# A Compressible “ Poor Man ’ ```A Compressible “Poor Man’s Navier−Stokes”
Discrete Dynamical System
Chetan Babu Velkur
vcbabu@uky.edu
J. M. McDonough
jmmcd@uky.edu
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506
Reported work supported by NASA EPSCoR
Presented at the 58’th Annual DFD Meeting of APS, Nov. 20-22, 2005
Chicago, IL
CFD
Motivation for New Approach to Turbulence
Modeling
• Direct Numerical Simulation (DNS): no modeling, permits interaction of turbulence with other physics on smallest scales. Run
times for DNS ∼
Ο(Re3).
• Reynolds Averaged Navier stokes (RANS): essentially all modelling, small-scale interactions impossible. Computes fairly quickly.
• Large−Eddy Simulation (LES): some modeling; but current models not able to treat details of SGS interactions. Run times for
LES ∼
•
Ο(Re2).
Filtering of equations as done in traditional LES leads to the fundamental problem of mapping:
Physics
Statistics
CFD
The New Approach ⎯ Use of Synthetic Velocity
•
Approach taken by Advanced CFD Group (UK)
•
Dependent variable decomposition same as LES.
•
Mimic physics of SGS fluctuations, i.e., u*, v*, w* are modeled.
•
Filter solutions not equations.
•
N.−S. equations now take the form,
U (x, t ) = ∼
u (x, t ) + u* (x, t )
.
∼
∼+ u*) ∇(u∼+ u*) = −∇( ∼
(u + u*) + (u
p + p*) + 1/Re Δ ( ∼
u + u*)
t
Directly compute ∼
u and ∼
p, model u* and p*.
U = u + u* and P = p∼ + p*
∼
•
CFD
The New Approach ⎯ Use of Synthetic Velocity (cont.)
•
Form used to find fluctuating velocity components,
u* = A M
A – amplitude factor deduced from an extension of Kolmogorov
theories.
M – modeled employing a discrete dynamical system (DDS) .
•
Formulation is applied at each discrete grid point and time level.
•
“The Poor man’s Navier− Stokes equations” (PMNS) in 2D.
an+1 = β1 an (1− an ) − γ1 an bn
bn+1 = β2 bn (1 − bn ) − γ2 an bn
•
These DDS are derived directly from momentum equations.
CFD
Use of PMNS equations in Turbulence Modeling
2-D Incompressible Turbulent Convection:
•
Experimental data taken from,
J. P. Gollub, S. V. Benson “Many routes to
turbulent convection,” J. Fluid Mech. 100,
pp. 449- 470, 1980.
•
Computed results from,
J. M. McDonough, J. C. Holloway and
M. G. Chong “ A discrete dynamical system model
of temporal flucuations in turbulent convection”
being prepared to be sent to J. Fluid Mech.
CFD
Derivation of 3 − D Compressible PMNS equations
• Scaling of the governing equations of compressible flow gives,
ρ t + ∇ ⋅ (ρU ) = 0
1
1
DU
ρ
=−
∇p +
∇τ ij
2
Dt
Re
γM
DE
1
1
1
(
)
(
)
pU
k
T
ρ
=−
∇
⋅
+
∇
⋅
∇
+
φ
2
Dt
Pe
Re
γM
⎛ ∂u i ∂u j
+
τ ij = μ ⎜⎜
⎝ ∂x j ∂xi
Equation of state:
∂u i
⎞
⎟ + δ ij λ∇ ⋅ U , φ = τ ij
⎟
∂x j
⎠
p = ρT
CFD
Derivation of 3 − D Compressible PMNS equations (cont.)
•
Assume Fourier representations for ρ, u, v, w, p, E
∞
∞
v (x,t) = ∑ ck(t) ϕk(x) , w (x,t) =k=∑-∞gk(t) ϕk(x)
u (x,t) = ∑bk(t) ϕk(x) ,
k=-∞
k=-∞
∞
ρ (x,t) = ∑ ak(t) ϕk(x) ,
k=-∞
∞
∞
∞
P (x,t) = ∑ ek(t) ϕk(x) , E (x,t) = ∑ hk(t) ϕk(x)
k=-∞
k=-∞
∞
assume ϕk to be orthonormal and in Co
•
Apply Galerkin procedure
• Substitute solution representations into governing equations .
