{\rtf1\adeflang1025\ansi\ansicpg936\uc2\adeff0\deff0\stshfdbch13\stshfloch0\stshfhich0\stshfbi0\deflang1033\deflangfe2052{\fonttbl{\f0\froman\fcharset0\fprq2{\*\panose 02020603050405020304}Times New Roman;}{\f13\fnil\fcharset134\fprq2{\*\panose 02010600030101010101}\'cb\'ce\'cc\'e5{\*\falt SimSun};}
{\f18\fmodern\fcharset136\fprq1{\*\panose 02020309000000000000}MingLiU{\*\falt \'b2\'d3\'a9\'fa\'c5\'e9};}{\f36\fnil\fcharset134\fprq2{\*\panose 02010600030101010101}@\'cb\'ce\'cc\'e5;}
{\f82\fmodern\fcharset136\fprq1{\*\panose 02020309000000000000}@MingLiU;}{\f179\froman\fcharset238\fprq2 Times New Roman CE;}{\f180\froman\fcharset204\fprq2 Times New Roman Cyr;}{\f182\froman\fcharset161\fprq2 Times New Roman Greek;}
{\f183\froman\fcharset162\fprq2 Times New Roman Tur;}{\f184\fbidi \froman\fcharset177\fprq2 Times New Roman (Hebrew);}{\f185\fbidi \froman\fcharset178\fprq2 Times New Roman (Arabic);}{\f186\froman\fcharset186\fprq2 Times New Roman Baltic;}
{\f187\froman\fcharset163\fprq2 Times New Roman (Vietnamese);}{\f311\fnil\fcharset0\fprq2 SimSun Western{\*\falt SimSun};}{\f361\fmodern\fcharset0\fprq1 MingLiU Western{\*\falt \'b2\'d3\'a9\'fa\'c5\'e9};}
{\f541\fnil\fcharset0\fprq2 @\'cb\'ce\'cc\'e5 Western;}{\f1001\fmodern\fcharset0\fprq1 @MingLiU Western;}}{\colortbl;\red0\green0\blue0;\red0\green0\blue255;\red0\green255\blue255;\red0\green255\blue0;\red255\green0\blue255;\red255\green0\blue0;
\red255\green255\blue0;\red255\green255\blue255;\red0\green0\blue128;\red0\green128\blue128;\red0\green128\blue0;\red128\green0\blue128;\red128\green0\blue0;\red128\green128\blue0;\red128\green128\blue128;\red192\green192\blue192;\red255\green153\blue0;}
{\stylesheet{\qj \li0\ri0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs21\lang1033\langfe2052\kerning2\loch\f0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 \snext0 Normal;}
{\*\cs10 \additive \ssemihidden Default Paragraph Font;}{\*
\ts11\tsrowd\trftsWidthB3\trpaddl108\trpaddr108\trpaddfl3\trpaddft3\trpaddfb3\trpaddfr3\trcbpat1\trcfpat1\tblind0\tblindtype3\tscellwidthfts0\tsvertalt\tsbrdrt\tsbrdrl\tsbrdrb\tsbrdrr\tsbrdrdgl\tsbrdrdgr\tsbrdrh\tsbrdrv
\ql \li0\ri0\widctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs20 \ltrch\fcs0 \fs20\lang1024\langfe1024\loch\f0\hich\af0\dbch\af13\cgrid\langnp1024\langfenp1024 \snext11 \ssemihidden Normal Table;}{
\s15\qc \li0\ri0\nowidctlpar\brdrb\brdrs\brdrw15\brsp20 \tqc\tx4153\tqr\tx8306\wrapdefault\aspalpha\aspnum\faauto\nosnaplinegrid\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs18\alang1025 \ltrch\fcs0
\fs18\lang1033\langfe2052\kerning2\loch\f0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 \sbasedon0 \snext15 \styrsid15098623 header;}{\s16\ql \li0\ri0\nowidctlpar
\tqc\tx4153\tqr\tx8306\wrapdefault\aspalpha\aspnum\faauto\nosnaplinegrid\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs18\alang1025 \ltrch\fcs0 \fs18\lang1033\langfe2052\kerning2\loch\f0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052
\sbasedon0 \snext16 \styrsid15098623 footer;}{\*\cs17 \additive \rtlch\fcs1 \af0 \ltrch\fcs0 \ul\cf2 \sbasedon10 \styrsid15098623 Hyperlink;}}{\*\latentstyles\lsdstimax156\lsdlockeddef0}{\*\pgptbl {\pgp\ipgp0\itap0\li0\ri0\sb0\sa0}}{\*\rsidtbl \rsid655764
\rsid811358\rsid1069012\rsid1074055\rsid1318786\rsid1462542\rsid2231290\rsid2847010\rsid2909350\rsid3146529\rsid3152368\rsid3153644\rsid3409549\rsid3869496\rsid4001084\rsid4197693\rsid4419642\rsid4477090\rsid5848812\rsid6441775\rsid6492656\rsid6493368
\rsid7939098\rsid8674677\rsid8869018\rsid8944438\rsid9449096\rsid10106844\rsid10236727\rsid10246904\rsid10505668\rsid10707375\rsid10897542\rsid11282186\rsid11941893\rsid12669821\rsid13312263\rsid13773176\rsid14707564\rsid14892288\rsid15098623\rsid15222873
\rsid15611782\rsid15665703\rsid15817367\rsid15866361\rsid16080730\rsid16195404\rsid16517661\rsid16546564\rsid16607898}{\*\generator Microsoft Word 11.0.0000;}{\info{\title Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression Ramp}{\subject Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression Ramp}{\author Document Search}
{\keywords Document Search}{\doccomm http://www.nuokui.com/pdf/bsBqaKwTJjLI.html}{\operator www.downhi.com}{\creatim\yr2010\mo9\dy28\hr22\min9}{\revtim\yr2014\mo4\dy4\hr1\min23}{\version26}{\edmins1077}{\nofpages1}{\nofwords27}{\nofchars168}{\*\manager http://www.downhi.com/}
{\*\company http://www.downhi.com/}{\*\category Document Search}{\nofcharsws186}{\vern24617}{\*\password 00000000}}{\*\xmlnstbl {\xmlns1 http://schemas.microsoft.com/office/word/2003/wordml}{\xmlns2 urn:schemas-microsoft-com:office:smarttags}}
\paperw11906\paperh16838\margl1134\margr1134\margt1134\margb1134\gutter0\ltrsect
\deftab420\ftnbj\aenddoc\donotembedsysfont1\donotembedlingdata0\grfdocevents0\validatexml1\showplaceholdtext0\ignoremixedcontent0\saveinvalidxml0\showxmlerrors1\formshade\horzdoc\dgmargin\dghspace180\dgvspace156\dghorigin1134\dgvorigin1134\dghshow0
\dgvshow2\jcompress\lnongrid\viewkind1\viewscale85\splytwnine\ftnlytwnine\htmautsp\useltbaln\alntblind\lytcalctblwd\lyttblrtgr\lnbrkrule\nobrkwrptbl\snaptogridincell\allowfieldendsel\wrppunct\asianbrkrule\rsidroot3869496\newtblstyruls\nogrowautofit
{\*\fchars
!),.:\'3b?]\'7d\'a1\'a7\'a1\'a4\'a1\'a6\'a1\'a5\'a8\'44\'a1\'ac\'a1\'af\'a1\'b1\'a1\'ad\'a1\'c3\'a1\'a2\'a1\'a3\'a1\'a8\'a1\'a9\'a1\'b5\'a1\'b7\'a1\'b9\'a1\'bb\'a1\'bf\'a1\'b3\'a1\'bd\'a3\'a1\'a3\'a2\'a3\'a7\'a3\'a9\'a3\'ac\'a3\'ae\'a3\'ba\'a3\'bb\'a3\'bf\'a3\'dd\'a3\'e0\'a3\'fc\'a3\'fd\'a1\'ab\'a1\'e9
}{\*\lchars ([\'7b\'a1\'a4\'a1\'ae\'a1\'b0\'a1\'b4\'a1\'b6\'a1\'b8\'a1\'ba\'a1\'be\'a1\'b2\'a1\'bc\'a3\'a8\'a3\'ae\'a3\'db\'a3\'fb\'a1\'ea\'a3\'a4}\fet0{\*\wgrffmtfilter 013f}\ilfomacatclnup0{\*\template
C:\\Documents and Settings\\Administrator\\\'d7\'c0\'c3\'e6\\doc.dot}{\*\ftnsep \ltrpar \pard\plain \ltrpar\qj \li0\ri0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0
\fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid14707564 \chftnsep
\par }}{\*\ftnsepc \ltrpar \pard\plain \ltrpar\qj \li0\ri0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0
\fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid14707564 \chftnsepc
\par }}{\*\aftnsep \ltrpar \pard\plain \ltrpar\qj \li0\ri0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0
\fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid14707564 \chftnsep
\par }}{\*\aftnsepc \ltrpar \pard\plain \ltrpar\qj \li0\ri0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0
\fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid14707564 \chftnsepc
\par }}\ltrpar \sectd \ltrsect\linex0\headery851\footery992\colsx425\endnhere\sectlinegrid312\sectspecifyl\sectrsid6493368\sftnbj {\headerr \ltrpar \pard\plain \ltrpar\s15\qc \li0\ri0\nowidctlpar\brdrb\brdrs\brdrw15\brsp20
\tqc\tx4153\tqr\tx8306\wrapdefault\aspalpha\aspnum\faauto\nosnaplinegrid\adjustright\rin0\lin0\itap0 \rtlch\fcs1 \af0\afs18\alang1025 \ltrch\fcs0 \fs18\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1
\af0\afs30 \ltrch\fcs0 \b\f13\fs30\cf6\insrsid1074055\charrsid1074055 \hich\af13\dbch\af13\loch\f13 Free Document Search and Download}{\rtlch\fcs1 \af0\afs30 \ltrch\fcs0 \b\fs30\cf6\loch\af13\insrsid6493368\charrsid1074055
\par }{\field{\*\fldinst {\rtlch\fcs1 \af0\afs32 \ltrch\fcs0 \f13\fs32\cf6\insrsid14892288 \hich\af13\dbch\af13\loch\f13 HYPERLINK "http://www.downhi.com/" }{\rtlch\fcs1 \af0\afs32 \ltrch\fcs0 \fs32\cf6\loch\af13\insrsid10707375\charrsid14892288
{\*\datafield
