An asymptotic description is derived for the interaction of a weak normal shock wave and a turbulent boundary layer along a plane wall. In the case studied the nondimensional friction velocity is small in comparison with the nondimensional shock strength, and the shock wave extends well into the boundary layer. Analytical results are described for the local pressure distribution and wall shear, and a criterion for incipient separation is proposed. A comparison of predicted pressures with available experimental data includes the effect of longitudinal wall curvature

Adamson, T. C., Jr.

Messiter, A. F.

English

NASA Technical Reports Server

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N79;20046
16
A STUDY OF THE INTERACTION OF A NORMAL SHOCK WAVE
WITH A TURBULENT BOUNDARY LAYER AT TRANSONIC SPEEDS_
A.F. Messiter and T.C. _l_on_ Jr.
The University of Michigan
SUMMARY
An asymptotic description is derived for the interac'_ion of a weak normal
shock wave and a turbulent boundary layer along a plane wall. in th_ case stud-
ied the nondlmenslonal friction velocity is small in comparison with the nondl-
menslonal shock s_rength, and the _hock wave extends well into the boundary
layer. Analytlcat results are described for the local pressure distribution
and wall shear, and a criterion for incipient separation is proposed. A compar-
ison of predicted pressures with available experimental data includes the effect
of longltudinal wall curvature.
INTRODUCTION
In transonic flows a transition from supersonic to subsonic speeds typi-
cally occurs through a shock wave which is normal to a solld boundary. Across
the boundary layer along this surface, the upstream fluid velocity decreases
from its value in the external flow _o zero at the wall, and the strength of an
incident shock wave must decrease to zeTo in the supersonic part of the boundary
layer; at the wall there is no discontinuity in pressure.
In the undisturbed turbulent boundary layer, viscous forces are important
only in a very thin wall layer having thickness of the order of the local vis-
cous length and velocity of the order of the friction velocity. In most of the
boundary layer the mean velocity profile is nearly uniform, with a decrease from
the external-flow value of the same order as the friction velocity. The flow
changes caused by a weak incident normal shock wave will depend on the relative
sizes of this velocity variation and the difference ,_ween the external-flow
Math number and one, i.e., on the ratio of the now mslonal friction velocity
to the nondimenslonal shock-wave strength. Asymp,.:.ic flow descriptions have
been derived for large (ref. i), moderate (refs. 2,3,4) and small (ref. 5)
values of this ratio, and the obllque-shock problem has also been studied (ref.6).
In each case the local mean pressure gradient and fluid deceleration are large
enough _hat the flow near the shock wave may be described as an inviscid rota-
tional flow in most of the boundary layer. In a thinner sublayer, changes in
the turbulent stresses do influence the changes in velocity, and here some model
must be chosen (e.g., a mlxing-length model) for describing these stresses. T_,e
displacement effect of the flow changes in this sublayer is small enough that
the dominant terms in the pressure are not affected. Thus the pressure is cal-
--- *This work was supported by NASA Langley Research Center, under Research
Grant NSG 1326.
271
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1979011859-271
https://ntrs.nasa.gov/search.jsp?R=19790011875 2020-03-20T18:41:35+00:00Z
culated from the inviscid-flow equations and then i8 substituted into the sub- d
layer equations for the calculation of changes in wall shear.
The present work is a continuation of the work of reference 5. Asymptotic
solutions are obtained in the limit as the nondimensional friction velocity and
shock-wave strength approach zero, such that the ratio of friction velocity to
shock strength also approaches zero, as does the distance from the wall to the
undisturbed sonic line divided by the boundary-layer thickness. The pressures
found in reference 5 are shown to be modified by higher-order and curvature
_ffects, and the calculation of wall shear is carried out, leading to a pro-
posed criterion for incipient separation.
SYMBOLS
cf skin-friction coefflci_ut
K local wall curvature, nondlmenslonal with L-I
L length of boundary layer, up to shock wave
Me Mach number in external flow ahead of shock
P,Pe,Po local static pressure, and external-flow static and stagnation
