Oi s
00
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM
No. 1189
THEORETICAL ANALYSIS OF STATIONARY POTENTIAL
FLOWS AND BOUNDARY LAYERS AT HIGH SPEED
By K. Oswatitsch and K. Wieghardt
TRANSLATION
"Theoretische Untersuchungen iiber stationare Potentialstromungen
und Grenzschichten bei hohen Geschwindigkeiten"
Lilienthal-Gesellschaft fiir Luftfahrtforschung Bericht S 13/l.Teil, pp. 7-24
''^^^P^
Washington
April 1948
DOCUME^n•SDEPARTMEN^
H'
NATIONAL ADVISOr.y COM'lITTEE FOR AERONAUTICE
TECHNICAL MEMORANDUM EO. 1189
THEORETICAL ANALYSIS OF STATIONARY POTENTIAL
FLOWS ALT) BOTOIDARY LAYERS AT HIGH SPEED*
I^ K. Oswatltacli and K. Wieghardt
The present report consista of two po.rtB. The first part
deals with the two-dimensional stationary flow in the presence
of local supersonic zones. A nLUtierical method of integration
of the etiuatiora of gas dynamics is developed. Proceednng from
solutions at great distance from the "bod,'' the flow pattern is
calciaated step "by step, Accordin^iy the related body form is
ottained at the end of the calculation.
The second part treats the relationship between the dis-
placement thickness nf laminar and turbulent boundary layers
and the pressure distribution at high speeds. The stability
of the boundary layer is investigated, resulting in basic
differences in the behevior of subsonic and supersonic flows.
Lastly, the decisive iaiportance of the boundary layer for the
pressure distribution, particularly for thin profiles, is demon-
strated.
PART I
.NOTATION
p pressure
p density
T absolute tempera tvre
K ^ ratio of specific heats
*'Theoretische Untersuchungen 'aber otationaro Potentialstrdmungen
und Gren::0chichten bei hohen Geschv;indigLoiten. " Lilienthal-
Gesellschaft fur Lrftfalirtforschung Bericat S IS/l-Teil, pp. T-f'^.
2 NACA TM Wo« II89
\x coefficient of friction
w velocity vector
V magnlt-ode of velociiy
u, v velocity components
$ velocity potential
c velocity of sound
c* critical velocity of aoijnd
Ma = w/c Mach number
F = 1 - Ma^
9 = w/p stream density
B.e* Eeynolds nuinter of the displacement thickness
S "boundary -layer thickness
S* displacement thickness
■0 momentum thickness
f stream filament section
r radius of curvature of the stream line
N normal to the stream line
E maximum "bump elevation
E related radius of curvature
Suhscript n refers to the conditions in the free -stream region,
subscript a to the outer flow, subscript w to the wall. Subscript
m refers to the chamber ">r drum values in the phase quantities and in
the velocity to the highest obtainable value. The quantities of state
in part I are made dinenaionle33 by the chamber quantitios and the
velocities by the highest velocity obtainable.
KACA TM No. IISQ
!> A H T I
1. NOTES OJM THE CEARi^.GTEEISTICS OF COivff'REGSIlU.E POTENTIAL FLOWC
The equation of gas dynamics is derived ty means of the energy
equation rather than the adiabatlc equation as customary. A simple
formula is obtained for the straam densjty which is valid in a vide
range ahout the critical velocity of sound. By applying this formula
a simplified equation of gas dynamics is derived v/hich in the tran-
sition zone from subsonic to supersonic, for sm:ill velocity compouents
V, describes the pi'ccesses very accurately. Lastly, the problem of
flow around a cylindrical bodv, ayraaietrical in two directions, is
analyzed. It is found that, from a certain flow velocity on, located
above the critical velocity, no maxira-om velocity can occur at the
point of maximum body thickness.
In the description, of a gas flow the most (general case involves
six unlcnown functions, namely, the tlxree compononts of the velocity
and the tliree phase quantities of the g,ao, the pressure p, the
density p, and the absolute temperature T. The, equation of state
of the gas permits the temperature to be expressed in terms of the
pressure and density, thus leaving five unknown functions for the
calculation of which the three Sulor equations and the continuity
equation are available. For tne missing equation it is customary
to use the adiabatlc curve to eliminate the pressure and density from
the equation ana so arrive at an equation between the velGC?ty com-
ponents, that is, the so-called equation of gas dynamics. However,
it appears to be unimov.-n that for the derivation of the equation of
gas dynamics the aosi:a.iption of the adiabatlc is not necessary at all,
but that the use of the energy theorem itself is sufficient. This
derivation is briefly carried out in the fo.llowing, ^/hile having
recourse to the vector method.
