NASA Technical Reports Server (NTRS) 20010004101: CFD Validation Studies for Hypersonic Flow Prediction

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by  NASA Technical Reports Server (NTRS)

AIAA Paper 2001-1025
CFD Validation Studies for Hypersonic
Flow Prediction
Peter A. Gnoffo
NASA Langley Research Center
Hampton, Virginia 23681-0001
39th AIAA Aerospace Sciences
Meeting & Exhibit
8-11 January 2001 / Reno, NV
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CFD Validation Studies for Hypersonic Flow Prediction
Peter A. Gnoffo _
[email protected]
NASA I,angley Research Center
Hampton, VA 23681-0001
Abstract
A series of experiments to measure pressure and heating for code validation involving hypersonic, lamina,',
separated flows was conducted at the C',.dspan-University at Buffalo Research Center (CUBRC) in the l.arge Energy
National Shock (LENS) tunnel. The experimental data serves as a tbcus for a code validation session but are not
available to the authors until the conclusion of this session. The first set of experiments considered here involve
Mach 9.5 and Mach! !.3 N2 flow over a hollow cylinder-flare with 30°flare angle at several Reynolds numbers
sustaining laminar, separated flow. Truncated and extended flare configurations are considered. The second set of
experiments, at similar conditions, involves flow over a sharp, double cone with fore-cone angle of 25'_and aft-cone
angle of 55". Both sets of experiments involve 30" compressions. Location of the separation point in the numerical
simulation isextremely sensitive to the level of grid refinement inthe numerical predictions. The numerical
simulations also show a significant influence of Reynolds number on extent of separation. Flow unsteadiness was
easily introduced into the double cone simulations using aggressive relaxation parameters thai normally promote
convergence.
Nomenclature
pressure coefficient, 2(p-p_)/p_V_ 2 Subscripts
Cp
L reference length from leading edge to junction art attachment point, zero shear
M Mach number i stream wise coordinate direction
.,11 molecular weight, kg/k-mole j normal coordinate direction
P pressure, N/m2 sep separation point, zero shear
Pr Prandfl number w wall surface conditions
q_ wall heat transfer rate, W/m 2 free stream conditions
Re p_V_/I_, Reynolds number, ml
S arc length, mm Table Abbreviations
St hypersonic approximation to Stanton number, A adapted grid. shock aligned
2q_/p_V. _ C curve fit collision cross section
T temperature. K HCEF hollow cylinder, extended flare
Tv vibrational temperature, K HCTF hollow cylinder, truncated flare
LL,_r velocity components in xand r directions, NS Navier-Stokes
respectively, m/s PG perfect gas
W_ free stream velocity, m/s S Sutherland's law
x,r cylindrical coordinates, cm SDB sharp, double cone
g eigenvalue limiter TE thermal equilibrium
Y specific heat ratio TLNS thin-layer Navier-Stokes
bt viscosity, kg/m-s TN thermal nonequilibrium
unadapted grid
P density, kg/m _ [r
Senior Research lingineer. Aerotherm¢_ynamics Brallch. Associate Felh,w AIAA
Copyright _,:32001 by the AmericaJ_ Institute of Aeronautics mid Astronautics. Inc. N_ c_pyright is asserted m the United States under I'ille 17.
U. S. Code. The U.S. (}ovcmmcnt has aroyally frec license to exercise all rights under the copyright claimed hcrcm I_r (;ovcrnmcntal Purp_scs
All other rights are reserved by the copyright owner.
