NASA Technical Reports Server (NTRS) 19930018273: Investigations on entropy layer along hypersonic hyperboloids using a defect boundary layer

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INVESTIGATIONS ON ENTROPY LAYER .ALONG HYPERSONIC HYPERBOLOIDS _J-_ C_(_-J
USING A DEFECT BOUNDARY LAYER
N9 3 o
J. Ph. Brazier, B. Aupoix and J. Cousteix
ONERA 'CERT- Dtpartement A_rothermodynamique
2 Avenue E. Belin B.P. 4025 - . /_
31055 Toulouse Cedex {France)
A defect approach coupled with matched asymptotic ex- Forhypersonicreentryflows,theReynoldsnumber isof-
pansions is used to derive a new set of boundary layer equa- tenmoderateathighaltitudesb,ecauseofthelowdensityof
tions. This method ensures a smooth matching of the bound- air.The boundary layersarethusthickand can be ofthe
ary layer with the inviscid solution. These equations are same orderofmagnitudeastheentropylayer,and theinvis-
solved to calculate boundary layers over hypersonic blunt bod- cidflowquantitiecsanundergoimportantvariationbsetween
ies, involving the entropy gradient effect. Systematic compar- thewalland theedgeoftheboundary layer.Becauseofthe
isons axe made for both axisymmetric and plane flows in sev- hypothesisofReynoldsnumber tendingtowardsinfinityt,he
eral cases with different Mach and Reynolds numbers. After boundarylayerisassumed tobeverythininVan Dyke'sthe-
a brief survey of the entropy layer characteristics, the defect oryand theinviscidflowgradientsarerepresentedonlyby
boundary layer results are compared with standard boundary theirwallvalue.So thesecondorderexpansioncannotensure
layer and full Navier-Stokes solutions. The entropy gradient agood matchingoftheboundary layerwiththeinviscifdlow
effects are found to be more important in the axisymmetrlc iftheinviscipdrofileasrenotlinear,andtheinfluencoefthe
case than in the plane one. The wall temperature has a great externalvorticitoyn theskinfrictioannd thewallheatflux
influence on the results through the displacement effect. Good isnotcorrectleystimated.
predictions can be obtained with the defect approach over a
cold wall in the nose region, with a first order solution. How- Defect approach
ever, the defect approach gives less accurate results far from
the nose on axisymmetric bodies because of the thinning of
Decomposition
the entropy layer.
To ensurea smooth matchingatany orderwhatevertheex-
ternalflow,a defectapproachhas been used,coupledwith
Introduction asymptoticexpansions[3].Intheboundary layerregion,the
variablesareno longerthe physicalvariablesb,ut the dif-
A blunt body in hypersonic flow is preceded by a bow ferenceofthem withtheexternalsolution(Le Balleur[91).
shock wave, detached in front of the nose. This strong curved We considera steadytwo-dimensionalflowofidealgas.The
shock wave induces an entropy gradient in the shock layer. variablesp,u,v,p and T standforthedensity,tangential
For an inviscid flow, the entropy gradient is related to the and normalcomponent ofthevelocityp,ressureand temper-
vorticity through Crocco equation : ature.The equationsarewrittenina system oforthogonal
curvilineacroordinates(_,77)where_ representsthecurvilin-
curlVAV= -gradH,+TgradS earabscissaalongthebody and 7/isthedistancetothewall.
Allthevariableasremade dimensionlessby referencintghem
Therefore, velocity and temperature gradients also exist in totheupstreamvaluesp= and Uo=,thenoseradiusRo and
an inviscid shock layer. The standard boundary layer theory To= U_/C r.Sowe write:
of Prandtl cannot take into account normal gradients out-
side of the boundary layer. Van Dyke proposed an enlarged P = PE + PD
theory called higher-order boundary layer theory based on
matched asymptotic expansions for high Reynolds numbers
v = v8 + vo- vE(LO)
[11, 12]. Two expansions corresponding to different approxi-
mations of the Navier-Stokes solutions arebuilt. One of them
P =PZ +PD
called outer expansion is valid far from the wall, where the
viscous effects are negligible. The other one called inner ex- T=TE+TD
pansion describes the boundary layer where the viscous effects
where thesubscriptE standsfor the external variablesand
are dominating. The Prandtl boundary layer equations then
thedefectvariablesarelabelledD. The term vE(_,0)has
represent the first term of an expansion in powers of a small
been added tokeep theconditionVD(_,O) = 0 at thewall
pararneter. The external flow normal gradients are accounted
whatevertheva3ueofvE.
