computational methods for the euler equations · 2020. 12. 30. · euler cfd 3-d, compressible (no...
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Computational Methods for the Euler Equations
Before discussing the Euler Equations and computational methods for them, let’s look at what we’ve learned so far: Method Assumptions/Flow type 2-D panel 2-D, Incompressible, Irrotational Inviscid Vortex lattice 3-D, Incompressible, Irrotational Inviscid, Small disturbance Potential method 3-D, Subsonic compressible, Irrotational, Inviscid, Prandtl-Glauert Small disturbance Euler CFD 3-D, Compressible (no ∞M limit), Rotational, Shocks, Inviscid The only major effect missing after this week will be viscous-related effects.
2-D Euler Equations in Integral Form Consider an arbitrary area (i.e. a fixed control volume) through which flows a compressible inviscid flow: ≡nv outward pointing normal (unit length) ≡dS elemental (differential) surface length
jdxidydSnvvv −=
Note: Path around surface is taken so that interior of control volume is on left.
y
x c
dS
Cδ
nv
dSnv
dy
dx
dS
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Conservation of Mass
Cinmassof
dAdtdchangeofrate
dACinMass
Cofoutflowmassofrate
Cinmassofchangeofrate
C
C
∫∫
∫∫=⇒
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛
ρ
ρ
0
where fluidofdestiny≡ρ
Now, the rate of mass flowing out of C : Mass flow out of ∫ =•=
CudSnuC
δρ vvv velocity vector
Conservation of x-momentum Recall that: total rate of change momentum ∑= forces For x-momentum this gives:
∑=⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛− Cofout
momflowxofrateCinmomentumx
ofchangeofrate Forces in x-direction
∫∫∫ ∑=•+CC
dSnuuudAdtd
δρρ vv Forces in x-direction
Now, looking closer at x-forces, for an inviscid compressible flow we only have pressure (ignoring gravity). Recall pressure acts normal to the surface
∫ •−=∑⇒C
dSinpxinsForceδ
vv
∫∫∫ =•+CC
dSnudAdtd
δρρ 0vv ⇒
dS
dSnpv−
Into surface
Normal to surface
Gives x-direction
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Conservation of y-momentum This follows exactly the same as the x-momentum: Conservation of Energy Recalling your thermodynamics:
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛Cto
addedheatCinfluid
ondoneworkCinenergyof
changeofratetotal
For the Euler equations, we ignore the possibility of heat addition.
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛Cofoutflow
energyofrateCinenergy
ofchangeofrateCinenergyof
changeofratetotal
The total energy of the fluid is:
)(21 22 vueE ++= ρρρ
Note: Tce v= where ≡vc specific heat at constant volume Static temperature So,
∫∫ ∫ •+=⎟⎟⎠
⎞⎜⎜⎝
⎛
C C
dSnuEEdAdtd
Cinenergyofchangeofratetotal
δ
ρρ vv
The work done on the fluid is through pressure forces and is equal to the pressure forces multiplied by (i.e. acting in) the velocity direction: ( ) ( )∫ •−=
C
dSunpworkδ
vv
∫∫ ∫ ∫ •−=•+C C C
dSinpdSnuuudAdtd
δ δ
ρρvvvv
∫∫ ∫ ∫ •−=•+C C C
dSjnpdSnuvvdAdtd
δ δ
ρρvvvv
Total energy
Internal energy Kinetic
energy
Pressure force
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⇒ Summary of 2-D Euler Equations
∫∫ ∫ ∫
∫∫ ∫ ∫
∫∫ ∫ ∫
∫∫ ∫
•−=•+
•−=•+
•−=•+
=•+
C C C
C C C
C C C
C C
dSnnpdSnuEEdAdtd
dSjnpdSnuvvdAdtd
dSinpdSnuuudAdtd
dSnudAdtd
δ δ
δ δ
δ δ
δ
ρρ
ρρ
ρρ
ρρ
vvvv
vvvv
vvvv
vv 0
These are often written very compactly as:
( )∫∫ ∫ =•++c sc
dsnjGiFUdAdtd 0vvv
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
≡
Evu
U
ρρρρ
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+≡
uHuv
puu
F
ρρρ
ρ2
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+≡
vHpv
vuv
G
ρρ
ρρ
2
ρpEenthalpytotalH +≡≡
Ideal gas: ⎥⎦⎤
⎢⎣⎡ +−−== )(
21)1( 22 vuERTp ρργρ
A Finite Volume Scheme for the 2-D Euler Eqns. Here’s the basic idea: (1) Divide up (i.e. discretize) the domain into simple geometric shapes (triangles
and quads)
∫∫ ∫ ∫ •−=•+C C C
dSunpdSnuEEdAdtd
δ δ
ρρ vvvv
Conservative state vector
Flux vector for x-direction
Flux vector for y-direction
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Looking at this small region: Cell 0 is surrounded by cells 1, 2, & 3. i.e. cell 0 has 3 neighbors: cell 1, 2, & 3. Nearest neighbors
9
0
10 8
2 17
6 5
4 3
1112
13
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(2) Decide how to place the unknowns in the grid. (a) Cell-centered: cell-average values of the conservative state vector are stored for each cell. (b) Node-based: point values of the conservative state vector are stored at each node. The debate still rages about which of these options is best. We will look at cell- centered schemes because these are easiest (although not necessarily the best). Also, they are very widely used in the aerospace industry. (3) Approximate the 2-D integral Euler equation on the grid to determine the chosen unknowns.
