An advanced film-cooling application is the thermal protection of the nozzle extension of high-performance rocket engines. The extension wall needs be protected from the hot supersonic thrust gas, very much like the combustion chamber or nozzle throat region, where the flow is, however, sub- or transonic. A reliable cooling modeling for practical applications requires benchmark results for generic cases with accurate flow-field details. To this end, fundamental investigations of the interaction between the thrust and cooling gas have been performed for flat-plate flow using high-order direct numerical simulations for the first time. A cool secondary gas is injected through a vertical slot of height s in a backward-facing step. The thrust-gas flow is steam (gaseous H2O) at Mach 3.3 with a turbulent boundary layer, and a laminar supersonic stream of cool helium is injected. The influence of the coolant mass flow rate is investigated by varying the blowing ratio F or the injection height s at a fixed cooling-gas temperature and Mach number. Several previously unknown effects are found fostering correlation model evolution of the film cooling, inter alia that the upstream wall temperature needs be taken into account and how the turbulent Prandtl and Schmidt number distributions are in the field, essential for improved Reynolds-averaged Navier–Stokes simulations.

1.
M.
Keller
and
M. J.
Kloker
, “
Effusion cooling and flow tripping in laminar supersonic boundary-layer flow
,”
AIAA J.
53
,
902
(
2015
).
2.
M.
Keller
and
M. J.
Kloker
, “
Direct numerical simulation of foreign-gas film cooling in supersonic boundary-layer flow
,”
AIAA J.
55
,
99
111
(
2017
).
3.
K. A.
Juhany
,
M. L.
Hunt
, and
J. M.
Sivo
, “
Influence of injectant Mach number and temperature on supersonic film cooling
,”
J. Thermophys. Heat Transfer
8
,
59
67
(
1994
).
4.
B.
Aupoix
,
A.
Mignosi
,
S.
Viala
,
F.
Bouvier
, and
R.
Gaillard
, “
Experimental and numerical study of supersonic film cooling
,”
AIAA J.
36
,
915
923
(
1998
).
5.
M.
Konopka
,
M.
Meinke
, and
W.
Schröder
, “
Large-eddy simulation of shock/cooling-film interaction
,”
AIAA J.
50
,
2102
2114
(
2012
).
6.
P.
Jiang
,
Z.
Liao
,
Z.
Huang
,
Y.
Xiong
, and
Y.
Zhu
, “
Influence of shock waves on supersonic transpiration cooling
,”
Int. J. Heat Mass Transfer
129
,
965
974
(
2019
).
7.
N.
Christopher
,
J. M. F.
Peter
,
M. J.
Kloker
, and
J. P.
Hickey
, “
DNS of turbulent flat-plate flow with transpiration cooling
,”
Int. J. Heat Mass Transfer
157
,
119972
(
2020
).
8.
C.
Yuan
,
J.
Li
,
Z.
Jiang
, and
H.
Yu
, “
Experimental investigation of liquid film cooling in hypersonic flow
,”
Phys. Fluids
31
,
046101
(
2019
).
9.
W.
Zhou
,
H.
Chen
,
Y.
Liu
,
X.
Wen
, and
D.
Peng
, “
Unsteady analysis of adiabatic film cooling effectiveness for discrete hole with oscillating mainstream flow
,”
Phys. Fluids
30
,
127103
(
2018
).
10.
S. M. H. B.
Abadi
,
S.
Zirak
, and
M. R.
Zargarabadi
, “
Effect of pulsating injection and mainstream attack angle on film cooling performance of a gas turbine blade
,”
Phys. Fluids
32
,
117102
(
2020
).
11.
M.
Hombsch
and
H.
Olivier
, “
Film cooling in laminar and turbulent supersonic flows
,”
J. Spacecr. Rockets
50
,
742
753
(
2013
).
12.
R. J.
Goldstein
, “
Film cooling
,”
Adv. Heat Transfer
7
,
321
379
(
1971
).
13.
A. M.
Cary
and
J. N.
Hefner
, “
Film-cooling effectiveness and skin friction in hypersonic turbulent flow
,”
AIAA J.
10
,
1188
1193
(
1972
).
14.
R. J.
Goldstein
,
E. R. G.
Eckert
,
F. K.
Tsou
, and
A.
Haji-Sheikh
, “
Film cooling with air and helium injection through a rearward-facing slot into a supersonic air flow
,”
AIAA J.
4
,
981
985
(
1966
).
15.
C.
Song
and
C.
Shen
, “
Effects of lip thickness on the flowfield structures of supersonic film cooling
,”
J. Thermophys. Heat Transfer
33
,
599
605
(
2019
).
16.
C.
Song
and
C.
Shen
, “
Effects of feeding pressures on the flowfield structures of supersonic film cooling
,”
J. Thermophys. Heat Transfer
32
,
648
658
(
2018
).
17.
C.
Song
and
C.