for in the second order term, which is a small perturbation
Expansionsarethenwrittenusingthesame smallparam-
of the first order solution. SeverM other second order effects,
etere asVan Dyke :
like the wall curvature, the displacement or the rarefied gas
effects are brought into evidence. The mdn advantage of this 1 p=u= P,,o
systematic method is to give not only the equations but also c=_ Re= /_(To)
the matching conditions between the different zones.
The external functions depends on the coordinates ((, 77). The - state :
outer expansions read :
p_ = ,y
= u,(_,_), eu_((,_) + --- The symbol r represents the distance from the wall to the
symmetry axis, with j = 0 for plane and j = 1 for axisym-
= v_(_,7) + _t_(_,,7) + ---
metric bodies. As in Prandtl equations, the wall curvature
vr(_,,) = P:((, 7) + _P_(_,,7) + --- appear in the first-order equations only through the trans-
Verse curvature radius in the c0ntinuity equation. The second-
= ,_:((,,?) + _n2(_,,?) + ---
order equations are small-perturbations of the above ones plus
source terms due to curvature effects, like in Van Dyke theory.
= T,((,_) + eT2((,r/) + .-.
Matchinl_ conditions
In the inner region, a stretched normal coordinate _/= r//_ is
Each expansion must satisfy the boundary conditions corre-
used for the defect variables :
sponding to its own validity domain. The upstream conditions
_,o(_,_) =_,(_,_) + _2(_,#) + ... are to be applied to the outer expansion and the wall condi-
tions to the inner one. The missing conditions are obtained
v_(_,_) = _:(_,#) + c_(_,#) + ...
by matching the inner and outer expansions. At the edge of
pD(_,_) = _(_,#) + _(_,,_) + "" the boundary layer, we can write :
p_,(_,,_)= _(_,,_) + _,_(_,_) + ... ---+U E
U ----_ UE
TD((,r/) = h((,£7) + e_((,_) .+ "'"
P -'*PE
The expansion for v must be shifted to avoid the degeneracy
p --*p_
of the continuity equation. These expansions are then brought
T --* T_
into the Navier-Stokes equations, and terms of like power of
are equated. and so for the defect variables--:
Equations up..-.,, 0
In the outer region, the defect variables are null and the equa-
.n --._s(_, 0)
tions for the outer flow are exactly the same as for Van Dyke's
theory, i.e. Euler equations. Concerning the inner region, one Po -_ 0
must first bring the above expansions into t_eNavier-Stokes pn -"*0
equations, then substract the external equations, and at last
Tn "-' O
equate same powers of _. For practical convenience, the inner
equations can then be rewritten in outer coordinates, using r/
Thus at first order : ....
instead of ¢/_and replacing £h and ,Suby :
lira u_ = 0
_(_,_) = _,(_,_) -_(_,_) = _(_,_) q.-._
y,(_,0) = 0
Then the following first-order equations are obtained : lim p: = 0
r/.-.oo
- continuity :
lim _l = 0
tim p_ = 0
- _-momentum :
0%: 0U__2 The conditions on 7, P and T are not independant since they
p: )u: ! are linked through the state equation. The condition on v is
not a boundary condition for the inner expansion but it gives
the wall condition for the outer flow.