Introduction Mach number and Reynolds number are derived fi'om
A series of numerical simulations were conducted those quantities using appropriate thermodynamic and
of experiments performed in the LENS facility for the transport property relations for molecular nitrogen in
purpose of code validation under hypersonic the simulation code. All experiments were conducted
conditions _.The experiments were conceived to in the Calspan-University at Buffalo Research Center
challenge simulation capabilities under conditions of (CUBRC) in the Large Energy National Shock (LENS}
large-scale separation while minimizing complicating tunnel _.Nominal flow conditions were at Mach 9.5 and
factors associated with turbulence, gas chemistry, and Mach 11.5 with Reynolds numbers per meter from
three-dimensional end-effects. Experimental data will 140000 to 360000 in nitrogen. Steady, laminar,
not be released until numerical simulations are reported axisymmetric flow was reported for all tests considered
on January ! 1,20012. here. Experimental data includes surface pressure and
The validation exercise here is similar to the First heating. This data will only be made available at the
Europe-US High Speed Flow Field Database conclusion of the conference session on January 1I,
Workshop _.The hollow cylinder, truncated flare 2001-'.
configuration was featured in that workshop where
both the original experiment in R5Ch and several
computational simulations were reported. More
recently, experimental data and numerical simulations
were reported +of two, double-cone (25°/35 °and
25"/50 °) shock-shock interaction problems. The
effective 10" compression case produced an Edney 5
Type VI interaction, similar to the interaction observed
here for the hollow-cylinder flare. The effective 25'>
compression case produced an Edney Type V
interaction (with some allowance for viscous flow _ii):°'d+reli_ / _ _ lS.O_9"_)
features} similar to the interaction produced here by the
/ /_P" s-s* ...._--I
25"/55 '_sharp, double cone.
The test conditions are closely related to the 11
problem of predicting control surface effectiveness and v'..,,f _i ,I _ I
heating at large deflection angles for access to space
vehicles. The problem has been computationally
investigated, for example, on the Space Shuttle 6 and
X-337. The axisymmetric flow of the present test Figure 1 Sharp, double cone (SDC} and computational
problems enables a much more comprehensive grid domain bounding pressure contour solution.
convergence study than possible on these more
complex configurations.
Models
Schematics of the models and representative
pressure fields are shown in Figures 1-3. Flow is from
left to right. The sharp, double cone (SDC) model is
shown in figure 1.The hollow cylinder, extended flare
(HCEF) model is shown in figure 2 and the hollow
cylinder, truncated flare (HCTF) is shown in figure 3.
The leading edge of the hollow cylinder is sharp. Flow
through the hollow cylinder isdesigned to pass through
the model without influencing the external flow. The
.... ,0.o8
extended flare may allow larger separation as
compared to the truncated flare in some cases where
the shock impingement moves past the truncation
height.
Test Conditions Figure 2: Hollow cylinder, extended flare (HCEF) and
Test conditions are presented inTable 1. computational domain bounding pressure contour
Fundamental quantities inTable ! (velocity, density, solution.
and temperature) are taken from the Calspan report;
2
thermal nonequilibrium (TN) were also exercised for
some simulations. In these options the gas is modeled
as single species molecular nitrogen: the TE and TN
specifications invoke thermodynamic curve fits for
heat capacity and enthalpy as functions of temperature.
Temperature in these tests is not high enough to
promote significant dissociation of nitrogen. Transport
properties in both the TE and TN options are derived
from curve fits (C) of collision cross-sections for
molecular nitrogen.
actual model
-Feeder Block /J extends pest Thermal nonequilibrium is modeled using a two-
b4ockboundary
temperature model. Translational and rotational
energies are chm'acterized by temperature T.
_%- 10.17 ."4 5.75 Vibrational and electronic energies are characterized
3.25
14.50 _- by temperature Tv. In the test problems considered
here, vibrational and electronic energy modes are
nearly frozen (except within the recirculation regions)
making the constant yperfect gas model agood
Figure 3: Hollow cylinder, truncated flare (HCTF) and
approximation for the equation of state.