00d0c9ea79f9bace118c8200aa004ba90b0200000003000000e0c9ea79f9bace118c8200aa004ba90b4e00000068007400740070003a002f002f00770065006e00640061006e0067002e0064006f00630073006f0075002e0063006f006d002f000000795881f43b1d7f48af2c825dc485276300000000a5ab0000000000}}
}{\fldrslt {\rtlch\fcs1 \af0\afs32 \ltrch\fcs0 \cs17\f13\fs32\ul\cf2\insrsid3146529\charrsid14892288 \hich\af13\dbch\af13\loch\f13 http://www.downhi.com/}}}\sectd \linex0\endnhere\sectdefaultcl\sftnbj {\rtlch\fcs1 \af0\afs32 \ltrch\fcs0
\fs32\cf6\loch\af13\insrsid6493368\charrsid15098623
\par }}{\*\pnseclvl1\pnucrm\pnstart1\pnindent720\pnhang {\pntxta \dbch .}}{\*\pnseclvl2\pnucltr\pnstart1\pnindent720\pnhang {\pntxta \dbch .}}{\*\pnseclvl3\pndec\pnstart1\pnindent720\pnhang {\pntxta \dbch .}}{\*\pnseclvl4\pnlcltr\pnstart1\pnindent720\pnhang
{\pntxta \dbch )}}{\*\pnseclvl5\pndec\pnstart1\pnindent720\pnhang {\pntxtb \dbch (}{\pntxta \dbch )}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang {\pntxtb \dbch (}{\pntxta \dbch )}}{\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang {\pntxtb \dbch (}
{\pntxta \dbch )}}{\*\pnseclvl8\pnlcltr\pnstart1\pnindent720\pnhang {\pntxtb \dbch (}{\pntxta \dbch )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang {\pntxtb \dbch (}{\pntxta \dbch )}}\pard\plain \ltrpar\qj \li0\ri0\sl180\slmult0
\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid6493368 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 {\rtlch\fcs1 \af0
\ltrch\fcs0 \insrsid6493368\charrsid1074055 \loch\af0\hich\af0\dbch\f13 \'a1\'a1\'a1\'a1}{\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid6493368\charrsid1074055
\par }\pard \ltrpar\qc \li0\ri0\sl180\slmult0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid1318786 {\rtlch\fcs1 \af0\afs36 \ltrch\fcs0 \b\fs36\insrsid1318786\charrsid1074055 \hich\af0\dbch\af13\loch\f0 Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression Ramp
\par }\pard \ltrpar\qj \li0\ri0\sl180\slmult0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid1074055 {\rtlch\fcs1 \af0 \ltrch\fcs0 \fs24\insrsid6493368\charrsid1074055 \loch\af0\hich\af0\dbch\f13 \'a3\'ba}{\rtlch\fcs1 \af0
\ltrch\fcs0 \fs24\insrsid6493368\charrsid1074055
\par }\pard \ltrpar\qc \li0\ri0\sl180\slmult0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid3869496 {\field{\*\fldinst {\rtlch\fcs1 \af0 \ltrch\fcs0 \fs24\insrsid1074055 \hich\af0\dbch\af13\loch\f0
\hich\af0\dbch\af13\loch\f0 HYPERLINK "http://www.downhi.com/"\hich\af0\dbch\af13\loch\f0 }{\rtlch\fcs1 \af0 \ltrch\fcs0 \fs24\insrsid13719882\charrsid1074055 {\*\datafield
00d0c9ea79f9bace118c8200aa004ba90b0200000003000000e0c9ea79f9bace118c8200aa004ba90b4600000068007400740070003a002f002f007700770077002e0064006f0077006e00680069002e0063006f006d002f000000795881f43b1d7f48af2c825dc485276300000000a5ab0000}}}{\fldrslt {
\rtlch\fcs1 \af0 \ltrch\fcs0 \cs17\fs24\ul\cf2\insrsid3869496\charrsid1074055 \hich\af0\dbch\af13\loch\f0 http://www.nuokui.com/pdf/bsBqaKwTJjLI.html}{\rtlch\fcs1 \af0 \ltrch\fcs0 \cs17\fs24\ul\cf2\insrsid6493368\charrsid1074055
\par }\pard \ltrpar\qj \li0\ri0\sl180\slmult0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid6493368 }}\pard\plain \ltrpar\qj \li0\ri0\sl180\slmult0
\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid6493368 \rtlch\fcs1 \af0\afs24\alang1025 \ltrch\fcs0 \fs21\lang1033\langfe2052\kerning2\loch\af0\hich\af0\dbch\af13\cgrid\langnp1033\langfenp2052 \sectd
\linex0\headery851\footery992\colsx425\endnhere\sectlinegrid312\sectspecifyl\sectrsid6493368\sftnbj {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid6493368\charrsid1074055
\par }{\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid6493368\charrsid1074055
\par }\pard \ltrpar\qj \li0\ri0\sl360\slmult1\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid15866361 {\rtlch\fcs1 \af0 \ltrch\fcs0 \fs24\insrsid3869496\charrsid1074055 \hich\af0\dbch\af13\loch\f0
Page 1
Direct Numerical Simulation of Supersonic Turbulent Boundary
Layer over a Compression Ramp\par
M. Wu\par
∗\par
and M. P. Martin\par
†\par
Princeton University, Princeton, New Jersey 08540\par
DOI: 10.2514/1.27021
A direct numerical simulation of shock wave and turbulent boundary layer interaction for a 24 deg compression
ramp configuration at Mach 2.9 and Re 2300 is performed. A modified weighted, essentially nonoscillatory scheme is
used. The direct numerical simulation results are compared with the experiments of Bookey et al. [Bookey, P. B.,
Wyckham, C., Smits, A. J., and Martin, M. P., “New Experimental Data of STBLI at DNS/LES Accessible Reynolds
Numbers,” AIAA Paper No. 2005-309, Jan. 2005] at the same flow conditions. The upstream boundary layer, the
mean wall-pressure distribution, the size of the separation bubble, and the velocity profile downstream of the
interaction are predicted within the experimental uncertainty. The change of the mean and fluctuating properties
throughout the interaction region is studied. The low frequency motion of the shock is inferred from the wall-
pressure signal and freestream mass-flux measurement.\par
Nomenclature\par
a
=
speed of sound
Cf
=
skin friction coefficient
Cr\par
k\par
=
optimal weight for stencil k
f
=
frequency
fs
=
frequency of shock motion
ISk
=
smoothness measurement of stencil k
Lsep
=
separation length
M
=
freestream Mach number
p
=
pressure
qk
=
numerical flux of candidate stencil k
Re
=
Reynolds number based on
Re
=
Reynolds number based on
r
=
number of candidate stencils in WENO
SL
=
dimensionless frequency of shock motion
T
=
temperature
u
=
velocity in the streamwise direction
v
=
velocity in the spanwise direction
w
=
velocity in the wall-normal direction
x
=
coordinate in the streamwise direction
y
=
coordinate in the spanwise direction
z
=
coordinate in the wall-normal direction
=
99% thickness of the incoming boundary layer
=
displacement thickness of the incomi http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ng boundary
layer
=
momentum thickness of the incoming boundary layer
=
density
!k
=
weight of candidate stencil k\par
Subscripts\par
w
=
value at the wall
1
=
freestream value\par
Superscript\par
=
nondimensional value\par
I. Introduction\par
MANY aspects of shock wave and turbulent boundary layer\par
interaction (STBLI) are not fully understood, including the
dynamics of shock unsteadiness, turbulence amplification and mean
flow modification induced by shock distortion, separation and
reattachment criteria as well as the unsteady heat transfer near the
separation and reattachment points, and the generation of turbulent
mixing layers and underexpanded jets in the interaction region,
especially when they impinge on a surface. Yet, STBLI problems are
of great importance for the efficient design of scramjet engines and
control surfaces in hypersonic vehicles. A more profound
understanding of STBLI will lead to flow control methodologies
and novel hypersonic vehicle designs.
Different canonical configurations have been used in STBLI
studies. The compression ramp configuration has been studied
extensively experimentally, and there are numerous experimental
data available for this configuration. For example, Settles et al. [1–3]
studied 2-D/3-D compression ramp and sharp fin STBLI problems in
detail. Dolling et al. [4,5] studied the unsteadiness for compression
ramp configurations, and Selig [6] studied the unsteadiness of STBLI
and its control for a 24 deg compression ramp. Recently, Bookey
et al. [7] performed experiments on a 24 deg compression ramp
configuration with flow conditions accessible for direct numerical
simulation (DNS) and large eddy simulation (LES), which provides
valuable data for the validation of our simulations.
