Single-Velocity, Multi-Component Single-Velocity, Multi-Component Turbulent Transport Models for Turbulent Transport Models for Interfacial Instability-Driven Flows 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 8 th 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 1
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 2
An averaged description of turbulent transport and An averaged description of turbulent transport and mixing is needed due to the very wide range of mixing is needed due to the very wide range of spatio-temporal scales in turbulent mixing layers spatio-temporal scales in turbulent mixing layers • Direct numerical simulation (DNS) cannot attain parameter regimes of 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 • Transport models are based on closing terms in the density-weighted averaged equations – Reynolds stress tensor – Density and energy flux • These quantities are modeled using an eddy viscosity approximation ICF supernova Oleg Schilling IWPCTM-12/01 3
Single-velocity formulations of multi- Single-velocity formulations of multi- component flow are significantly less complex component flow are significantly less complex than multiple-velocity formulations 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 4
The derivation of the single-velocity equations The derivation of the single-velocity equations begins with the full, N -fluid equations expressing begins with the full, N -fluid equations expressing mass, momentum, and energy conservation mass, momentum, and energy conservation • In compact form, these equations are ( r labels each fluid): r � � r � � J �� � t � � r � � r � � r � S � r � F � � � x � – where the fields, fluxes, forces, and sources are � r v � r � r � r v � r v � � r v � r � p r � �� � � �� r r � r � � r � r , J �� � � r e r � � r e r � p r � v � rad, r r � v � r � �� r � � � � r � r v � � , r � r � r r � � � R r 0 � r g � r R r v � r � r � F � , S � H r R r � � r g � v � r 0 � r R r 0 Oleg Schilling IWPCTM-12/01 5
These fields are defined so that summing These fields are defined so that summing appropriate expressions over each fluids appropriate expressions over each fluids recovers the non-reacting, single-fluid equations recovers the non-reacting, single-fluid equations • The quantities ρ r , v α r , U r , ϕ r , Φ α rad,r , Φ r , g α , R r , and H r 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 p r � p r � � r , U r � r r r � v � � r � 2 � r � v � � v � r � � r � �� � �� � x � � � � x � d � x � 2 2 � U r � � m r H r � g � x v r e r � • Consistency with the single-fluid equations is obtained with the constraints N J �� N � r � � r r � �� � , � J �� � r � 1 � r � 1 N F � N S � r � F � r � S � , � r � 1 � r � 1 N R r � 0 , r R r � 0 N � � � r � 1 � r � 1 Oleg Schilling IWPCTM-12/01 6
The single-velocity equations are obtained by The single-velocity equations are obtained by decomposing the velocity into a mean velocity decomposing the velocity into a mean velocity plus a diffusion velocity plus a diffusion velocity • Introduce the local mass fraction of fluid r N m r � x , t � � 1 � r m r � x , t � � , � r � 1 � • Write the velocity of fluid r as N m r v r v r � v � V r , v � � r � 1 where V r is the diffusion velocity, which expresses the molecular transport caused by the concentration gradient in fluid r • The identity N m r V r � 0 � r � 1 is central to the derivation of the single-velocity equations Oleg Schilling IWPCTM-12/01 7
The single-velocity equations are a The single-velocity equations are a consequence of the previous identities 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 � J �� � F � � S � � � t � �� � � � � x � • The fields and fluxes are � v � 0 � D � v � v � � p � �� � � �� � v � � � �� J �� � �� � � � rad e � e � � e � p � v � � v � � �� � � � J � � �� v � � � � J � � �� where the last term in J αβ depends on the diffusion velocity and must be modeled Oleg Schilling IWPCTM-12/01 8
The forces, sources, and other quantities are The forces, sources, and other quantities are defined as follows defined as follows • The forces and sources are 0 0 0 � g � 0 0 N F � � S � � � � r � 1 H r R r 0 � g � v � 0 0 0 • The total density, pressure, radiative flux, viscous stress tensor, dynamic viscosity, and bulk viscosity are N p r � � r , U r � N � r N � � p � � r � 1 rad � � r � 1 rad, r � � � r � 1 � � � v � � v � � � 2 � � v � � �� � � � �� � x � � � � x � d � x � r r r � V � � r � 2 � r � V � N � r � V � � � r � 1 � �� � x � � � � x � d � x � N � r , N � r � � � r � 1 � � � r � 1 Oleg Schilling IWPCTM-12/01 9
The diffusive fluxes are defined as follows The diffusive fluxes are defined as follows • The multi-component viscous diffusion stress tensor is N m r V � r V D � � � � r � 1 � �� • The diffusive energy flux is N m r e r V � e � � � r � 1 r J � • The diffusive scalar flux is N m r � r V � r J � � � � � r � 1 Oleg Schilling IWPCTM-12/01 10
The averaged equations are obtained by The averaged equations are obtained by introducing the Favre-Reynolds introducing the Favre-Reynolds decompositions and averaging decompositions and averaging • The Favre-Reynolds decompositions are r � x , t � � � r � x , t � � � � r � x , t � �� � � � � p r � x , t � � p r � x , t � � p r � x , t � � � r � x , t � � � r � x , t � � � r � x , t � � • The Favre average is �� � � � � � � • The Favre-averaged multi-component fluid dynamics equations are � J �� � t � � � � F � � S � � � � � � � x � Oleg Schilling IWPCTM-12/01 11
The Favre-averaged fields and fluxes are The Favre-averaged fields and fluxes are defined as follows defined as follows • The fields and fluxes are � v � � � � � � � � e � � � � � 0 � � v � �� v � D �� � � v � � � v � � v � � p � �� � � � �� � � D �� � � �� �� � � �� � �� J �� � rad � � e � p � v � e �� � � e �� v � � � � e � p �� v � � � v � � J � rad �� � J � �� � v � �� � �� �� � � � �� � �� � � � � � � � � � � � v � � � � J � � �� � � � �� v � � �� � J � �� � � Oleg Schilling IWPCTM-12/01 12
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