computational domain bounding pressure contour
solution. Grid: The computational domain for each
configuration is included in figures 1-3. The outer
boundary of each domain is initialized with straight
Numerical Algorithm lines from a point just upstream and above the leading
Inviscid: The Langley Aerothermodynamic edge of the model to an outflow boundary. The
Upwind Relaxation Algorithm (LAURA) s''_is used for position of the boundary is designed to tully contain
all simulations in this report. Key elements of LAURA the shock. Grid lines emanating from the body of the
include Roe's averaging _uand Yee's Total Variation SDC to the outer boundary are straight and parallel, at
Diminishing (TVD) _zformulation of second-order, an angle normal to a reference line extending from the
inviscid fluxes. Yee's TVD formulation has been found leading to trailing edge. The HCEF and HCTF
to be exceptionally robust and Courant-number- configurations utilize a curvilinear grid that is normal
independent using point-implicit relaxation for to the body. Both the SDC and sharp leading edge
hypersonic flow simulations. 3"he TVD algorithm uses hollow cylinder flare utilize a feeder block. The lower
a non-linear, minmod function as a flux limiter that boundary of the feeder block is tbrmed by the flow axis
maintains second-order accuracy away from exttema in the case of the SDC configuration and is an
but can admit limit cycles in the convergence process, extension of the cylinder surface in the HCEF and
particularly in the vicinity of captured shocks. This HCTF configurations.
occurrence usually manifests itself as a stalling of An exponential stretching function is used to
convergence at a very low error norm+ essentially a distribute grid lines in the j direction from the body
benign tinging in the solution at a level that has no across the boundary layer and the shock. Cell Reynolds
impact on aerothermodynamic quantities. However, the numbers less than 5 (HCEF, HCTF) or 11(SDC) and
sharp, double cone test problem has proven to be more maximum stretching factors less than or equal to 1.I1
challenging then the typical application: the ringing has throughout the domain are within the LAURA
more profound consequences as will be discussed in parameter space for which heating in attached flows is
the results section. expected to be grid converged. The experience base for
Viscous: Viscous flux is computed using central stream wise resolution required to get grid converged
differences. A thin-layer, Navier-Stokes (TLNS) separated flows is not sufficiently developed tbr apfiori
formulation is applied in both iandj coordinate estimates of grid convergence.
directions. This formulation includes all but the cross In the hollow cylinder configurations a shock
derivative terms of the Navier-Stokes (NS) equation alignment algorithm in LAURA was used to bring the
set. The complete NS equation set is also applied. outer boundary in closer to the shock and make best
Comparisons of TLNS and NS simulations show use of _'id resources. The alignment procedure in the
almost identical results for the test problems. SDC applications induced large-scale instabilities that
Gas Model: Perfect gas simulations for nitrogen did not damp out: the shock position would oscillate
specify _'= 1.4, ,,R = 28.018, Pr = 0.7 !, and Sutherlands and waves would bounce off the outer boundary before
law (S) for viscosity, U_1.3998 10¢'T_/'-/(T + it was re-adapted. Consequently, larger numbers of
106.667). Options for thermal equilibrium (TE) and
3
grid in thej direction are required. The ranges of grid the feeder block. Extrapolation is used across the
resources used are defined in Tables 2-4. predominantly supersonic outflow boundary. The./" = 1
Boundary Conditions: Boundary conditions in boundary of the feeder block uses an axis boundary
LAURA require definition of variables within pseudo condition in the case of the SDC and an extrapolation
cell centers across the boundary. No-slip, cold-wall inthe case of the hollow cylinder flares. Computed
boundary conditions are used at the surface. Fixed, velocities remained parallel to this boundary using this
supersonic, inflow boundary conditions are applied at specification.
the outer (j = .J,,,,,Oboundary and the i = 1boundary of
Table I: Test Problems
Model Run V_,m/s p_,kflm _ T_,K Tw, K M= Re=,ml
HCEF 8 2667. 0.001206 132.8 296.7 11.35 359600.
HCEF 9 2566. 0.000845 121.1 296.7 11.44 264830.
HCEF Ii 2609. 0.000507 128.9 297.2 11.27 152010.
HCEF 14 2432. 0.000794 156.1 295.6 9.55 185800.
HCTF 18 2661. 0.001175 130.6 295.6 11.42 355210.
SDC 24 2737. 0.001247 200.6 295.6 9.48 263790.