In contrast with numerous experimental data, there are few
detailed numerical simulations such as DNS and LES. Numerical
simulations of STBLI have been mainly confined to Reynolds
a http://www.nuokui.com/pdf/bsBqaKwTJjLI.html veraged Navier–Stokes simulation (RANS) due to the limitation of
computational resources. However, RANS is shown not capable of
predicting the wall pressure or the heat flux within a satisfactory
accuracy for shock interactions. Settles et al. [2] compared
experimental results with those of a one-equation model RANS for
the compression ramp configuration and showed that there were
significant differences in the wall-pressure distribution when the
flow was separated. Zheltovodov [8] showed that the state-of-the-art
RANS models do not give accurate predictions for strong STBLI.
The unsteady nature of STBLI problems is believed to account for
the discrepancies between RANS and experiments. DNS and LES of
STBLI have existed for less than a decade. Knight et al. [9] compiled
a summary of existing LES for the compression ramp configuration
and concluded that LES did not predict the wall pressure or the
separation length accurately in separated flows. In 2000, Adams [10]
performed the first DNS for an 18 deg compression ramp flow at\par
Received 3 August 2006; revision received 27 December 2006; accepted
for publication 27 December 2006. Copyright � 2007 by the authors.
Published by the American Institute of Aeronautics and Astronautics, Inc.,
with permission. Copies of this paper may be made for personal or internal
use, on condition that the copier pay the $10.00 per-copy fee to the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include
the code 0001-1452/07 $10.00 in correspondence with the CCC.\par
∗Graduate Student, Mechanical and Aerospace Engineering Department.\par
Student Member AIAA.\par
†Assistant Professor, Mechanical and Aerospace Engineering Department.\par
Member AIAA.\par
AIAA JOURNAL
Vol. 45, No. 4, April 2007\par
879\par
Page 2
M ˆ 3 and Re ˆ 1685. Because of the lack of experimental data at
the same flow conditions, Adams was not able to draw definite
conclusions by c http://www.nuokui.com/pdf/bsBqaKwTJjLI.html omparing his DNS with higher Reynolds number
experiments. The same is true for the LES of STBLI induced by a
compression corner of Rizzetta and Visbal [11]. In 2004, Wu and
Martin [12] performed DNS for a 24 deg compression ramp
configuration. The DNS results were compared with experiments
from Bookey et al. [7] at the same flow conditions. Significant
discrepancies were found in the size of the separation bubble and the
mean wall-pressure distribution [13]. Given the stringent constrains
in grid size for affordable DNS of STBLI, the numerical dissipation
of the original WENO (weighted essentially nonoscillatory) method
[14,15] was found responsible for these discrepancies [16].
Experiments of STBLI have shown evidence of large scale, slow
shock motion. The characteristic time scale for the motion is of the
order of 10 =U1, which is 1 order of magnitude greater than the
characteristic time scale of the incoming boundary layer. Dussauge
et al. [17] compiled frequencies that were found in experiments for
different configurations and found that the dimensionless frequency
of the shock motion is mainly between 0.02 and 0.05. The
dimensionless frequency is defined as
SL ˆ fsLsep=U1
(1)
A complete physical explanation of the low frequency motion
remains an open question. Andreopoulos and Muck [18] studied the
shock unsteadiness of a compression ramp configuration. They
concluded that the shock motion is driven by the bursting events in
the incoming boundary. Recently, Ganapathisubramani et al. [19]
proposed that very long coherent structures of high and low
momentum are present in the incoming boundary layer and are
responsible for the low frequency motion of the shock. These
structures can be as long as 40 and meander in the spanwise
direction. Despite the existence of large scale slow motion of the
shock that is found in experiments, no evidence has been reported in
previous numerical simulations.
In this paper, we present http://www.nuokui.com/pdf/bsBqaKwTJjLI.html new DNS data for a 24 deg compression
ramp STBLI configuration. The governing equations and flow
conditions are presented in Secs. II and III, respectively. The
modifications to the original WENO method are described in Sec. IV,
and the accuracy of the DNS data by comparison against
experimental data [7] at the same conditions is reported in Sec. V.
The shock motion, including evidence of low frequency motion, is
described in Sec. VI.\par
II. Governing Equations\par
The governing equations are the nondimensionalized conserva-
tive form of the continuity, momentum, and energy equations in
curvilinear coordinates. The working fluid is air, which is assumed to
be a perfect gas.
@U
@t
‡
@F
@
‡
@G
@
‡
@H
@
ˆ 0
(2)
where
U ˆ J
8
>>>><
>>>>:
u
v
w
e
9
>>>>=
>>>>;
;
F ˆ Fc ‡ Fv
(3)
and
Fc ˆ Jr
8
>>>><
>>>>:
u 0
u u 0 ‡ p s \par
x\par
v u 0 ‡ p s \par
y\par
w u 0 ‡ p s \par
z\par
… e ‡ p †u 0
9
>>>>=
>>>>;
Fv ˆ Jr
8
>>>>>>>>>><
>>>>>>>>>>:
0
\par
xxsx ‡
xys
y ‡
xzs
z\par
\par
yxsx ‡
yys
y ‡
yzs
z\par
\par
zxsx ‡
zys
y ‡
zzs
z\par
… \par
xxu ‡
xyv ‡
xzw†s
x ‡\par
… \par
yxu ‡
yyv ‡
yzw†s
y ‡\par
… \par
zxu ‡
zyv ‡
zzw†s
z\par
q \par
x s
x\par
q \par
y s
y\par
q \par
z s
z\par
9
>>>>>>>>>>=
>>>>>>>>>>;
(4)
and
s \par
x ˆ x=r
;\par
u 0 ˆ u s \par
x ‡ v s
y ‡ w s
z\par
r \par
ˆ\par
2\par
x ‡ 2
y ‡ 2
z\par
q
(5)
In curvilinear coordinates, flux terms G and H have similar forms as
F. \par
ij is given by the Newtonian linear stress–strain relation:\par
\par
ij ˆ\par
1
Re
2 S \par
ij\par
2
3
ijS \par
kk\par
(6)
The heat flux terms q \par
j are given by Fourier law:\par
q \par
j ˆ\par
1
Re
k @T
@x \par
j\par
(7)
The dynamic viscosity is computed by Sutherland’s law:
ˆ 1:458 10 6T3=2=…T ‡ 110:3†
(8)
The nondimensionalization is done by ˆ = 1, u http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ˆ u=U1,
e ˆ e=U2\par
1, p ˆ p= 1U2
1, and T ˆ T=T1, and ˆ = 1.\par
Incoming boundary layer thickness is used as the characteristic
length scale.\par
III. Flow Configuration\par
Figure 1 shows an inviscid flow schematic for the present STBLI
configuration. The incoming flow conditions are listed in Table 1,
including the reference experiment of Bookey et al. [7] for the same
flow.
To minimize numerical errors in the computation of Jacobian
matrices, we generate the grid using analytical transformations.
Details about the transformation can be found in Wu and Martin [12].
A sample grid is plotted in Fig. 2. The grid is clustered near the corner
in the streamwise direction and near the wall in the wall-normal
direction. The size of the computational domain is shown in Fig. 3.
There are 9 and 7 upstream and downstream of the corner in the
streamwise direction, 2:2 in the spanwise direction, and 5 in the
wall-normal direction. The number of grid points used is 1024
160 128 in the streamwise, spanwise, and wall-normal directions,
respectively. The largest and smallest grid spacings in the streamwise
direction are x‡ ˆ 7:2 and x‡ ˆ 3:4, respectively, with grid
points clustered near the corner. The grid spacing in the spanwise
direction is y‡ ˆ 4:1. In the wall-normal direction at the inlet, the
first grid is at z‡ ˆ 0:2 and there are 28 grid points within z‡ < 20.\par
Shock
Flow
24o\par
Fig. 1 Inviscid flow schematic for the compression ramp case.
880\par
WU AND MARTIN\par
Page 3
IV. Numerical Method and Boundary Conditions\par
A third-order accurate low-storage Runge–Kutta method is used
for the time integration, and a fourth-order accurate central standard
finite difference scheme is used to compute the viscous flux terms.
The incoming boundary layer is generated as in Martin [20]. The
rescaling method developed by Xu and Martin [21] is used to
generate the inflow condition. The recycling http://www.nuokui.com/pdf/bsBqaKwTJjLI.html station is located at 4:5
downstream of the inlet. Figure 4 plots the autocorrelation of u0 in the
streamwise direction. The correlation decreases to 0.1 in about 1:2 .
In turn, we find that the recycling station can be located as close as 2
downstream of the inlet without affecting the statistics of the
boundary layer. The data indicate that there is no forcing frequency
imposed by the rescaling method, as discussed further in Sec. VI.
Supersonic outflow boundary conditions are used at the outlet and
the top boundary. We use a nonslip condition at the wall, which is
isothermal. The wall temperature is set to 307 K. Details about initial
and boundary conditions can be found in [12,13]. To compute the
convective flux terms, we modify a fourth-order bandwidth-
optimized WENO [15] method by adding limiters [22]. Later, we
present a brief description of the original WENO method and how the
limiters are used.