The wall conditions for the inner flow are :
- _/-momentum :
_ _
- energy :
T=7'l+t_+¢(T_+t_) =Tw
hence :
u:((,0) = -g:(_,0)
-,(6, 0) = 0
t,((,0) = T, - T:(_,0)
Discussion the two Euler + boundary layer methods. Euler calculations
Thanks to the small perturbation approach, the calculations are made with a code from ONERA [14]. Standard bound-
of external flow and boundary layer are uncoupled and can ary layer solutions are obtained using a program developed
be performed separa£ely provided that a specified sequence in DERAT [2]. Only first-order boundary layer are presented
is respected. First order external problem must be solved here since second-order outer flow solutions are not yet avail-
first, then first order internal, second order external, and so able. Several second-order calculations using Van Dyke's the-
on. The defect boundary layer equations are parabolic and ory have been made on a hypersonic blunt body i1, 4, 5, 6, 71.
can be solved by space marching at a very low cost, like the
standard Prandtl equations.
The conditions at the edge of the boundary layer ensure Axisymmetric hyperboloid
a smooth merging of the boundary layer into the inviscid flow Past a hyperboloid, the shock wave curvature decrease lastly
whatever the inviscid profiles. From a theoretical point of and the entropy field tends to be uniform, except for the
streamlines near the wall, which crossed the strongly curved
view, it can be shown that the defect expansions are consistent
with Van Dyke's ones by the fact that at a given order they shock wave at the nose. In this case, the entropy layer is
differ only by terms which are higher-order in Van Dyke's characterized by a non-zero normal gradient at the wall-and
theory. a decreasing thickness towards the rear, since the mass-flow
Using the above conditions, the first order y-momentum is constant in the entropy layer and the circumference of the
equation reduces to body increases (fig. 1 left). The entropy values at the wall
and at the edge of the entropy layer remain constant because
pl =0
the wall is a streamline and outside of the entropy layer the
So, the pressure in the first-order boundary layer is every-
flow is isentropic. So the normal entropy gradient at the wall
where equal to the local inviscid flow pressure, instead of its
deeply increases downstream. The shock layer is thinner than
wall value like in Van Dyke's theory. in the plane case. Far from the nose, the flow is similar to
a flow past a sharp cone except in the entropy layer, whose
aspect is quite similar to a viscous boundary layer (fig. 2).
Applications
Boundary layer profiles are displayed on figures 4 to 7 for
the Mach 23.4 case. Longitudinal velocity profiles are plotted
To experiment the defect approach, several cases have been on figure 4 at a distance of nine nose radius from the stagna-
selected for a blunt body in a hypersonic flow of ideal gas. tion point. One can see on this figure the important velocity
The general shape of the body is a plane or axisymmetric gradient at the wall in the inviscid flow. This gradient dimin-
hyperboloid, defined by the nose radius and the angle of the ishes distinctly between the wall and the boundary layer edge.
asymptotes, at zero degree incidence. The numerical data are So even with a second-order expansion, Van Dyke's method
given by Shinn, Moss and Simmonds [10] for a hyperboloid could not give a good matching, since it considers only the
equivalent to the windward symmetry line of the U.S. space wall value of the gradient. In this case, it would widely over-
shuttle. Two points of the reentry trajectory of the STS-2 estimate the skin friction (Adams [1]). Due to the very low
flight are considered here : wall temperature compared to the inviscid flow one, the dis-
placement effect is quasi-null and the Navier-Stokes solution
Reentry trajectory-FlightSTS-2 recasts exactly the inviscid profile in the outer region. In this
Mach M_ 26.6 23.4 case, the agreement is quite good with the first-order defect
time (s) 250 650 boundary layer. A composite profile has been plotted also,
altitude (km) 85.74 71.29 using the additive composite expansion (Van Dyke [13]) con-
nose radius R0 (m) 1.322 1.253 structed with the first order inner and outer expansions. It
asymptotes half-angle (o) 41.7 40.2 gives good results for the longitudinal velocity, slightly differ-
pressurep_ (Pa) 0.3634 4.0165 ent of the defect ones.