SDC 28 2664. 0.000655 185.6 293.3 9.59 144010.
Table 2: LAURA Cases for Run 8
Case (I xJ) ei Viscous State x_/L As_p, mm x,tt/L As,,, mm
1 272 x96, U 0.300 TLNS, S PG 0.5203 0.722 1.3132 0.757
2 272 x 96, U 0.001 TLNS, S PG 0.5202 0.722 1.3127 0.757
3 544 x 96, U 0.300 TLNS, S PG 0.5269 0.362 1.3112 0.380
4 272 x 96, A 0.300 TLNS, S PG 0.4650 0.734 1.3419 0.782
5 272 x 96, A 0.001 TLNS, S PG 0.4810 0.732 1.3322 0.775
6 544 x 96, A 0.001 TLNS, S PG 0.4736 0.367 1.3357 0.389
7 544 x 96, A 0.001 NS, S PG 0.4708 0.367 1.3363 0.389
8 1088 x96, A 0.001 NS, S PG 0.4705 0.184 1.3373 0.195
9 1088 x 96, A* 0.001 NS, S PG 0.4689 0.183 1.3329 0.194
10 544 x 96, A 0.001 TLNS, C TE 0.5342 0.362 1.2994 0.376
11 544 x 96, A 0.001 TLNS, C TN 0.4794 0.366 1.3266 0.385
12 1512 x 192, A 0.001 NS, S PG 0.4410 0.184 1.3475 0.098
"Avery small error in I. equal to 0.28 mm out of 101.7016 mm was found and corrected after Case 8.
Table 3: LAURA Cases for Run 18
Case (I xJ) e_ Viscous State x_e/L As_p, mm Xatt/L mSau,
1 272 x96, U 0.001 NS, S PG 0.5036 0.727 1.3170 0.771
2 272 x96, A 0.001 NS, S PG 0.4754 0.731 1.3330 0.781
3 544 x96, A 0.001 NS, S PG 0.4861 0.365 1.3280 0.389
4 1088 x96, A 0.001 NS, S PG 0.4840 0.182 1.3284 0.194
5 2176 x 192, A 0.001 NS, S PG 0.4714 0.090 1.3352 0.096
Hollow Cylinder Flare Results Grid convergence for the HCTF configuration in Run
Overview: Five test conditions involving the 18was also investigated as defined in Table 3.
hollow cylinder flare are defined in Table I. The Pressure contours and streamlines for Run 8,
most comprehensive set of tests were executed for Case 12on the HCEF are shown in figure 4 over the
Run 8on the HCEF configuration to investigate cylinder and in figure 5over the flare. The 30°flare
issues of grid convergence and effects of numerical sets up an oblique shock. The high post shock
parameters and gas models on the computed results pressure is felt upstream through the boundary layer
as defined in Table 2. Case 1was initialized with and induces separation. A weaker oblique shock sets
uniform flow. In all subsequent cases, solutions were up ahead of the separation point and the stronger flare
initialized using earlier case converged solutions. shock moves further downsueam with the
4
reattachment point. An equilibrium condition is observed. There is a local minimum in shear beneath
established with separation at x/L = 0.441 and this upwelling but there is no embedded, counter-
reattachment at x/L = 1.348. (In the case of the rotating vortex in this simulation.
truncated tiare, the equilibrium reattachment point In Tables 2and 3 the values of X,_p/Land Xatt/L
may be constrained by the location of the truncation indicate locations of zero surface shear. The
point). The separation shock overtakes the leading separation point Ibllows the initial pressure rise by
edge shock at approximately x/L = 0.9 in figure 4. approximately 3mm in Run 8. The variables As_,.r
Grid resolution in the vicinity of this interaction is and As_ttinthe tables indicate the stream wise mesh
approximately 0.1 mm in the stream wise direction spacing across the separation and re-attachment
and 0.2 mm inthe normal direction. The shocks points, respectively.
appear to merge into a single shock with the current Case 12uses the densest grid but suffered some
available resolution. This separation shock intersects ringing of the flux limiter so that the error norm
the flare shock at x/L = 1.42. Resolution in the stalled (order 10'). Case 9 results have the best
vicinity of this interaction is approximately 0.1 mm combination of residual convergence (order 10.7)and
in the stream wise direction and 0.03 mm in the grid density. Using either Case 9 or 12 results as a
normal direction. reference in Table 2, it is evident that grid adaptation
incoarser grids provides earlier separation and better
agreement with the reference than unadapted results.