In WENO methods, the numerical fluxes are approximated by the
weighted sum of fluxes on the candidate stencils. Figure 5 plots a
sketch of the WENO three-point candidate stencils. The numerical
flux can be expressed as
fi‡1\par
2\par
ˆ
Xr\par
kˆ0\par
!kqr\par
k\par
(9)
where qr\par
k are the candidate fluxes at (i ‡ 1=2) and !k are the weights.\par
The weights are determined by the smoothness on each candidate
stencil, where the smoothness is measured by
ISk ˆ
X\par
r 1
mˆ1\par
Z xi‡1=2\par
xi 1=2\par
… x†2m 1
@mqr\par
k\par
@xm
2
dx
(10)
Thus, larger weights are assigned to stencils with smaller ISk. For the
three-point per candidate stencil WENO shown in Fig. 5, Taylor
expansion of the above equation gives
ISk ˆ …f0\par
i x†2‰1 ‡ O… x2†Š\par
(11)
This means that in smooth regions for a well-resolved flowfield
[meaning f0\par
i is O…1†], ISk is of the order of x2, while for a\par
discontinuity,ISk is of the order of 1. Details about the formulation of
WENO methods can be found in Jiang and Shu http://www.nuokui.com/pdf/bsBqaKwTJjLI.html [23] and Martin et al.
[15], for example.
Previous work on WENO methods has been focused on
maximizing the bandwidth resolution and minimizing the dissipation
of the candidate stencils, that is, optimizing the linear part of WENO
methods. Examples include Weirs [14] and Martin [15]. The
numerical dissipation inherent in such methods can be avoided by
increasing the mesh size, which results in accurate results for
isotropic turbulence and turbulent boundary layers [15]. In stringent
problems such as STBLI, increasing the grid size is not affordable
and the numerical dissipation inherent in original WENO methods
precludes obtaining accurate results [16].
To mitigate the problem, we add limiters in the smoothness
measurement [22], namely, absolute limiter and relative limiter. The
definitions are shown in Eqs. (12) and (13), respectively,\par
7δ
9δ
4.5δ
2.2δ
5
δ\par
Fig. 3 Size of the computational domain for the DNS.\par
x/δ
z/δ\par
-5
0
5
0
2
4
6
8\par
Fig. 2 Sample grid for the DNS.
Table 1 Conditions for the incoming turbulent boundary layer
M
Re
, mm
, mm
Cf
, mm
1, kg=m3
U1, m=s
T1, K
Experiment [7]
2.9
2400
0.43
2.36
0.00225
6.7
0.074
604.5
108.1
DNS
2.9
2300
0.38
1.80
0.00217
6.4
0.077
609.1
107.1\par
∆x/δ
〈u
′(x)u
′(x
∆
x)〉/u\par
rm
s2\par
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1\par
Fig. 4 Autocorrelation of u0 in the streamwise direction at z ˆ 0:1 in
the incoming boundary layer for the DNS.
i
i-2
i 3
i-1
i 1
i 2
i 1/2
q1
q0
q3
q2
Fig. 5 Sketch of WENO candidate stencils with three points per
candidate.\par
WU AND MARTIN\par
881\par
Page 4
!k ˆ
Cr\par
k; if max…ISk† < AAL\par
!k; otherwise
(12)
!k ˆ
Cr\par
k; if max…ISk†= min…ISk† < ARL\par
!k; otherwise
(13)
where Cr\par
k are the optimal weights [15] and AAL and ARL are the\par
thresholds for the limiters. It is found that the relative limiter is more
general and less problem dependent [22]. In contrast, the relative
limiter defi http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ned by Eq. (13) is method dependent, that is, WENO
methods with different candidate stencil sizes have different
threshold values in the relative limiter. Thus, we define an alternative
relative limiter:
!k ˆ
Cr\par
k; if max…TVk†= min…TVk† < ATV
RL\par
!k; otherwise
(14)
where TVk stands for the total variation on each candidate stencil.
This new definition allows for consistent threshold values of about 5
in the relative limiter, independently of the stencil size. The improved
performance of the limiter (14) for the fourth-order bandwidth-
optimized WENO scheme is illustrated by computing the Shu–Osher
problem [24]. The initial conditions are
8
><
>:
\par
l ˆ 3:86;\par
\par
r ˆ 1 ‡ 0:2 sin…5x†\par
u \par
l ˆ 2:63;\par
u \par
r ˆ 0\par
p \par
l ˆ 10:33; p
r ˆ 1\par
(15)
Figure 6 plots the results at t ˆ 1:8. The line is the converged
numerical result using 1600 grid points. The square and triangle
symbols are the computed results with and without the relative
limiter, respectively, using 200 grid points. It is clear that the result
with the relative limiter is much better in the high frequency
fluctuation region, where the resolution is poor.
We also find that the absolute limiter alone improves our DNS
results for the compression ramp case [16]. In what follows, we show
that using both the limiters at the same time reduces the numerical
dissipation further, thereby improving the DNS results. According to
the definition of both the limiters, we see they have different effects
on reducing the numerical dissipation. To show the effects of
applying the limiters more clearly, 2-D nonlinearity index contour
plots computed in the wall-normal direction for the DNS of the
compression ramp case are shown in Fig. 7. The nonlinearity index is
defined as [25]
NI
ˆ
1
…r…r‡1††1=2
Xr\par
kˆ0\par
‰1=…r‡1†Š ‰…!k=Cr\par
k†=\par
Pr\par
kˆ0…!k=Cr
k†Š\par
‰1=…r‡1†Š
2 1=2
(16)
where r is the number of candidate http://www.nuokui.com/pdf/bsBqaKwTJjLI.html stencils. The nonlinearity index
has a value in the range of [0, 1]. The magnitude of NI indicates how
much dissipation is added by WENO. The smaller NI is, the less
dissipation is added. Ideally, NI should be zero everywhere except
for regions near discontinuities. Figure 7a shows that without any
limiter, the nonlinearity index has high values in a very large region
of the computational domain. Because in WENO methods,
numerical fluxes are computed in characteristic space, the NI values
plotted here are also computed in characteristic space for the
characteristic equation with eigenvalue equal to u ‡ a. The average
NI value is about 0.5. With the absolute limiter added, the dissipation
is reduced greatly, as shown in Fig. 7b. The average NI value is 0.09.
The same plot with the relative limiter is shown in Fig. 7c. The
average NI value is also about 0.09 for this case. With both the
relative and absolute limiters, as shown in Fig. 7d, the average NI
value is 0.02, indicating that the numerical dissipation is further\par
x*
ρ*\par
-0.5
0
0.5
1
1.5
2
2.5
3.0
3.5
4.0
4.5
exact
no limiter
relative limiter\par
Fig. 6 Density distribution att ˆ 1:8 for the Shu–Osher’s problem with
and without the relative smoothness limiter.
Fig. 7 Nonlinearity index for the compression ramp case: a) without limiters, b) with the absolute limiter, c) with the relative limiter, and d) with both the
relative and absolute limiters from DNS.
882\par
WU AND MARTIN\par
Page 5
reduced. In the DNS, we apply both limiters. However, the
simulation can be unstable, and we find that this can be avoided by
changing the relative limiter to
!k
ˆ
Cr\par
k; if max…TVk†= min…TVk† < ATV
RL and max…TVk† < BTV
RL\par
!k; otherwise
(17)
The additional threshold value BTV\par
RL guarantees enough dissipation\par
whenever max…TVk† is larger than the threshold. The threshold
values are AAL ˆ 0:01, ATV\par
RL ˆ 5, and BTV
RL ˆ 0:2 in the DNS. A stu http://www.nuokui.com/pdf/bsBqaKwTJjLI.html dy\par
of WENO methods including limiters for DNS of compressible
turbulence is given in Taylor et al. [22].\par
V. Accuracy of the DNS\par
DNS statistics are gathered using 300 flowfields with time
intervals equal to 1 =U1. Figure 8 plots the spanwise energy
spectrum of u at z‡ ˆ 15 for the incoming boundary layer. The
Reynolds number for the DNS is relatively low. Therefore no
obvious inertial range is observed in the spectrum. Over five decades
of decay are observed in the energy and no pileup of energy due to
numerical error is observed in the high frequency range. The DNS
results are compared with the experiments of Bookey et al. [7].
Figure 9 plots the mean wall-pressure distribution. Repeatability
studies [6] indicate an experimental uncertainty of about 5%. The
DNS data predict the wall-pressure distribution within the
experimental uncertainty. Figure 10 plots the nondimensionalized
size of the separation bubble versus Reynolds number. In the DNS,
the separation and reattachment points are defined as the points
where the mean skin friction coefficient changes sign. The
experimental value is inferred from surface oil visualization. The
error on the experimental value is hard to quantify from this
technique and it can easily be 10%,\par
‡\par
which corresponds to the error
bar in Fig. 10. The empirical envelope is from Zheltovodov et al. [26]
who correlated the size of the separation bubble for a large set of
experimental data. The characteristic length is defined as [26]
Lc ˆ
M3
p2
ppl
3:1
(18)
where p2 is the downstream inviscid pressure, and ppl is the plateau
pressure computed according to the empirical formula by Zukoski
[27]
ppl ˆ p1
1
2
M ‡ 1
(19)
The data points for the DNS and the reference experiment both lie
within the empirical envelope. The difference between them is about
10%. The predicted separation and reattachment points are at x ˆ
3 and x ˆ 1:3 , respectively (the corner is locate http://www.nuokui.com/pdf/bsBqaKwTJjLI.html d at x ˆ 0). In the
experiment of Bookey et al., the separation and reattachment points
are at x ˆ 3:2 and x ˆ 1:6 , respectively.
Figure 11a plots velocity profiles from the DNS and the
experiments of Bookey et al. [7] in the incoming boundary layer.
Figure 11b plots velocity profiles4 downstream of the corner, where
the velocity is nondimensionalized by that at the boundary layer
edge. There is a 5% uncertainty in the experimental measurement for
the boundary layer thickness,� as shown in the error bar. For both the
upstream and downstream data, the agreement is within 5%.