pressures ahead of shock, nondimenslcnal with sonic pressure ahead
of shock
Re,R_ Reynolds number based on L and on sxternal-flow quantities or sonic
values
U,Ue local velocity and external-flow velocity, nondlmenslonal with
critical sound speed
uT nondim_usional frictiou velocityi u_ = _ U cf
X,x X/L, (M_- 1)-1/2 X/6; X = coordlnat_ along wall
¥,y,_ Y/L, Y/6. (M_- 1)-1/2 y/uT; Y - coordinate normal to wall
y ratio of specific heats
boundary-layer thickness ahead of shock, nondlmensional wltb L
E Ue-i
< von K_rm_n constant, taken equal to 0.41
n pressure-gradient parameter in velocity profile
z wall shear _treos, nondimensional with undisturbed valueW
!
272
0 -859-272....................... -;i979
p.-' ¥ / J
l
Ii°
PRESSURE DISTRIBUTION
In the velocity defect layer, y = Y/_ = 0(i), where 6 = O(uT), and ahead
of the shock wave U ~ I + E + uT u01(Y), as in figure i. We take UOl = K-IIny
- K-I H (i + cos _ y) for 0 < y < i and uol = 0 for y > 1 (e.g., ref. 7), where
K = 0.41 and for zero pressure gradient H is about 0.5. The sonic llne is lo-
cated at y = exp (2H - <e/UT) , and for uT << E this value is small. In the
very thin wall layer Y = O(u_ 1 Re-l), and the velocity U = O(uT) has the same |
form as for an incompressible flow having density and viscosity coefficient • !
equal to the wall values in the actual flow. For uT-I Re -I << Y << _ a mixing-
length model is used for the shear stress, and the velocity profile obtained is
required to match asymptotically with both the defect-layer and the wall-layer
profiles. These matching conditions provide a relationship among the quantities
_, UT, and Re. A second such relationship is found from the momentum integral
equation, in the manner described for incompressible flow, e.g., in reference 7,
with quadratic ter_m in uT retained. For simplicity the total enthalpy is
assumed uniform.
I
For Y = 0(6) immediately downstream of the shock wave, the differential
equations and shock jump conditions (expanded for uT  0,£ + 0, and u_/E + 0)
show that pressure changes O(u T) occur over a distance x =(H_ - I)-I/2"X/6
= 0(i). The mean velocity profile has a term (i + e)-I plus terms O(uT) con-
taining a known rotational part and an unknown Irrotational part (fig. 2). The
shock wave is located at x = O(uT/_) , and so the shock Jump conditions are ex-
: panded in Taylor series about x = O. For a first approximation Laplace's equa-
tion in the variables x,y is solved in a quarter plane by distributing fluid
sources along x = 0. For (x2 + y2)i/2greater than about 2, it is found that the
solution is nearly that due to a poin t source which represents the change in
boundary-layer displacement thickness (as also noted in refs. 3,4). Two correc-
tion terms, while formally of higher order, are important numerically at realis-
tic values of uT and £. One represents changes in vorticlty along a streamline
of the mean flow; the other accounts for the difference between streamlines and
lines y = constant in the calculation of the rotational part of U, and also
enters in the irrotational part through the boundary condition at x = 0. The
wall pressure with these effects _ncluded is
Pw/Pe = 1 + 2ye +y(2y-l) 2+ ...
(1+ u 2+3.18n+1.5H2 C u°l(n)+ {1+
2 i-_ 1 + _ } x x2 + n2 (I)
The solution (i) is logarithmically infinite as x + 0, as must be expected since
the overall pressure change is O(e) rather than O(uT). Numerical solution of
the nonlinear transonic small-dlsturbance equations would be required to obtain
the correct form in a region near x = 0 which is small if uT/e << i. _or a wall
with longitudinal curvature, the pressure gradient in the inviscid flow behind a
normal shock wave has a logarithmic singularity described by
AP = 4YK {x6 in6 x_+ y2 -I
- y6 (tan Zx - 4)} (2)
273
/ \ :- ,....,"
f
The boundary-layer interaction effect for x = 0(I) is not large enough to change
this expression, and so a curvature term can simply be added as a correction to
equation (I).
The pressure distributions found by adding equations (i) and (2) are com-
pared in figure 3 with experimental results from reference 8. For Me = 1.322
and Re = 9.4 x 105, other parameters are calculated as £ = .247, uT = .050,
= .0182, and cf = .0032. The shock-wave position found from the data is the
origin for the pressure at 15 mm., where y _ 4.1. The origin for the wall
pressure is shifted upstream through _x = 2.87, equal to the displacement of the
shock wave calculated using the fact that the shock-wave slope is proportional
to the y-component of velocity at x = O. In the experiments the wall was a