Shen
, “
Effects of feeding Mach numbers on the flowfield structures of supersonic film cooling
,”
J. Thermophys. Heat Transfer
33
,
264
270
(
2019
).
18.
H.
Foroutan
and
S.
Yavuzkurt
, “
Numerical simulations of the near-field region of film cooling jets under high free stream turbulence: Application of RANS and hybrid URANS/large eddy simulation models
,”
J. Heat Transfer
137
,
011701
(
2015
).
19.
G. B.
Yepuri
,
A. B.
Talanki Puttarangasetty
,
D. K.
Kolke
, and
F.
Jesuraj
, “
Effect of RANS-type turbulence models on adiabatic film cooling effectiveness over a scaled up gas turbine blade leading edge surface
,”
J. Inst. Eng. (India): Ser. C
99
,
393
400
(
2018
).
20.
K.
Abdullah
,
O. H.
Abdulguad
,
A. N.
Mohammed
, and
Z.
Suleiman
, “
Comparison of RANS and U-RANS for flat plate film cooling
,”
Appl. Mech. Mater.
773–774
,
353
357
(
2015
).
21.
R. A.
Seban
, “
Effects of initial boundary-layer thickness on a tangential injection system
,”
J. Heat Transfer
82
,
392
393
(
1960
).
22.
G.
Olsen
,
R.
Nowak
,
M.
Holden
, and
N.
Baker
, “
Experimental results for film cooling in 2-D supersonic flow including coolant delivery pressure, geometry, and incident shock effects
,” in
28th Aerospace Sciences Meeting
(AIAA,
1990
), pp.
1
9
.
23.
M. S.
Holden
and
K.
Rodriguez
, “
Experimental studies of shock-wave/wall-jet interaction in hypersonic flow
,”
Technical Report No. NASA-CR-195844
,
Calspan-UB Research Center
,
1994
.
24.
M.
Konopka
,
M.
Meinke
, and
W.
Schröder
, “
Large-eddy simulation of shock-cooling-film interaction at helium and hydrogen injection
,”
Phys. Fluids
25
,
106101
(
2013
).
25.
W.
Peng
and
P.-X.
Jiang
, “
Influence of shock waves on supersonic film cooling
,”
J. Spacecr. Rockets
46
,
67
73
(
2009
).
26.
W.
Peng
,
X.-K.
Sun
, and
P.-X.
Jiang
, “
Effect of coolant inlet conditions on supersonic film cooling
,”
J. Spacecr. Rockets
52
,
1456
1464
(
2015
).
27.
W.
Peng
,
X.-K.
Sun
,
P.-X.
Jiang
, and
J.
Wang
, “
Effect of continuous or discrete shock wave generators on supersonic film cooling
,”
Int. J. Heat Mass Transfer
108
,
770
783
(
2017
).
28.
M.
Keller
,
M. J.
Kloker
, and
H.
Olivier
, “
Influence of cooling-gas properties on film-cooling effectiveness in supersonic flow
,”
J. Spacecr. Rockets
52
,
1443
1455
(
2015
).
29.
A.
Gülhan
and
S.
Braun
, “
An experimental study on the efficiency of transpiration cooling in laminar and turbulent hypersonic flows
,”
Exp. Fluids
50
,
509
(
2011
).
30.
S.
Ludescher
and
H.
Olivier
, “
Experimental investigations of film cooling in a conical nozzle under rocket-engine-like flow conditions
,”
AIAA J.
57
,
1172
1183
(
2019
).
31.
J. A.
Majeski
and
R. A.
Weatherford
, “
Development of an empirical correlation for film-cooling effectiveness
,”
AIAA Paper No. 88-2624
,
1988
.
32.
J. L.
Stollery
and
A. A. M.
El-Ehwany
, “
A note on the use of a boundary-layer model for correlating film-cooling data
,”
Int. J. Heat Mass Transfer
8
,
55
65
(
1965
).
33.
T.
Kanda
,
F.
Ono
,
M.
Takahashi
,
T.
Saito
, and
Y.
Wakamatsu
, “
Experimental studies of supersonic film cooling with shock wave interaction
,”
AIAA J.
34
,
265
271
(
1996
).
34.
J. M. F.
Peter
and
M. J.
Kloker
, “
Preliminary work for DNS of rocket-nozzle film-cooling
,” in
Deutscher Luft-und Raumfahrtkongress DLRK
(Deutsche Gesellschaft für Luft- und Raumfahrt – Lilienthal-Oberth e.V.,
2017
), Vol.
DLRK-2017-450178
, pp.
1
7
.
35.
J. O.
Hirschfelder
,
C. F.
Curtiss
, and
R. B.
Bird
,
Molecular Theory of Gases and Liquids
(
John Wiley & Sons, Inc
.,
1954
).
36.