For example, contrast Case 4with Case 1and Case 6
with Case 3.
1.2
Residual convergence: In all cases except Case
1.1
12, the L_,error norm dropped to order 10-_'or lower.
I
Case 12 residual convergence stalled at an L2error
0.9 norm of order 10_for 128 hours of single processor
0.8 RI2000 CPU time. In contrast, Case 8 required 100
0.7 point-implicit relaxation steps and 5660 line-implicit
0.6 relaxation steps for 16.3 CPU hours to drop the L:
0.5 error norm to 3.1 107.(In general, these test cases
0.4 with large separation and fine stream wise grids are
more susceptible to ringing of the flux limiter and
0.3
0 0.5 1 require smaller Courant numbers to reduce ringing.
wL The sharp double cone results section includes
additional discussion on this issue of stalled
Figure 4: Pressure contours (flooded) and streamlines convergence.)
over cylinder part of HCEF for Run 8,Case 12. Eigenvalue limiter: The eigenvalue limiter
provides positive definite dissipation in the upwind
scheme when Roe averaged eigenvalues on an iface
are less than 2e,. (Limiters in thej direction spanning
the boundary layer utilize an additional reduction
factor.) The limiting is only engaged in regions where
there is flow reversal or near-sonic velocity.
Expansion shocks are admitted without the limiter.
Previous experience with attached, fully supersonic
flow indicates that the smaller limiter provides more
accurate solutions on coarser grids while the larger
values of the limiter enhance solution robustness.
Little effect is seen in the unadapted grid between
Case 1and 2. A 1.6 mm difference (approximately
two stream wise cells) in the separation point is
1 1.5
WL observed for the adapted, coarse grid result (Case 4
versus Case 5). The larger eigenvalue limiter in this
comparison provides better agreement with the Case
Figure 5: Pressure contours (flooded) and streamlines
12reference, in contradiction to previous experience
over flare part of HCEF for Run 8, Case 12.
as noted above. Subsequent cases retain use of the
In figure 4. a slight upwelling of the streamline smaller value (e, = .001 )in keeping with prior
in the reverse flow region near the wall at x/L = 0.9 is experience. The separation point for Run 12on the
finesgtridsoccuwr ithinthesamecellforc,=0.001
or"0.300.
1.4
Physical models: The additional cross derivative
1.3 ÷
terms included in Case 7 for the NS equation set
1.2
provide insignificant (within 1stream wise cell) 1.1
II
difference in location of the separation point as 1 I'
0.9 F
compared to the TLNS equation set of Case 6.
0.8
The thermal equilibrium option for single species u"o.r
nitrogen engages the curve fits for heat capacity, 0.6
including effects of vibrational excitation ignored in 0.5
the perfect gas (PG) model. It also engages curve fits 0.4
0.3 ........ Perlecl Gas Case 6
for collision cross section (C) to compute transport - Thermal Eq. Ca:me10
0.2
"-_ Themlal Non. Eq Case 11
properties rather than using Sutherland's law (S) and 0.1
constant Prandtl number in the PG model. When %
0.5 1 1.5 2 2.5
vibrational excitation is included under conditions of
thermal equilibrium (Case 10) adecrease in
separation extent is observed (-6 mm) as compared Figure 6: Surface pressure coefficient (global view)
to the PG mode (Case 6). However. when thermal over HCEF from Run 8showing effect of
nonequilibrium effects are included (Case 11) the thermodynamic model.
vibrational temperature stays relatively low,
vibrational energy modes are not significantly
populated, and the constant yapproximation of the 0.1
PG model is more accurate. In this situation, the
0.09
onset of separation for the TN model (Case 11)is ........ Perfecl Gas Case 6
0.08
only about 0.6 mm delayed as compared to the PG Thermal Eq. Caso 10
0.07 __ -- ..... Thermal Non. Eq Case 11 __
model (Case 6).
