Velocity and Temperature Boundary- Layer Modeling Using ...
Transcript of Velocity and Temperature Boundary- Layer Modeling Using ...
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 1Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Velocity and Temperature Boundary-Layer Modeling Using Averaged
Molecule cluster Transport Equations
R. Groll
Center of Applied Space Technology and MicrogravityUniversity of Bremen, Am Fallturm, D-28359 Bremen
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 2Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
LCharacteristic
layer thickness
Micro- und macroscopic approach
Micro channel flow | continuum approch
Molecular number density inside transsonic nozzle
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 3Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Modification of parabola near the wall
u Gas
-uW
all
Ls
Slip Length
Slip Velocity
x2
x1
Poiseuille flow of high Knudsen numbers
Micro channel flow | slip velocity
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 4Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
0 00
limV
NnV
Number densityMolecule numberControl volume 0
NV
Mean velocity ( , , )i i Bn f t x d
Molecule velocity i
1i i iu n
n Number-density weighted
''i i i Velocity deviation
x
Molecule number density and statistical filters
Statistical modeling| Filtering
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 5Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Continuity of molecular diffusion
'
'
jj
i j i jj i
nn Dx
n D n nx x
Velocity correlation
32
Sc=1
23
k kB
i i
k kk k
m
mmn Tk
u mnD
n mp mE d
Analogy of macroscopic values
0 0
'
j
j jj j
j jj j
nDx
n nn nt x t x
n n nt x x
Statistical modeling | Velocity correlations
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 6Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
jj
E n nE n n
0 0j jj j
n nn nt x t x
Molecule velocity ξ
Number-density weighted averaging
Momentum transport equation
2
0
2
1
i i j i i j i jj j j
ji kij
j j i k
ijj i j k k
nn n n nt x t x x
Dnx x x x
D n n n nx n x x x x
UnsteadyConvectionSourceDiffusionConversion
Statistical modeling | Momentum transport
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 7Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Turbulence analogy
12 k k
2
2
2 1 2
ij
j i ki j ij ij ij
i j k k i j k
D n n n n Dn x x x x x x x
Energy transport equation
Terms of kinetic dissipation and other energetic production
Depending on velocity and number density gradients
Energy Diffusion without density gradients works with Pr=κ/(κ +2) = 1/γ < 1 .
compressionconvection dissipationdiffusion
2
2 20
2 1
k ij j ij
j j j j k j
i
j i i i j
n n n n Dn n nt x t x x x x x
D n n D nDnx x n x x n x
conversion
Statistical modeling | Energy transport
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 8Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Thermodynamic consistence of transport equations
Constitutive equation
Variation energy / intrinsic energy
compressionconvection
1 kj
j k
ue eu Ts et x x
Compression Energy increase Expansion Entropy increase
Only forγ>1
2
de Tds pdve nTs n n p v
Entropy increase
32
2
dissipation conversion diffusion
1i iij
j j i i j i j j j
D n n n nnTs n D Dnx x n x x x x x x x
Statistical modeling | thermodynamic consistance
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 9Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
**
2/3 2/30 0
111 /
w w ww
ww
LKnL
Boundary conditions
Well-knownwall shear stress
23w
w wL L
Boundary Conditions | Wall shear stress
* *
2* * *00
20 0 0 0
2123 3 3
w
w w w ww
w wKnc
L L p Kn cL p L
2/33
11 /
w ww ww w w
L LL L L
Modeled influence of large mean-free-paths on wall shear stress
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 10Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Kn=0.05 Kn=1.00
*
0*
dp Kndx
* *0
0 0
*0
0 0
ii
u pu pHp p
xKn xL L
Pressure gradient condition:
Non-dimensional values:
Boundary Conditions | Velocity profiles
Computational resuts
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 11Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Conclusions
Algebraic model describes scalar diffusion coefficient
(→ diffusion modelling)
For high Knudsen numbers : Diffusion isdescribed by formulation of dilute gases
(→ slip modelling)
Continuity, momentum and ernergy are calculated by transport equations ofnumber density n and the first and second statistical moments of themolecular velocity. (→ statistical modelling)
Conclusions | Scale-transitioning Analogy
*2/3
11 wL Kn
Analogy of macroscopic values
32
23
k k
m
mE d
jj
E n nuE n n
( , , )i i Bn f t x d
1
2 k ke
IUVSTA 2011| Leinsweiler, May 16th 2011
R. Groll | 12Center of Applied Space Technology and Microgravity
Computational Thermofluid Dynamics Division
Many thanks for your attention!