compressible flows 1 - lth · issues related to compressible flows ... compressible flow. 6. ......
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Compressible Flows 1
Compressible flow
What if you have a supersonic wind tunnel and way too much time on your hands..?
3
Issues related to compressible flows
•Governing equations might change character
• Shocks• Preferential direction of propagation•Boundary conditions• Small timesteps (compared to
incompressible)
4
p
v1
Compressible flow?
• Compressibility:
• τ : property of the fluid• Water: 5x10-10 m2/N @1atm• Air: 10-5 m2/N @1atm
v2
dpdv
v1
−=τ Tdpdv
v
−=
1τ
sdpdv
v
−=
1τ
dpd τρρ =
5
Compressible flow
6
Speed of sound
• What is sound?• da, dp,dρ,dT: small
dTTddppdaa
++++
ρρT
pa
ρ
( )( )( )( )22 )( daaddppap
ddaaa
++++=+
++=
ρρρ
ρρρρd
dpa =
For a calorically perfect gas: RTa γ=
7
Limit of compressibility?( )
xu
xu
xu
xu
∂∂
≈∂∂
+∂∂
=∂
∂ ρρρρxu
xu
∂∂
<<∂∂ ρρ
i.e.
Can be written as:
VdVd
<<ρρ
VV =
ρdadp 2=
speed of sound
From Bernoulli: VdVdp ρ−=
11 22
2
22 <<⇔<<⇔<< MaV
Vdp
adp
ρρ
Mach number 3.0≤M
Usually the limit is set to:
8
Flow regimes0.3 < M < 1
Subsonic M < 1 M < 1M > 1
Transonic
M < 1 M > 1
SupersonicWhich is the most difficultto compute?
M<0.3 Incompressible
M>>1 Hypersonic
9
Examples
[www.eng.vt.edu/fluids/msc/gallery/gall.htm]
10
Examples
[www.eng.vt.edu/fluids/msc/gallery/gall.htm]
11
Examples
ExamplesPrismatic airfoil
Inviscid flowGrid refinement
Coarse grid Refinement 1 Refinement 2
Coarse: Cdp=0.0500R1: Cdp=0.0518R2: Cdp=0.0530
ExamplesPrismatic airfoilInviscid vs viscous
InviscidCdp=0.0500
ViscousCdp=0.0500Cd,tot=0.0539
ExamplesPrismatic airfoilWall refinement
coarse refined
Coarse: Cdp=0.0500, Cd,tot=0.0539Refined:Cdp=0.0531, Cd,tot=0.0569
ExamplesPrismatic airfoilWall refinement
Converged: Cdp=0.0500, Cd,tot=0.0539Non-converged:Cdp=0.0495, Cd,tot=0.0533
Residual 10-8 Residual 10-3
NACA0015 Cd=0.14
Prisma Cd=0.058
17
Characteristic parameters
• Characteristic sound speed
• Characteristic Mach number
p, ρ, T, V, M
p*, ρ∗, T*, V*, M=1
V*=V+dVdQ=0
** RTa γ=
*/* aVM =
11lim
)1(12 *
2*
2
−+
=−−
+=
∞→ γγ
γγ M
M
MM
18
Total/stagnation parameters
• Stagnation sound speed
• Total density
p, ρ, T, V, M
p0, ρ0, T0, V=0
V*=V+dVds=0
00 RTa γ=
000 / RTp=ρ
19
Isentropic flow relations
• Why?• Compute parameters
after/before shock waves
• Boundary conditions• …
1211
00
12100
20
211
211
211
−−
−−
−+=
=
−+=
=
−+=
γγ
γ
γγ
γγ
γρρ
γ
γ
MTT
MTT
pp
MTT
20
(Quasi) 1D flows
• Assume• Very thin boundary layers• Small rate of change in
area• Large curvature
x
y( )xh
( )xR
( )yxV ,
1<<dxdh
( ) ( )xRxh <<
x
y( )xh
( )xV
21
Isentropic flow with area changes
Continuity ( ) ( ) ( ) constant== mxAxVx ρ
Differentiate the continuity and momentum equations
22
21
10
0
Vdp
MAdA
VdV
dadp
VdVdpA
dAVdVd
ρρ
ρ
ρρ
−=−
=⇒
=
=+
=++
22
Isentropic flow with area changes
22 11
Vdp
MAdA
VdV
ρ−=
−=
0>dA
0<dA
1<M 1>M
00
<>
dpdV
00
<>
dpdV
00
><
dpdV
00
><
dpdV 0=dA ∞→dV
12 =M
?