Figure 12 plots mass-flux turbulence intensities at different
streamwise locations for the DNS. Downstream of the interaction,
we see that the maximum of the mass-flux turbulence intensity is
amplified by a factor of 5, which is consistent with the number 4.8
that Selig et al. [28] found in experiments. Notice that the
experiments of Selig et al. are at a much higher Reynolds number
(Re ˆ 85; 000). However, the Mach number and ramp angle are the
same. Therefore the pressure rise throughout the interaction region is
the same. Assuming that the mass-flux turbulence intensity
amplification is mainly a function of pressure rise, it is reasonable to
make the above comparison.
Figure 13 plots Van Driest transformed mean velocity profiles at
different streamwise locations. Near the inlet of the computational
domain (x ˆ 8 ), the profile agrees well with the log law in the
logarithmic region. The profile does not change at x ˆ 4:1 , which
is about 1 upstream of the separation location. Downstream of the
interaction, the profiles show characteristic dips in the logarithmic\par
kyδ
E\par
u’\par
(k\par
y,x\par
,z
)/U\par
∞
2\par
50
100 150200
10-8
10-7
10-6
10-5
10-4
10-3
(kδ)-5/3\par
Fig. 8 Spanwise energy spectrum of u at z‡ ˆ 15 in the incoming
boundary layer for the DNS.\par
x/δ
P\par
w\par
/P\par
∞\par
-5
0
5
10
1.0
1.5
2.0
2.5
3.0
3.5
4.0 http://www.nuokui.com/pdf/bsBqaKwTJjLI.html
4.5
5.0
Bookey et al.
DNS\par
Fig. 9 Mean wall-pressure distribution from DNS and experimental
data, error bars at 5%.\par
Reδ
L se\par
p\par
/L\par
c\par
104
105
106
107
0
5
10
15
20
25
30
Empirical envelope
Bookey et al.
DNS\par
Fig. 10 Size of the separation bubble from DNS and experimental data,
error bars at 10%.\par
‡Smits, A., private communication, 2006.
�Smits, A., private communication, 2006.\par
WU AND MARTIN\par
883\par
Page 6
region, which is consistent with what Smits and Muck [29] found in
higher Reynolds number (Re ˆ 85; 000) experiments.\par
VI. DNS Results\par
Figure 14 is an instantaneous isosurface contour plot of the
magnitude of pressure gradient jrpj ˆ 0:5 for the DNS. It shows the
3-D shock structure. Except for the foot of the shock, which is inside
the boundary layer edge, the shock is quite flat in the spanwise
direction. Also a few shocklets that merge into the main shock are
visible downstream of the corner. They are formed due to the
compression at the reattachment point.
Figure 15 plots an instantaneous numerical schlieren plot, in
which the variable is defined as
NS ˆ c1 exp‰c2…x xmin†=…xmax
xmin†Š
(20)
where x ˆ jr j, and c1 and c2 are constants. We use c1 ˆ 0:8 and
c2 ˆ 10 in our analysis. This transformation enhances small density
gradients in the flowfield and resembles schlieren in experiments. As
shown in Fig. 15, the main shock wrinkles and the shock foot
penetrates into the boundary layer. A few shocklets emanate from the
edge of the boundary layer downstream of the interaction and they
merge into the main shock eventually. The turbulence structures in
the incoming boundary layer and downstream of the interaction are\par
z/δ
〈u
〉/U\par
∞\par
0
0.5
1
1.5
0.0
0.2
0.4
0.6
0.8
1.0
DNS
Bookey et al.\par
a)\par
z/δ
〈u
〉/U\par
e\par
0
0.5
1
1.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DNS
Bookey et al.\par
b)\par
Fig. 11 Velocity profiles in the incoming boundary layer a http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ) and 4 downstream of the corner b) from DNS and experimental data. The error bar
indicates a 5% error in the measurement for the boundary layer thickness.\par
z/δ
(ρ
u)′\par
rms\par
/ρ\par
∞\par
U\par
∞\par
0
0.5
1
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
-8.0δ
-4.1δ
2.0δ
4.2δ
6.1δ
6.9δ\par
Fig. 12 Mass-flux turbulence intensities at different streamwise
locations for the DNS.\par
z
〈u
〉 VD\par
100
101
102
103
0
5
10
15
20
25
30
-8.0δ
-4.1δ
2.0δ
4.2δ
6.1δ
6.9δ
2.44log(z ) 5.1\par
Fig. 13 Van Driest transformed mean velocity profiles at different
streamwise locations for the DNS.
Fig. 14 Contour plot of the magnitude of pressure gradient,
jrpj ˆ 0:5P1= , showing the 3-D shock structure for the DNS.\par
x/δ
z/δ\par
-5
0
5
0
2
4
6
8\par
Fig. 15 Instantaneous numerical schlieren plot for the DNS.\par
x/δ
z/δ\par
-5
0
5
0
2
4
6
8\par
Fig. 16 Time and spanwise averaged numerical schlieren plot for the
DNS.
884\par
WU AND MARTIN\par
Page 7
clearly seen. Downstream of the interaction, the gradients are
steeper, showing the turbulence amplification due to theflow through
the shock. Figure 16 plots the time and spanwise averaged numerical
schlieren. The turning of the flow at the separation bubble upstream
of the corner, results in the first portion of the main shock, which is at
29 deg and corresponds to an 11 deg turning angle. Near the
reattachment point, the flow is turned again by the ramp wall. The
compression waves can also be seen in Fig. 16. These waves are the
averaged shocklets shown in Fig. 15. They merge into the main
shock at a location of about 4 downstream of the corner and change
the angle of the main shock. The second part of the shock has an angle
of about 37 deg, which is still less than that of an inviscid shock angle
(43 deg). This is because the computational domain is not long
enough to let the shock evolve further. Notice that the shock appears
thicker in Fig. 16, indicating the motion o http://www.nuokui.com/pdf/bsBqaKwTJjLI.html f the main shock.\par
A. Evolution of the Boundary Layer\par
As shown in Fig. 13, streamwise velocity profiles change greatly
throughout the interaction region. Figure 17 plots three velocity
profiles at different streamwise locations using outer scales. For the
profile at x ˆ 1:9 , which is inside the separation region, the
velocity profile is very different from that at the inlet. It has a linear
behavior. Downstream of the interaction, at x ˆ 6:1 , the boundary
layer profile is not recovered. Also notice that there is no visible
oscillation near the shock, which means that the limiters presented in
Sec. III do not affect the good shock-capturing properties of WENO.
Turbulent fluctuations are amplified through the interaction
region. Figure 18 plots four components of the Reynolds stresses at
different streamwise locations. Downstream of the interaction, all the
components are amplified greatly. In particular, components u0u0
and v0v0 are amplified by factors of about 6, as shown in Figs. 18a
and 18b. Component w0w0 is amplified by a factor of about 12.
Component u0w0 has the largest amplification factor of about 24. As
being discussed in the previous section, mass-flux turbulence
intensity is amplified by a factor of about 5. Figure 19 plots the time-
averaged TKE (turbulent kinetic energy) in the streamwise-wall-
normal plane. In the incoming boundary layer, the TKE level is low
and the maximal value occurs very close to the wall. The TKE is
amplified through the interaction region. Inside the separation bubble
near the ramp corner, the TKE level is low. Downstream of the
interaction, the TKE is greatly amplified.
Morkovin’s SRA (strong Reynolds analogy) is well known for
compressible turbulent boundary layer flows. The SRA relations are
given by
T02
p
~T
ˆ …
1†M2
u02
p
~u
(21)
RuT ˆ
u0T0
u02
p
T02
p
ˆ const
(22)
where a tilde in the equations denotes Favre average. Figure 20
shows http://www.nuokui.com/pdf/bsBqaKwTJjLI.html T0\par
rms ~u=…
\par
1†M2u0\par
rms\par
~T and RuT at different streamwise
locations. Upstream of the separation region, Fig. 20a, the SRA
relations are satisfied except in the very near wall region and the
region close to the boundary layer edge. Figures 20b and 20c show
the data inside the interaction region. We observe that the SRA
relations are still valid in the outer part of the boundary layer
(z > 0:5 ). While in the near wall region, the SRA cannot be applied.
