computational fluid dynamics cfd · 2015. 4. 1. · computational fluid dynamics cfd 1. something...
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Computational Fluid Dynamics
CFD
1
Something to discuss:
Why do we need CFD?
Are there alternative ways?
Two views on CFD
”We computer advocates don’t
think wind tunnels will become
useless. They will always be
great places to store computer
printout in” D.R Chapman,
1991
”CFD is like a dog walking on its hind
legs; it’s doing it badly, but it’s
amazing that it can do it at all” Anon
ExamplesDiesel engine simulation
- Scania D12 mod. optical engine
- 2500bar injection pressure
- 6bar IMPEg (20% load)
- N-heptane fuel
- 45°, 90° and 135° inter-jet angle
- 2.18 Swirl ratio
- 15.1:1 Compression ratio
Model assumptions:
- Primary atomization not modeled
- Particle size distribution
- Constant wall temperature
- Top-hat rail-pressure profile
- Valve motions not included
ExamplesDiesel engine simulation
Sample Results
Instantaneous axial velocity field
Traces from blades
Wake downstream
the tower
H
Staggered configuration
DY=0 D
DY=0.36 D
DY=0.71 D
DY=1.43 D
Staggered configuration
Mean velocity Rms of fluctuation
Relation to Hemodynamics
• Blood flow rate (Re)
• Wall shear-stresses- Magnitude
- Spatial- and temporal-fluctuations
– Basically local in character
– Intermediate arteries i.e. Re-dependent
– Near bifurcations
Velocity Field
WSS-Magnitude
Closed building
•4 wind directions
•60 m/s
•k-e RNG
•770000 cells
•0.5 m node distance close to
surface
•Domain size 1000X1000X500
m
Closed building
Path lines
0 deg. 90 deg.45 deg.
Open building
•2 blade configurations
•4 m/s
•k-e RNG
•800000 cells
•0.5 m node distance close to
surface
•Domain size 1000X1000X500 m
Open building
Case 1 Case 2
Axial Compressor
Aeroelastic simulation of compressorblades using ANSYS 12
Axial Compressor
Mean Mach number
Axial Compressor
Aeroelastic simulation of compressorblades using ANSYS 12
Axial Compressor
Deformation of the blade tip
0i
i
x
u
i
j
ij
j
i
jij
jiiF
xx
u
xx
p
x
uu
t
u
1
Models for
turbulence,
combustion etc.
Geometry
Mathematical
description Results
For example speed, pressure, temperature Numerical
methods
Governing equations
and boundary
conditions
Discretisation,
choice of grid
System of
algebraic
equations
Equation
system solver
Approximate
solution
Mathematical
description of
physical ”reality”
FV, FD, FE?
All these steps introduces errors!
How can we guarantee that the approximate
solution is close to the exact one and close to
”reality”
Governing equations
The governing equations will depend on what assumptions can
be made regarding the flow. For example is it incompressible or
compressible?
The flow situation will determine the character of the system of
equations. This will in turn influence the choice of numerical
method.
Discretisation and grid
Questions:
•How complex is the geometry?
•What accuracy is required? Grid quality?
•What about stability?
•Grid refinement?
Examples
Prismatic airfoil
ExamplesPrismatic airfoil
Inviscid flow
Grid refinement
Coarse grid Refinement 1 Refinement 2
Coarse: Cdp=0.0500
R1: Cdp=0.0518
R2: Cdp=0.0530
ExamplesPrismatic airfoil
Inviscid vs viscous
Inviscid
Cdp=0.0500
Viscous
Cdp=0.0500
Cd,tot=0.0539
ExamplesPrismatic airfoil
Wall refinement
coarserefined
Coarse: Cdp=0.0500, Cd,tot=0.0539
Refined:Cdp=0.0531, Cd,tot=0.0569
ExamplesPrismatic airfoil
Wall refinement
Converged: Cdp=0.0500, Cd,tot=0.0539
Non-converged:Cdp=0.0495, Cd,tot=0.0533
Residual 10-8 Residual 10-3
System of equation solver
Questions:
•Speed of the computation?
•What accuracy is required?
•What about stability?
Approximate solution
Questions:
•Is the solution physically reasonable? Conservation?
•How to determine the accuracy of the solution?