• Commute differentiation and summation.
• Calculate Galerkin inner products.
• Reduce to a single wave vector.
•
Construct forward Euler discretisation, rearrange to define bifurcation parameters
to get PMNS equations.
CFD
3− D Compressible PMNS equations
• PMNS equations for 3-D compressible flow:
a n +1 = a n − γ 11 a n b n − γ 12 a n c n − γ 13 a n g n
[
]
[
]
[
]
(
)
1
1 ⎡1
1
1
n
n
n
n
n⎤
α
b
ψ
e
χ
b
ς
c
ς
g
−
+
+
+
1
1
1
2
⎥⎦
3
3
an
a n ⎢⎣ 3
(
)
1
1
n
n
α
c
ψ
e
−
+
2
an
an
)
1
1 ⎡1
1
1
n
n
n
n
n⎤
α
c
ψ
e
χ
g
ς
b
ς
c
−
+
+
+
3
3
2
3
⎥
3
3
an
a n ⎢⎣ 3
⎦
b n +1 = β 1b n 1 − b n − γ 21b n c n − γ 22 b n g n + αb n +
c n +1 = β 2 c n 1 − c n − γ 31b n c n − γ 32 c n g n + αc n +
(
g n +1 = β 3 g n 1 − g n − γ 41b n g n − γ 42 c n g n + αg n +
h n +1 = h n − γ 51 h n b n − γ 52 h n c n − γ 53 h n g n −
( ) + (c ) + (g ) ⎤⎥ +
+
1 ⎡4 n
ξ1 b
a n ⎢⎣ 3
+
1
an
2
n 2
n 2
⎦
[
1
1
⎡1
⎤
n
n
χ
c
ς
b
ς3g n ⎥
+
+
2
⎢3 2
3
3
⎣
⎦
]
1
1
n n
n n
n n
η
e
b
+
η
e
c
+
η
e
g
+
κT n
1
2
3
n
n
a
a
( ) + (b ) + (g ) ⎤⎥ +
1 ⎡4 n
ξ2
c
a n ⎢⎣ 3
2
n 2
n 2
⎦
( ) + (b ) + (c ) ⎤⎥
1 ⎡4 n
ξ3
g
a n ⎢⎣ 3
2
n 2
n 2
⎦
⎡ ⎛2 n n⎞
⎛2 n n⎞
⎛ 2 n n ⎞⎤
⎢λ1 ⎜ 3 b c ⎟ + λ 2 ⎜ 3 c g ⎟ + λ1 ⎜ 3 b g ⎟⎥
⎠
⎝
⎠
⎝
⎠⎦
⎣ ⎝
CFD
Bifurcation Parameters
•
3− D compressible PMNS equations has thirty six bifurcation parameters.
•
Bifurcation parameters can be grouped into three families corresponding to whether they depend on Re, M or strain rates.
•
Is there a closure problem?
•
All bifurcation parameters are calculated using the resolved part
of LES and hence known before hand.
CFD
Time Series and Power Spectral Density
TIMESERIES
PSD
Periodic
Subharmonic
TIMESERIES
PSD
NoisySubharmonic
NoisyPhase lock
NoisyQuasi-Periodic
with fund.
Phase lock
Quasi-periodic
NoisyQuasi-Periodic
without fund.
CFD
Regime Maps (Bifurcation Diagrams)
⎛ τz k2⎞
⎟
β1 = 4 ⎜1 −
⎜
Re ⎟⎠
⎝
1
γ 11 = τ Blm
,k
ψ1 =
τ k1
γ M2
CFD
Summary
•
Shortcomings of present day turbulence models was presented.
•
Use of synthetic velocity in turbulence modeling was presented.
•
Derivation of 3-D compressible PMNS was presented.
•
3-D compressible PMNS produces all non-trivial types of behavior as observed for the incompressible case.