The location of the last plot in Fig. 20d is 6:1 away from the ramp
corner, which is very close to the outlet. The two quantities show a\par
z/δ
〈u
〉/U\par
∞\par
0
1
2
3
0
0.2
0.4
0.6
0.8
1
1.2
-8.0δ
-1.9δ
6.1δ\par
Fig. 17 Velocity profiles at three different streamwise locations for the
DNS.\par
z/δ
〈ρ
u
′u
′〉/〈ρ
〉U\par
∞
2\par
10-2
10-1
100
0.00
0.05
0.10
-8.0δ
-4.1δ
-1.9δ
1.0δ
4.2δ
6.1δ\par
z/δ
〈ρ
v
′v
′〉/〈ρ
〉U\par
∞
2\par
10-2
10-1
100
0.00
0.01
0.02
0.03
0.04
0.05
-8.0δ
-4.1δ
-1.9δ
1.0δ
4.2δ
6.1δ\par
z/δ
〈ρ
w
′w
′〉/〈ρ
〉U\par
∞
2\par
10-2
10-1
100
0.00
0.01
0.02
0.03
0.04
-8.0δ
-4.1δ
-1.9δ
1.0δ
4.2δ
6.1δ\par
z/δ
〈ρ
u
′w
′〉/〈ρ
〉U\par
∞
2\par
10-2
10-1
100
0.00
0.01
0.02
0.03
0.04
0.05
-8.0δ
-4.1δ
-1.9δ
1.0δ
4.2δ
6.1δ\par
a)
b)
c)
d)\par
Fig. 18 Reynolds stresses at different streamwise locations for the
DNS.\par
x/δ
z/δ\par
-5
0
5
0
2
4
6
8
10
0.01 0.02 0.04 0.05 0.07 0.08 0.10\par
Fig. 19 Contours of the time-averaged TKE u0u0=2 1U2\par
1 level for\par
the DNS.
z/δ\par
0
0.2
0.4
0.6
0.8
1
0.0
0.5
1.0
1.5
2.0
T’rmsu/((γ-1)M2u’rmsT)
-RuT
~
~\par
z/δ\par
0
0.2
0.4
0.6
0.8
1
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
T’rmsu/((γ-1)M2u’rmsT)
-RuT
~
~\par
z/δ\par
0
0.2
0.4
0.6
0.8
1
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
T’rmsu/((γ-1)M2u’rmsT)
-RuT
~
~\par
z/δ\par
0
0.2
0.4
0.6
0.8
1
0.0
0.5
1.0
1.5
2.0
T’rmsu/((γ-1)M2u’rmsT)
-RuT
~
~\par
a)
b)
c)
d)\par
Fig. 20 SRA Eqs. (21) and (22) at different streamwise locations for the
DNS: a) x http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ˆ 5:7 ; b) x ˆ 2 ; c) x ˆ 2 ; d) x ˆ 6:1 .\par
WU AND MARTIN\par
885\par
Page 8
trend of going back to their values upstream of the interaction;
however, these still deviate from the SRA relations. This indicates
that the boundary layer is not fully recovered to equilibrium within
the computational domain. In fact, Martin et al. [30] pointed out that
it may take 22–30 for the boundary layer to recover downstream of
the interaction in our case.
Figure 21 plots the isosurface of the discriminant of the velocity
gradient tensor, which is a quantity used to identify vortical
structures in incompressible flows [31]. The level shown in Fig. 21 is
1 10 5 normalized by the maximal value. Figures 22 and 23 are
zoomed in views of Fig. 21 upstream and downstream of the ramp
corner, respectively. Upstream of the interaction, coherent structures
are observed. Near the corner where the interaction takes place, the
structures are more chaotic and of smaller extent. There are two
possible reasons accounting for the change of the structure extent.
First, the structures can be chopped by the strong shock and become
smaller. Second, fluids are compressed through the shock, and the
vortical structures are also compressed and become smaller.
Downstream of the interaction the structures are still small and
chaotic. Near the outlet of the computational domain, they start to
show a trend of going back to their original size and shape upstream
of the corner.\par
B. Shock Motion and Wall-Pressure Fluctuation\par
Experiments have shown evidence of large scale, slow shock
motion. Ganapathisubramani et al. [19] proposed that very long
structures of uniform momentum in the incoming boundary layer are
responsible for the slow motion. There have been many experimental
studies on the turbulent structure of supersonic boundary layers [32–
36]. In particular, Ganapathisubramani et al. [37] have shown
evidence of the existence of very long s http://www.nuokui.com/pdf/bsBqaKwTJjLI.html tructures in supersonic
boundary layers. For the signal length that they considered, they
observed structures as long as 8 . In our DNS, we have only 9
upstream of the corner. However, using the Taylor hypothesis as it is
done experimentally [19], the DNS data also exhibit these very long,
meandering regions of low momentum. Figure 24 plots contours of
normalized mass flux in the logarithmic region (z ˆ 0:2 ) from the
DNS. The rake signal is reconstructed using Taylor’s hypothesis and
a convection velocity of 0:76U1. Notice that the aspect ratio of x to y
is 0.067 in the figure. The presence of these long structures in the
DNS data shows that they are an inherent part of a turbulent boundary
layer. In addition, we observe evidence of the low frequency shock
motion, as shown later.
The shock motion can be inferred from the wall-pressure signal or
from monitoring the mass flux in the freestream, for example.
Figure 25 plots wall-pressure signals versus time at different\par
Fig. 21 Isosurface of the discriminant of the velocity gradient tensor
for the DNS. Isosurface value is 10 5 that of the maximum value.
Fig. 22 Isosurface of the discriminant of the velocity gradient tensor
upstream of the ramp corner for the DNS. Zoomed visualization of
Fig. 21.
Fig. 23 Isosurface of the discriminant of the velocity gradient tensor
downstream of the ramp corner for the DNS. Zoomed visualization of
Fig. 21.\par
x/δ
y/δ\par
0
50
100
150
200
0
1
2\par
510
480
450
420
390\par
u(m/s)\par
Fig. 24 Rake signal at z= ˆ 0:2. The x axis is reconstructed using
Taylor’s hypothesis and a convection velocity of 0:76U1. Data are
averaged along the streamwise direction in 4 .\par
tU∞/δ
P\par
w\par
/P\par
∞\par
0
100
200
300
1.0
1.5
2.0
2.5
-6.9δ
-2.98δ (mean separation point)
-2.18δ\par
Fig. 25 Wall-pressure signals at different streamwise locations for the
DNS.
886\par
WU AND MARTIN\par
Page 9
streamwise locations. The length of http://www.nuokui.com/pdf/bsBqaKwTJjLI.html the signals is about 300 =U1.
For the signal at x ˆ 2:18 , the wall pressure shows a range of
frequencies, including a low frequency mode. The magnitude of the
signal varies from about 1.2 to 2.0 in a periodic manner, indicating
that the shock is moving upstream and downstream around that point.
It should be pointed out that the intermittent character of the wall-
pressure signals from the DNS is not as strong as that observed in
higher Reynolds number experiments. This may be due to the
Reynolds number difference. When the Reynolds number is low,
which is the case for the current DNS, viscous effects are more
prominent, and the shock does not penetrate into the boundary layer
as deeply as for higher Reynolds number cases. In fact, it is observed
from the DNS data that the shock is diffused into a compression fan-
type structure near the shock foot region. Figure 26 plots the energy
spectra for the same wall-pressure signals. To avoid overlapping, the
spectra for the signals at x ˆ 2:98 and x ˆ 2:18 are multiplied
by 103 and 106, respectively. In the incoming boundary (x ˆ 6:9 ),
the most energetic frequency is around0:1–1U1= . However, for the
other two signals, the most energetic frequency is much lower. At
x ˆ 2:98 , the spectrum has a peak at frequency equal to
0:007U1= , which corresponds to a time scale of 140 =U1. For the
signal at x ˆ 2:18, the most energetic frequency ranges from
0:007U1= to 0:01U1= , corresponding to a time scale of
100–140 =U1. The dimensionless frequency computed from
Eq. (1) is between 0.03 to 0.043 for the last two signals, which is
consistent with what Dussauge et al. [17] found based on
experimental data. Recall that the recycling station is located at 4:5
downstream of the inlet in the DNS. It is doubted that this can impose
a forcing frequency of about 0:2U1= on the flow. Figure 26 shows
that none of the signals has a dominant frequency near this specific
value. Figure 27 plots t http://www.nuokui.com/pdf/bsBqaKwTJjLI.html he intermittency function computed from wall
pressure. It is defined as the fraction of time that the wall pressure at a
location is greater than a threshold. Here the threshold value used is
1:2P1. The inverse maximum slope of the intermittency function is
1:7 . The intermittency profile shifts in the streamwise direction with
different threshold value. However, its shape is not affected much by
the threshold.
The motion of the shock can also be measured in the freestream.
For example, Weiss and Chokani [38] used mass-flux signals along
the streamwise direction at a location of 1:5 away from the wall.
Figure 28 plots three mass-flux signals measured in the experiments
of Weiss and Chokani [38]. The signal measured at the mean shock
location shows an intermittent character. We use the same method by
Weiss and Chokani. The mass-flux signals are measured at different
streamwise locations with a distance of 2 away from the wall.
Figure 29 plots three mass-flux signals normalized by the freestream
quantities. The characteristics of the signals are similar to those
observed in Fig. 28. The solid line is a signal measured at a location
upstream of the shock. The magnitude of mass flux is about 1.1 for
this signal. The dash-dotted line is a signal measured downstream of
the shock. The mass flux fluctuates around 1.8. The dotted line data
are measured inside the shock motion region. The magnitude of the
signal varies between that of the solid line and dash-dotted line,
indicating that the shock moves upstream and downstream of this
point. Notice that in Fig. 28 the length of the signals is about\par
fδ/U∞
E\par
p/p
∞
2\par
10-2
10-1
100
101
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
-6.9δ
-2.98δ
-2.18δ\par
Fig.26 Spectra of wall-pressure signals at different streamwise
locations for the DNS. The spectra for the signals at x ˆ 2:98 and
x ˆ 2:18 are multiplied by 103 and 106, respectively.\par
x/δ
In
te
rm
itte
http://www.nuokui.com/pdf/bsBqaKwTJjLI.html n
c
y
fu
n
c
tio
n\par
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
0
0.2
0.4
0.6
0.8
1\par
Fig. 27 Intermittency function computed from wall pressure for the
DNS.
