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

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

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Rebecca Bertsch

Advisor: Dr. SharathGirimajiMarch 29, 2010

Supported by: NASA MURI and Hypersonic Center

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

- 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-Stokes

Sub-grid Modeling

RANS Modeling

Bousinessq approach

ARSM reduction

DNS

LES

Application

Background

Second moment closure

Decreasing Fidelity of Approach

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

- 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.

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

(Mass)

(Momentum)

(Energy)

Substitutions:

Apply averaging principle and decompose density

- Subtract mean from instantaneous
- Apply homogeneity condition(shear flow only)
- Apply linear approximations.

- Easier to solve in Fourier space
- Apply Fourier transform to variables
- PDEs become ODEs

DNS

R-RDT

F-RDT

Good overall agreement

DNS

R-RDT

F-RDT

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

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

*Verification of behavior found in Cambon et. al.

- Validation of method and verification of previous results complete.
- New investigations of three-stage physics follows.

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

*Acoustic timescale first presented in Lavin et al.

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

- Regime 1:
- Regime 2:
- Regime 3:

Shear time aligns 1st regime, constant Mg value.

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

First regime over by 4 shear times.

Second regime aligns in mixed time.

Pressure plays an insignificant role in 1st regime.

Zero pressure fluctuations.

Dilatational and internal energy stay at initial values.

No flow-thermodynamic interactions.

Pressure works to nullify production in 2nd regime.

Pressure fluctuations build up.

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

Equi-partition is achieved as will be seen later.

Rapid pressure strain correlation settles to a constant value

Production nearly insensitive to initial Mg value.

- Energy growth rates nearly independent of Mg.
- p’(total) =p’(poisson) + p’(acoustic wave).

- 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.

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

R-RDT

F-RDT

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

F-RDT preserves entropy, R-RDT does not

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- Conclusions

Initial temperature fluctuations delay onset of second regime.

KE evolution influenced by initial Mt only weakly

Dilatational energy maintains dominant role longer.

Balance of energies nearly independent of initial Mt value

Initial Temperature fluctuation

Initial Turbulent Mach number

1st transition heavily dependent on temperature fluctuations

Initial Temperature fluctuation

Initial Turbulent Mach number

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

- 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.

- Introduction
- RDT Linear Analysis of Compressible Turbulence
- Method
- 3-Stage Evolution of Flow Variables
- Evolution of Thermodynamic Variables
- Effect of Initial Thermodynamic Fluctuations

- 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.

- Explains the physics of three-stages.
- Role of initial thermodynamic fluctuations quantified.
- Aided in improving to compressible turbulence modeling.

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

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