23
Nozzle flow
•ConvergentAVm ρ=
maxmm = when 1=M**** VAm ρ=
Further decreasing pb will not change the mass flow since
1max =M
( )( )
( )21
00*
1121
21
21
0*
11
0***
max
12
12
12
RTA
RTAVAm
ργ
γ
γγρρ
γγ
γ
−+
−
+
=
=
+
+
==
24
Nozzle flows
•Convergent-divergent11
22
2
* 211
121 −
+
−+
+=
γ
γ
γγ
MMA
A
1211
00
12100
20
211
211
211
−−
−−
−+=
=
−+=
=
−+=
γγ
γ
γγ
γγ
ρρ
γ
γ
MkTT
MTT
pp
MTT
25
Normal shock relations
1
1
1
1
1
1
1
ˆ
Msh
pAVρ
2
2
2
2
2
2
2
ˆ
Msh
pAVρ
iu
01020102
*1
*2121212
121221 11
TTppAATTVVppssMM
=<>><>>><>
ρρ
26
Governing equations
•Are the equations different in different regimes?
RTp
VeE
ufx
ux
puxTk
xq
xEu
tE
fxx
px
uutu
xu
t
iij
iji
j
j
jjj
j
ij
ij
ij
jii
i
i
ρ
ρτ
ρρρ
ρτρρ
ρρ
=
+=
+∂∂
+∂∂
−
∂∂
∂∂
+=∂
∂+
∂∂
+∂∂
+∂∂
−=∂
∂+
∂∂
=∂∂
+∂∂
2
0
2
27
Euler equations
• Inviscid
RTp
VeE
ufx
puxTk
xq
xEu
tE
fxp
xuu
tu
xu
t
iij
j
jjj
j
iij
jii
i
i
ρ
ρρρρ
ρρρ
ρρ
=
+=
+∂∂
−
∂∂
∂∂
+=∂
∂+
∂∂
+∂∂
−=∂
∂+
∂∂
=∂∂
+∂∂
2
0
2
28
Euler equations – conserved form0=
∂∂
+∂∂
+∂∂
+∂∂
zH
yG
xF
tU
+
=
2
2Ve
wvu
U
ρ
ρρρρ
+
+
+=
puVeu
uwuv
pu
u
F
2
2
2
ρ
ρρρ
ρ
+
+
+=
pvVev
vwpv
uvv
G
2
2
2
ρ
ρρ
ρρ
+
+
+
=
pwVew
pw
vwuww
H
2
2
2
ρ
ρ
ρρρ
29
Character of the Euler equations
Subsonic Sonic Supersonic
Steady Elliptic Parabolic Hyperbolic
Unsteady Hyperbolic Hyperbolic Hyperbolic
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 conditionsExamples: 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
Flux vector splitting schemes
•Determining the correct flux at the cell faces
• FVS splits the flux contributions into a positive and a negative part
• Two examples shown here: Steger-Warming and van Leer splitting
33
Flux vector splitting schemes
34
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
+𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑
= 0
�𝛿𝛿𝑣𝑣
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
𝑑𝑑𝑑𝑑 + �𝛿𝛿𝑣𝑣
𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 +
𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 0
�𝛿𝛿𝑣𝑣
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
𝑑𝑑𝑑𝑑 + �𝑠𝑠
𝜕𝜕𝑑𝑑𝑑𝑑 − 𝜕𝜕𝑑𝑑𝜕𝜕 = 0
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
𝛿𝛿𝑑𝑑 + �𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠
𝜕𝜕∆𝑑𝑑 − 𝜕𝜕∆𝜕𝜕 = 0
