Hypersonic flow over spherical dome protuberances was investigated to determine increased pressure and heating loads to the surface. The configuration was mathematically modeled in a time-dependent three-dimensional analysis of the conservation of mass, momentum (Navier-Stokes), and energy equations. A boundary mapping technique was used to obtain a rectangular parallelepiped computational domain, a MacCormack explicit time-split predictor-corrector finite difference algorithm was used to obtain solutions. Results show local pressures and heating rates for domes one-half, one, and two boundary layer thicknesses high were increased by factors on the order of 1.4, 2, and 6, respectively. Flow over the lower dome was everywhere attached while flow over the intermediate dome had small windward and leeside separations. The higher dome had an unsteady windward separation region and a large leeside separation region. Trailing vortices form on all domes with intensity increasing with dome height. Discussion of applying the results to a thermally bowed thermal protection system are presented

Olsen, G. C.

Smith, R. E.

English

NASA Technical Reports Server

· NASA Technical Memorandum 84656 
(NASA-TM-846556) ANA~ISIS OF lEROTHEa~AL 
.LOADS ON· SPHERICAL DOllE PBOTUBERABCES (~ASA) 
13 p He A02/MF A01 CSCL.20D 
ANALYSIS OF AEROTHERMAL LOADS ON SPHERICAL 
DOME PROTUBERANCES 
GEORGE C. OLSEN AND R. E. SMITH 
APRIL 1983 
'NJ\SJ\ 
National Aeronautics and 
Space Administration 
l.MgIey Research Center 
HClmpton, Virginia 23665 
!l83-22545 
Unclas 
G3134 03381 
https://ntrs.nasa.gov/search.jsp?R=19830014274 2020-03-21T04:35:24+00:00Z
" 
ANALYSIS OF AEROTHERMAL LOADS ON SPHERICAL DOME PROTUBERANCES ORIGINAL PAGE i9 
Of POOR QW\lITY 
George C. Olsen 
R. E. Smith 
NASA Langley Research Center 
Hampton. Virginia 
Abstract 
Hypersonic flow over spheri ca 1 dome pt'otuber-
ances was investigat~d to determine increased 
pressure and heating loads to the surface. The 
confi gu rat i on was mathemat i ca lly modeled ina t i me-
dependant three-dimensional analysis of the 
conservation of mass. momentum (Havier-Stokes). and 
energy equations. A boundary mappi ng technique was 
used to obtain a reGtangular parallelepiped compu-
tational domain, and a MacCormack explicit time-
split predictor-corrector finite difference algo-
rithm was used to obtain solutions. Results show 
local pressures and heating rates for domes one-
half, one, and two boundary layer thicknesses high 
were increased by factors on the order of 1.4. 2, 
and 6, respectively. However, because lee-side 
pressure and thermal loads were reduced the two 
lower height domes did not experience any net 
increase in total loads. The total loads on the 
higher dome were increased by twenty-five percent. 
Flow over the lower dome \~as everywhere attached 
while flow over the interrllediate dome had small 
windward and leeside separations. The higher dome 
had an unsteady windward separation region and a 
large leeside separation region. Trail ing vortices 
form on an domes with illtensity increasing wit;l 
dome height. Discussions of applying the results 
to a thermally bowed thermal protection system are 
presented. 
e 
h 
K 
k 
l.; ,Ln ,Lr; 
P 
Rg 
r 
T 
t 
u, v. w 
V 
x. y, z 
t5fp 
e: 
~, n. I; 
Nomenclature 
specific heat at constant volume (eq. 5) 
i nterna I energy 
dome height above flat plate 
therrna I conduct i vity of ai r 
variable grid concentration 
parameter (eq. 14) 
differential operators (eq. 19) 
pressure 
gas constant (eq. 4) 
radius measured from dome 
centerline 
radius of flat-plate/dome 
intersect ion 
te~erature 
time 
ve 1 oei ty components 
functional relationship between 
the physical and computational 
domains (eqs. 13 and 15) 
coordinate directions 
flat plate boundary-layer 
thickness at x= x 
one-half the fille! width 
transformed coordinates 
n 
II 
P 
Tij 
Subscripts 
max 
c 
redistributed transformed 
coordinate (eq. 14) 
viSCOSity (eq. 6) 
density 
stress tensor (eq. 2) 
maximum coordinate value 
coordinate value at dome 
centerline 
I nt roduct ion 
Protuberances in the form of large-diameter 
small-height spherical domes often occur on hyper-
sonic vehicles as a result of hardware design or 
thermal bowing> MYpersonic flow over these protu-
berances can produce increases 'I n I oca 1 wa 11 
pressures and heating rates. Interest in this flow 
