

2

1

0

)2N(Nx

NNN

333

222

111

F

F

F

F

ADB000

0

0ADB0

00ADB

000ADB


Now, in 2-D, the linearization procedure will result in:

NN

1N

222

11

GF

H

0

HGF

HG

Where each Fi, Gi, Hi are themselves block tri-diagonal systems as in the quasi 1-D problem. In other words,

etc.

M,gM,g

1M,g

2g2g2g

1g1g

1

DB

A

0

ADB

AD

G


• At this point, we have a choice:– We can solve the full system as is, i.e. an

(M+N)x(M+N) linear sparse system.

Or– We can split the operator and apply the

Alternating Direction Implicit (ADI) method to reduce the 2-D operator to a product of 1-D operators in each of the coordinate directions and solve each alternately, at each time step.


Douglas-Gunn ADI scheme

• Recall that after Crank-Nicolson differencing in time, linearization, and discretization of the spatial derivatives, we have:

• Split the operator by implicit factorization, approximate to either order (t) or (t)2 as in the original discretization errors.

nn1n DD)t(D2

)t(D

2

)t(I

D2

)t(D

2

)t(I

D2

)t(ID

2

)t(ID

2

)t(D

2

)t(I


• Note that:

• Thus, operator splitting yields:

• Defining , we have:

2)t(OD

2

)t(D

2

)t(ID

2

)t(ID

2

)t(I

nnn1nRDD)t(D

2

)t(ID

2

)t(I

n1nD

2

)t(I

nnRDD)t(D

2

)t(I


• Note that now, the solution of

and

is identical to solving two equivalent quasi 1-D problems in each of the coordinate directions and .

• The Douglas-Gunn ADI scheme after implicit factorization can be shown to be unconditionally stable in 3-D as well as long as the convective term

is absent, but is conditionally stable with the convective term present.

n1nD

2

)t(I

nRD

2

)t(I

VV


• The conditional stability of this scheme worsens and may vanish if there are periodic boundary conditions.

• A virtue of the Douglas-Gunn ADI approach is that the same boundary conditions used for

can be be used for . This is a result of consistent splitting of the operator. Other operator splitting schemes exist that are inconsistent - the same BCs used for

cannot be used for .

n1n

n1n


• In the present case study problem, our governing equations are of the form:

• Applying the LBI method to this equation yields:

• The Douglas-Gunn operator splitting then yields:

0y

G

x

F

t

)(H

nn1nRD

2

)t(D

2

)t(A

nn1n1RD

2

)t(AAD

2

)t(A


• Still, no matrix inversion is required.

• The solution procedure can be implemented as follows:

– Set

solve for in “xinv”

– Next, solve

for n+1 - n in “rinv”

n1n1D

2

)t(AA

"xset"

n

"xset"

RD2

)t(A

"rset"n1n

"rset"

AD2

)t(A


2-D LBI Code with Douglas-Gunn operator splitting


Sample solutions for 2-D LBI (x-y geometry)


Key features

• 10,000 time steps at t = 10-4 (non-dimensional)• then for 100,000 time steps at t = 2x10-3 (non-

dimensional).– Lref = 1 cm.; Pref = 1.013 x 105 Pa; Tref = 300 K

• 260 x 50 grid (x * r)• Artificial Dissipation: or 2

2

x

2

2

r

)r(1.0exp,r

)x(T2

10x25.02

exp,x

x

2T10x25.0

3imp,x

x

)r(2

103

imp,r

2-d conservation of momentum (contd.)

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