temperature T® (K) 199 205 The corresponding profiles for the temperature are shown
velocityU® (m/s) 7530 6730
on figure 5. The defect profile is in rather good agreement
density p_ (kg/m s) 6.35 10-6 6.80 10-s
with the Navier-Stokes solution, but in this case the compos-
reference temperature To (K) 56321 44900 ite expansion written with Van "Dyke's first order solutions
gives very bad results and does not improve the inner solu-
Reynolds number Re = p_,,U_Ro 183.55 1865.65
_(T0) tion. This is due to the negative slope at the wall for the
smallparameter _ = Re -x/2 0.074 0.023 inviscid temperature. Figures 6 and 7 show the velocity and
temperature profiles at twenty-one nose radius. The growing
Reynolds number Re_ = p_U®Ro 4792 42374 boundary layer has overlapped a larger part of the entropy
layer. Because of the constant total enthalpy, the positive
velocity gradient at the wall induces a negative temperature
The Prandtl number isassumed tobe constant and equal
gradient. In spite of this, the wall heat flux is increased by
to0.725. The ratioofspecificheats_/is 1.4.The wall tem-
the vorticity, as well as the skin friction, as can be seen on the
perature isfixedand equal to 1500 K. The viscositylaw is
figures 8and 9. But the increase is far more important for the
Sutherland's.No comparison with experimental data ispos- wall friction than for the flux. The defect approach underes-
siblesincetherealgaseffectsare notyetincluded. So Navier- timates slightly these quantities but gives better predictions
Stokessolutions[8]have been taken asreference,tocompare than the standard boundary layer.
-s
14
3.11
3.08
3,05
12 12 ; 302
2.99
2.96
2.93
10- 10
2.90
2.87
2,84
2.81
8-
2.78
2.75
2.72
6- 269
4-
2
O_
0 2 4 6 8
plane
Figure 1: Entropy level in the shock layer -Mach 23.4
0.9-
0.7_
0,6_
0.5 m
I="
0.4 m
0.3 a
0.2_
0,0 I I I I I I _1 0.0 I I I I
-3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -_7 -3.0 -2.9 -2.8 -2.7
S $
Figure2:Entropyprofil-eaxisymmetriccase Figure 3: Entropyprofile - plane case Mach 23.4,_ = 9
Mach 23.4,_ = 9
.... Euler 1st order 0.18- ..... Euler 1st order
0.18- --- Van Dyke 1st order --- Van Dyke 1st order 0
.... composite Istorder
0.16- ..... cdoemfepctos1itseto1rsdteorrder 0.16- defect 1storder
O Navier-Stokes l O Navier-Stokes
0.14-
0.14-
0.12- 0.12-
i !
t_0.10- !
_0.10-
l I!
I
0.08- I
0.08 - l
i pl I
•i i'
0.06- i i
0.06- "/4 iI .... 1 /
/I ".,
/ /s/ :
0.04- / ,,'"_ - 0.04-
l// f_'J ."
0.02- ""'"...
0.02- _ """
0._ -- i l ! I
0.00 : - I I I I : I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4
T
g
Figure 4: Longitudinal velocity profiles Figure5:Temperatureprofiles
Mach 23.4,Y_ = 1500K, _ = 9 M_h 23.4,T= = 1500K,_ = g
..... Euler 1st order ..... Euler 1st order
--- Van Dyke 1storder --- Van Dyke 1storder
0.2_l" defect 1st order 0.25- defect 1storder
0.20- 0.20-
0.15-
0.15- ¢-
¢=
0.10-
0.10- I
/
/
/
. I : ",. //
/ :" ". Sf
jl " 0.05-
0.05- t" :" j_,' "%
p/' ."
/,P .." ==.,,- "%
j._ ." .,w_ %,
s.o .."