0.06
Comparison of cases 6, 10, and !! for surface
o.os
pressure coefficient and Stanton number are
"'f- s"
presented in Figures 6-9. They confirm on aglobal 0.04
and detailed basis the near equivalence of the PG and 0.03 .,,'_ //
TN models for conditions of Run 8. 0.02 . /"/ . /j"
Influence of Mach Number and Reynolds
0.01
Number: Runs 8, 9,and 11exhibit a variation in
0 L0.4I5 L , 0.5I J _0./_
Reynolds number at approximately constant Mach
x/l.
number 11.3. Surface pressure coefficient and
Stanton number for these three runs using consistent
Figure 7: Detail from figure 6 around sepm'ation
grids, gas model, and numerical parameters from point.
Case 9 are compared in figures 10-14. The extent of
separation increases with increasing Reynolds
number. The over-expansion on the flare is mole
2
pronounced at the higher Reynolds number (Run 8).
Pa_ecg Gas Case 6
The influence of Mach number is investigated by 1.75 Thermal Eq. Caas 10
comparing Run 14 at Mach 9.5 to Runs 9 and 11at Therrml Non. Eq Case 11
1.5
Mach 11.4. The free stream Reynolds numbers for
Runs 9and 11 bound the Run 14Reynolds number. 1.25 /f
The extent of separation for Run 14 isalso bounded o"
by the extent of separation for Runs 9 and 11, as
0.75;.//_
shown in figures 10-14. The effect of Mach number
over this limited range appears to be much less 0.5 -.."_""
significant than the effect of Reynolds number.
0.25
q:3 135 i i i
1.4 1.45
Figure 8: Detail from figure 6around attachment
point.
0.6 _ --
...... Perlecl Gas Cm 6
| ---_ Thermal Eq. Case 10 0.575 "
0.04 i j Thermal Non. Eq Case 11
0.55 _----
0,03
i I'
! 0.525 ,l_1,_1.<.d_"
0.02 Run M ReIm-_
o.5-Wh--14 ,_,/ 8 11.35 359600
9 11.44 264830
tx ,,t
: _/ ..... 11 11.27 152010
0.01 0.475 _ J 14 9.55 185800
% , 1 , c I o.4q15_ ,
0.5 1 1.5 2.5 1.6 1.7 1.8 1.9 2
x/l. ]Ul-
Figure 9: Stanton number (global view) over HCEF Figure 12: Detail from figure 11 showing magnitude
from Run 8 showing effect of thermodynamic model. of over-expansion and reflected waves.
Run M Re, m-I
0.1 -- 8 11.35 359600 -- ,_ - 0.01 .........
: 9 11.44 264830 i_','/ Run M Re, m"
0,09 - .... 11 11.27152010 , _dl'_Ii 0.009 8 11.35 359800 • i
_ 14 9.55 1_800 _',/ - - - 9 1l.Z_l 254830 _ it
0.08 0.008 - ---- 11 11.27 152010_-
---- __
0.07 __ .......... . ÷ 0.007 : 14 9.55 185800 _//
i i /
0.06 0.006
' I /' /" '
0= 0.05 I /1 i _io.oo5 ",."" .... "!i;/ "/
i/
0.04 0.004 -_ _ "_'-._ 11 . !? t/
0.03 i j / 0.003 • .... .1"1"_
z / J* [
0.02 0.002 . _.. _. _..... _;:zt
_.-..7...7.TJZ-.J:--- I...... , ,, •_, \_.. "_"..-_.a_.,///
0.01 0.001
,J .;It i,Ii ,ll, Lilt , ,J, ,,] _ L, I' ',J ',I
OU.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
xa_ x/I.
