Skin friction $(cf)$ and heat transfer (St) predictions were made for a turbulent boundary layer over randomly rough surfaces at Reynolds number of $1×106$. The rough surfaces are scaled models of actual gas turbine blade surfaces that have experienced degradation after service. Two different approximations are used to characterize the roughness in the computational model: the discrete element model and full 3D discretization of the surface. The discrete element method considers the total aerodynamic drag on a rough surface to be the sum of shear drag on the flat part of the surface and the form drag on the individual roughness elements. The total heat transfer from a rough surface is the sum of convection on the flat part of the surface and the convection from each of the roughness elements. Correlations are used to model the roughness element drag and heat transfer, thus avoiding the complexity of gridding the irregular rough surface. The discrete element roughness representation was incorporated into a two-dimensional, finite difference boundary layer code with a mixing length turbulence model. The second prediction method employs a viscous adaptive Cartesian grid approach to fully resolve the three-dimensional roughness geometry. This significantly reduces the grid requirement compared to a structured grid. The flow prediction is made using a finite-volume Navier-Stokes solver capable of handling arbitrary grids with the Spalart-Allmaras $(S‐A)$ turbulence model. Comparisons are made to experimentally measured values of $cf$ and St for two unique roughness characterizations. The two methods predict $cf$ to within $±8%$ and St within $±17%$, the RANS code yielding slightly better agreement. In both cases, agreement with the experimental data is less favorable for the surface with larger roughness features. The RANS simulation requires a two to three order of magnitude increase in computational time compared to the DEM method and is not as readily adapted to a wide variety of roughness characterizations. The RANS simulation is capable of analyzing surfaces composed primarily of roughness valleys (rather than peaks), a feature that DEM does not have in its present formulation. Several basic assumptions employed by the discrete element model are evaluated using the 3D RANS flow predictions, namely: establishment of the midheight for application of the smooth wall boundary condition; $cD$ and Nu relations employed for roughness elements; and flow three dimensionality over and around roughness elements.

1.
Bons
,
J. P.
,
Taylor
,
R.
,
McClain
,
S.
, and
Rivir
,
R. B.
, 2001, “
The Many Faces of Turbine Surface Roughness
,”
ASME J. Turbomach.
0889-504X,
123
(
4
), pp.
739
748
.
2.
Taylor
,
R. P.
, 1990, “
Surface Roughness Measurements on Gas Turbine Blades
,”
ASME J. Turbomach.
0889-504X,
112
(
1
), pp.
175
180
.
3.
,
F.
, and
Suzuki
,
M.
, 1993, “
External Heat Transfer Enhancement to Turbine Blading due to Surface Roughness
,” ASME Paper No. 93-GT-74.
4.
Blair
,
M. F.
, 1994, “
An Experimental Study of Heat Transfer in a Large-Scale Turbine Rotor Passage
,”
ASME J. Turbomach.
0889-504X,
116
(
1
), pp.
1
13
.
5.
Guo
,
S. M.
,
Jones
,
T. V.
,
Lock
,
G. D.
, and
Dancer
,
S. N.
, 1998, “
Computational Prediction of Heat Transfer to Gas Turbine Nozzle Guide Vanes With Roughened Surfaces
,”
ASME J. Turbomach.
0889-504X,
120
(
2
), pp.
343
350
.
6.
Suder
,
K. L.
,
Chima
,
R. V.
,
Strazisar
,
A. J.
, and
Roberts
,
W. B.
, 1995, “
The Effect of Adding Roughness and Thickness to a Transonic Axial Compressor Rotor
,”
ASME J. Turbomach.
0889-504X,
117
, pp.
491
505
.
7.
Ghenaiet
,
A.
,
Elder
,
R. L.
, and
Tan
,
S. C.
, “
Particles Trajectories through an Axial Fan and Performance Degradation due to Sand Ingestion
,” ASME Paper No. 2001-GT-497.
8.
Schlichting
,
H.