_,'J . .''° "%-.°°
0.00
0.00 I I I I '"" I I I I I I'
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 O.1 0.2 0.3 0.4
T
U
Figure 6: Velocity profles Mach 23.4, T,, = 1500 K, _= 21 Figure7:Temperatureprofiles
Mach 23.4,T.,= 1500K, ( = 21
--- Van Dyke 1st order
-- defect 1st order
o Novier-stokes
bOo00 o
I
0 2O
--" Van Dyke 1st order
0.006-
-- defect 1st order
o Novier-Stokes
0.005-
E
0.004-
X
0,003-
0.002-
0.001-
0.000
I I I I
0 5 10 15 20
distance along the body
Figure 9: Wall heat flux on the hyperboloid - Mach 23.4, T. = 1500K
/,
..... Euler1storder ..... Euler1storder
--- Van Dyke1storder --- Van Dyke 1storder
0.12- -- defect1storder 0.12- -- defectls| order
0 Nav_er-Stokes 0 O Nav_er-Stokes
0.10- 0.I0-
o
¢-
o
0.08- 00_
0.06- 0.06-
/
,,/ O /
0.04-
0.04-- //,,,/// O_ .
j/,e OJ
0.02- ,.,// _ _ ." 0.02-
,,_"-O_,, _ - ..
o.oo( i i I ""f 1 0.00 i t !
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4
U T
Figure 10: Longitudinal velocity profiles Figure 11: Temperature profiles
Mach 23.4, T, = 15000 K, ( = 4 Mach 23.4, T. = 15000 K, ( = 4
Figures 10 and 11 show the velocity and temperature pro- the slope at tee wall of the velocity profile. But no Navier-
files with an arbitrary temperature of ten times the temper-
Stokes solution is yet available on such a large domain.
ature of the preceding case. The displacement effect is then The corresponding skin friction and wall heat flux are
far more important and it is obvious on these figures that the
shown on figures 16 and 17. As forecast from the velocity
Navier-Stokes solution is shifted from the Euler solution in
profiles, the defect approach improves greatly the standard
the outer zone. So the first-order boundary layer methods
boundary layer result, but widely overestimates the skin fric-
give poor results and a second-order calculation seems to be
tion on the rear of the body. The predictions concerning the
necessary. wall heat flux seem to be more reliable.
The velocity and temperature profiles at nine nose radius
abscissa in the Mach 26.6 case are presented on figures 12 Plane hyperbola
and 13. Because of the lower density, the Reynolds number Let us now consider a plane hyperbola in the same conditions
is small and the boundary layer is about twice as thick as in of hypersonic flows. On figure 1-right are displayed the en-
the Mach 23.4 case. So a large part of the entropy layer is tropy levels in the inviscid shock layer. The main difference
overlapped by the boundary layer. The inviscid velocity and with the a--'isymmetric case is that now the entropy gradient
temperature gradients at the edge of the boundary layer are is null at the wall (Van Dyke [12]). Figure 3 shows entropy
far weaker than their wall values. Due to the high value of the profile across the shock layer. The entropy gradient layer is
expansion parameter _, the second order effects are more im- thus located at a short distance above the wall. So the ve-
portant and a slight displacement efl'ect is visible between the locity and temperature gradients in the inviscid flow are null
Euler and Navier-Stokes profiles outside the boundary layer. at the wall as well, and their influence will be significant only
The agreement between the Navler-Stokes and defect profiles with a very thick boundary layer. Moreover, far downstream,
is rather good, but the shear at the wall is a bit too high the flow can be assimilated to a parallel flow and the entropy
for the later one. Note that because of the negative inviscid layer'sthicknessremains constantwhereas inthe a.xisymmet-
temperature gradient st the wall, the Van Dyke's composite riccasetheentropy layergetsthinnertowards therearpartof
expansion gives again poor results on the temperature profile. the body. Thus theentropy gradientremains bounded. Since
Figures 14 and 15 show the saxne quantities at twenty- itisnullatthe wall,itsinfluenceon the skinfrictionand the
one nose radius from the nose. The entropy layer is now com- heat fluxwillnow be farlessimportant.
pletely included into the boundary layer, and the gradients On figures18 and 19 areplottedthevelocityand temper-
in the entropy layer become higher than those of the viscous atureprofileson the Mach 23.4hyperbola atnine nose radius
boundary layer.So the hypothesis of neglecting the viscous abscissa.The inviscidgradientsarehardly visibleoutsidethe
effectsinthe externalflowdoes not hold any longerand the boundary layerand allthe methods givethe same results.