Fig. 28 Mass-flux signals at different streamwise locations from Weiss
and Chokani [38]: a) sensor positioned upstream of the shock; b) sensor
positioned at the mean shock location; c) sensor positioned downstream
of the shock.\par
tU∞/δ
ρu/ρ\par
∞\par
U\par
∞\par
0
100
200
300
0.8
1.0
1.2
1.4
1.6
1.8
2.0\par
upstream of the shock (x=-2.9δ)
inside shock motion region (x=0.8δ)
downstream of the shock (x=1.5δ)\par
Fig. 29 Mass-flux signals at different streamwise locations at z ˆ 2 for
the DNS.\par
WU AND MARTIN\par
887\par
Page 10
1200 =U1, which is nearly 4 times longer than that from the DNS.
Also the Mach number in the experiments is 3.5, which is greater than
that in the DNS. Figure 30 plots the energy spectra for the three DNS
mass-flux signals. The measurement inside the shock motion region
is dominated by much lower frequencies relative to those in the other
two signals. The spectrum peaks in a frequency range of
0:007–0:013U1= , which corresponds to a time scale of
77–140 =U1. This is consistent with the result obtained from the
wall-pressure analysis. Notice that the mass-flux signals have much
lower resolution than that of the wall-pressure signals shown in
Fig. 25. This is because the wall-pressure signals in Fig. 25 were
recorded at each time step during the simulation, while the mass-flux
signals were obtained using data that were saved at large time
intervals.
The scale of the shock motion can be quantified by the
intermittency function proposed by Weiss and Chokani [38]. It is
defined as the fraction of time that the shock resides upstream of the
measurement location. Thus the intermittency function is 0=1 if a
location is always upstream/downstream of the shock. Instantaneous
massflux is used to determine whether a given location is up http://www.nuokui.com/pdf/bsBqaKwTJjLI.html stream or
downstream of the shock. When the instantaneous mass flux is
greater than some threshold value, the location is said to be
downstream of the shock, and vice versa. The average of the
upstream and downstream mass flux is used as the threshold.
Figure 31 plots the intermittency function versus streamwise
location. For reference, the experimental result from Weiss and
Chokani [38] is also plotted. Notice that the experimental data points
are shifted in the streamwise direction to make the center of the DNS
and experimental intermittency function align with each other.
Define the intermittent length of the shock motion as the inverse
maximum slope of the intermittency function. Thus, for the DNS, the
intermittent length is 0:47 . For Weiss and Chokani’s experiments,
the intermittent length is about 0:2 .
It is known that large scale shock motion produces high level wall-
pressure fluctuations. Figure 32 plots the normalized wall-pressure
fluctuation versus streamwise location. There are two peaks present.
The first one is at x ˆ 2:3 , which is downstream of the mean
separation point. It has a magnitude of about 13.5%. The second peak
is located at about x ˆ 0:8 with a magnitude of about 11.5%. The
magnitude of the first peak is lower than that of higher Reynolds
number experiments. For example, Dolling and Murphy [4]
measured a peak value of about 20%. Currently, no experimental
data at the same flow conditions are available for comparison.\par
VII. Conclusions\par
A DNS of a 24 deg compression ramp configuration is performed.
Applying limiters to the smoothness measurement in the WENO
scheme reduces the numerical dissipation. In particular, using a
combination of absolute and relative limiters is very effective. The
DNS data predict the experiments with a satisfactory accuracy for the
upstream boundary layer, mean wall-pressure distribution, size of the
separation bubble, velocity profile downstream of the int http://www.nuokui.com/pdf/bsBqaKwTJjLI.html eraction,
and mass-flux turbulence intensity amplification.
Numerical schlieren and 3-D isosurfaces of jrpj reveal the
structures of the shock system. Turbulence intensities are amplified
greatly through the interaction region. In particular, mass-flux
turbulence intensity is amplified by a factor of about 5. Reynolds
stress components are greatly amplified with amplification factors of
about 6–24. As summarized by Smits and Muck [29], there are a few
mechanisms that account for turbulence amplification. Across the
shock, the turbulence level is increased due to the Rankine–Hugoniot
jump conditions and nonlinear coupling of turbulence, vorticity, and
entropy waves [39]. The unsteady shock motion also pumps energy
from the mean flow into the turbulent fluctuations. In addition, the
concave streamline curvature near the ramp corner makes the flow
unstable and amplifies the turbulence level [40]. SRA relations are
satisfied in the incoming boundary layer. However, in a large
neighborhood of the interaction region, the relations are found not
valid, especially in the near wall region (z < 0:5 ). This indicates that
the boundary layer has not fully recovered to equilibrium
downstream of the interaction within the computational domain.
Wall-pressure and mass-flux signals including spectral analysis
indicate that there is a low frequency motion of the shock with a
characteristic time scale of about 77–140 =U1, which is consistent
with that found in experiments. The magnitude of the shock motion is
quantified by the intermittency function computed from mass-flux
signals in the freestream. The intermittent length defined as the
inverse of the maximum slope of the intermittency function is 0:47
in the DNS. Dolling and Or [5] found the amplitude of the shock
motion of about 0:8 at higher Reynolds number experiments. The
physical mechanism that drives the low frequency motion in the DNS
remains to be studied.\par
Acknowledgments http://www.nuokui.com/pdf/bsBqaKwTJjLI.html \par
This work is supported by the U.S. Air Force Office of Scientific
Research under grants AF/F49620-02-1-0361 and AF/9550-06-1-\par
fδ/U∞
E\par
ρu\par
/(ρ
U\par
∞\par
)2\par
10-2
10-1
10-6
10-5
10-4
10-3
10-2
upstream of the shock
inside shock motion region
downstream of the shock\par
Fig. 30 Spectra of mass-flux signals at different streamwise locations
for the DNS.
x/δ
Inte
rm
itte
nc
y
func
tion\par
-0.5
0
0.5
1
1.5
0
0.2
0.4
0.6
0.8
1
DNS
Weiss & Chokani\par
Fig. 31 Intermittency function computed from mass flux for the DNS.\par
x/δ
p
′ rms
/P\par
w\par
-5
0
5
0.04
0.06
0.08
0.10
0.12
0.14\par
Fig. 32 Normalized wall-pressure fluctuation distribution for the DNS.
888\par
WU AND MARTIN\par
Page 11
0323. The authors would like to acknowledge A. J. Smits for useful
discussions during the assessment of the accuracy of the numerical
data.\par
References\par
[1] Settles, G. S., Vas, I. E., and Bogdonoff, S. M., “Details of a Shock-
Separated Turbulent Boundary Layer at a Compression Corner,” AIAA
Journal, Vol. 14, No. 12, 1976, pp. 1709–1715.
[2] Settles, G. S., Fitzpatrick, T., and Bogdonoff, S. M., “Detailed Study of
Attached and Separated Compression Corner Flowfields in High
Reynolds Number Supersonic Flow,” AIAA Journal, Vol. 17, No. 6,
1979, pp. 579–585.
[3] Settles, G. S., Perkins, J. J., and Bogdonoff, S. M., “Investigation of
Three-Dimensional Shock/Boundary-Layer Interactions at Swept
Compression Corners,” AIAA Journal, Vol. 18, No. 7, 1980, pp. 779–
785.
[4] Dolling, D. S., and Murphy, M. T., “Unsteadiness of the Separation
Shock Wave Structure in a Supersonic Compression Ramp Flowfield,”
AIAA Journal, Vol. 21, No. 12, 1983, pp. 1628–1634.
[5] Dolling, D. S., and Or, C. T., “Unsteadiness of the Shock Wave
Structure in Attached and Separated Compression Corner Flow Fields,”
AIAA Paper 83-1715, July 1983.
[6] Selig, M. S., “Unsteadiness of Shock Wave/Turbulent Boundary Layer
Interaction http://www.nuokui.com/pdf/bsBqaKwTJjLI.html s with Dynamic Control,” Ph.D. Thesis, Princeton
University, Princeton, NJ,1988.
[7] Bookey, P. B., Wyckham, C., Smits, A. J., and Martin, M. P., “New
Experimental Data of STBLI at DNS/LES Accessible Reynolds
Numbers,” AIAA Paper 2005-309, Jan. 2005.
[8] Zheltovodov, A. A., “Advances and Problems in Modeling of
Shockwave Turbulent Boundary Layer Interactions,” Proceedings of
the International Conference on the Methods of Aerophysical
Research, Institute of Theoretical and Applied Mechanics,
Novosibirsk, Russia, 2004, pp. 149–157.
[9] Knight, D., Yan, H., Panaras, G. A., and Zheltovodov, A., “Advances in
CFD Prediction of Shock Wave Turbulent Boundary Layer
Interactions,” Progress in Aerospace Sciences, Vol. 39, Nos. 2–3,
2003, pp. 121–184.
[10] Adams, N. A., “Direct Numerical Simulation of Turbulent Boundary
Layer along a Compression Ramp at M ˆ 3 and Re ˆ 1685,” Journal
of Fluid Mechanics, Vol. 420, Oct. 2000, pp. 47–83.
[11] Rizzetta, D., and Visbal, M., “Large-Eddy Simulation of Supersonic
Compression-Ramp Flow by High-Order Method,” AIAA Journal,
Vol. 39, No. 12, 2001, pp. 2283–2292.
[12] Wu, M., and Martin, M. P., “Direct Numerical Simulation of
Shockwave/Turbulent Boundary Layer Interactions,” AIAA Pa-
per 2004-2145, June 2004.
[13] Wu, M., Taylor, E. M., and Martin, M. P., “Assessment of STBLI DNS
Data and Comparison against Experiments,” AIAA Paper 2005-4895,
June 2005.
[14] Weirs, G. V., “A Numerical Method for the Direct Simulation of
Compressible Turbulence,” Ph.D. Thesis, University of Minnesota,
Minneapolis, MN, 1998.