The Euler equations
Integrate over a control volume δv
Use Green’s theorem
Discretise
Steger-Warming splitting
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
= 0
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+ 𝐴𝐴𝜕𝜕𝑈𝑈𝜕𝜕𝜕𝜕 = 0
𝐴𝐴 =𝜕𝜕𝜕𝜕𝜕𝜕𝑈𝑈
𝑇𝑇−1𝐴𝐴𝑇𝑇 = Λ
Consider the 1D Euler equation:
Using the Jacobian
it can be written as
The Jacobian can be reformulated using its eigenvalues and eigenvectors. Here T-1 is a matrix whose rows are the left eigenvectors of A and Λ is a diagonal matrix containing the eigenvalues of A 35
Steger-Warming splitting
36
𝜆𝜆1 = 𝑢𝑢
𝜆𝜆2 = 𝑢𝑢 + 𝑎𝑎
𝜆𝜆3 = 𝑢𝑢 − 𝑎𝑎
𝜕𝜕 = 𝐴𝐴𝑈𝑈 = 𝑇𝑇Λ 𝑇𝑇−1
𝜕𝜕 = 𝜕𝜕+ + 𝜕𝜕− = 𝐴𝐴+𝑈𝑈 + 𝐴𝐴−𝑈𝑈
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕+
𝜕𝜕𝜕𝜕 +𝜕𝜕𝜕𝜕−
𝜕𝜕𝜕𝜕 = 0
The flux can be written in terms of the Jacobian and U as
Now the flux is split into a positive and a negative contribution
Hence, we can write the 1D Euler equation as
Λ = Λ+ + Λ−Likewise, the eigenvalues can be split
The eigenvalues for the 1D case are:
Steger-Warming splitting
37
𝜕𝜕− =12𝜌𝜌𝛾𝛾𝑢𝑢 − 𝑎𝑎
1𝑢𝑢 − 𝑎𝑎
12𝑢𝑢 − 𝑎𝑎 2 +
12
3 − 𝛾𝛾𝛾𝛾 − 1
𝜕𝜕+ =12𝜌𝜌𝛾𝛾
2𝛾𝛾 − 1 𝑢𝑢 + 𝑎𝑎2 𝛾𝛾 − 1 𝑢𝑢2 + 𝑢𝑢 + 𝑎𝑎 2
𝛾𝛾 − 1 𝑢𝑢3 +12 𝑢𝑢 + 𝑎𝑎 3 +
12𝑎𝑎
2 3 − 𝛾𝛾𝛾𝛾 − 1 𝑢𝑢 + 𝑎𝑎
Λ+ =𝑢𝑢
𝑢𝑢 + 𝑎𝑎0
Λ− =0
0𝑢𝑢 − 𝑎𝑎
Λ+ = Λ Λ− = 0Assuming the flow is in the positive x-direction, For the supersonic caseAnd for the subsonic case
The fluxes in the subsonic case:
Steger-Warming splitting
38
𝑈𝑈𝑖𝑖𝑛𝑛+1 = 𝑈𝑈𝑖𝑖𝑛𝑛 −Δ𝑡𝑡Δ𝜕𝜕
𝜕𝜕𝑖𝑖+1/2 − 𝜕𝜕𝑖𝑖−1/2
𝜕𝜕𝑖𝑖+1/2 = 𝜕𝜕+ + 𝜕𝜕− 𝑖𝑖+1/2
𝜕𝜕𝑖𝑖+1/2+ = 𝜕𝜕𝑖𝑖+ 𝜕𝜕𝑖𝑖+1/2
− = 𝜕𝜕𝑖𝑖+1−
The discrete equation to solve becomes:
For a first order discretisation:
Note that ’+’ is a backward and ’-’ a forward discretisation
Steger-Warming splitting:• Performs better than central differencing techniques• Captures shocks well but oscillations will occur at sonic
conditions• Causes some problems close to stagnation points• This is due to the split fluxes nor being continously
differentiable
van Leer flux splitting
39
Addresses the problems asociated with Steger-Warming splitting at sonic transition and stagnation points.The following conditions must then be met to esure standard upwinding differences at supersonic flows:1. 𝜕𝜕 = 𝜕𝜕+ + 𝜕𝜕−
2. 𝜕𝜕𝐸𝐸+
