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

Fluctuations

Rebecca Bertsch

Advisor: Dr. Sharath GirimajiMarch 29, 2010

Supported by: NASA MURI and Hypersonic Center

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

• 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• 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.

Application

BackgroundNavier-Stokes

Bousinessq approach

ARSM reduction

Second moment closure

LES

Sub-grid Modeling RANS Modeling

DNS

Decreasing Fidelity of Approach

ARSM reduction

2-eqn. ARSM

Averaging Invariance

Application

7-eqn. SMC

Transport Processes

Linear Pressure

Effects: RDT

Nonlinear pressure effects

Spectral and dissipative processes

2-eqn. PANS

Navier-Stokes Equations

Objectives

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

2. Explain physics of three stage evolution of flow parameters

3. Investigate role of pressure in each stage of turbulence evolution

4. Investigate dependence of regime transitions

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

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

(Mass)

(Momentum)

(Energy)

Reynolds vs. Favre-averagingApproach R-RDT(Previous work)

Easier

F-RDT(Current Study)

More appropriate for compressible flow

Averaging Unweighted: Weighted:

Moments 2nd order: 3rd order:

# of PDEs 25 64

Decomposition of variables

Substitutions:

Mass`

Momentum

Energy

Mean field Governing Eqns.

Mass

Mom.

Energy

Apply averaging principle and decompose density

Path to Fluctuating Field Eqns.

• Subtract mean from instantaneous• Apply homogeneity condition(shear flow only)

• Apply linear approximations.

Mass

Mom.

Energy

Linear F-RDT Eqns. for Fluctuations

Physical to Fourier Space

• Easier to solve in Fourier space• Apply Fourier transform to variables

• PDEs become ODEs

xti

ketutxu

ˆ,"

xti

ketTtxT

ˆ,"

xti

kettx

'' ˆ,

ijj

i uixu

ˆ

Mass

Momentum

Energy

Evolution of

Homogeneous shear flow eqns.

Final moment equations

iijjjim

jmi

m

ijm

ji uTuRiuTu

Ri

x

Uuu

xU

uudt

uud ~~~~

**

iijjjim

jmi

m

ijm

ji TiRTiRx

U

xU

dt

d ~~~~

**

iijjjim

jmi

m

ijm

ji TRiuTuiR

x

Uu

xU

udt

ud ~~~~

**

*~~

immjm

im

i uiTRi

xU

udtud

*~~

immjm

im

i iTiRxU

dtd

** ~1~~

mimjm

im

i uuTiTRi

xU

udtud

** ~1~~

mimjm

im

i uTiTiRxU

dtd

*~1 mmm uTi

dtd

*mmmidt

d

*~1 mmm uuTi

dtd

Important ParametersInput Gradient Mach number

Turbulent Mach Number

Temperature Fluctuation Intensity

Output Turbulent Kinetic Energy

Turbulent Polytropic Coefficient

Equi-partition Function

Timescales Shear time

Acoustic time

Mixed time

RTSM g

ckM t

TTTs"

St

0at

tSaMSt og

2""iiuuk

2

2

''

''

ppp

n

0"2

"2~~2 TTc

uu

v

Validation- b12 Anisotropy Component DNS R-RDT

F-RDT

ijji

ij k

uub

31

2

""

Good overall agreement

Validation- KE Growth Rate DNS R-RDT

F-RDT

dtdk

Sk1

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

Peel-off from burger’s limit clear; shows regime transition.*Verification of behavior found in Cambon et. al.

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

Regimes of Evolution

• Regime 1:

• Regime 2:

• Regime 3:

gMSt 2~0

312~ 0 atM g

3at

Evolution of Gradient Mach NumberShear time aligns 1st regime, constant Mg value.

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

Evolution of Turbulent Mach Number

First regime over by 4 shear times.

Second regime aligns in mixed time.

Three Regime Physics: Regime 1

Pressure plays an insignificant role in 1st regime.

)(rijij

ji Pdt

uud

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

Pressure works to nullify production in 2nd regime.

)(rijij

ji Pdt

uud

Three Regime Physics: Regime 2Pressure 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

Rapid pressure strain correlation settles to a constant value

Three Regime Physics: Regime 3

Production nearly insensitive to initial Mg value.

Three Regime Physics: Regime 3

• Energy growth rates nearly independent of Mg.

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

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.

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

R-RDT F-RDT

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

F-RDT preserves entropy, R-RDT does not

2

2

''

''

ppp

n

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

Initial temperature fluctuations delay onset of second regime.

KE: Initial Turbulent Mach Number

KE evolution influenced by initial Mt only weakly

Equi-Partition Function: Initial Temperature Fluctuation

Dilatational energy maintains dominant role longer.

0"2

"2~~2 TTc

uu

v

Equi-Partition Function: Initial Turbulent Mach Number

Balance of energies nearly independent of initial Mt value

0"2

"2~~2 TTc

uu

v

Regime 1-2 Transition

Initial Temperature fluctuation

Initial Turbulent Mach number

1st transition heavily dependent on temperature fluctuations

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

• 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.

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• 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

1. Explains the physics of three-stages.

2. Role of initial thermodynamic fluctuations quantified.

3. Aided in improving to compressible turbulence modeling.

References1. S. B. Pope. Turbulent Flows. Cambridge University Press, 2000.

2. 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.

3. 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.

4. 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.

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

6. 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.

7. 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.

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

9. 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.

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

11. 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.

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

13. 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…