A Hamilton–Jacobi differential equation is used to naturally and smoothly (via Dirichlet boundary conditions) set turbulence length scales in separated flow regions based on traditional expected length scales. Such zones occur for example in rim-seals. The approach is investigated using two test cases, flow over a cylinder at a Reynolds number of 140,000 and flow over a rectangular cavity at a Reynolds number of 50,000. The Nee–Kovasznay turbulence model is investigated using this approach. Predicted drag coefficients for the cylinder test-case show significant (15%) improvement over standard steady RANS and are comparable with URANS results. The mean flow-field also shows a significant improvement over URANS. The error in re-attachment length is improved by 180% compared with the steady RANS k-ω model. The wake velocity profile at a location downstream shows improvement and the URANS profile is inaccurate in comparison. For the cavity case, the HJ–NK approach is generally comparable with the other RANS models for measured velocity profiles. Predicted drag coefficients are compared with large eddy simulation. The new approach shows a 20–30% improvement in predicted drag coefficients compared with standard one and two equation RANS models. The shape of the recirculation region within the cavity is also much improved.

References

References
1.
Baldwin
,
B. S.
, and
Lomax
,
H.
,
1978
, “
Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows
,” AIAA Paper No. 78-257.
2.
Baldwin
,
B. S.
, and
Barth
,
T. J.
,
1990
, “
A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows
,” NASA Technical Memorandum No. 102847.
3.
Spalart
,
P. R.
, and
Allmaras
,
S. R.
,
1992
, “
A One-Equation Turbulence Model for Aerodynamic Flows
,” AIAA Paper No. 92-0439.
4.
Secundov
,
N.
,
Strelets
,
M.
Kh.
, and
Travin
,
A. K.
,
2001
, “
Generalization of vt-92 Turbulence Model for Shear-Free and Stagnation Point Flows
,”
ASME J. Fluids Eng.
,
123
, pp.
11
15
.10.1115/1.1341196
5.
Nikuradse
,
J.
,
1932
,
Gesetzmasigkeit der Turbulenten Stromung in glatten Rohren
, Vol.
356
,
VDI-Verlang
,
Berlin
.
6.
Klostermeier
,
C.
,
2008
, Investigation Into the Capability of Large Eddy Simulation for Turbomachinery Design Ph.D. thesis, University of Cambridge, Cambridge, UK.
7.
Leschziner
,
M. A.
,
2000
, “
Turbulence Modelling for Separated Flows With Anisotropy-Resolving Closures
,”
Philos. Trans. R. Soc. London
,
358
, pp.
3247
3277
.10.1098/rsta.2000.0707
8.
Aupoix
,
B.
, and
Spalart
,
P. R.
,
2003
, “
Extensions of the Spalart-Allmaras Turbulence Model to Account for Wall Roughness
,”
Int. J. Heat Fluid Flow
,
24
, pp.
454
462
.10.1016/S0142-727X(03)00043-2
9.
Fares
,
E.
, and
Schroder
,
W.
,
2002
, “
A Differential Equation to Determine the Wall Distance
,”
Int. J. Numer. Methods Fluids
,
39
, pp.
743
762
.10.1002/fld.348
10.
Spalding
,
D. B.
,
1972
, “
A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives
,”
Int. J. Numer. Methods Eng.
,
4
, pp.
551
561
.10.1002/nme.1620040409
11.
Launder
,
B. E.
,
Reece
,
G. J.
, and
Rodi
,
W.
,
1975
, “
Progress in the Development of a Reynolds Stress Turbulence Closure
,”
J. Fluid Mech.
,
68
(
3
), pp.
537
566
.10.1017/S0022112075001814
12.
Tucker
,
P. G.
,
Rumsey
,
C. L.
,
Spalart
,
P. R.
,
Bartels
,
R. E.
, and
Biedron
,
R. T.
,
2005
, “
Computations of Wall Distances Based on Differential Equations
,”
AIAA J.
,
43
(
3
), pp.
539
549
.10.2514/1.8626
13.