plate having longitudinal curvature which can be inferred from the measured
pressures immediately behind the shock wave. The change AP/Po in a y-distance
of 15 mm. was taken as .015, giving K = .21. The theoretical curves also in-
elude a constant streamw±se pressure gradient caused by cross-section area
change, estimated from the data as AP/Po = .03 in a distance of 50 mm. The
parameter H was taken equal to the constant-pressure value H = 0.5, which leads
to a predicted location of the undisturbed sonic line at y = .36. If instead
= 0.4, the _:ediction is y = .30, whereas the measured value is y = .29; this
change in H would influence the pressures only slightly. The wall pressures are
in excellent agreement for x greater than about 3; the re-_on x < 3 requiring
solution of nonlinear equations is relatively large in this case, because the
Reynolds number is not high enough for the upstream sor.r line to be really
close to the wall. The very beginning of the rise in wall pressure can be ex-
pressed by an exponential function with known rate of decay but unknown multipli-
cative constant, as shown in figure 3 with orJBin chosen tentatively for good
agreement with the data. By comparison with umerical results of ref. 2, it
should be possible to estimate this constant. The pressures at 15 nun. from the
wall at, in good agreement except for small x, where the error probably arises
from approximating the shock-wave location by x = 0; adding the appropriate
second-order term is expected to improve the agreement.
WALL SHEAR
The above description of mean-flow perturbations in terms of disturbances
i-an invlscid rotational flow does not remain valid as y -_ 0. For y = Y/6
= O(u T EI/2), it is easllv, found that turbulent stresses as well as pressure and
inertia terms must be retained in the boundary-layer momentum equation. This
thinner layer (still much thicker than the viscous wall layer) has been calle
Reynolds stress sublayer in reference i and a blending layer in reference 2.
Double expansions as uT  0and £ + 0 are assumed for the velocity components,
pressure, and Reynolds stress, with the latter represented by a simple mixing-
length model, chosen for analytical convenience In the belief that the results
may not depend strongly on the model used. The wall shear stress has the form
Tw(X) =l+ae+ {a(a-l)E2+...+UTTl(X) +cu {(In_i/2u ) _l£(X)+_ll(X)} +.... (3)
where a = constant. This expanslon_ like that for the pressure, requires modifi-
cation when x is small. The y-component of velocity and the related displacement i
274
1979011859-274
effect of th_s sublayer are small enough that the previous calculation of pres-
sure is unaffected, and the boundary condition v = 0 at y = 0 used for y = 0(i)
is thereby verified. With use of the solution for pressure evaluated as y _ O,
it is found that Tl(X) is proportional to the pressure perturbation, and there-
fore decreases monotonically with increasing x; also T., is constant. The term
TII(X) is obtained as a lengthy expression including terms proportional to the
pressure perturbation and a posttiCe te_m proportional to in x (ref. 9). Since
the pzessure perturbation decreases to zero as x increases (i.e., is small for
_i/2_ << X << i), it follows that Tw has a minimum. Thus for typical values of
Me and Re figure 4 shows that Tw first decreases sharply and then rises again
slowly. At a given x, Tw decreases as E increases and increases as Re increases.
This behavior is in general agreement with existing experimental results.
If it is assumed that equation (3) remains a good approximation even when
changes in Tw are not small as originally assumed, s criterion for incipient
separation can be proposed. For example, curve III in figure 4 is drawn for
Re* ffi106 and Me = 1.149, corresponding to E ffi0.12; if E is increased to 0.2,
the minimum value of Tw decreases to zero, and this is taken as an indication of
incipient separation. Values of Me corresponding to this condition, namely that
_w ffi0 and dTw/dx = 0 simultaneously, are plotted against Re* in figure 5, and
show a slow increase in the shock strength required for separation as Re in-
creases. As a first step toward studying flows with separation, a numerical
solution was also carried out for a momentum equation appropriate for describing
a slender low-speed separation bu_ble having length Ax = 0(i). Terms _:ere
retained to represent the laminar and turbulent stresses and the pressure gra-
dient, which was considered known from the previous solutions because the dis-
