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Rapidly Sheared Compressible Turbulence: Characterization of Different Pressure Regimes and Effect of Thermodynamic Fluctuations. Rebecca Bertsch Advisor: Dr. Sharath Girimaji March 29, 2010 Supported by: NASA MURI and Hypersonic Center. Outline . Introduction

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Rapidly Sheared Compressible Turbulence: Characterization of Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Rebecca Bertsch

Advisor: Dr. SharathGirimajiMarch 29, 2010

Supported by: NASA MURI and Hypersonic Center


Outline
Outline Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Progress
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Motivation
Motivation Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Compressible stability, transition, and turbulence plays a key role in hypersonic flight application.

  • Hypersonic is the only type of flight involving flow-thermodynamic interactions.

  • Crucial need for understanding the physics of flow-thermodynamic interactions.


Navier Different Pressure Regimes and Effect of Thermodynamic Fluctuations-Stokes

Sub-grid Modeling

RANS Modeling

Bousinessq approach

ARSM reduction

DNS

LES

Application

Background

Second moment closure

Decreasing Fidelity of Approach


Transport Processes Different Pressure Regimes and Effect of Thermodynamic Fluctuations

2-eqn. ARSM

7-eqn. SMC

Navier-Stokes Equations

Spectral and dissipative processes

Nonlinear pressure effects

ARSM reduction

Averaging Invariance

2-eqn. PANS

Application

Linear Pressure Effects: RDT


Objectives
Objectives Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Verify 3-stage evolution of turbulent kinetic energy (Cambon et. al, Livescu et al.)

  • Explain physics of three stage evolution of flow parameters

  • Investigate role of pressure in each stage of turbulence evolution

  • Investigate dependence of regime transitions

  • *Previous studies utilized Reynolds-RDT, current study uses more appropriate Favre-RDT.


Progress1
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Inviscid conservation equations
Inviscid Different Pressure Regimes and Effect of Thermodynamic Fluctuations Conservation Equations

(Mass)

(Momentum)

(Energy)


Reynolds vs favre averaging
Reynolds vs. Favre-averaging Different Pressure Regimes and Effect of Thermodynamic Fluctuations


Decomposition of variables
Decomposition of variables Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Substitutions:


Mean field governing eqns
Mean field Governing Eqns. Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Apply averaging principle and decompose density


Path to fluctuating field eqns
Path to Fluctuating Field Eqns. Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Subtract mean from instantaneous

  • Apply homogeneity condition(shear flow only)

  • Apply linear approximations.


Linear f rdt eqns for fluctuations
Linear F-RDT Eqns. for Fluctuations Different Pressure Regimes and Effect of Thermodynamic Fluctuations


Physical to fourier space
Physical to Fourier Space Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Easier to solve in Fourier space

  • Apply Fourier transform to variables

  • PDEs become ODEs


Homogeneous shear flow eqns
Homogeneous shear flow eqns. Different Pressure Regimes and Effect of Thermodynamic Fluctuations


Final moment equations
Final Different Pressure Regimes and Effect of Thermodynamic Fluctuationsmoment equations


Important parameters
Important Parameters Different Pressure Regimes and Effect of Thermodynamic Fluctuations


Validation b 12 anisotropy component
Validation- b Different Pressure Regimes and Effect of Thermodynamic Fluctuations12 Anisotropy Component

DNS

R-RDT

F-RDT

Good overall agreement


Validation ke growth rate
Validation- KE Growth Rate Different Pressure Regimes and Effect of Thermodynamic Fluctuations

DNS

R-RDT

F-RDT


Progress2
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Three stage behavior shear time
Three-stage Behavior: Shear Time Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Peel-off from burger’s limit clear; shows regime transition.

*Verification of behavior found in Cambon et. al.


Status before current work
Status Before Current Work Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Validation of method and verification of previous results complete.

  • New investigations of three-stage physics follows.


Three stage behavior acoustic time
Three-stage Behavior: Acoustic Time Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Three-stages clearly defined; final regime begins within 2-3 acoustic times.

*Acoustic timescale first presented in Lavin et al.


Three stage behavior mixed time
Three-stage Behavior: Mixed Time Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Three-stages clearly defined; onset of second regime align.


Regimes of evolution
Regimes of Evolution Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Regime 1:

  • Regime 2:

  • Regime 3:


Evolution of gradient mach number
Evolution of Gradient Mach Number Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Shear time aligns 1st regime, constant Mg value.

Mg(t) reaches 1 by 1 acoustic time regardless of initial value.


Evolution of turbulent mach number
Evolution of Turbulent Mach Number Different Pressure Regimes and Effect of Thermodynamic Fluctuations

First regime over by 4 shear times.

Second regime aligns in mixed time.


Three regime physics regime 1
Three Regime Physics: Regime 1 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Pressure plays an insignificant role in 1st regime.


Three regime physics regime 11
Three Regime Physics: Regime 1 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Zero pressure fluctuations.

Dilatational and internal energy stay at initial values.

No flow-thermodynamic interactions.


Three regime physics regime 2
Three Regime Physics: Regime 2 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Pressure works to nullify production in 2nd regime.


Three regime physics regime 21
Three Regime Physics: Regime 2 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Pressure fluctuations build up.

Dilatational K. E. and I. E. build up.

Equi-partition is achieved as will be seen later.


Three regime physics regime 3
Three Regime Physics: Regime 3 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Rapid pressure strain correlation settles to a constant value


Three regime physics regime 31
Three Regime Physics: Regime 3 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Production nearly insensitive to initial Mg value.


Three regime physics regime 32
Three Regime Physics: Regime 3 Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Energy growth rates nearly independent of Mg.

  • p’(total) =p’(poisson) + p’(acoustic wave).