[15] Martin, M. P., Taylor, E. M., Wu, M., and Weirs, V., “A Bandwidth-
Optimized WENO Scheme for the Effective Direct Numerical
Simulation of Compressible Turbulence,” Journal of Computational
Physics, Vol. 220, No. 1, 2006, pp. 270–289.
[16] Wu, M., and Martin, M. P., “Assessment of Numerical Methods for
DNS of Shockwave/Turbulent Boundary Layer Interaction,” AIAA
Paper 2006-0717, Jan. 2006 http://www.nuokui.com/pdf/bsBqaKwTJjLI.html .
[17] Dussauge, J. P., Dupont, P., and Devieve, J. F., “Unsteadiness in Shock
Wave Boundary Layer Interactions with Separation,” Aerospace
Science and Technology, Vol. 10, No. 2, 2006, pp. 85–91.
[18] Andreopoulos, J., and Muck, K. C., “Some New Aspects of The Shock-
Wave/Boundary-Layer Interaction in Compression-Ramp Flows,”
Journal of Fluid Mechanics, Vol. 180, July 1987, pp. 405–428.
[19] Ganapathisubramani, B., Clemens, N. T., and Dolling, D. S., “Effects of
Upstream Coherent Structures on Low-Frequency Motion of Shock-
Induced Turbulent Separation,” AIAA Paper 2007-1141, Jan. 2007.
[20] Martin, M. P., “DNS of Hypersonic Turbulent Boundary Layers. Part 1:
Initialization and Comparison with Experiments,” Journal of Fluid
Mechanics (to be published).
[21] Xu, S., and Martin, M. P., “Assessment of Inflow Boundary Conditions
for Compressible Turbulent Boundary Layers,” Physics of Fluids,
Vol. 16, No. 7, 2004, pp. 2623–2639.
[22] Taylor, E. M., Wu, M., and Martin, M. P., “Optimization of Nonlinear
Error Sources for Weighted Essentially Non-Oscillatory Methods in
Direct Numerical Simulations of Compressible Turbulence,” Journal of
Computational Physics (to be published).
[23] Jiang, G., and Shu, C., “Efficient Implementation of Weighted ENO
Schemes,” Journal of Computational Physics, Vol. 126, No. 1, 1996,
pp. 202–228.
[24] Shu, C.-W., and Osher, S., “Efficient Implementation of Essentially
Non-Oscillatory Shock-Capturing Schemes, II,” Journal of Computa-
tional Physics, Vol. 83, No. 1, 1989, pp. 32–78.
[25] Taylor, E., and Martin, M., “Stencil Adaption Properties of a WENO
Scheme in Direct Numerical Simulations of Compressible
Turbulence,” Journal of Scientific Computing (to be published).
[26] Zheltovodov, A. A., Sch�lein, E., and Horstman, C., “Development of
Separation in The Region Where a Shock Interacts with a Turbulent
Boundary Layer Perturbed by Rarefaction Waves,” Journal of Applied
Mechanics and Technical Phys http://www.nuokui.com/pdf/bsBqaKwTJjLI.html ics, Vol. 34, No. 3, 1993, pp. 346–354.
[27] Zukoski, E., “Turbulent Boundary Layer Separation in Front of a
Forward Facing Step,” AIAA Journal, Vol. 5, No. 10, 1967, pp. 1746–
1753.
[28] Selig, M. S., Andreopoulos, J., Muck, K. C., Dussauge, J. P., and Smits,
A. J., “Turbulent Structure in a Shock Wave/Turbulent Boundary-
Layer Interaction,” AIAA Journal, Vol. 27, No. 7, 1989, pp. 862–869.
[29] Smits, A. J., and Muck, K. C., “Experimental Study of Three Shock
Wave/Turbulent Boundary Layer Interactions,” Journal of Fluid
Mechanics, Vol. 182, Sept. 1987, pp. 291–314.
[30] Martin, M. P., Smits, A. J., Wu, M., and Ringuette, M., “The
Turbulence Structure of Shockwave and Boundary Layer Interaction in
a Compression Corner,” AIAA Paper 2006-0497, 2006; also Journal of
Computational Physics (to be published).
[31] Blackburn, H. M., Mansour, N. N., and Cantwell, B. J., “Topology of
Fine-Scale Motions in Turbulent Channel Flow,” Journal of Fluid
Mechanics, Vol. 310, March 1996, pp. 269–292.
[32] Samimy, M., Arnette, S. A., and Elliott, G. S., “Streamwise Structures
in a Turbulent Supersonic Boundary Layer,” Physics of Fluids, Vol. 6,
No. 3, 1994, pp. 1081–1083.
[33] Smith, M. W., and Smits, A. J., “Visualization of the Structure of
Supersonic Turbulent Boundary Layers,” Experiments in Fluids,
Vol. 18, No. 4, 1995, pp. 288–302.
[34] Spina, E. F., Donovan, J. F., and Smits, A. J., “On the Structure of High-
Reynolds-Number Supersonic Turbulent Boundary Layers,” Journal of
Fluid Mechanics, Vol. 222, Jan. 1991, pp. 293–327.
[35] Cogne, S., Forkey, J., Miles, R. B., and Smits, A. J., “The Evolution of
Large-Scale Structures in a Supersonic Turbulent Boundary Layer,”
Proceedings of the Symposium on Transitional and Turbulent
Compressible Flows, ASME Fluids Engineering Division, Fairfield,
NJ, 1993.
[36] Dussauge, J. P., and Smits, A. J., “Characteristic Scales for Energetic
Eddies in Turbulent Supersonic Boundary Layers,” Proceedings http://www.nuokui.com/pdf/bsBqaKwTJjLI.html of the
Tenth Symposium on Turbulent Shear Flows, Pennsylvania State
University, University Park, PA, 1995.
[37] Ganapathisubramani, B., Clemens, N. T., and Dolling, D. S., “Large-
Scale Motions in a Supersonic Turbulent Boundary Layer,” Journal of
Fluid Mechanics, Vol. 556, June 2006, pp. 271–282.
[38] Weiss, J., and Chokani, N., “Quiet Tunnel Experiments of Shockwave/
Turbulent Boundary Layer Interaction,” AIAA Paper 2006-3362,
June 2006.
[39] Anyiwo, J. C., and Bushnell, D. M., “Turbulence Amplication in
Shock-Wave Boundary-Layer Interaction,” AIAA Journal, Vol. 20,
No. 7, 1982, pp. 893–899.
[40] Bradshaw, P., “The Effect of Mean Compression or Dilatation on the
Turbulence Structure of Supersonic Boundary Layers,” Journal of
Fluid Mechanics, Vol. 63, April 1974, pp. 449–464.\par
N. Clemens
Associate Editor\par
WU AND MARTIN\par
889
}{
\rtlch\fcs1 \af0 \ltrch\fcs0 \fs24\insrsid6493368\charrsid1074055
\par }\pard \ltrpar\qj \li0\ri0\sl180\slmult0\nowidctlpar\wrapdefault\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0\pararsid6493368 {\rtlch\fcs1 \af0 \ltrch\fcs0 \insrsid1074055
\par
\par
\par }{\rtlch\fcs1 \af0\afs30 \ltrch\fcs0 \fs30\cf17\dbch\af18\insrsid1074055\charrsid1074055 \hich\af0\dbch\af18\loch\f0 Free Document Search Engine. support all pdf,DOC,PPT,RTF,XLS,TXT\hich\af0\dbch\af18\loch\f0 ,Ebook! \hich\af0\dbch\af18\loch\f0 F
\hich\af0\dbch\af18\loch\f0 ree\hich\af0\dbch\af18\loch\f0 \hich\af0\dbch\af18\loch\f0 download! You can search all kind of documents!}{\rtlch\fcs1 \af0\afs30 \ltrch\fcs0 \fs30\cf17\dbch\af18\insrsid6493368\charrsid1074055 \hich\af0\dbch\af18\loch\f0 }{
\rtlch\fcs1 \af0\afs30 \ltrch\fcs0 \fs30\cf17\insrsid1074055\charrsid1074055
\par }{\field\fldedit{\*\fldinst {\rtlch\fcs1 \af0\afs28 \ltrch\fcs0 \fs28\cf11\insrsid14892288\charrsid1074055 \hich\af0\dbch\af13\loch\f0 HYPERLINK "http://www.downhi.com/"}{\rtlch\fcs1 \af0\afs28 \ltrch\fcs0 \fs28\cf11\insrsid10707375\charrsid1074055
{\*\datafield
00d0c9ea79f9bace118c8200aa004ba90b0200000003000000e0c9ea79f9bace118c8200aa004ba90b4e00000068007400740070003a002f002f00770065006e00640061006e0067002e0064006f00630073006f0075002e0063006f006d002f000000795881f43b1d7f48af2c825dc485276300000000a5ab0000000000}}
}{\fldrslt {\rtlch\fcs1 \af0\afs28 \ltrch\fcs0 \cs17\fs28\ul\cf2\insrsid14892288\charrsid1074055 \hich\af0\dbch\af13\loch\f0 http://www.downhi.com/}}}\sectd
\linex0\headery851\footery992\colsx425\endnhere\sectlinegrid312\sectspecifyl\sectrsid6493368\sftnbj {\rtlch\fcs1 \af0\afs28 \ltrch\fcs0 \fs28\cf11\insrsid6493368\charrsid1074055
\par }{\rtlch\fcs1 \af0\afs28 \ltrch\fcs0 \fs28\cf11\insrsid15098623\charrsid1074055
\par }}