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Computational Methods for the Euler Equations
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Let’s look in detail at step (3): Cell-average unknowns:
( )( )( ) ⎥
⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0
0
0
0
0
Evu
U
ρρρρ
( )( )( ) ⎥
⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1
1
1
1
1
Evu
U
ρρρρ
.............2 =U ...........3 =U
Specifically, we define 0U as:
∫∫≡00
01
CUdA
AU where
⎩⎨⎧
≡≡
00
0
0
cellofareaAcellC
Now, we apply conservation eqns:
( )∫∫ ∫ =•++0 0
0C C
dSnjGiFUdAdtd
δ
vvv
The time-derivative term can be simplified a little:
∫∫ =0
00C dt
dUAUdA
dtd
The surface flux integral can also be simplified a little: ( ) ( )
( )
( )∫∫
∫∫
•++
•++
•+=•+
a
c
c
b
b
aC
dSnjGiF
dSnjGiF
dSnjGiFdsnjGiF
vvv
vvv
vvvvvv
0δ
f b e
a c
d
12
0
3
Cells: 0,1, 2, 3 Nodes: a, b, c, d, e, f
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Combining these expressions: Now, we make some approximations. Let’s look at the surface integral from
ba → :
( )∫ •+b
adsnjGiF vvv
The normal can easily be calculated since the face is a straight line between nodes a & b. Recall, the unknowns are stored at all centers. So, what would be a logical approximation for : ( )∫ =•+
b
a abdSnjGiF ???vvv
Option #1= Option #2= Note: Option #1 ≠ option #2 in general. There is very little difference in practice between these options. Let’s stick with:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) caca
a
c caca
bcbc
c
b bcbc
ababab
b
aab
snjGGiFFdSnjGiF
SnjGGiFFdSnjGiF
SnjGGiFFdSnjGiF
∆•⎥⎦⎤
⎢⎣⎡ +++=•+≡ℑ
∆•⎥⎦⎤
⎢⎣⎡ +++=•+≡ℑ
∆•⎥⎦⎤
⎢⎣⎡ +++=•+≡ℑ
∫
∫
∫
vvvvv
vvvvvv
vvvvvv
3030
2020
1010
21
21
21
21
21
21
Where
( )( )( )( )33
22
11
00
UFFUFFUFFUFF
≡≡≡≡
( )( )( )( )33
22
11
00
UGGUGGUGGUGG
≡≡≡≡
( ) ( )
( )∫
∫ ∫=•++
•++•++
a
c
b
a
c
b
dSnjGiF
dSnjGiFdSnjGiFdt
dUA
0
00
vvv
vvvvvv
b
ac2 1 0 3
abnv
No approximations so far!
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Finally, we have to approximate dt
dUA 00 somehow. The simplest approach is
forward Euler:
000 =ℑ+ℑ+ℑ+ cabcabdt
dUA
And n
abℑ etc. are defined as:
( ) ( )( )( )nn
nn
ababnnnnn
ab
UFF
etcUFF
SnjGGiFF
11
00
1010
.21
21
≡
≡
∆•⎥⎦⎤
⎢⎣⎡ +++≡ℑ vvv
For steady solution, basic procedure is to make a guess of U at 0=t and then iterate until the solution no longer changes. This is called time marching. Question What assumptions have we made in developing our 2-D Euler Equation Finite Volume Method?
001
00 =ℑ+ℑ+ℑ+
∆−+
nca
nbc
nab
nn
tUU
A
Where ( )no
no tUU ≡ and iterationntntn ≡∆≡ ,