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 Velkurvcbabu@uky.edu

J. M. McDonoughjmmcd@uky.edu

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.