Simulation Measurement
Governing
equations
Governing equationsDifferentialrelationer
System of equations:
VfVppVVTkqDt
De
pfDt
VD
0
V
t
Mass
Momentum
Energy
Governing equations
System of equations:
ii
i
i
i
ijj
iii
i fux
pu
x
u
x
Tk
xt
q
x
Eu
t
E
j
ij
j
i
j
jii
xx
pf
x
uu
t
u
0
i
i
x
u
t
Mass
Momentum
Energy
Governing equations
ii
i
i
i
ijj
iii
i fux
pu
x
u
x
Tk
xt
q
x
Eu
t
E
j
ij
j
i
j
ij
i
xx
pf
x
uu
t
u
0
i
i
i
ix
u
xu
t
Mass
Momentum
Energy
Non-conserved forms
Governing equations
Conserved form
ii
i
i
i
ijj
iii
i fux
pu
x
u
x
Tk
xt
q
x
Eu
t
E
j
ij
j
i
j
jii
xx
pf
x
uu
t
u
0
i
i
x
u
t
Mass
Momentum
Energy
Governing equationsConserved form
i
j
iji Jx
F
t
U
Mass
Momentum
Energy
Jx
F
t
U
i
i
Jx
F
t
U
j
j
E
uU i
jkk
j
jj
ijijji
i
ux
TkpuEu
puu
u
F
t
qfu
fJ
kk
i
0
Classification of
PDEs
Classification of PDEs
A comment on characteristic curves
Characteristic curves are curves along which signals
are propagated.
Example, an initial value problem.
)()0,(
0
0 xuxu
x
uc
t
u
By using the chain rule one
can find:
cdt
du
dt
dxby defined lines along 0
or,
ctxu on constant
The ”constancy of u” is the
signal carried along the
characteristic curves
Solution: )(, 0 ctxutxu
x
t
Charateristic curves
Classification of PDEs
22222
11111
fy
vd
x
vc
y
ub
x
ua
fy
vd
x
vc
y
ub
x
ua
P
dx
dy
Differentials:
dyy
vdx
x
vdv
dyy
udx
x
udu
Classification of PDEs
dv
du
f
f
y
vx
v
y
ux
u
dydx
dydx
dcba
dcba
A
2
1
2222
1111
00
00
dydx
dydx
dcbf
dcbf
B
00
00
2222
1111
A
B
x
u
Characteristic curves correspond to 0A
21221122112212
1221 dxdbdbdxdycbcbdadadycacaA
Classification of PDEs
21221122112212
1221 dxdbdbdxdycbcbdadadycacaA
0122112211221
2
1221
dbdb
dx
dycbcbdada
dx
dycaca
0
2
c
dx
dyb
dx
dya
a
acbb
dx
dy
2
42
Three situations:
04
04
04
2
2
2
acb
acb
acb Hyperbolic, two real characteristics
Parabolic, one real characteristic
Elliptic, no real characteristic
Classification of PDEs
Example: 2D inviscid steady flow of a compressible gas under small perturbations.
0''
0''
1 2
v
v
y
u
y
v
x
uMa
'
'
vv
uUu
0 ;1 ;1 ;0
1 ;0 ;1
2222
1112
1
dcba
dcbMaa
Characteristic equation:
01221121221
2
1221
cba
dbdbdx
dycbdada
dx
dycaca
Classification of PDEs
Characteristic equation:
01221121221
2
1221
cba
dbdbdx
dycbdada
dx
dycaca
1
0
1 2
c
b
Maa 011
22
dx
dyMa
12
14
1
12
2
2
Ma
Ma
Madx
dyCompare:
a
acbb
dx
dy
2
42
hyperbolic 1
elliptic 1
2
2
Ma
Ma
Classification of PDEs
Hyperbolic
PDomain of
dependence Region of influence
Charateristic lines
y
xExamples: Inviscid supersonic flow, unsteady inviscid flow
Classification of PDEs
Parabolic
PDomain of
dependence Region of influence
y
x
Known boundary conditions
Known boundary conditions
Examples: Steady boundary layer flow, unsteady heat conduction
Classification of PDEs
Elliptic
P
y
x
Every point influences all
other points
Examples: Steady subsonic inviscid flow, incompressible inviscid flow