arsm -asfm reduction
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
• 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:
(1) (( , i
Mixing Process steps
Large - Scale 2)Molecular - Scale scalar due to turbulent, u ) (fuel & oxidizer)
6444444444447444444444448
Turbulent Stirring Molecular Mixing Chemical Reaction
Velocity Field(ARSM),
Scalar Flux Field(ASFM),
[Mona]
[Carlos] [Gaurav]
Scalar Dissipation Rate,
iu
i ju u
Turbulent Stirring Molecular Mixing Chemical Reaction
Turbulent mixing
iu
Differential Transport eq.
Reduced Differential algebraic
Modeling
Weak Equilibriumassumption
Representation theory
,( , , , , , , , )i i i j i j ,i gu F u u U Θ k k M
Scalar Flux molding approaches
2
2
6 .
3 .
1 . / 2
1 . / 2
11 .
i j
i
i
eqn s u u
eqn su
eqn k u
eqn k
Total eqn s
Constitutive Relations
ii
unormalized scalar flux
kk
:
: mm
k knormalized gradeint mean scalark x
ktime scale ratio r
k
/:
/
i i
2
TKE k ( u u )/2
half scalar variance : k ( )/2
:ξ
i i m g tTrans(ξ )= f ( Θ r,Ma ,Ma )mn mnS ,W , ,
( , , )mn mn mn mnb b S W rARSM:
ij( , , )i mn mn g t jS W r,Ma ,Ma D
Weak equilibrium assumption
3/2
( . )gS kMa Pressureeffect vs Inertia effecta
:tMa Turbulent Mach number
ASFM with variable Pr_t effect
ξi i m g tTrans(ξ )= f ( Θ r,Ma ,Ma )mn mn mnb ,S ,W , ,
Algebraic Scalar Flux modeling approach:
Algebraic Scalar Flux modeling approach (step-by-step)
Step (1) the evolution of passive scalar flux
ii i i
Du DiDt P i i j j j j iu u u U P
ii
ukk
( ) 1-2
i i iKD i
i iDt k k kk
D P PP/ 2i ik u u 2 / 2k
Step (2) Assumptions: • the isotropy of small scales • weak equilibrium condition, advection and diffusion terms 0
0i ( ) 0
D iiDt
D
12
i iKi k k kk
P PP
Step (3) Pressure –scalar gradient correlation i
Algebraic Scalar Flux modeling approach (step-by-step)
Step (3) Modeling Pressure-scalar gradient correlation i
1. High Mg- pressure effect is negligible.
2. Intermediate Mg - pressure nullifies inertial effects.
3. Low Mg – Incompressible limit
[Craft & Launder, 1996]
( )ri i P 0iDu
Dt
( ) ( )r s θi θi θiΠ Π Π
( ) :rθiModifying Π
(s)θiΠ unmodified .
( )1 5 2 3 4( ) jr i
i k i j j i jk j i j
UUkc c u u c u c u c u uk x k x x x
( ) 0ri
0iu
Step (4) Applying ARSM by Girimaji’s group
Algebraic Scalar Flux modeling approach (step-by-step)
Step (4) using ARSM developed by Girimaji group, ( , )ij ij ij ijb b S W
,1 1, (1 )2θ i ij j i, j j ij i j i i
k kN ξ = b Θ +U ξ N b Uε r
2 3
i j ijij
u ub
k
[Wikström et al, 2000])3
T
ijθ4i ij , j
θ
ij
1- cξ = ( )(b ΘN
1444444442444444443
3 21 1 1 22 2 ( ) 0N GN Q R N GQ R
/,/
θ1
21 θ5 θ4
22 θ5 θ4 S W
1 1G = 2c - tr bS -1-2 r
1R = 4( - c ) 1- c tr bΘ2
k1R = 4( - c ) 1- c tr (c S +c W)bΘ r2 k
T
θ4i j
θ
ij
1- c( )u u
N
: = Tensorial eddy diffusivity
,( , , , , , )ij TT ij ij ij j gS W k k M
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
,Prp t
i eff eff lami t
cu k k
x
Ti ijj
ux
Preliminary Validation of the Model
2 /Pr
pi lam i
t
C C ku k
Isentropic relations (compressible flows)
y
x
Fast stream Tt1 = 295 K, M=2.01 Pressure inlet
0.025
- 0.025X=0 X=0.5 X=0.1 X=0.15 X=0.2 X=0.25 X=0.3
slow stream Tt2 = 295 K, M=1.38 Pressure inlet
Geometry of planar mixing layer
for both free-stream inletsthe turbulent intensity =0.01 %, turbulent viscosity ratio = 0.1
Fast stream
Slowstream
Pressure-inletPtot,1
Pstat,1
Pressure-inletPtot,1
Pstat,1
U1M1T1
U2M2T1
Pressure-outletTout
NRBC: avg bd. press.
