CCFFDDAdvanced Computational Fluid Dynamics
A Compressible A Compressible ““Poor ManPoor Man’’s s NavierNavier StokesStokes””Discrete Dynamical SystemDiscrete Dynamical System
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Chetan Babu [email protected]
J. M. [email protected]
Reported work supported by NASA EPSCoR
Presented at the 58’th Annual DFD Meeting of APS, Nov. 20-22, 2005
Department of Mechanical EngineeringUniversity of Kentucky, Lexington, KY 40506
Chicago, IL
CCFFDDAdvanced Computational Fluid Dynamics
Motivation for New Approach to Turbulence Modeling
• Direct Numerical Simulation (DNS): no modeling, permits interac-tion of turbulence with other physics on smallest scales. Runtimes for DNS ∼ Ο(Re3).
• Reynolds Averaged Navier stokes (RANS): essentially all model-ling, small-scale interactions impossible. Computes fairly quickly.
• Large−Eddy Simulation (LES): some modeling; but current mod-els 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 funda-mental problem of mapping:
Physics Statistics
CCFFDDAdvanced Computational Fluid Dynamics
The New Approach ⎯ Use of Synthetic Velocity
• Approach taken by Advanced CFD Group (UK) ∼
• Mimic physics of SGS fluctuations, i.e., u*, v*, w* are modeled.
• Filter solutions not equations.
• N.−S. equations now take the form,
(u u*)t
(u u*) ∇(u u*) = −∇( p p*) 1/Re Δ ( u u*)∼ ∼ ∼ ∼ ∼+ + + + + + +
• Directly compute u and p, model u* and p*.
• Dependent variable decomposition same as LES.
U (x, t ) u (x, t ) u* (x, t ) = +∼
U u u*= + and P p p*= +
.
∼
∼ ∼
CCFFDDAdvanced Computational Fluid Dynamics
• Form used to find fluctuating velocity components,
u* = A M
A – amplitude factor deduced from an extension of Kolmogorovtheories.
M – modeled employing a discrete dynamical system (DDS) .
• Formulation is applied at each discrete grid point and time level.
The New Approach ⎯ Use of Synthetic Velocity (cont.)
• These DDS are derived directly from momentum equations.
• “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
CCFFDDAdvanced Computational Fluid Dynamics
Use of PMNS equations in Turbulence Modeling
2-D Incompressible Turbulent Convection:
• Experimental data taken from,
J. P. Gollub, S. V. Benson “Many routes toturbulent 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 modelof temporal flucuations in turbulent convection”being prepared to be sent to J. Fluid Mech.
CCFFDDAdvanced Computational Fluid Dynamics
• Scaling of the governing equations of compressible flow gives,
Equation of state:
Derivation of 3 D Compressible PMNS equations
( ) 0=⋅∇+ Ut ρρ
ijRep
MDtDU τ
γρ ∇+∇−=
112
( ) ( ) φγ
ρRe
TkPe
pUMDt
DE 1112 +∇⋅∇+⋅∇−=
Uxu
xu
iji
j
j
iij ⋅∇+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
= λδμτj
iij x
u∂∂
= τφ,
Tp ρ=
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CCFFDDAdvanced Computational Fluid Dynamics
• Assume Fourier representations for ρ, u, v, w, p, E∑ gk(t)ϕk(x)
• Apply Galerkin procedure • Substitute solution representations into governing equations .
k=-∞∑ hk(t)ϕk(x)
∑bk(t)ϕk(x) ,∞
k=-∞u (x,t) = v (x,t) =∑ ck(t)ϕk(x) ,
k=-∞
∞w (x,t) =
k=-∞
∞
ρ (x,t) =∞
k=-∞
∞∑ ak(t)ϕk(x) , P (x,t) =∑ ek(t)ϕk(x) , E (x,t) =
k=-∞
∞
Derivation of 3 D Compressible PMNS equations (cont.)
assume ϕk to be orthonormal and in Co∞
• 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.
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CCFFDDAdvanced Computational Fluid Dynamics
• PMNS equations for 3-D compressible flow:
3 D Compressible PMNS equations
nnnnnnnn gacabaaa 1312111 γγγ −−−=+
( ) [ ] ⎥⎦⎤
⎢⎣⎡ +++−++−−−=+ nnn
nnn
nnnnnnnnn gcb
aeb
abgbcbbbb 211122211
1
31
31
31111 ςςχψααγγβ
( ) [ ] ⎥⎦⎤
⎢⎣⎡ +++−++−−−=+ nnn
nnn
nnnnnnnnn gbc
aec
acgccbccc 322232312
1
31
31
31111 ςςχψααγγβ
( ) [ ] ⎥⎦⎤
⎢⎣⎡ +++−++−−−=+ nnn
nnn
nnnnnnnnn cbg
aec
aggcgbggg 323342413
1
31
31
31111 ςςχψααγγβ
[ ] nn
nnnnnnn
nnnnnnnn Ta
gecebea
ghchbhhh κηηηγγγ 11321535251
1 +++−−−−=+
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥⎦⎤
⎢⎣⎡ +++⎥⎦
⎤⎢⎣⎡ +++⎥⎦
⎤⎢⎣⎡ +++
2223
2222
2221 3
41341
341 nnn
nnnn
nnnn
n cbga
gbca
gcba
ξξξ
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+ nnnnnn
n gbgccba 3
232
321
121 λλλ
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CCFFDDAdvanced Computational Fluid Dynamics
• 3 D compressible PMNS equations has thirty six bifurcation pa-rameters.
• Is there a closure problem?
• All bifurcation parameters are calculated using the resolved partof LES and hence known before hand.
Bifurcation Parameters
• Bifurcation parameters can be grouped into three families cor-responding to whether they depend on Re, M or strain rates.
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CCFFDDAdvanced Computational Fluid Dynamics
Time Series and Power Spectral Density
Periodic
Subharmonic
Phase lock
Quasi-periodic
Noisy-Subharmonic
Noisy-Phase lock
Noisy-Quasi-Periodic
with fund.
Noisy-Quasi-Periodicwithout fund.
Broadband-with fund.
Broadband-with dif. fund.
Broadband-without fund.
TIMESERIES PSD TIMESERIES PSD
CCFFDDAdvanced Computational Fluid Dynamics
Regime Maps (Bifurcation Diagrams)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−=
Rekz 2
1 14τ
β
1,11 klmBτγ =
21
1 Mk
γτ
ψ =
CCFFDDAdvanced Computational Fluid Dynamics
Summary
• Shortcomings of present day turbulence models was presented.
• Derivation of 3-D compressible PMNS was presented.
• Use of synthetic velocity in turbulence modeling was presented.
• 3-D compressible PMNS produces all non-trivial types of beha-vior as observed for the incompressible case.