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ARSM -ASFM reduction

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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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Presentation Transcript
slide1

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

slide2

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

slide3

Turbulent mixing

Velocity Field

(ARSM),

Scalar Dissipation Rate,

Turbulent Stirring

Molecular Mixing

Chemical Reaction

[Gaurav]

[Carlos]

Scalar Flux Field

(ASFM),

[Mona]

slide4

Scalar Flux molding approaches

Modeling

Differential Transport eq.

Constitutive Relations

Weak Equilibrium

assumption

Representation theory

Reduced Differential

algebraic

slide5

Algebraic Scalar Flux modeling approach:

ARSM:

Weak equilibrium assumption

ASFM with variable Pr_teffect

slide6

Algebraic Scalar Flux modeling approach (step-by-step)

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

slide7

Algebraic Scalar Flux modeling approach (step-by-step)

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

slide8

Algebraic Scalar Flux modeling approach (step-by-step)

Step (4) using ARSM developed by Girimaji group,

[Wikström et al, 2000]

: = Tensorial eddy diffusivity

slide9

Preliminary Validation of the Model

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

slide10

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

slide11

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

slide12

Post-processing

  • 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

slide13

Post-processing

  • Flux components
  • Constant-/variable-Cμ
  • Tensorial eddy diffusivity
  • Streamwisescalar flux:
  • Transversal scalar flux:.
  • Thickness growth rate [ongoing]
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

slide17

Fast stream

Slow stream

Normalized Temp Profile

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

slide18

Fast stream

Slow stream

Eddy diffusivity profile

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

slide19

Scalar flux components

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

Streamwise scalar flux @ x=0.2

Fast stream

Slow stream

slide20

Scalar flux components

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

Transversal scalar flux @ x=0.2

Fast stream

Slow stream

slide21

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

Toward outlet

Fast stream

Slow stream

slide22

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)

slide24

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)

slide25

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)

slide26

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)

slide27

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)

slide28

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
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