, 1936, “
Experimental Investigation of the Problem of Surface Roughness
,”
Ing.-Arch.
0020-1154,
VII
(
1
).
9.
,
J.
, 1933, “
Laws for Flows in Rough Pipes
,”
VDI-Forschungsh.
0042-174X,
361
.
10.
Cebeci
,
T.
, and
Chang
,
K. C.
, 1978, “
Calculation of Incompressible Rough-Wall Boundary Layer Flows
,”
AIAA J.
0001-1452,
16
(
7
), pp.
730
735
.
11.
Boyle
,
R. J.
, 1994, “
Prediction of Surface Roughness and Incidence Effects on Turbine Performance
,”
ASME J. Turbomach.
0889-504X,
116
, pp.
745
751
.
12.
Aupoix
,
B.
, and
Spalart
,
P. R.
, 2002, “
Extensions of the Spalart-Allmaras Turbulence Model to Account for Wall Roughness
,” ONERA Technical Report No. TP 2002-173.
13.
Simpson
,
R. L.
, 1973, “
A Generalized Correlation of Roughness Density Effects on the Turbulent Boundary Layer
,”
AIAA J.
0001-1452,
11
, pp.
242
244
.
14.
Sigal
,
A.
, and
Danberg
,
J.
, 1990, “
New Correlation of Roughness Density Effect on the Turbulent Boundary Layer
,”
AIAA J.
0001-1452,
28
(
3
), pp.
554
556
.
15.
Bons
,
J. P.
, 2002, “
St and Cf Augmentation for Real Turbine Roughness With Elevated Freestream Turbulence
,”
ASME J. Turbomach.
0889-504X,
124
(
4
), pp.
632
644
.
16.
Taylor
,
R. P.
, 1983, “
A Discrete Element Prediction Approach for Turbulent Flow Over Rough Surfaces
,” Ph.D. dissertation, Mississippi State University, Mississippi State, MS.
17.
Finson
,
M. L.
, 1982, “
A Model for Rough Wall Turbulent Heating and Skin Friction
,” AIAA Paper No. 82-0199.
18.
,
J. C.
, and
Hodge
,
B. K.
, 1971, “
The Calculation of Compressible Transitional Turbulent and Relaminarizational Boundary Layers over Smooth and Rough Surfaces Using an Extended Mixing-Length Hypothesis
,” AIAA Paper No. 77-682, (1977).
19.
Lin
,
T. C.
, and
Bywater
,
R. J.
, 1980, “
The Evaluation of Selected Turbulence Models for High-Speed Rough-Wall Boundary Layer Calculations
,” AIAA Paper No. 80-0132.
20.
McClain
,
S. T.
,
Hodge
,
B. K.
, and
Bons
,
J. P.
, 2004, “
Predicting Skin Friction and Heat Transfer for Turbulent Flow over Real Gas-Turbine Surface Roughness Using the Discrete-Element Method
,”
ASME J. Turbomach.
0889-504X,
126
, pp.
259
267
.
21.
Bogard
,
D. G.
,
Schmidt
,
D. L.
, and
Tabbita
,
M.
, 1998, “
Characterization and Laboratory Simulation of Turbine Airfoil Surface Roughness and Associated Heat Transfer
,”
ASME J. Turbomach.
0889-504X,
120
(
2
), pp.
337
342
.
22.
Barlow
,
D. N.
, and
Kim
,
Y. W.
, 1995, “
Effect of Surface Roughness on Local Heat Transfer and Film Cooling Effectiveness
,” ASME Paper No. 95-GT-14.
23.
Bons
,
J. P.
, 2005, “
A Critical Assessment of Reynolds Analogy for Turbine Flows
,”
ASME J. Heat Transfer
0022-1481,
127
, pp.
472
485
.
24.
Mills
,
A. F.
, 1992,
Heat Transfer
,
1st ed.
, Irwin, IL.
25.