Case 2 Case 3r Case 4 Case 5
R= U2/ U1 0.57 0.25 0.16 0.16
s =ρ2/ ρ1 1.55 0.58 0.60 1.14
Mr 0.91 1.44 1.73 1.97
M1, M2 1.91, 1.36 2.22, 0.43 2.35, 0.3 2.27, 0.38
Tt1, Tt2, (K) 578, 295 315, 285 360, 290 675,300
U1, U2, (m/s) 700, 399 561, 142 616, 1000 830,131
P(kPa) Inlet pressure [G&D, 1991] 49 53 36.05 32
Ps(kPa) inlet pressure [simulation] 56,49.5 57 40 37
Schematic of planar mixing layer
•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.
2norm
T TTT
•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
t i i i j eff j i ij heffkinetic energy work pressure work
E u E u p k T u S
144444424444443
23ij eff j i i j k k ijeff
u u u
12v i iE c T u u
Post-processing
•Flux components1. Constant-/variable-Cμ
2. Tensorial eddy diffusivity Streamwise scalar flux:
Transversal scalar flux:.
Post-processing
•Thickness growth rate [ongoing]
1 1 2 ,2 1 1 ,1θ4
θ
1- cu ( ) u u u uN
2 2 2 ,2 2 1 ,1θ4
θ
1- cu ( ) u u u u
N
,i eff iu k
1-a) Standard k-ε model with constant-Cμ
Case -5Mr = 1.97
Case -3rMr = 1.44
Case -4Mr = 1.73
Normalized Temp Contours
Case -2Mr = 0.91
2norm
T TTT
1-a) Standard k-ε model with constant-Cμ
Case -5Mr = 1.97
Case -3rMr = 1.44
Case -4Mr = 1.73
Bounded Normalized Temp Contours
Case -2Mr = 0.91
0 1normT
Normalized Temp Profile 1-a) Standard k-ε model with constant-Cμ
0( )y yb
Fast stream
Slow stream
2norm
T TTT
Eddy diffusivity profile 1-a) Standard k-ε model with constant-Cμ
Fast stream
Slow stream
0( )y yb
effk
Scalar flux components 1-a) Standard k-ε model with constant-Cμ
Streamwise scalar flux @ x=0.2
Fast streamSlow stream
0( )y yb
1u
Scalar flux components 1-a) Standard k-ε model with constant-Cμ
Transversal scalar flux @ x=0.2
Fast streamSlow stream
0( )y yb
2u
Eddy diffusivity profile for case 5 (Mr=1.97), @ different stations
Fast stream
Slow stream
0( )y yb
effk
Toward outlet
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
1-b) Standard k-ε model with variable Cμ (Mg effect)
Normalized Total Temp Profile @ x=0.02
2norm
T TTT
1-a) Standard k-ε model with constant-Cμ
1-b) Standard k-ε model with variable Cμ (Mg effect)
Fast stream
Slow stream
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
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 1u
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 2u
• 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 state1.Normalized mean stream-wise velocity2.Normalized mean temperature
Convergence issues