𝜕𝜕𝑈𝑈must have all eigenvalues ≥ 0
3. 𝜕𝜕𝐸𝐸−
𝜕𝜕𝑈𝑈must have all eigenvalues ≤ 0
Restrictions in order to eliminate problems with the Steger-Warming scheme (1-3) and achieve uniqueness of the splitting (4):
1. 𝜕𝜕± must be continuous and 𝜕𝜕+ = 𝜕𝜕 𝑓𝑓𝑓𝑓𝑓𝑓 𝑀𝑀 ≥ 1
𝜕𝜕− = 𝜕𝜕 𝑓𝑓𝑓𝑓𝑓𝑓 𝑀𝑀 ≤ −12. The components must exhibit the same symmetry as the original
flux vector in terms of Mach number3. The Jacobians 𝜕𝜕𝐸𝐸
±
𝜕𝜕𝑈𝑈must be continuous and have one eigenvalue
vanish for 𝑀𝑀 < 14. 𝜕𝜕± must be a polynomial in Mach number of lowest possible order
Flux difference splittingThe Roe scheme
40
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
= 0
𝑈𝑈 𝜕𝜕, 0 = �𝑈𝑈𝐿𝐿 𝜕𝜕 < 0𝑈𝑈𝑅𝑅 𝜕𝜕 > 0
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+ �̂�𝐴𝜕𝜕𝑈𝑈𝜕𝜕𝜕𝜕 = 0
�̂�𝐴 = �̂�𝐴 𝑈𝑈𝐿𝐿,𝑈𝑈𝑅𝑅�̂�𝐴 𝑈𝑈𝐿𝐿,𝑈𝑈𝑅𝑅 → 𝐴𝐴
𝜕𝜕𝑅𝑅 − �𝜕𝜕𝑖𝑖+12
+ �𝜕𝜕𝑖𝑖+12
− 𝜕𝜕𝐿𝐿 = �̂�𝐴+ + �̂�𝐴− 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
The Roe scheme is a method to determine the flux by solving a linear problem
The Riemann problem for the fluxes in the 1D Euler equation:
Roe’s linear approximation:
Note that the Jacobian is replaced by the matrixAway from the disturbance this matrix should goto the Jacobian, i.e.
The flux difference may be written as:
Flux difference splittingThe Roe scheme
41
𝜕𝜕𝑅𝑅 − �𝜕𝜕𝑖𝑖+12
+ �𝜕𝜕𝑖𝑖+12
− 𝜕𝜕𝐿𝐿 = �̂�𝐴+ + �̂�𝐴− 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
�𝜕𝜕𝑖𝑖+12
= 𝜕𝜕𝑅𝑅 − �̂�𝐴+ 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
�𝜕𝜕𝑖𝑖+12
= 𝜕𝜕𝐿𝐿 + �̂�𝐴− 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
�𝜕𝜕𝑖𝑖+12
=12
𝜕𝜕𝑅𝑅 + 𝜕𝜕𝐿𝐿 − �̂�𝐴 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
�𝜕𝜕𝑖𝑖+12
=12 𝜕𝜕𝑖𝑖 + 𝜕𝜕𝑖𝑖+1 − �𝑇𝑇𝑖𝑖+1/2 �Λ𝑖𝑖+1/2 �𝑇𝑇𝑖𝑖+1/2
−1 𝑈𝑈𝑅𝑅 − 𝑈𝑈𝐿𝐿
We have now two expressions for the cell face flux
Assuming symmetry one may use the average
A first order formulation of the flux for the Roe scheme. Note that one as before utilises the eigenvalues and eigenvectors to determine the ’Jacobian’:
AUSMAdvection Upstream Splitting Method
42
This technique is similar to van Leer splitting. However, while the convective terms are transported by a convection velocity, the pressure is transported by acoustic waves. Hence, it is reasonable to treat these terms separately.