Schultz
,
D. L.
, and
Jones
,
T. V.
, 1973, “
Heat-transfer Measurements in Short-duration Hypersonic Facilities
,” Advisory Group for Aerospace Research and Development, Report No. 165, NATO, Belgium.
26.
Gatlin
,
B.
, and
Hodge
,
B. K.
, 1990
An Instructional Computer Program for Computing the Steady, Compressible Turbulent Flow of an Arbitrary Fluid Near a Smooth Wall
, 2nd printing,
Mississippi State University Press
,
Mississippi State, MS
.
27.
Wang
,
Z. J.
, and
Chen
,
R. F.
, 2002, “
Anisotropic Solution-Adaptive Viscous Cartesian Grid Method for Turbulent Flow Simulation
,”
AIAA J.
0001-1452,
40
, pp.
1969
1978
.
28.
Roe
,
P. L.
, 1983, “
Approximate Reimann Solvers, Parameter Vectors, and Difference Schemes
,”
J. Comput. Phys.
0021-9991,
43
, p.
357
.
29.
Venkatakrishnan
,
V.
, 1995, “
Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters
,”
J. Comput. Phys.
0021-9991,
118
, pp.
120
130
.
30.
Wang
,
Z. J.
, 1998, “
,”
Comput. Fluids
0045-7930,
27
(
4
), pp.
529
549
.
31.
Chen
,
R. F.
, and
Wang
,
Z. J.
, 2000, “
Fast, Block Lower-Upper Symmetric Gauss-Seidel Scheme for Arbitrary Grids
,”
AIAA J.
0001-1452,
38
(
12
), pp.
2238
2245
.
32.
Spalart
,
P. R.
, and
Allmaras
,
S. R.
, 1992, “
A One-Equation Turbulence Model for Aerodynamic Flows
,” Paper No. AIAA-92-0439.
33.
Spalart
,
P. R.
, and
Allmaras
,
S. R.
, 1994, “
A One-Equation Turbulence Model for Aerodynamic Flows
,”
Rech. Aerosp.
0034-1223,
1
, pp.
5
21
.
34.
Bons
,
J. P.
, and
McClain
,
S. T.
, 2004, “
The Effect of Real Turbine Roughness and Pressure Gradient on Heat Transfer
,”
ASME J. Turbomach.
0889-504X,
126
, pp.
385
394
.
35.
White
,
F. M.
, 1991,
Viscous Fluid Flow
,
2nd ed.
,
McGraw-Hill
,
New York
.
36.
Kays
,
W. M.
, and
Crawford
,
M. E.
, 1993,
Convective Heat and Mass Transfer
,
3rd ed.
,
McGraw-Hill
,
New York
.
37.
Schlichting
,
H.
, 1979,
Boundary Layer Theory
,
7th ed.
,
McGraw-Hill
,
New York
.
38.
Dipprey
,
D. F.
, and
Sabersky
,
R. H.
, 1962, “
Heat and Momentum Transfer in Smooth and Rough Tubes at Various Prandtl Numbers
,”
Int. J. Heat Mass Transfer
0017-9310,
6
, pp.
329
353
.
39.
Wassel
,
A. T.
, and
Mills
,
A. F.
, 1979, “
Calculation of Variable Property Turbulent Friction and Heat Transfer in Rough Pipes
,”
ASME J. Heat Transfer
0022-1481,
101
, pp.
469
474
.
40.
Taylor
,
R. P.
, and
Hodge
,
B. K.
, 1993, “
A Validated Procedure for the Prediction of Fully-Developed Nusselt Numbers and Friction Factors in Pipes with 3-Dimensional Roughness
,”
J. Enhanced Heat Transfer
1065-5131,
1
, pp.
23
35
.
41.
Kithcart
,
M. E.
, and
Klett
,
D. E.
, 1997, “
Heat Transfer and Skin Friction Comparison of Dimpled Versus Protrusion Roughness
,” NASA Report No. N97-27444, pp.
328
336
.