Oleg Schilling University of California, Lawrence Livermore National - - PowerPoint PPT Presentation

Oleg Schilling IWPCTM-12/01 1

Single-Velocity, Multi-Component Turbulent Transport Models for Interfacial Instability-Driven Flows Single-Velocity, Multi-Component Turbulent Transport Models for Interfacial Instability-Driven Flows

Oleg Schilling

University of California, Lawrence Livermore National Laboratory P.O. Box 808, L-22, Livermore, CA 94551 (925) 423-6879, schilling1@llnl.gov

Presented at the 8th International Workshop on the Physics of Compressible Turbulent Mixing California Institute of Technology, Pasadena, CA 9-14 December 2001

This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48

Oleg Schilling IWPCTM-12/01 2

Outline of presentation Outline of presentation

• Motivation

– The need for turbulent transport and mixing models – Single- vs. multiple-velocity, multi-component fluid formulations

• Derivation of the Favre-Reynolds averaged single-velocity equations
• Two-equation turbulence models

– The general K-Z model – The K-ε model – Derivation of consistent K-l, K-ω, and K-τ models

• Work in progress: a priori model tests

– Determination of model coefficients from experimental data – Determination of model coefficients from simulation data

• Conclusions

Oleg Schilling IWPCTM-12/01 3

An averaged description of turbulent transport and mixing is needed due to the very wide range of spatio-temporal scales in turbulent mixing layers An averaged description of turbulent transport and mixing is needed due to the very wide range of spatio-temporal scales in turbulent mixing layers

• Direct numerical simulation (DNS) cannot attain parameter regimes
• f interest for astrophysical and inertial confinement fusion (ICF)

applications

• Large-eddy simulation (LES) is not yet sufficiently developed
• Interim solution: turbulent transport and mixing models, which have

similarities with LES

ICF supernova

• Transport models are based
• n closing terms in the

density-weighted averaged equations – Reynolds stress tensor – Density and energy flux

• These quantities are modeled

using an eddy viscosity approximation

Oleg Schilling IWPCTM-12/01 4

Single-velocity formulations of multi- component flow are significantly less complex than multiple-velocity formulations Single-velocity formulations of multi- component flow are significantly less complex than multiple-velocity formulations

• Single-velocity, multi-component fluid formulations:

– Equations systematically derived from reacting flow theory – Equations have nearly the same form as the single-fluid, compressible fluid dynamics equations – Additional fluxes involving a diffusion velocity are present – The diffusion velocity is obtained, and these fluxes are expressed in terms of a mass diffusion flux

• Multiple-velocity, multi-component fluid formulations:

– Require multiple advection terms equal to number of fluids – Require fluid dynamic fields for every fluid, so the number of equations to model and solve is large – Require phenomenological modeling of interfacial source terms arising from interfacial averaging: drag, added mass terms, etc.

Oleg Schilling IWPCTM-12/01 5

The derivation of the single-velocity equations begins with the full, N-fluid equations expressing mass, momentum, and energy conservation The derivation of the single-velocity equations begins with the full, N-fluid equations expressing mass, momentum, and energy conservation

• In compact form, these equations are (r labels each fluid):

– where the fields, fluxes, forces, and sources are

• t r

r J

r r

x

F

r S r

r

r

r r v

r

r er r r , J

r

• r v

r

r v

r v r pr r

r e r prv

r v r r rad,r

r r v

r ,r

F

r

r g , S

r

Rr v

r Rr

Hr Rr r gv

r

r R r

Oleg Schilling IWPCTM-12/01 6

These fields are defined so that summing appropriate expressions over each fluids recovers the non-reacting, single-fluid equations These fields are defined so that summing appropriate expressions over each fluids recovers the non-reacting, single-fluid equations

• The quantities ρr, vαr, Ur, ϕr, Φα rad,r, Φr, gα, Rr, and Hr are the density,

velocity, internal energy, scalar, radiative flux, scalar flux, acceleration, reaction rate, and heat of formation

• The pressure, viscous stress tensor, and total energy are
• Consistency with the single-fluid equations is obtained with the

constraints

pr prr,Ur

• r

r

v

r

x v

r

x

• r 2r

d

• v

r

x

er

v r

2

2 Ur mr Hr g x

r1

N r r

,

r1

N J r

J

r1

N F r F

,

r1

N S r S

r1

N Rr 0 ,

r1

N r Rr 0

Oleg Schilling IWPCTM-12/01 7

The single-velocity equations are obtained by decomposing the velocity into a mean velocity plus a diffusion velocity The single-velocity equations are obtained by decomposing the velocity into a mean velocity plus a diffusion velocity