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 = 0
𝜕𝜕 =𝜌𝜌𝑢𝑢𝜌𝜌𝑢𝑢2𝜌𝜌𝜌𝜌𝑢𝑢
+0𝑝𝑝0
= 𝜕𝜕 𝑎𝑎 +0𝑝𝑝0
𝜌𝜌 = 𝜕𝜕 +𝑝𝑝𝜌𝜌
𝜕𝜕1/2𝑎𝑎 = 𝑢𝑢1/2
𝜌𝜌𝜌𝜌𝑢𝑢𝜌𝜌𝜌𝜌 𝐿𝐿/𝑅𝑅
= 𝑀𝑀1/2
𝜌𝜌𝑎𝑎𝜌𝜌𝑎𝑎𝑢𝑢𝜌𝜌𝑎𝑎𝜌𝜌 𝐿𝐿/𝑅𝑅
∗ 𝐿𝐿/𝑅𝑅 = �∗ 𝐿𝐿 𝑖𝑖𝑓𝑓 𝑀𝑀1/2 ≥ 0∗ 𝑅𝑅 𝑓𝑓𝑡𝑡𝑜𝑜𝑜𝑓𝑓𝑜𝑜𝑖𝑖𝑜𝑜𝑜𝑜
Again 1D Euler equation
Divide the flux in a convective part and a pressure part
The flux at the cell face is evaluated using values on either side of the shock
AUSM
43
𝑀𝑀1/2 ≈ 𝑀𝑀𝐿𝐿+ + 𝑀𝑀𝑅𝑅
−
𝑀𝑀± =±
14𝑀𝑀 − 1 2 𝑖𝑖𝑓𝑓 𝑀𝑀 ≤ 1
12𝑀𝑀 ± 𝑀𝑀 𝑓𝑓𝑡𝑡𝑜𝑜𝑜𝑜𝑜𝑖𝑖𝑜𝑜𝑜𝑜
𝑝𝑝1/2 = 𝑝𝑝𝐿𝐿+ + 𝑝𝑝𝑅𝑅−
𝑝𝑝± =
𝑝𝑝2 1 ± 𝑀𝑀 𝑖𝑖𝑓𝑓 𝑀𝑀 ≤ 1
𝑝𝑝2𝑀𝑀 ± 𝑀𝑀
𝑀𝑀 𝑓𝑓𝑡𝑡𝑜𝑜𝑜𝑓𝑓𝑜𝑜𝑖𝑖𝑜𝑜𝑜𝑜
The convection Mach number can be calculated using van Leer splitting
The pressure is also found by considering splitting
AUSM
44
𝜌𝜌𝑢𝑢𝜌𝜌𝑢𝑢𝑢𝑢 + 𝑝𝑝𝜌𝜌𝑢𝑢𝜌𝜌 1/2
= 𝑀𝑀1/212
𝜌𝜌𝑎𝑎𝜌𝜌𝑎𝑎𝑢𝑢𝜌𝜌𝑎𝑎𝜌𝜌 𝐿𝐿
+𝜌𝜌𝑎𝑎𝜌𝜌𝑎𝑎𝑢𝑢𝜌𝜌𝑎𝑎𝜌𝜌 𝑅𝑅
−12 𝑀𝑀1/2
𝜌𝜌𝑎𝑎𝜌𝜌𝑎𝑎𝑢𝑢𝜌𝜌𝑎𝑎𝜌𝜌 𝐿𝐿
−𝜌𝜌𝑎𝑎𝜌𝜌𝑎𝑎𝑢𝑢𝜌𝜌𝑎𝑎𝜌𝜌 𝑅𝑅
+0
𝑝𝑝𝐿𝐿+ + 𝑝𝑝𝑅𝑅−0
The final expression for the flux at the cell face:
Numerical dissipation term
Boundary conditions for the Euler equations
45
Boundary conditions for the Euler equations are depenent on the character of the equations. In the hyperbolic case the number of conditions that is needed depends on the direction of information transport over the boundary.
