1. ARSM -ASFM reduction RANSLESDNS 2-eqn. RANS Averaging Invariance Application DNS 7-eqn. RANS Body force effects Linear Theories: RDT Realizability, Consistency Spectral and non- linear theories 2-eqn. PANS Near-wall treatment, limiters, realizability correction Numerical methods and grid issues Navier-Stokes Equations Dr. Girimaji Research Road map

2. Need for a new approach to modeling the scalar flux considering compressibility effects  Mg effect Application: Turbulent combustion/mixing in hypersonic aircrafts Objective Physical sequence of mixing: Turbulent StirringMolecular MixingChemical Reaction

3. Velocity Field (ARSM), Scalar Flux Field (ASFM), [Mona] [Carlos] [Gaurav] Scalar Dissipation Rate, Turbulent StirringMolecular MixingChemical Reaction Turbulent mixing

4. Differential Transport eq. Reduced Differential algebraic Modeling Weak Equilibrium assumption Representation theory Scalar Flux molding approaches Constitutive Relations

5. ARSM: Weak equilibrium assumption ASFM with variable Pr_t effect Algebraic Scalar Flux modeling approach:

6. 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

7. Algebraic Scalar Flux modeling approach (step-by-step) Step (3) Modeling Pressure-scalar gradient correlation 1. High Mg- pressure effect is negligible. 2. Intermediate Mg - pressure nullifies inertial effects. 3. Low Mg – Incompressible limit [Craft & Launder, 1996] Step (4) Applying ARSM by Girimaji’s group

8. Algebraic Scalar Flux modeling approach (step-by-step) Step (4) using ARSM developed by Girimaji group, [Wikström et al, 2000] : = Tensorial eddy diffusivity

9. 1. 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 2.Variable tensorial diffusivity Preliminary Validation of the Model

10. Isentropic relations (compressible flows) y x Fast stream T t1 = 295 K, M=2.01 Pressure inlet 0.025 - 0.025 X=0X=0.5X=0.1X=0.15X=0.2X=0.25X=0.3 slow stream T t2 = 295 K, M=1.38 Pressure inlet Geometry of planar mixing layer for both free-stream inlets the turbulent intensity =0.01 %, turbulent viscosity ratio = 0.1

11. Fast stream Slow stream Pressure-inlet P tot,1 P stat,1 Pressure-inlet P tot,1 P stat,1 U1M1T1U1M1T1 U2M2T1U2M2T1 Pressure-outlet T out NRBC: avg bd. press. Case 2Case 3rCase 4Case 5 R= U2/ U10.570.250.16 s =ρ2/ ρ11.550.580.601.14 Mr0.911.441.731.97 M1, M21.91, 1.362.22, 0.432.35, 0.32.27, 0.38 Tt1, Tt2, (K)578, 295315, 285360, 290675,300 U1, U2, (m/s)700, 399561, 142616, 1000830,131 P(kPa) Inlet pressure [G&D, 1991]495336.0532 Ps(kPa) inlet pressure [simulation]56,49.5574037 Schematic of planar mixing layer

12. 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 total temperature, 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 Post-processing

13. Flux components 1.Constant-/variable-Cμ 2.Tensorial eddy diffusivity Streamwise scalar flux: Transversal scalar flux:. Post-processing Thickness growth rate [ongoing]

14. 1-a) Standard k-ε model with constant-Cμ

15. Case -5 Mr = 1.97 Case -3r Mr = 1.44 Case -4 Mr = 1.73 Normalized Temp Contours Case -2 Mr = 0.91 1-a) Standard k-ε model with constant-Cμ

16. Case -5 Mr = 1.97 Case -3r Mr = 1.44 Case -4 Mr = 1.73 Bounded Normalized Temp Contours Case -2 Mr = 0.91

17. Normalized Temp Profile 1-a) Standard k-ε model with constant-Cμ Fast stream Slow stream

18. Eddy diffusivity profile 1-a) Standard k-ε model with constant-Cμ Fast stream Slow stream

19. Scalar flux components 1-a) Standard k-ε model with constant-Cμ Streamwise scalar flux @ x=0.2 Fast stream Slow stream

20. Scalar flux components 1-a) Standard k-ε model with constant-Cμ Transversal scalar flux @ x=0.2 Fast stream Slow stream

21. Eddy diffusivity profile for case 5 ( Mr=1.97), @ different stations Fast stream Slow stream Toward outlet

22. Comparing Scalar flux components, Axial vs. Transversal for Mr-1.8 (case5) and Mr 0.97 (case2) 1-a) Standard k-ε model with constant-Cμ

23. 1-b) Standard k-ε model with variable Cμ (Mg effect)

24. Normalized Total Temp Profile @ x=0.02 1-a) Standard k-ε model with constant-Cμ 1-b) Standard k-ε model with variable Cμ (Mg effect) Fast stream Slow stream

25. 1-a) Standard k-ε model with constant-Cμ 1-b) Standard k-ε model with variable Cμ (Mg effect) Eddy Diffusivity Profile @ x=0.02

26. 1-a) Standard k-ε model with constant-Cμ 1-b) Standard k-ε model with variable Cμ (Mg effect) Streamwise scalar flux @ x=0.02

27. 1-a) Standard k-ε model with constant-Cμ 1-b) Standard k-ε model with variable Cμ (Mg effect) Transversal scalar flux @ x=0.02

28. 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 1.Normalized mean stream-wise velocity 2.Normalized mean temperature Convergence issues