• Introduce the local mass fraction of fluid r
• Write the velocity of fluid r as

where Vr is the diffusion velocity, which expresses the molecular transport caused by the concentration gradient in fluid r

• The identity

is central to the derivation of the single-velocity equations

mrx,t

r

• ,

r1

N mrx,t 1

vr v Vr , v r1

N mr vr

r1

N mr Vr 0

Oleg Schilling IWPCTM-12/01 8

The single-velocity equations are a consequence of the previous identities The single-velocity equations are a consequence of the previous identities

• Substituting the velocity decomposition into the multi-component

equations, summing, and using the previous identities gives the single-velocity equations

• The fields and fluxes are

where the last term in Jαβ depends on the diffusion velocity and must be modeled

• t

J x

F S

• v

e

• J

v v v p e pv v

rad

v

• D

J

e

J

Oleg Schilling IWPCTM-12/01 9

The forces, sources, and other quantities are defined as follows The forces, sources, and other quantities are defined as follows

• The forces and sources are
• The total density, pressure, radiative flux, viscous stress tensor,

dynamic viscosity, and bulk viscosity are

F g S g v r1

N

Hr Rr

r1

N r

p r1

N prr,Ur

v x v x

• 2

d

• v

x

r1

N r V

r

x V

r

x

• r 2r

d

• V

r

x

• rad r1

N rad,r

r1

N r ,

r1

N r

Oleg Schilling IWPCTM-12/01 10

The diffusive fluxes are defined as follows The diffusive fluxes are defined as follows

• The multi-component viscous diffusion stress tensor is
• The diffusive energy flux is
• The diffusive scalar flux is
• D r1

N mr V r V

J

e r1 N mr er V r

J

r1 N mr r V r

Oleg Schilling IWPCTM-12/01 11

The averaged equations are obtained by introducing the Favre-Reynolds decompositions and averaging The averaged equations are obtained by introducing the Favre-Reynolds decompositions and averaging

• The Favre-Reynolds decompositions are
• The Favre average is
• The Favre-averaged multi-component fluid dynamics equations are
• r x,t

r x,t r x,t

prx,t p rx,t prx,t rx,t rx,t rx,t

• t

J x

F S

Oleg Schilling IWPCTM-12/01 12

The Favre-averaged fields and fluxes are defined as follows The Favre-averaged fields and fluxes are defined as follows

• The fields and fluxes are
• v

e

• J

v v v p

• D

e p v v

rad

J

e

v J

• D v

v

• p v

v rad J e e v

• J

v

Oleg Schilling IWPCTM-12/01 13

The Favre-averaged forces and sources are defined as follows The Favre-averaged forces and sources are defined as follows

• The forces and sources are
• At large Reynolds numbers, the viscous stress terms and diffusive

fluxes are assumed to be negligible compared to the Reynolds stress tensor and turbulent fluxes

F g S g v

• v

gr Hr R r Hr Rr

Oleg Schilling IWPCTM-12/01 14

A gradient diffusion approximation is usually used to model the turbulent stresses and fluxes A gradient diffusion approximation is usually used to model the turbulent stresses and fluxes

• The gradient diffusion approximation is
• The eddy viscosity

is determined by the solution of transport equations for two turbulence variables K (= E’’) and

• vj

v x j t

• ij vi

vj vi vj

• 2 E ij

3 2 t

• S kj

kj 3

• v l

x l

vj

x j t

Z CZ E

m

• n

t C E

m2n n

Z

1

n

Oleg Schilling IWPCTM-12/01 15

The turbulent kinetic energy equation is closed as follows The turbulent kinetic energy equation is closed as follows

• The unclosed turbulent kinetic energy equation is
• Use the gradient diffusion approximation to close the diffusion term

and density flux, and (Mat is the turbulent Mach number)

• t

E

• x j

E v j

• forceproduction

vi

vi gi turbulentdiffusion

• x j

v 2 v j

• 2

p vj

vi ij vi ij D

• meanvelocityproduction
• v i
• x j

vi

vj

• pressurework

vi

p x i

• pressure
• dilatation

p v j

• x j
• kineticenergydissipation

rate

ij

v i

• xj ij

D v i

• x j

p v i

• x j 2 ij
• v i

x j 3 Mat Mat 2

vi

v i

• t
• x i

Oleg Schilling IWPCTM-12/01 16

The modeled turbulent kinetic energy dissipation rate transport equation is obtained as follows The modeled turbulent kinetic energy dissipation rate transport equation is obtained as follows