configuration and co~utation of the pressure and 
heating rate distributions on them has increased 
because of proposed alternative metallic Thermal 
Protection Systems (TPS) to replace the fragile 
Reusable Surface Insulation (RSI) on the original 
Space Shuttle. 1,2,3 Titanium and/or superalloy 
designs that are weight competitive with RSI and 
offer the advantages of mechanical fastening. 
longer life, and increased damage resistance have 
been fabricated. 
However, the use of metallic TPS can introduce 
some physical phenomena not present in the RSI 
system. One such phenom§non is thermal bowing of 
the metallic TPS panels. Bowing can occur because 
metals have relatively large coefficients of 
thermal expansion and because in th~ course of 
performing their function as a thermal protection 
system the TPS panels nust sustain large tempera-
ture gradients through their thickness. These 
conditions lead to large thermal expansion of the 
outer portion of the panel and small thermal expan-
sion of the inner portion of the panel. The square 
panel s are he I d down at the corners but otherwi se 
thermal expansion is unrestrained. As a result the 
center of the panel bows out into the stream so 
that the vehiclerooldline is altered. An array of 
panels would take on a quilted configuration. 
Analytical and experimental studies of two-
dimensional flol1l over a wavy.,wall are available in 
the literature, however. extrapolation of these 
results to the three-dimens10nal flow over an array 
of bowed panels or spherical dtlme protuberances 
results in an inaccurate description of the 
phenomena. The analysis must be done in three-
dimensions to adequately predict the physical 
flows. No analytical studies of a configuration 
resembling the thermally bowed TPS were found in 
the literature. probably because large amounts of 
computer time and storage are requi red to perform 
the analysis. Consequently. this, preliminary study 
was undertaken to determine if the increased 
pressure and heating effects were significant and 
if the problem required in-depth consideration. 
Thermally bowed panels were modeled by the generic 
configuration of spherical domes. Numerical gr'ld-
ding and comput~r storage limitations prohibited 
modeling an array of full scale TPS panels so a 
single row of 4.6 inch diameter domes was selected 
to be co~atible with thegriddin,g requirements. 
Dome heights of 0.1, 0.2, and 0.4 inches were 
considered in a flow field with a laminar-boundary-
layer thickness of 0.2 inches. Free stream 
condit~ons are for ~lach 7 flight at 120,000 ft. 
altitude. The analysis of this flow configur~tion 
will yield dimensionally similar solutions. 
However, the proper balancing of the similarity 
parameters to ascertain that the small scale pro-
cesses in the boundary layer will fall into the 
similarity pattern requires additional a~alytica,l 
and experimental data not yet available. 
Therefore. no extension of the results of this 
stuqy to larger protuberances will be made at this 
time. 
The analysis employs a three-dimensional 
formulation of the time dependent equations for 
conservation of mass, momentum, and energy. A 
MacCormack explicit time-split predictor-corrector 
finite difference algorithm was used to obtain a 
steaqy-state solution on Langley's Cyber 203 vector 
processing computer. The results of those solu-
tions are presented in this paper us1n9 state-of· 
the-art computer graphics techniques. 
A. comprehensive experimental program to verify 
the results obtained in this analysis is scheduled 
for a later time. Some preliminary small scale 
tests have provided qualitative correlation with 
some of the predicted phenomena; no experimental 
data will be presented in this paper. 
AnalYsis 
Governing Equations 
Aerothermal loads on spherical domed protuber-
ances were computed by' mathemat i ca l1y mode 11 ng the 
flow field with the continuum mechanics equations 
of motion. Air, the fluid medium, was modeled as a 
compressible, viscous, thermally conducting gas. 
Limiting assumptions include considering air as an 
isotropic, newtonian ideal gas in local thermo-
qynamic equi 11 brium. The three-dimensi ona 1 
t iJ1lf-dependant system of equat ions descri bi ng th1 s 
model include the continuity equation (conservation 
of mass), the Navi er-Stokes equat ions of mot i on 
(conservation of momentum), the energy equation (conservation of energy). and the ideal gas 