Tucker
,
P. G.
,
2003
, “
Differential Equation Based Wall Distance Computation for DES and RANS
,”
J. Comput. Phys.
,
190
(
1
), pp.
229
248
.10.1016/S0021-9991(03)00272-9
14.
Cantwell
,
B.
, and
Coles
,
D.
,
1983
, “
An Experimental Study of Entrainment and Transport in the Turbulent Near Wake of a Circular Cylinder
,”
J. Fluid Mech.
,
136
, pp.
321
374
.10.1017/S0022112083002189
15.
Grace
,
S. M.
,
Dewar
,
G. W.
, and
Wroblewski
,
D. E.
,
2004
,
“Experimental Investigation of the Flow Characteristics Within a Shallow Wall Cavity for Both Laminar and Turbulent Upstream Boundary Layers,”
Exp. Fluids
,
36
, pp.
791
804
.10.1007/s00348-003-0761-3
16.
Jefferson-Loveday
,
R. J.
,
2008
, “
Numerical Simulations of Unsteady Impinging Jet Flows
,” Ph.D. thesis, Swansea University, Swansea, UK.
17.
Nee
,
V.
, and
Kovasznay
,
S. G.
,
1969
, “
Simple Phenomenological Theory of Turbulent Shear Flows
,”
Phys. Fluids
,
12
(
3
), pp.
473
484
.10.1063/1.1692510
18.
Tucker
,
P. G.
,
2011
,
Computation of Unsteady Turbomachinery Flows: Part I—Progress and Challenges,”
Prog. Aerosp. Sci.
,
47
, pp.
522
545
.10.1016/j.paerosci.2011.06.004
19.
Antionia
,
R. A.
, and
Browne
,
L. W. B.
,
1987
, “
Conventional and Conditional Prandtl Number in a Turbulent Plane Wake
,”
Int. J. Heat Mass Transfer
,
30
(
10
), pp.
2023
2030
.10.1016/0017-9310(87)90083-4
20.
Rebollo
,
M. R.
,
1972
, “
Analytical and Experimental Investigation of a Turbulent Mixing Layer of Different Gases in a Pressure Gradient
,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
21.
Versteeg
,
H. K.
, and
Malalasekera
,
W.
,
1995
,
An Introduction to Computational Fluid Dynamics
,
Prentice-Hall
,
Englewood Cliffs, NJ
, Chap. 3.
22.
Wilcox
,
D. C.
,
1988
, “
Reassessment of the Scale Determining Equation for Advanced Turbulence Models
,”
AIAA J.
,
26
(
11
), pp.
1299
1310
.10.2514/3.10041
23.
Menter
,
F. R.
,
1993
, “
Zonal Two Equation k − ω Turbulence Models for Aerodynamic Flows
,” AIAA Paper No. AIAA-93-2906.
24.
Wieselsberger
,
C.
,
Betz
,
A.
, and
Prandtl
,
L.
,
1923
,
Versuche uber den Widerstand gerundeter und kantiger Korper
,
Ergebnisse AVA Gottingen
,
Gottingen, Germany
, pp.
22
32
.
25.
Birch
,
S. F.
,
1997
, “
The Role of Structure in Turbulent Mixing
,” AIAA Paper No. 97-2636.
26.
Lukovic
,
B
.,
2002
, “
Modeling Unsteadiness in Steady Simulations With Neural Network Generated Lumped Deterministic Source Terms
,” Ph.D. thesis, University of Cincinnati, Cincinnati, OH.
27.
Gharib
,
M.
, and
Roshko
,
A.
, “
The Effect of Flow Oscillations on Cavity Drag
,”
J. Fluid Mech.
,
177
, pp.
501
530
.10.1017/S002211208700106X
28.
Jefferson-Loveday
,
R. J.
, and
Tucker
,
P. G.
,
2010
, “
LES Impingement Heat Transfer on a Concave Surface
,”
Numer. Heat Transfer, Part A
,
58
, pp.
247
271
.10.1080/10407782.2010.505166
You do not currently have access to this content.