J. M. F.
Peter
and
M. J.
Kloker
, “
Numerical simulation of film cooling in supersonic flow
,” in
Future Space-Transport-System Components under High Thermal and Mechanical Loads
, edited by
N. A.
Adams
,
W.
Schröder
,
R.
Radespiel
,
O. J.
Haidn
,
T.
Sattelmayer
,
C.
Stemmer
, and
B.
Weigand
(
Springer International Publishing
,
2021
), pp.
79
95
.
37.
A.
Babucke
,
M.
Kloker
, and
U.
Rist
, “
DNS of a plane mixing layer for the investigation of sound generation mechanisms
,”
Comput. Fluids
37
,
360
368
(
2008
).
38.
M.
Keller
and
M. J.
Kloker
, “
Direct numerical simulations of film cooling in a supersonic boundary-layer flow on massively-parallel supercomputers
,” in
Sustained Simulation Performance 2013
, edited by
M. M.
Resch.
,
W.
Bez
,
E.
Focht
,
H.
Kobayashi
, and
Y.
Kovalenko
(
Springer International Publishing
,
2013
), pp.
107
128
.
39.
M. J.
Kloker
, “
A robust high-resolution split-type compact FD scheme for spatial direct numerical simulation of boundary-layer transition
,”
Appl. Sci. Res.
59
,
353
377
(
1997
).
40.
D. V.
Gaitonde
and
M. R.
Visbal
, “
Padé-type higher-order boundary filters for the Navier–Stokes equations
,”
AIAA J.
38
,
2103
2112
(
2000
).
41.
C.
Bogey
,
N.
de Cacqueray
, and
C.
Bailly
, “
A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations
,”
J. Comput. Phys.
228
,
1447
1465
(
2009
).
42.
J.
Poggie
,
N. J.
Bisek
, and
R.
Gosse
, “
Resolution effects in compressible, turbulent boundary layer simulations
,”
Comput. Fluids
120
,
57
69
(
2015
).
43.
C.
Wenzel
,
B.
Selent
,
M. J.
Kloker
, and
U.
Rist
, “
DNS of compressible turbulent boundary layers and assessment of data-/scaling-law quality
,”
J. Fluid Mech.
842
,
428
468
(
2018
).
44.
E.
Touber
, “
Unsteadiness in shock-wave/boundary-layer interactions
,” Ph.D. thesis (
University of Southampton
,
2010
).
45.
D. B.
Spalding
and
S. W.
Chi
, “
The drag of a compressible turbulent boundary layer on a smooth flat plate with and without heat transfer
,”
J. Fluid Mech.
18
,
117
143
(
1964
).
46.
F. M.
White
,
Viscous Fluid Flow
, 3rd ed. (
McGraw-Hill
,
2006
).
47.
S. E.
Guarini
,
R. D.
Moser
,
K.
Shariff
, and
A.
Wray
, “
Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5
,”
J. Fluid Mech.
414
,
1
33
(
2000
).
48.
S.
Pirozzoli
and
M.
Bernardini
, “
Turbulence in supersonic boundary layers at moderate Reynolds number
,”
J. Fluid Mech.
688
,
120
168
(
2011
).
49.
Actual F = 0.586 44 for the matched-pressure case C-II.
50.
A.
Bukva
,
K.
Zhang
,
N.
Christopher
, and
J.-P.
Hickey
, “
Assessment of turbulence modeling for massively-cooled turbulent boundary layer flows with transpiration cooling
,”
Phys. Fluids
33
,
095114
(
2021
).
51.
S.
Pirozzoli
and
F.
Grasso
, “
Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M = 2.25
,”
Phys. Fluids
18
,
065113
(
2006
).
52.
D.
Ahlman
,
G.
Brethouwer
, and
A. V.
Johansson
, “
Direct numerical simulation of a plane turbulent wall-jet including scalar mixing
,”
Phys. Fluids
19
,
065102
(
2007
).
53.
D. C.
Wilcox
,
Turbulence Modelling for CFD
, 3rd ed. (
DCW Industries
,
2006
).
54.
G.
He
,
Y.
Guo
, and
A. T.
Hsu
, “
The effect of Schmidt number on turbulent scalar mixing in a jet-in-crossflow
,”
Int. J. Heat Mass Transfer
42
,
3727
3738
(
1999
).
55.
Y.
Tominaga
and
T.
Stathopoulos
, “
Turbulent Schmidt numbers for CFD analysis with various types of flowfield
,”
Atmos. Environ.
41
,
8091
8099
(
2007
).
56.
P.
Marquardt
,
M.
Klaas
, and
W.
Schröder
, “
Experimental investigation of the turbulent Schmidt number in supersonic film cooling with shock interaction
,”
Exp. Fluids
61
,
160
(
2020
).
57.
L. M.
Mack
, “
Boundary-layer linear stability theory
,”
AGARD Report No. 709
,
Jet Propulsion Laboratory
,
1984
.
You do not currently have access to this content.