equation (thermoqynamic state equation). The~e 
equations have been derived in the literature and 
are shown below written in cartesian tensor form 
where repeated indices indicate a summation over 
a(pwi ) the range of indices (i.e., -ax--- for i • 1. 2. 3 
i apw1 apw2 apw3 is -ax ~ ax- + ax-)' In addition wlo w2. W3 
123 
are the velocity components u, v. w, and Xl. X2, X3 
are the coordinate directions x. y. z. The 
Kronecker delta. 6iji equals 1 when i a j and 
equals zero ~en i , j. 
ORIGINAL PA~~ ~S 
OF POOR QUALITY 
Continuity equation. 
for 1 .. 1. 2. 3 
Navier-Stokes equations of motion. 
.. 0 
where 
for i, j. k .. 1. 2. 3 
(results il1l three equations. one for .each 
coordinate direction) 
Energy equation. 
for i, j. k. .. 1. 2. 3 
Ideal gas state equation. 
(1) 
(2) 
P .. pRgT (4) 
These equations along with the ideal gas 
relationship. 
e = cvT 
and the Sutherland viscosity formula. 
T3/2 -8 lb sec 
p = 2.270 i + 198.6 x 10 ~
(5 ) 
(6 ) 
constitute a set of five partial differential 
equations and one algebraic equation in six 
dependant variables: three velocity components, u. 
V. and Wi density. p; te~erature. Ti and pressure, 
P. 
The physical domain of the problem is shown in 
figure 1. The thermally bowed TPS is modeled as a 
series of dOfnes on a flat plate. Because of 
symmetry planes at the center of each dome and 
between domes. only one-half of a dome need be 
modeled. The dome was specified by its height, h. 
above the flat plate and its radius. rot at the 
dome/flat plate intersection. The intersection 
between the dome and flat plate was made 
mathematically smooth by modeling a fillet region 
Ilt ro..:!: E with a Hermite polynomial. 
Bound~ry Conditions 
A boundary layer profile was input as the 
upstream boundary plane data and held constant 
throughout the computations. The boundary layer 
profile was determined using a two-dimensional 
'!:\ 
boundary layer analysis6 for flow over a flat 
plate. The flow configuration is shown In figure I 
and free stream conditions for Mach 7 flight at an 
altitude of 120,000 ft. are listed in Table I. The 
boundary layer profi les for the starting plane are 
shown in figurt! 2. These profiles, along with a 
crossflow velocity, w, of zero were impressed on 
the stdrtillY plant! (x = 0) and held constant 
thrCluuhout the cOlllputations. For a flat plate with 
no dOIll!! the boundary layer height, l'if ' at the 
centc," of the three-dimensional regioR (x = xc) is 
ll.Z illclws. 1I0llle twiyhts, h, with ratios h/ofp = 
O.!J, I, ilml 2 were investigated. The ratio of dome 
ratiius to boundary layer height, ro/~p, was held 
constdnt ilt 11.5. 
lioth side boundaries are planes of synlnetry on 
which lhe dependent Vilriables are deter'';ined by 
'1uadr,ll Ie extrapolation frolll interior I'uints so 
that lhe cross flow gradients, il/ilz, of all state 
vari.lhles are zero and the cross flow velocity, w, 
is Zero. The uppt!r boundary (lnd downstream 
bound.!ri es arc specifi cd as ,no change boundari es, 
i • e., set ellua I to the plane next to them. The 
flat-plate/dome surface is a no-slip boundary with 
it constant wall temperature of 1440°1{. This wall 
temperature is both the upper use temperature and 
the flat plate radiation equilibrium temperature 
for the input boundary conditions for a titanium 
TPS assuilliny a high emittance coating and zero 
thermal diffusivity. 
Computational Ilomain 
In order to generate a boundary fitted 
coordinate system and provide increased resolution 
in the boundary layer. the physical domain (fig. 1) 
was transformed into a rectangular parallelepiped 
computational domain ~sing a two-boundary grid 
generation technique. The computational dDmain, 
shown in fi gu re 3, has equa ny spaced gri d 1 i nes in 
the r., 11. and r. di recti ons. 
Transformat i on of the governi n9 differentia I 
equations, eqs. 1, 2, and 3. to the computational 
doma in requ i res rep I acement of a 11 spa t i a I 
derivatives. n /il Xi for i = 1, 2, 3, with the 
following, 
ilC 
il () JI. (7) 
:iT. = nc rrr 
1 JI. 1 
where CJI. for JI. = 1, 2. 3 are the transformed 
coordinates r., nand r., i.e., 
n il ar, il illl a ar; :rx. = 'ilf: ii"f."' + nn ax.- + ar. ax-:-' The resulting 
1 1 1 1 
equations of Illation are, 
Continuitr, 
!l.e. ilt + 
il(PW i ) (lCJI. 
aC JX = 0 
JI. 
Havier Stokes equations of motion 
(8) 
(9) 
3 
ORIGINAL PAG~ nss 