defectboundary layerprobably givesoverestimated valuesfor When the Reynolds number islower,the matching ofthe
..... Euler 1st order Euler 1storder
--- Van Dyke 1storder ram, Van Dyke 1storder
O O
defect 1storder m defect 1storder
0.5-- 0.51
O Navier-Stokes 0 Navier-Stokes
O
0.4-- 0.4"
O O
¢"0.3- _" 0.3-
I
0.2- I
0.2-- I
/
O /
/
///
o.1
i "s O / ::
ol- ,.,/" o _ .'" z
_,' ".%
=
Is "S'S _ ........_,..''".--
0.0_" _ I I I I I I O.C _- I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0,1 0.2 0.3 0.4
T
U
Figure 12: Longitudinal velocity profles Figure 13: Temperature profiles
:=:
Mach 26.6, T,, = 1500 K, _ = 9 Mach 26.6, T,, = 1500 K, _ = 9
z
..... Euler 1storder I Euler 1storder !I
I I
0.8" --- Van Dyke 1storder II 0.8" --- Van Dyke 1storder II D
-- defect 1storder I defect 1storder I
I I
I
0,7- 0.7- I
I
I
0.6_ 0.6"
0.5_ 0.5--
¢.
0.4- 0.4-
I
0.3-- 0.3 _ /
/
I /
I I
/
0,2-- 0.2 _ jf
/
//
// j_
0.1 O.t_
f
t "Jtl / /
0.0 I_ I I I '"" ";|'- I O.C i i I 'r'-
0.0 0+1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4
T
U
Figure 14: Longitudinal velocity profiles Figure 15: Temperature profiles
Mzch 26.6, T. = 1500 K, ( = 21 Mach 26.6, T., = 1500 K, _ = 21
0.025
--- Van Dyke 1st order
-- defect 1st order
0
o Navier-Stokes
0.020-
0o
0.015- 00__
._u %\ VuO000
%
%
® 0.010-
0.005-
0.000
I I I I
0 5 10 15 2O
distance dang the body
Figure 16: Skin friction on the hyperboloid - Mach 26.6, T,_= 1500 K
0
--" Van Dyke 1st order
-- defect 1st order
0.020-
o Navier-Stokes
0.015-
0 o
0.010-
0.005-
0.000
I I I I
0 5 10 15 2O
distance along the body
Figure 17: Wall heat flux on the hyperboloid - Mach 26.6, T= = 1500 K
0.30- ..... Euler 1storder I 0.30- ..... Euler 1st order
----- Van Dyke 1st order I --- Van Dyke lsi order
I
defect 1st order I -- defect 1storder
I
0.25- O Navier-Stokes II 0.25- O Navier-Stokes
0.20- 0.20-
0.15- _'0.15-
0.10- 0.10-
0.05- 0.05 -
0.00 0.00 I I I I
0.00 0,05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2 0.3 0.4
U T
Figure 18: Longitudinsl velodty profiles Figure 19: Temperature profiles
Mach 23.4, 7',== I500 K, _ = 9 Mach 23.4, T. = 1500 K, ( = 9
..... Euter 1storder / ..... Euler 1st order
--- Van Dyke 1storder / --- Van Dyke 1,storder =
/
-'_ composite 1st order ] --_ composite 1storder
/
0.8-- -- defect 1st order ] 0.8" -- defect 1stor¢_r
A
0 Navier-Slokes
0.6" 0.6"
0 Navier-Stokes. }
:l
:1
I
I . f.
II :: 0.4- I:
I :
I :
/
0.2 _
0.0 ;'- I I I I 0.0 .- I I I I
0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4
u T
Figure 20: Longitudinal velocity profiles Figure 21: Temperature profiles
Mach 26.6, T. = 1500K, _ = 9 Mach 26.6, T., = 1500 K, _ = 9