• The turbulent kinetic energy dissipation rate equation is obtained by

multiplying the turbulent kinetic energy equation by ε/K and a dimensionless constant for each term:

• t
• x j

v j

• force production

C0

E

vi

vi gi mean velocity production

C1

E

• v i
• x j

vi

vj

• kinetic energy dissipation rate

C2

• 2

E

• turbulent diffusion
• x j

t

• x j
• pressure work

C3

E vi p x i pressure-dilatation

C4

E p v i

• x i

Oleg Schilling IWPCTM-12/01 17

The modeled Z transport equation is obtained from the K and ε ε ε ε equations as follows The modeled Z transport equation is obtained from the K and ε ε ε ε equations as follows

• Using the K and ε equations,
• t

Z

• x j

Z v j

• t

v j

• x j

Z Z

m E

• t

v j

• x j

E

n

• t

v j

• x j
• m Z

E

• t

E

• xj

E v j n Z

• t
• xj

v j

• Z

E m nCK

• t

E

• x j

E v j

Oleg Schilling IWPCTM-12/01 18

The turbulent diffusion term is transformed as follows The turbulent diffusion term is transformed as follows

• Substituting

it follows that

Z CZ E

m

1/n

Z

m E

• x j

t k E x j

• n
• x j

t

• x j
• m Z

E

• x j

t k E x j

• x j

t

• Z

x j m Z E E x j

• t

n Z x j m Z E E x j 1n Z Z x j m E E x j

Oleg Schilling IWPCTM-12/01 19

Finally, the modeled form of the Z transport equation is as follows Finally, the modeled form of the Z transport equation is as follows

• t

Z

• x j

Z v j

• t

v j

• x j

Z

• force production

CZ0 Z

E

vi

v i gi mean velocity production

CZ1 Z

E

• v i
• x j

vi

v j

• kinetic energy dissipation rate

CZ2 Z

E

• turbulent diffusion
• x j

t Z x j

m Z

E

• x j

t k E x j turbulent diffusion

m

• x j

t

• Z

E E x j

• t

n Z x j mZ E E x j 1n Z Z xj m E Ek

• x j
• pressure work

CZ3 Z

E vi p x i pressure-dilatation

CZ4 Z

E p v i

• x i

Oleg Schilling IWPCTM-12/01 20

The coefficients in the modeled Z transport equation are obtained from those in the ε ε ε ε equation The coefficients in the modeled Z transport equation are obtained from those in the ε ε ε ε equation

• The coefficients in the standard K-ε model are
• The coefficients in the Z equation are
• Different choices of m and n yield different 2-equation models:

–

with m = 0 and n = 1 (turbulent energy dissipation) – with m = 3/2 and n = -1 (turbulent lengthscale) – with m = -1 and n = 1 (turbulent frequency) – with m = 1 and n = -1 (turbulent timescale)

k 1.0 , 1.3 , C1 1.44 , C2 1.92 C0 C4 1.0 CZ0 m nC0 , CZ1 m nC1 , CZ2 m nC2 CZ3 m nC3 , CZ4 m nC4

E- E- E- E-

Oleg Schilling IWPCTM-12/01 21

The K-Z model simplifies for several special types of turbulent flows, which can be used to tune the model coefficients The K-Z model simplifies for several special types of turbulent flows, which can be used to tune the model coefficients

• Isotropic turbulence: power-law decaying solutions

– Production terms proportional to τij, the turbulent diffusion terms, and the mean velocity vanish

• Free shear flows (plane wake; mixing layer; plane, round, and

radial jet): far-field, self-similar, statistically-stationary solutions – Solutions depend on the similarity variable η = y/x – Turbulent boundary layers: power-law solutions in the logarithmic layer – Sufficiently far from the boundary, the eddy viscosity dominates the molecular viscosity and the advection terms are negligible

Oleg Schilling IWPCTM-12/01 22

The K-Z model equations have power-law solutions for isotropic turbulence The K-Z model equations have power-law solutions for isotropic turbulence

• The model equations reduce to coupled ordinary differential

equations

• The initial conditions are

and

• The corresponding solutions are
• Experimentally, K(t) ∝ t-1.34, which determines Cε2 (or CZ2)

dK dt dZ dt CZ2 Z K

K0 K0 Z0 Z0 CZK0

m0 n Kt K0

• 1

CZ2mn n K0 t n/CZ2mn

• 1 C2 1

K0 t 1/C21 Zt Z0

• Kt

K0 CZ2 Kt K0 mnC2

Oleg Schilling IWPCTM-12/01 23

The K-Z model equations have similarity solutions for free shear flows The K-Z model equations have similarity solutions for free shear flows

