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Navier-Stokes Equations. DNS. Body force effects. Linear Theories: RDT. 7-eqn. RANS. Realizability, Consistency. Spectral and non-linear theories. ARSM -ASFM reduction. 2-eqn. RANS. Averaging Invariance. 2-eqn. PANS. Near-wall treatment, limiters, realizability correction.

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Arsm asfm reduction

Navier-Stokes Equations

DNS

Body force effects

Linear Theories: RDT

7-eqn. RANS

Realizability, Consistency

Spectral and non-linear theories

ARSM -ASFM reduction

2-eqn. RANS

Averaging Invariance

2-eqn. PANS

Near-wall treatment, limiters, realizability correction

Numerical methods and grid issues

Application

Dr. Girimaji Research Road map

DNS

LES

RANS


Arsm asfm reduction

Objective

  • Need for a new approach to modeling the scalar flux considering compressibility effects Mg effect

  • Application: Turbulent combustion/mixing in hypersonic aircrafts

  • Physical sequence of mixing:

Turbulent Stirring

Molecular Mixing

Chemical Reaction


Arsm asfm reduction

Velocity Field

(ARSM),

Scalar Dissipation Rate,

Turbulent Stirring

Molecular Mixing

Chemical Reaction

[Gaurav]

[Carlos]

Scalar Flux Field

(ASFM),

[Mona]


Arsm asfm reduction

Modeling

Differential Transport eq.

Constitutive Relations

Weak Equilibrium

assumption

Representation theory

Reduced Differential

algebraic


Arsm asfm reduction

ARSM:

Weak equilibrium assumption

ASFM with variable Pr_teffect


Arsm asfm reduction

Step (1) the evolution of passive scalar flux

  • Step (2) Assumptions:

  • the isotropy of small scales

  • weak equilibrium condition, advection and diffusion terms  0

Step (3) Pressure –scalar gradient correlation


Arsm asfm reduction

Step (3) Modeling Pressure-scalar gradient correlation

High Mg- pressure effect is negligible.

Intermediate Mg - pressure nullifies inertial effects.

Low Mg – Incompressible limit

[Craft & Launder, 1996]

Step (4) Applying ARSM by Girimaji’s group


Arsm asfm reduction

Step (4) using ARSM developed by Girimaji group,

[Wikström et al, 2000]

: = Tensorial eddy diffusivity


Arsm asfm reduction

Standard k-ε model

1-a) with constant- Cμ =0.09

1-b) variable- Cμ with Mg effects which uses the linear ARSM [Gomez & Girimaji ]

Assume Pr_t = 0.85

Variable tensorial diffusivity


Arsm asfm reduction

Geometry of planar mixing layer

Isentropic relations (compressible flows)

0.025

Fast stream Tt1 = 295 K, M=2.01

Pressure inlet

slow stream Tt2 = 295 K, M=1.38

Pressure inlet

- 0.025

X=0

X=0.5

X=0.1

X=0.15

X=0.2

X=0.25

X=0.3

y

x

for both free-stream inlets

the turbulent intensity =0.01 %, turbulent viscosity ratio = 0.1


Arsm asfm reduction

Schematic of planar mixing layer

U1

M1

T1

Pressure-inlet

Ptot,1

Pstat,1

Pressure-outlet

Tout

NRBC: avg bd. press.

Fast

stream

Slow

stream

Pressure-inlet

Ptot,1

Pstat,1

U2

M2

T1


Arsm asfm reduction

  • Normalized mean total temperature

    The mean total temperature is normalized by initial mean temperature difference of

    two streams and cold stream temperature. Due to the Boundedness of the totaltemperature, the normalized value, in theory, should remain between zero and unity.

  • Eddy diffusivity (eddy diffusion coefficient)

For the approach (a), in which the turbulence model is the standard k-ε, the scalar diffusion on coefficient or eddy diffusivity is obtained by modeling the turbulent scalar transport using the concept of “Reynolds’ analogy” to turbulent momentum transfer. Thus, the modeled energy equation is given by


Arsm asfm reduction

  • Flux components

  • Constant-/variable-Cμ

  • Tensorial eddy diffusivity

  • Streamwisescalar flux:

  • Transversal scalar flux:.

  • Thickness growth rate [ongoing]


Arsm asfm reduction

1-a) Standard k-ε model with constant-Cμ


Case 5 mr 1 97

Normalized Temp Contours

Case -5Mr = 1.97

1-a) Standard k-ε model with constant-Cμ

Case -2Mr = 0.91

Case -3rMr = 1.44

Case -4Mr = 1.73


Case 5 mr 1 971
Case -5Mr = 1.97

Bounded Normalized Temp Contours

Case -2Mr = 0.91

Case -3rMr = 1.44

Case -4Mr = 1.73


Arsm asfm reduction

Fast stream

Slow stream

Normalized Temp Profile

1-a) Standard k-ε model with constant-Cμ


Arsm asfm reduction

Fast stream

Slow stream

Eddy diffusivity profile

1-a) Standard k-ε model with constant-Cμ


Arsm asfm reduction

Scalar flux components

1-a) Standard k-ε model with constant-Cμ

Streamwise scalar flux @ x=0.2

Fast stream

Slow stream


Arsm asfm reduction

Scalar flux components

1-a) Standard k-ε model with constant-Cμ

Transversal scalar flux @ x=0.2

Fast stream

Slow stream


Arsm asfm reduction

Eddy diffusivity profile for case 5 (Mr=1.97), @ different stations

Toward outlet

Fast stream

Slow stream


Arsm asfm reduction

1-a) Standard k-ε model with constant-Cμ

Comparing Scalar flux components, Axial vs. Transversal

for Mr-1.8 (case5) and Mr 0.97 (case2)


Arsm asfm reduction

1-b) Standard k-ε model with variable Cμ (Mg effect)


Arsm asfm reduction

Normalized Total Temp Profile @ x=0.02

Fast stream

Slow stream

1-a) Standard k-ε model

with constant-Cμ

1-b) Standard k-ε model

with variable Cμ (Mg effect)


Arsm asfm reduction

Eddy Diffusivity Profile @ x=0.02

1-a) Standard k-ε model

with constant-Cμ

1-b) Standard k-ε model

with variable Cμ (Mg effect)


Arsm asfm reduction

Streamwise scalar flux @ x=0.02

1-a) Standard k-ε model

with constant-Cμ

1-b) Standard k-ε model

with variable Cμ (Mg effect)


Arsm asfm reduction

Transversal scalar flux @ x=0.02

1-a) Standard k-ε model

with constant-Cμ

1-b) Standard k-ε model

with variable Cμ (Mg effect)


Arsm asfm reduction

Convergence issues

  • All simulations were continued until a self-similar profiles (for mean velocity and temperature) are achieved in different Mach cases.

  • Main Criterion to check convergence : imbalance of Flux (Mass flow rate ) across the boundaries (inlet & outlet) goes to zero. < 0.2%

  • Error-function profile  self-similarity state

  • Normalized mean stream-wise velocity

  • Normalized mean temperature