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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

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Outline

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

  • 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

  • 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

  • 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.


Outline

Navier-Stokes

Sub-grid Modeling

RANS Modeling

Bousinessq approach

ARSM reduction

DNS

LES

Application

Background

Second moment closure

Decreasing Fidelity of Approach


Outline

Transport Processes

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

  • 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

  • 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 Conservation Equations

(Mass)

(Momentum)

(Energy)


Reynolds vs favre averaging

Reynolds vs. Favre-averaging


Decomposition of variables

Decomposition of variables

Substitutions:


Mean field governing eqns

Mean field Governing Eqns.

Apply averaging principle and decompose density


Path to fluctuating field eqns

Path to Fluctuating Field Eqns.

  • 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


Physical to fourier space

Physical to Fourier Space

  • Easier to solve in Fourier space

  • Apply Fourier transform to variables

  • PDEs become ODEs


Homogeneous shear flow eqns

Homogeneous shear flow eqns.


Final moment equations

Final moment equations


Important parameters

Important Parameters


Validation b 12 anisotropy component

Validation- b12 Anisotropy Component

DNS

R-RDT

F-RDT

Good overall agreement


Validation ke growth rate

Validation- KE Growth Rate

DNS

R-RDT

F-RDT


Progress2

Progress

  • 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

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

  • 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

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

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


Regimes of evolution

Regimes of Evolution

  • Regime 1:

  • Regime 2:

  • Regime 3:


Evolution of gradient mach number

Evolution of Gradient Mach Number

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

First regime over by 4 shear times.

Second regime aligns in mixed time.


Three regime physics regime 1

Three Regime Physics: Regime 1

Pressure plays an insignificant role in 1st regime.


Three regime physics regime 11

Three Regime Physics: Regime 1

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

Pressure works to nullify production in 2nd regime.


Three regime physics regime 21

Three Regime Physics: Regime 2

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

Rapid pressure strain correlation settles to a constant value


Three regime physics regime 31

Three Regime Physics: Regime 3

Production nearly insensitive to initial Mg value.


Three regime physics regime 32

Three Regime Physics: Regime 3

  • Energy growth rates nearly independent of Mg.

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


Three regime conclusions

Three-regime conclusions

  • 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

  • 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 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

  • 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

Initial temperature fluctuations delay onset of second regime.


Ke initial turbulent mach number

KE: Initial Turbulent Mach Number

KE evolution influenced by initial Mt only weakly


Equi partition function initial temperature fluctuation

Equi-Partition Function: Initial Temperature Fluctuation

Dilatational energy maintains dominant role longer.


Equi partition function initial turbulent mach number

Equi-Partition Function: Initial Turbulent Mach Number

Balance of energies nearly independent of initial Mt value


Regime 1 2 transition

Regime 1-2 Transition

Initial Temperature fluctuation

Initial Turbulent Mach number

1st transition heavily dependent on temperature fluctuations


Regime 2 3 transition

Regime 2-3 Transition

Initial Temperature fluctuation

Initial Turbulent Mach number

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


Initial fluctuations conclusions

Initial fluctuations conclusions

  • 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

  • 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

  • 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

  • Explains the physics of three-stages.

  • Role of initial thermodynamic fluctuations quantified.

  • Aided in improving to compressible turbulence modeling.


References

References

  • 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.

  • 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…


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