Three regime conclusions
Three-regime conclusions Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Regime 1: Turbulence evolves as Burger’s limit; pressure insignificant.

  • Regime 2: Pressure works to nullify production; turbulence growth nearly zero.

  • Regime 3: Turbulence evolves similar to the incompressible limit.


Progress3
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Polytropic coefficient
Polytropic Different Pressure Regimes and Effect of Thermodynamic Fluctuations Coefficient

R-RDT

F-RDT

n≈γ according to DNS with no heat loss (Blaisdell and Ristorcelli)

F-RDT preserves entropy, R-RDT does not


Progress4
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Ke initial temperature fluctuation
KE: Initial Temperature Fluctuation Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Initial temperature fluctuations delay onset of second regime.


Ke initial turbulent mach number
KE: Initial Turbulent Mach Number Different Pressure Regimes and Effect of Thermodynamic Fluctuations

KE evolution influenced by initial Mt only weakly


Equi partition function initial temperature fluctuation
Equi Different Pressure Regimes and Effect of Thermodynamic Fluctuations-Partition Function: Initial Temperature Fluctuation

Dilatational energy maintains dominant role longer.


Equi partition function initial turbulent mach number
Equi Different Pressure Regimes and Effect of Thermodynamic Fluctuations-Partition Function: Initial Turbulent Mach Number

Balance of energies nearly independent of initial Mt value


Regime 1 2 transition
Regime 1-2 Transition Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Initial Temperature fluctuation

Initial Turbulent Mach number

1st transition heavily dependent on temperature fluctuations


Regime 2 3 transition
Regime 2-3 Transition Different Pressure Regimes and Effect of Thermodynamic Fluctuations

Initial Temperature fluctuation

Initial Turbulent Mach number

2nd transition occurs within 4 acoustic times regardless of initial conditions


Initial fluctuations conclusions
Initial fluctuations conclusions Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Turbulence evolution heavily influenced by temperature fluctuations.

  • Velocity fluctuations weakly influence flow.

  • Regime 1-2 transition delayed by temperature fluctuations.

  • Regime 2-3 transition occurs before 4 acoustic times.


Progress5
Progress Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Introduction

  • RDT Linear Analysis of Compressible Turbulence

    • Method

    • 3-Stage Evolution of Flow Variables

    • Evolution of Thermodynamic Variables

    • Effect of Initial Thermodynamic Fluctuations

  • Conclusions


Conclusions
Conclusions Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • F-RDT approach achieves more accurate results than R-RDT.

  • Flow field statistics exhibit a three-regime evolution verification.

  • Role of pressure in each role is examined:

    • Regime 1: pressure insignificant

    • Regime 2: pressure nullifies production

    • Regime 3: pressure behaves as in incompressible limit.

  • Initial thermodynamic fluctuations have a major influence on evolution of flow field.

  • Initial velocity fluctuations weakly affect turbulence evolution.


Contributions of present work
Contributions of Present Work Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • Explains the physics of three-stages.

  • Role of initial thermodynamic fluctuations quantified.

  • Aided in improving to compressible turbulence modeling.


References
References Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • S. B. Pope. Turbulent Flows. Cambridge University Press, 2000.

  • G. K. Batchelor and I. Proudman. "The effect of rapid distortion of a fluid in turbulent motion." Q. J. Mech. Appl. Math. 7:121-152, 1954.

  • C. Cambon, G. N. Coleman and D. N. N. Mansour. "Rapid distortion analysis and direct simulation of compressible homogeneous turbulence at finite Mach number." J. Fluid Mech., 257:641-665, 1993.

  • G. Brethouwer. "The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport, linear theory and direct numerical simulations." J. Fluid Mech., 542:305-342, 2005.

  • P.A. Durbin and O. Zeman. "Rapid distortion theory for homogeneous compressed turbulence with application to modeling." J. Fluid Mech., 242:349-370, 1992.

  • G. A. Blaisdell, G. N. Coleman and N. N. Mansour. "Rapid distortion theory for compressible homogeneous turbulence under isotropic mean strain." Phys. Fluids, 8:2692-2705, 1996.

  • G. N. Coleman and N. N. Mansour. "Simulation and modeling of homogeneous compressible turbulence under isotropic mean compression." in Turbulent Shear Flows 8, pgs. 269-282, Berlin:Springer-Verlag, 1993


References cont
References cont. Different Pressure Regimes and Effect of Thermodynamic Fluctuations

  • L. Jacquin, C. Cambon and E. Blin. "Turbulence amplification by a shock wave and rapid distortion theory." Phys. Fluids A, 5:2539, 1993.

  • A. Simone, G. N. Coleman and C. Cambon. "The effect of compressibility on turbulent shear flow: a rapid distortion theory and direct numerical simulation study." J. Fluid Mech., 330:307-338, 1997.

  • H. Yu and S. S. Girimaji. "Extension of compressible ideal-gas RDT to general mean velocity gradients." Phys. Fluids 19, 2007.

  • S. Suman, S. S. Girimaji, H. Yu and T. Lavin. "Rapid distortion of Favre-averaged Navier-Stokes equations." Submitted for publication in J. FLuid Mech., 2009.

  • S. Suman, S. S. Girimaji and R. L. Bertsch. "Homogeneously-sheared compressible turbulence at rapid distortion limit: Interaction between velocity and thermodynamic fluctuations."

  • T. Lavin. Reynolds and Favre-Averaged Rapid Distortion Theory for Compressible, Ideal Gas Turbulence}. A Master's Thesis. Department of Aerospace Engineering. Texas A \& M University. 2007.


Questions
Questions… Different Pressure Regimes and Effect of Thermodynamic Fluctuations


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