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
• 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.
Background
Navier-Stokes
Sub-grid Modeling
DNS
LES
Second
moment
closure
RANS Modeling
ARSM
reduction
Application
Decreasing Fidelity of Approach
Bousinessq
approach
Transport
Processes
Spectral and
dissipative
processes
Navier-Stokes
Equations
7-eqn. SMC
Linear
Pressure
Effects: RDT
ARSM
reduction
Nonlinear
pressure
effects
2-eqn. ARSM
Averaging
Invariance
2-eqn. PANS
Application
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-averaging
R-RDT(Previous work)
F-RDT(Current Study)
Easier
More appropriate for
compressible flow
Averaging
Unweighted:
Weighted:
Moments
2nd order:
3rd order:
# of PDEs
25
64
Approach
Decomposition of variables
Substitutions
Mass
`
Momentum
Energy
Mean field Governing Eqns.
Apply averaging principle and decompose density
Mass
Mom.
Energy
Path to Fluctuating Field Eqns.
• Subtract mean from instantaneous
• Apply homogeneity condition(shear flow only)
• Apply linear approximations.
Linear F-RDT Eqns. for Fluctuations
Mass
Mom.
Energy
Physical to Fourier Space
• Easier to solve in Fourier space
• Apply Fourier transform to variables
" 
ˆ i t x
u  x , t    u t e
k
 

i

T  x , t    Tˆ t e t x
k
 
' 
'
i t  x
  x , t    ˆ t e
"
k
ui
 i j uˆi
x j
• PDEs become ODEs
Homogeneous shear flow eqns.
Mass
Momentum
Energy
Evolution of
Final moment equations
~
~
d ui u j
U j
U i
R
R
*
*
~
~
 umu j
 ui um
 i  i u j   T u j   i  j ui   T ui  
dt
xm
xm


~
~
d  i j
U j
U i
*
*
~
~
  m j
  i m
 iR i  j   T  j   iR j  i   T  i  
dt
xm
xm
~
~
d  iu j
U j
U i
R
*
*
~
~
  mu j
  i um
 iR i u j   T u j   i  j  i   T  i  
dt
xm
xm







~
d ui 
U i
R
*
~
 um 
 i  j   T    i m  mui
dt
xm

~
di 
U
*
~
  m  i  iR j   T    i m  m i
dt
xm
~
d ui 
U i
R
*
~ *
~
 um 
 i  j   T   i  1 T  m ui um
dt
xm

~
di 
U i
*
~ *
~
  m 
 iR j   T   i  1 T  m  i um
dt
xm
d 
*
~
 i m   1 T um    m 
dt
d 
*
 i m  m    m 
dt
d 
*
~
 i  1 T  m um   um 
dt










Important Parameters
M g  S
Input
Output
RT
Mt  k c
Turbulent Mach Number
Ts  T " T
Temperature Fluctuation Intensity
k  u u 2
" "
i i
p' p' p 2
n
''  2

Timescales
St
Gradient Mach number
u2" u2"
~ ~
2cv T  T0 
St
at  0
M g  Sa  o t
Turbulent Kinetic Energy
Turbulent Polytropic Coefficient
Equi-partition Function
Shear time
Acoustic time
Mixed time
Validation- b12 Anisotropy Component
R-RDT
DNS
" "
i j
uu 1
bij 
  ij
2k 3
F-RDT
Good overall agreement
Validation- KE Growth Rate
DNS
R-RDT
F-RDT
1 dk

Sk dt
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:
0  St ~ 2 M g
• Regime 2:
0
~2
 at  1  3
Mg
• Regime 3:
at  3
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
First regime over by 4
shear times.
Second regime aligns
in mixed time.
Three Regime Physics: Regime 1
d ui u j
dt
 Pij   ij(r )
Pressure plays an insignificant role in 1st regime.
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
d ui u j
dt
 Pij   ij(r )
Pressure works to nullify production in 2nd regime.
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
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
p' p' p 2
n
''  2
F-RDT
n≈γ according to DNS with no heat loss (Blaisdell and Ristorcelli)
F-RDT preserves entropy, R-RDT does not
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
u2" u2"

~ ~
Temperature Fluctuation

2cv T  T0 
Dilatational energy maintains dominant role longer.
Equi-Partition Function: Initial
u2" u2"

~ ~
Turbulent Mach Number

2cv T  T0 
Balance of energies nearly independent of initial Mt value
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 RRDT.
• 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.
References
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turbulent motion." Q. J. Mech. Appl. Math. 7:121-152, 1954.
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direct simulation of compressible homogeneous turbulence at finite Mach
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turbulence and passive scalar transport, linear theory and direct numerical
simulations." J. Fluid Mech., 542:305-342, 2005.
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compressed turbulence with application to modeling." J. Fluid Mech., 242:349370, 1992.
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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.
9.
10.
11.
12.
13.
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 NavierStokes 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…