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 +
𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 = 0
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+ 𝐴𝐴𝜕𝜕𝑈𝑈𝜕𝜕𝜕𝜕 + 𝐵𝐵
𝜕𝜕𝑈𝑈𝑑𝑑𝑑𝑑 = 0
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡 + 𝑇𝑇Λ𝑎𝑎𝑇𝑇−1
𝜕𝜕𝑈𝑈𝜕𝜕𝜕𝜕 + 𝑆𝑆Λ𝑏𝑏𝑆𝑆−1
𝜕𝜕𝑈𝑈𝑑𝑑𝑑𝑑 = 0
The 2D Euler equations can be reformulated using eigenvectors and eigen values.
Boundary conditions for the Euler equations
46
𝜕𝜕𝑈𝑈𝜕𝜕𝑡𝑡
+ 𝑇𝑇Λ𝑎𝑎𝑇𝑇−1𝜕𝜕𝑈𝑈𝜕𝜕𝜕𝜕
+ 𝑆𝑆Λ𝑏𝑏𝑆𝑆−1𝜕𝜕𝑈𝑈𝑑𝑑𝑑𝑑
= 0
𝜕𝜕𝑊𝑊𝜕𝜕𝑡𝑡 + Λ𝑎𝑎𝑇𝑇−1
𝜕𝜕𝑊𝑊𝜕𝜕𝜕𝜕 + 𝑇𝑇−1𝑆𝑆Λ𝑏𝑏𝑆𝑆−1𝑇𝑇
𝜕𝜕𝑊𝑊𝑑𝑑𝑑𝑑 = 0
𝑊𝑊 = 𝑇𝑇−1𝑈𝑈
𝜆𝜆1,2 = 𝑢𝑢,𝑢𝑢𝜆𝜆3,4 = 𝑢𝑢 ± 𝑎𝑎
Assuming T-1 is constant:
W are called characteristic variables or Riemann variables
The eigenvalue for this case are
1 & 2 are the streamline charecteristics and 3 & 4 are the wave fronts
Boundary conditions for the Euler equations
47
𝜆𝜆1,2 = 𝑢𝑢,𝜆𝜆3,4 = 𝑢𝑢 ± 𝑎𝑎
Inflow and outflow boundary conditions
t
x
λ1,2
λ3
t
x
λ4
λ1,2
λ3 λ4
Supersonic inflow Subsonic inflow
Boundary conditions for the Euler equations
48
Inflow and outflow boundary conditions
t
x
λ1,2
λ3
t
x
λ4
λ1,2
λ3 λ4
Supersonic inflow Subsonic inflow
Boundary conditions for all variables
Boundary conditions for 1 variable
Boundary conditions for the Euler equations
49
49
t
x
λ1,2
λ3
t
x
λ4
λ1,2
λ3 λ4
Supersonic outflow Subsonic outflow
All values on the boundary can be extrapolated from the interior
Boundary conditions for no. of variables -1
Boundary conditions for the Euler equations
Wall boundary conditions
𝜕𝜕 𝜕𝜕, 𝑑𝑑 = 𝑑𝑑 − 𝑓𝑓 𝜕𝜕
𝑑𝑑 = 𝑢𝑢𝑑𝑑𝑓𝑓𝑑𝑑𝜕𝜕 = 𝑢𝑢
𝑑𝑑𝑑𝑑𝑑𝑑𝜕𝜕 𝑠𝑠𝑠𝑠𝑠𝑠𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎
𝜕𝜕𝑝𝑝𝜕𝜕𝑛𝑛 =
𝜌𝜌𝑑𝑑𝑡𝑡2
𝑅𝑅
𝜕𝜕𝑝𝑝𝜕𝜕𝑛𝑛 = 0
A correct BC at ai impermiable wall is that the material derivative of the wall should vanish
This gives the following BC:
Pressure on the wall may be extrapolated from the interior:
An alternative is to solve the normal momentum equation on the wall. At the wall pressure gradient is balanced by the cetrifugal force