OF POOR QUALITY 
where 
Energy Equation 
(10) 
Written in short form the systetll of partial differ-
ential equations is, 
ilU aF act llG aCt aH ac 
at + 3C, ax + aCt F + ~ az' ~ 0 (ll ) 
\~here, 
The 
J 
F 
G 
H 
pU 
puu - TU 
puv - TI2 
puw - TI3 
pv 
pUV - T12 
pVv - 1'22 
pVw - T32 
aT aCm 
pev - K ac ay· - (UTI2 + VT22 + WT32) 
III 
pW 
puw - 1'13 
pVW - ,TZ3 
pWW - 1'33 
ac 
K aT m ( ) pew - ~ F - UT13 + vT23 + WT33 
required Jacobian transform matrix, 
a~ a~ a~ 
[~ -ax ~ rz 1u 1u 1u (12) ax ay ilz 
~ ~ ]L 
ax ay az 
is determined by a linear mapping of the Uoundaries 
at n= 0 and n= 1 of the computation domain onto 
the boundaries of the physical domain at y = 0 and 
Y :: Ymax in the following form; 
y = y (';. 1. r;) n + Y (~. o. 1;) (1 - n). (13) 
Further transformation to increase resolution 1n 
the boundary layer region entails an exponential 
grid stretching in the n direction of the form. 
n .. 
ekn_l 
ek -1 
where k is a free parameter that adjusts grid 
spacing. Equation 13 becomes, 
(14 ) 
y .. y (E.:. 1, 1;) ii + y (E.:, 0, 1;) (1 - ii) (15) 
For the geometry of the subject problem the 
resulting Jacobian matrix is of the form. 
_1 
xmax 
a a 
J an an
 an (16) .. ax ~ Tz" 
a a _1 
zmax 
an an an 
where ax,~. and~, which vary for the flat 
plate. fillet. and dome regions as defined by the 
radius measured from the dome centerline. 
(17) 
are shown in the Appendix. 
These Jacobian elements are calculated and 
stored for each grid pOint in the domain thus 
allowin~ the solutions from the computational 
domain to be transformed to the physical domain. 
Substituting the transform Jacobian (eq. 16) 
into the short form of the equation system (eq. 11) 
~duces the system to the following, 
~ + .at _1_ + .aE. .an + .a§. .an + .ali .an + .ali _1_ = a 
at aE.: xmax an ax an ay an az a~ zmax 
(18) 
Numed ca 1 Procedure 
A finite-difference solution to the governing 
equations was attained using the second-order 
accurilte MacCormack expl i cit t~me-sp 1 it 
predictor-corrector al gorithm. The algorithm 
splits the differencing scheme into a series of 
one-dimensional operators. The operator in the 
principle flow direction. Lr• advances the 
computation by one time incfement while the 
operators in the other two coordinate directions. 
Lo and Le. are divided into two steps of one-half a 
tlme-increment each and applied symmetrically 
around the principle flow direction operator. This 
arrangement allows the prinCiple flow direction 
operator to run at a Courant. Friedricks. Lewy (CFL) number close to the optil1lJm value of one. 
The operators are applied serially as follows: 
• [Ln(¥)] [L~(~t)] [LE.:(At)] [Ld~t)] 
[Ln(~t)] (U1,j 'k) (19) 
Each operator has two steps, a predictor step that 
advances the solution by its time increment based 
on a backward spatial differencing of the 
one-dimensional equations and a corrector step that 
recalculates the advancement based on a forward 
spatial differencing of the predicted results and 
averages the result with the predictor. The 
predictor I1IJst therefore lag the corrector Qy a 
spatial increment in each coordinate direction. 
The combination of these two steps results in a 
second-order-accurate differencing scheme in time 
and space. As an example the first operatior. 
Ln(At/2) applied to the data at time step n Is as 
fo1lows, 
Predictor step: 
[/F~ - F~ ) :m \1 J J-l ax 
Corrector step: 
U*n+l = 1 lu + Un 
i,j,k 2 ~ i,j,k i,j,k 
(21) 
Where bars over the variables indicate predictor 
values and asterisks over the variables indicate an 
intermediate result in the serial application of 
operators as shown in equation 19. The other 
operators have similar form. 
Although a complete stability analysis of this 
algorithm is not available the CFL limit yields a 
conservative time step as follows, 
At < min [M + ill + hl 
- Ax Ay Az 
(22) 
+ c jOl + _1_ + ...L]-l 
Ax2 Ay2 Al2 
where c is the local speed of sound. 
In the formulation of this algorithm terms of 
order three and higher have been truncated. These 
terms are not significant in many flow regimes, 
but, when modeling high energy flow fields with 
strong shocks. as in this paper, they can be 
significant. In the I'egion of a shock. where 
Multiple pages missing from available version
(Pages 4 thru 13)


Analysis of aerothermal loads on spherical dome protuberances

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