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CIS/ME 794Y A Case Study in Computational Science & Engineering
2-D Conservation of Mass
u
dx
dy
dxux
u
v
dyvy
v
vectorvelocity);v,u(V
(x,y)
)1)(dy)(dx(t
)1)(dx(dyvy
v
)1)(dy(dxux
u
)1)(dx(v)1)(dy(ustoredmoutminm
0y
)v(x
)u(t
CIS/ME 794Y A Case Study in Computational Science & Engineering
2-D Conservation of Momentum
bodyFsurfaceF)z)(y)(x(dtVd
cesignificanphysicalitsofbecause
derivativeMaterialortialtanSubsthecalledisDt
DDt
VDyVv
xVu
tV
dtVd
vdtdyandu
dtdx,But
dtdy
yV
dtdx
xV
tV
dtVd),t,y,x(VVSince
CIS/ME 794Y A Case Study in Computational Science & Engineering
2-D Conservation of Momentum (contd.):Surface forces
dx
(x,y)
dy
)1)(dx(2
dyyP
P
)1)(dx(2
dyyP
P
)1)(dy(2
dxxP
P
)1)(dy(
2
dxxP
P
xy(dy)(1)
yx(dx)(1)
)1)(dy(dxxxy
xy
)1)(dx(dy
yyx
yx
CIS/ME 794Y A Case Study in Computational Science & Engineering
2-D Conservation of Momentum (contd.)
)1)(dx(dyx
yxyx)1)(dx(yx)1)(dy(dx
x
PP)1)(dy(P
xsurfaceF
Similarly,
)1)(dy(dxx
xyxy)1)(dy(xy)1)(dx(dy
y
PP)1)(dx(P
ysurfaceF
The body forces are expressed as:
)1)(dy)(dx(fF vbody
where is the body force per unit volume. For example,
vf
Bjforgf vv
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,
ixij
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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)
CIS/ME 794Y A Case Study in Computational Science & Engineering
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)
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
Recapitulation
So far, Formulation of case study:quasi 1-D compressible flow Numerical solution techniques
Steady vs. Time-marching to steady state Finite differences (FD). Time marching to steady state
(a) Explicit schemes (McCormack, FTBS) Easier to program Restricted to small t for stiff problems May not yield a solution at all for really stiff
systems.
CIS/ME 794Y A Case Study in Computational Science & Engineering
Recapitulation (contd.)
(b)Implicit schemes (LBI): Harder to program Allows use of larger t even for stiff problems May be the only way to find a solution for really
stiff systems
Finite elements (FE), time marching to steady state
(a)Linearization same as for LBI FD method
(b)well-suited for complex geometries.
CIS/ME 794Y A Case Study in Computational Science & Engineering
Recapitulation (contd.)
Both FD and FE techniques ultimately require solution of linear equations Mx = f
In the LBI method, M is a block tri-diagonal matrix
Solution of systems such as Mx = f using PETSc allows you to explore parallel solution vs. serial solution. implications for performance Iterative methods (ex. Conjugate gradient) are well-
suited to parallelization.
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
• 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)
CIS/ME 794Y A Case Study in Computational Science & Engineering
• The real domain
is then transformed into a rectangular computational domain, using coordinate transformation:
x
y or r
CIS/ME 794Y A Case Study in Computational Science & Engineering
• 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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
Applying the LBI method yielded:
or,
2D
t
n1n
x
n1n
n
n1n
xn1n
2D)t(
nx
n1nx D)t(D
2
)t(I
CIS/ME 794Y A Case Study in Computational Science & Engineering
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
CIS/ME 794Y A Case Study in Computational Science & Engineering
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.
N,gN,g
1N,g
2g2g2g
1g1g
1
DB
A
0
ADB
AD
G
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