Figure 10: Influence of Reynolds number and Mach Figure 13" Influence of Reynolds number and Mach
number on pressure coefficient over HCEF cylinder. number on Stanton number over HCEF cylinder.
0.06
1.4 _- Run M Re, m" Run M° "Re, m1
1.3 - ,8 11.35 359600 -- 8 11.35 359600
.... 9 11.44 264830 0.05 9 11.44 284830
1.2- ' /_ .... 11 11.27 152010 11 11.27 152010
t, It 14 9.55 185800 ,.\ .... 14 9.55 185800
1.1__ , I",'l
0.04
!/d N'
¢ 0.03
o.,i//t ! t_j "
'/d
0.02 /d
/) '/
0.6 :- if/ _11_ 0.01 _ '
t .
0"t]2'_.3 1:4 1.5 L1_ J1!7 i18 119" i '211' 2!2 _:2 '113' lt,I 1.5 '118' 1.7 118 '1:9'
x.,'L x/L
Figure 11" Influence of Reynolds number and Mach Figure 14: Influence of Reynolds number and Mach
number on pressure coefficient over HCEF flare. number on Stanton number over HCEF flare.
7
contours at the separation point are presented in
figure 16.
Grid Convergence: Grid convergence of surface
pressure coefficients and Stanton number using
equivalent physical models and numerical parameters
are documented in figures 17-20 for Run 8 (HCEF).
Figure 17 features a global view of Cpconvergence
as a function of Mid density. Figures 18and 19 focus
on the separation and attachment points. Only a
global view of Stanton number convergence as a
function of grid is presented because relative
differences follow the same trend as presented in
corresponding figures for Cp. Also, corresponding
1.4 1.,45 1.5 1.55 figures for Run 18 are similar to Run 8results and are
x/I. not presented. Runs 8and !8are at nearly equivalent
free stream conditions.
Figure 15: Surface pressure coefficient disu'ibution
and overlay of pressure coefficient contours from
Run 8 (black) and Run 18(red) around peak pressure.
1 272 x 96 C_e 5
0,1 ....... 544 x 96 Case 6
0.09 0.8 _ _ _ - 1088x 96 Case 9
o_ 1512 x192 Case 12
00..1018_ 0.6_- • - --
0.07
B_ck- Extended Flare - Run 8
0.06 Red .Trunc,aledFlare. Run1/1 0,4
::: / /
o o,.... 1....,i,.... ,Is.... 3
Z',,,/,/ ,, ,, x;L
"_.35 0.4 ' 0.45 ' 0.5 0.55
x/L Figure 17: Surface pressure coefficient as function of
grid resolution for Run 8 over HCEF.
Figure 16: Surface pressure coefficient distribution
and overlay of pressure coefficient contours from
Run 8(black) and Run 18 (red) around peak pressure
point. 0.07
f
Truncated Flare Effect: The attachment point for 0.06
Run 8, the largest Reynolds number test, on the
0.05
extended flare is situated approximately 8mm ,5
upstream of the corresponding truncated flare 0.04
expansion corner. The peak pressure for Run 8 ! ./ I
U 0.03
below the shock-shock interaction islocated on the
extended flare at the equivalent location of the
0.02
expansion corner on the truncated flare. Run !8over
the truncated flare has equivalent Mach and Reynolds 0.01 , i 544 x96 Citie 6
i --- 1068x 96 Case 9
numbers as Run 8. The attachment point for Run 18
0 -- 1512 x192 Case 12
is constrained by the expansion comer; flow relief
0,45 O.5 0.25
through expansion at this corner reduces the extent of x/L
separation. This point is illustrated in figure 15 in
which pressure contours in the vicinity of the
Figure 18: Detail of figure 17 around separation
attachment point for Run 8 (Case 12) and Run 18
point.
(Case 5) are compared. The corresponding pressure
8