ANALYSIS OF “POOR MAN’S NAVIER−STOKES”
AND THERMAL ENERGY EQUATIONS FOR
HIGH-RAYLEIGH NUMBER TURBULENT CONVECTION
J. M. McDonough
Departments of Mechanical Engineering and Mathematics
University of Kentucky, Lexington, KY 40506-0503
E-mail: jmmcd@uky.edu
Website URL: http://www.engr.uky.edu/~acfd
Acknowledgments: Reported work supported in part by grants from AFOSR and
NASA-EPSCoR:
Presented at American Physical Society 58 th Division of Fluid Dynamics Meeting, Chicago, IL,
Nov. 20−22, 2005
University of Kentucky Advanced CFD & Immersive Visualization Lab
CF
D
BACKGROUND/MOTIVATION
Efficient LES Models Able to Account for Interactions of Physics on
Small Scales Still in Their Infancy
Physically-Realistic Temporal Fluctuations Required for Such Models
Implies Need to Retain as Much of Governing Equations as Possible
within Framework of Efficient Model
Possible Approach: use symbol (or, more accurately, a pseudo-differential operator) of governing equations
Present Work Describes Way to Do This, and Demonstrates Outcomes in Terms of Comparisons with Actual Physics
OUTLINE OF PMNS + THERMAL ENERGY EQ. DERIVATION
Poor Man’s Navier-Stokes (PMNS) (McDonough & Huang, Int. J. Numer.
Meth. Fluids, 2004) Plus Thermal Energy Equations Obtained from Following Sequence of Steps
Invoke Leray projection to remove ∇p from momentum equations
Assume Fourier representations for dependent variables u,v,T with basis
functions exhibiting complex exponential-like differentiation properties
Construct Galerkin ODEs
∞
k 2
(2)
(1)
⋅
2
ak = −∑ Almk al + Almk am bl − —— a k
Re
l,m=1
∞
k 2
bk = −∑ Blmk bl + Blmk am bl − —— bk − Gr
—2 ck
Re
l,m=1
Re
⋅
(1)
2
(2)
∞
k 2
(2)
(1)
⋅
ck = −∑ Clmk am cl + Clmk bm cl − —— ck ,
Pe
l,m=1
k = 1, … , ∞
PMNS/Thermal Energy Eq. Derivation (Cont.)
Decimate infinite system to single wavevector
2
(1)
(2)
k
a⋅k + Almk ak2 + Almk ak bk = −—– ak
Re
⋅
(1)
k 2
(2)
Gr
bk + Blmk bk + Blmk ak bk = −—–
b
−—
ck
k
2
Re
2
Re
(1)
(2)
k 2
⋅
c k + Clmk ak ck + Clmk bk ck = −—– ck
Pe
Remark: right-hand sides are symbols, and left-hand sides similar to
pseudo-differential operators
Use simple Euler time integrators (and suppress indices)
a
(n+1)
b
(n+1)
c
(n+1)
=a
(n)
=b
(n)
=c
(n)
k 2 (n)
−τ
[ Re a
−τ
[
[
k 2 (n)
Re
(1)
b
(
(1)
(b )
+A
+B
(n) (n)
2
(1)
a (n)
−τ C a c +C
(n)
(2)
)
2
]
+ A a (n) b (n)
(2)
(2)
(n) (n)
+B a
b
Re
k 2
] ⁄ (1+ —
Pe )
b (n) c (n)
]
Gr
+—
2 c
PMNS/Thermal Energy Eq. Derivation (Cont.)
Employ transformations of May (Nature, 1976) and McDonough & Huang
(Int. J. Numer. Meth. Fluids, 2004) to obtain final form of discrete dynamical system (DDS)
a (n+1) = β1 a(n)(1−a (n)) − γ12 a(n) b(n)
b (n+1) = β2 b(n)(1−b (n)) − γ21 a(n) b (n) + αT c (n)
(
c (n+1) = − γuT a
(n+1)
+ γvT b
2-D PMNS+Thermal
Energy Equations
) c ⁄ (1+βT ) + c0
(n+1)
(n)
Total of 9 bifurcation parameters: β1 , β2 , βT , γ12 , γ 21 , γuT , γvT , αT , c0
In a complete LES all computed “on the fly” based on resolved-scale
Physical Interpretation of
γij s Treated in (McDonough et al., JoT, 2003)
Other Parameters to Be Discussed Here
DIMENSIONLESS PARAMETERS
Scaling Employed Leads to Re, Pe, Gr/Re 2
Must Relate These to Bifurcation Parameters β s, and α
But Equations Solved (i.e., evaluated) on Small Local Subdomains, Implying Dimensionless Parameters Need Rescaling
h 2 ∇u
Re → Reh ≡ 
,
ν
h 2 ∇u
Pe → Peh ≡ 
,
ν
βgδTh 3
Gr → Grh ≡ 
ν2
Rescaled parameters related to PMNS eqs. bifurcation parameters as
β1,2 = 4
(
τk2
1 − 
Reh
),
βT = Pe h ,
αT = Grh /Reh2
Desirable to Express These in Terms of Original (un-rescaled) Dimensionless Parameters to Permit Direct Comparisons with Experiment
BIFURCATION PARAMETERS ‫ض‬PHYSICAL PARAMETERS
Define Three Constants
2
C 1 = τ k , C 2,i = ∆ui h i ,
C3 = ( ∑
i
hi2
)[ ∑(∆ui ⁄ hj ) ]
2
i,j
Then It Can Be Shown That
C1 / C 2,i
Re =  ,
1 −βi /4
and
C1
Pe = 
C 3 βT
It Can Also Be Shown That Using Ra = GrPr Leads to
C12
αT
Ra = 
τ 
C2,i C3 βT (1 −β i /4)
Finally, It Can Be Shown That
1
Pr = 
β (1 − β 2 /4) ,
T
and
C 1 /C 3 〈b,c〉
Nu = 1 +  
βT ∆ c/h 2
1/2
RESULTS
Previously Have Shown Ability of PMNS + Thermal Energy Equation
to Match Experimental Results of Gollub & Benson (JFM, 1980)
Results mainly qualitative
Time series produced by model
resemble those of experiments
as model parameter related to
Ra is increased
But does demonstrate ability of
symbol/pseudo-differential oprator based models to replicate
temporal physics
Here, Additional More Quantitative Results Associated with Reproducing Experimental Correlations Are Presented
RESULTS (Cont.)
Comparisons Made with Cioni et al. Data (JFM, 1997)
High-Ra Convection with Pr = 0.025
SUMMARY AND CONCLUSIONS
Outline Given for Derivation of Discrete Dynamical System Proposed for Use as Part of LES-Like SGS Models
Correspondence Between Model Bifurcation Parameters and
Those of Original Governing Equations Demonstrated
Possibility to Accurately Match Experimental Data Shown in
Terms of Both Temporal Fluctuations and Global Quantities
Suggests Form of Model Able to Retain Considerable Physical
Detail Despite Its Simplicity and Efficiency
Hence, Models of This Form May Be Useful as SGS Models for
LES (and maybe RANS), and Also for Real-Time Control