• The model equations reduce to (and are solved by transforming to

the similarity variable) where r = 1 corresponds to a round jet and r = 0 otherwise, and the shear stress is

v x

• x v y
• y

K xy

v x y 1 yr

• y

yr t

k K y

v x

• x v y
• y

Z CZ1 Z

K t v i x j 2

CZ2 Z

K

1

yr

• y

yr t

• Z

y

• mZ

Kyr

• y

yr t

k K y

m

yr

• y

yr t

• Z

K K y

• t

n Z y mZ K K y 1n Z Z y m K K y

xy t

v x y

Oleg Schilling IWPCTM-12/01 24

The K-Z model equations have similarity solutions for the mixing layer The K-Z model equations have similarity solutions for the mixing layer

• The solutions have the form

where v = v1 – v2 is the velocity difference between the two streams

v xx,y v v x Kx,y v2 K Zx,y CZ v2m Km

v3 x nn

Oleg Schilling IWPCTM-12/01 25

The K-Z model equations have solutions consistent with the law-of-the-wall in bounded flows The K-Z model equations have solutions consistent with the law-of-the-wall in bounded flows

• The Reynolds-averaged and K-Z equations reduce to
• The solutions have the form (where vτ is the friction velocity, κ is

the von Kármán constant, and C, D, CZ are constants)

• y

t

v x y

t

v x y 2

• y

t K K y

CZ1 Z

K t v x y 2 CZ2 Z K

• y

t

• Z

y

m Z

K

• y

t k K y

m

y t

• Z

K K y

t

n Z y m Z K K y 1n Z Z y m K K y

v x

v logy C ,

K Dv

2

, Z CZ Dm v

2m3n 1 y n

Oleg Schilling IWPCTM-12/01 26

Application to asymptotically self-similar Rayleigh-Taylor mixing Application to asymptotically self-similar Rayleigh-Taylor mixing

• The turbulence production term is of the form
• Assume that at sufficiently late times, the scaling of the mixing

layer width is and that turbulence variables are proportional to this lengthscale and the corresponding velocity scale

• Then, K, ε, and Z are

PZ CZ1 Z

E ij

• v i

x j

ht Atgt2 E

v2 2

• dh

dt 2 4Atg2 t2 Ek

• t

8Atg2 t Z CZ E

m

• n

Atg2m2n t2mn

Oleg Schilling IWPCTM-12/01 27

In Rayleigh-Taylor mixing the eddy viscosity and Reynolds stress tensor scale as follows In Rayleigh-Taylor mixing the eddy viscosity and Reynolds stress tensor scale as follows

• The eddy viscosity scales as
• The Reynolds stress scales as
• Therefore, if the Favre-averaged strain-rate tensor dimensionally

scales as (qij is dimensionless) then and

t E

m2n n

Z

1

n Atg2 t3

ij 2 E ij

d 2 t

• S ij

ij d

• v k

x k

2 Atg2 t2

4 d ij

• Sij

ij d

• v k

x k

t

• Sij

1 2

• vi

xj

• v j

x i

qij

1 ht dht dt

• 1

t

ij 2 Atg2 t2

5 d ij qij

PZ Atg2mn t2mn1

5 d ij qij qij

Oleg Schilling IWPCTM-12/01 28

Conclusions Conclusions

• The methodology presented here provides a systematic and self-

consistent approach to the derivation of 2-equation turbulent transport models – This provides an improved l transport equation – Also provides a consistent expression for the diffusion and cross diffusion terms, which are important in many flow (e.g., near a boundary)

• Several canonical turbulent flows can be used to reduce the model

equations and specify model parameters before application to interfacial-instability induced turbulence

• The general Z equation is consistent with the t2 scaling of the mixing

layer width

• Both an ω and a τ equation were derived as alternatives to the ε and

l equation – τ may be a better physical variable than ε and l

Oleg Schilling IWPCTM-12/01 29

Work in progress Work in progress

• Completion of solutions for canonical turbulent flows
• Completion of solutions for Rayleigh-Taylor instability-induced

turbulence

• Commencement of examination of model parameters and forms of

modeled terms using high-resolution DNS data

• Eventually, application to Richtmyer-Meshkov instability-induced

turbulence