placement effect of the bubble was sufficlently small. Velocity profiles
obtained are plotted against 9 in figure 6.
CONCLUDING REMARKS
The favorable comparison with experiment shown in the figures for the latter
part of the pressure rise seems to imply that the asymptotic representation
given here, with the addition of second-order terms as noted, does correctly
include the most impo£tant features of the local flow field. A study of other
second-order terms is continuing, and an analytical =olution has been derived
for the correction necessary in the flow behind a s]," : wave in a circular pipe;
a comparison with existing experimental results is .ng carried out. It is
anticipated that a comparison with numerical resu_, of Melnick and Grossman
will suBgest a simple curve fit, in terms of a coolu_nate nondimenslonal with
the distance to the sonic llne, to fill in the ,_II pressure in the region near
the origin. It should now be possible to incorporate these _nalytical representa-
tions for the local presst_e and velocity into potentlal-flow calculations of
the transonic flow past an airfoil. The predictions for wall shear, and for the
Mach number at which separation first occurs, show the expected trends, and
should be checked for quantitative accuracy against existing experimental data.
It would seem especially importan_ to continue the attempt at developing a
rational theory of separation for turbulent boundary layers.
275
'"........... 1979011859-275
RI_I_EES
1. Adsason, T.C., Jr.; and Feo, A.: IDteractlon between a Shock Wave and a
Turbulent Boundary Layer in Transonic Plow. SlAM J. Appl. Hath., vol.
29, 1975, _p. 121-145.
2. Halntk, P.E.; and Gro_.-wn, B.: _Jmlysie of the Interaction of a Weak
Noraa£ Shock Nave with a _._rbu!ent Boundary Layer. AIAA Paper No.
74-598, 1974.
3. Halnik, R.E.; and Grossnan, B.: Further Developments in an Analysis of the
_nCeract£on of a WeakNot_al Shock Wave_rlth a Turbulent Boundary Layer.
SyaposimaTran3eonicum II, K. Oswa¢itsch and D. Rues, eds. Springer-
i Verlag, 1976, pp. 262-272.
4. Helnik, R.E.; and Grossman, B.: Interactions of Normal Shock Waves with
Turbulent Bo, ndary Layers at Transonic Speeds. Transonic Flow Problems
in Turboemchtnery, T.C. Adamson, Jr., and M,J. Platzer, eds. Plenum
_. Press, 1977, pp. 415-433.
5. Adamson, T.C., Jr.; and Messiter, A.F.: Normal Shock Nave - Turbulent
Boundary Layer Interactions in Transonic Flow near Separation. Tran-
sonic Flow Problems in Turbomachinery, T.C. Adamson, Jr., and M.P.
Platzer, eds. Plenum Press, 1977, pp. 392-414.
6. Adamson, T.C., Jr.,: The Structure of Shock Nave - Turbulent Boundary Layer
Interactions in Transonic Plow. S3naposium Transsonicum II, K. Oswatttsch
and D. Rues, eds. Springer-Verlag, 1976, 7P. 244-251.
7. Cebeci, T.; and Smith, A.M.O.: Analysis of Turbulent Boundary Layers.
Academic Press, 1974.
i_ 8. Ackeret, J.; Feldmann, F.; and Rott, N.: Untersuchungen an Verdtchtung-
i st_ssen und Grenzschlchten in schnell bewegten Gasen. Mittellungen aus
dem Instltut f_r Aerodynamlk, ETH Z_rlch, Nr. i0, 1946. Translated as
: NACA TM 1113, 1947.
9. Liou, M.S.: Asymptotic Analysis _£ Interaction between a Normal Shock Nave
i and a Turbulent Boundary Layer in Transonic Flow. Ph.D. Thesis,
i_ Department of Aerospace Engineering, The University of Michigan, 1977.
}
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2 ..
k .... . 1979011859-276
9Y
•---- I -_-_ E --- E = u,-II
I
': Iu m °-_- cti 2
I
I
, uz.ccE¢cl
i--" "-- Olu.r )t
8=0 (ur ) I _'I
I -
, _,_ _ U =1+_:+U,rUm(Y/S)I --( "-""/11 "ll[/'l[Ir/ " I /llr/,'/// /If Ill ////lllllll///l_"
I
"WoII Loyer U
Figure I.- Velocity proglle In undlstuTbed boundary layer.
y = Y/8
Approaches
U: I+ _: NormolShock
M =Me>! o$ y --, oD
y=l
U-" _"_ +_rUo_ (yl+uTu_ Ix,y)
_' U~I+E+uru°'(Y) ShockWove:Xs"Olur/_')
ur <<E<(I
e--o |
' er Region:y-Ole -x_/ur) "
I
i ,'///i ,, _///_,_///),j ,)//,,,,//t/,,,_'_ _,_,,,,/i/,_//;,"r'/-_
I x"(M'2"II"_X/8
I Figure 2.- Representation of outer flow field. ? I
r 27"/
_'Cl ,
1979011859-277
,.°I_w
!
O.
z. M,• IJ25,Re",t0T
1T.Me- 1.149,Re "10
111'.M, • 1.149, Re*"I06
c i £
0 5 40 15( M2- t)"/ZX/u_ " 1
Figure 4.- Predicted wall shear stress.
278
_%.,.
o .
•. . ............. ::J
1979011859-278
E,o.25,Re•_0s
1.0-
9
, -_ U.O.01 I Separation
0.5
0 I 2 2.5 4
(Me2-! )VZX/u,
Figure 6.- Calculated velocLty pro£ile8 in a slender separated-flow
resion.
!
279
" ..... 1979011859-2-79


A study of the interaction of a normal shock wave with a turbulent boundary layer at transonic speeds

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