Pressure, density, and temperature are maae nondimensional by
the corresponding "cnambcr quantities" p^j^, p^^^, Tj^; that is, the
quantities of state at velocity 0, and all occurring velocities
and velocity compoiients by the maxi.mtMi obtainable velocity, that
is, the velocity at pressure 0. VJith v denoting the velocity
vector, c the specific heat at constant pr'-^ss^ii'e, and k the
ratio of the specific heats this r.iaximu.'n velocity is
„ 2 -. 2c T = 2—^^ ^
.1. \ - y^
NACA TM N(;» 1189
The eg.uatlon of energy of an ideal gag In gtatlonary flow has
expressed ncrjilmftnslcnally - the following Bimple form
w2 + |=l - (1.1)
and the corresponding continuity condition reads
div w + — grad p = (1.2)
p o
The oft repeated c[\iantlty
2
n
X - 1
has a physical significance; it indicates the degrees of freedom
of a molec-ule. For air n == 5 is very exact.
The EiJ-er eqi.iation is then written as
/ v^ \ 1 , ,
fgrad ^ - w X rot w) xn = grad v (1.3)
The quantity xn enters the equation through the nondlmenslonal
notation.
Pressiire and density are eliminated hy forming the gradient of
the energy equation, thus ohtaining
- grad w^ + — grad p + (w - l) — grad p
Scalar aultiplication hy w and application of (1.2) and (I.3)
gives the equa.tion of gas dynamics
(1 - w^) div w + nw grad ^4 = C (l.U)
This equation is written here in a form where the velocity of bound
is already eliminated and only the flow velocity itself is present.
Ve will use this equation in the next section.
For an insight into the potential flow properties in the speed
range of Ma = 1 the ,junt derived equation is much too com-olicated.
So the processes in a flow filament are enalyzed, unsteady variations
NACA TM No. II89 -^
and friction procee'^es excluded so that adiatatic chajiges of state
ccn be assrjiied- The enerjy equation (l.l) then affords a connection
■between velocity and density, and. the stream density wp can be
represented as a fmiction of the velocity v:
G(v) - pw = w(l - w2)rV'2 _ (1.5)
It is known that this function reaches a rnaximmn. at the point where
the velocity is exactly eqp.al to the velocity of sound. This
particular point is generally denoted as the critical speed of
sound c*. With f as the section of a flow filament tJie continuity
equation x'eads f9 - const. The speed c* is therefore characterized
by the fact that a fJ.oT' filament for Jais value of vf reaches a
siKiillest pcseible cross sect;Lon. For w> c'**' ."■- for w < c* the
flow filament section is greatei".
Less familiar is the smallnes.^ of the str'-^am density changes
e
over a very substantial speed range. To indicate it ~q^ is
represented for x - l.UO in the range of 0.;;)C* < w < 1.5c* in
figure 1. QutLiitity '■■'* denotes the value of Q for w = c '^ , the
same applies to the derivatives of l? . This characteristic of the
stream density is of decisive significance for the effect of the
boundary layer on th3 flow, as vril]. be shown elsowhere.
Consider the function 9 in the vicinity of the maximum
developed and sjgnify its derivative with t)*^ ^\m> ^^^' ^°^^
It is fovind that the parabola
- i]^- (1.6)
is already sufficiently acciu^ate for a wide speed range. This
approximation is indicated by dashes in figure 1. The calculation
for the coefficient of the quadratic term ^^ives the cimpj^e result
1 .^ ^^w K + 1 n + 1 /, ^^
The equation (I.6) serves in good stead for the derivation of a
simplified dynamic gas equation for two-dimensional flows on the
limiting assumptions that the y component of velocity w,
signified by v, is small compared to the velocity of so-'and iiiid that
u, the X component of the velocity, does not differ too much
from the velocity of sound. The stream density wp can be
MCA TM Wo. 1189
replaced by up = 0(u) and the equation of continuity (1.2) on
applying the same cmicsicns as effected with respect to the terms
with the factor v for the derivation of the Prandtl law, can te
written as
P
lb,, b^ u ^ cu
=
The coefficient of -^^ depends only on u; it is simply a dif-
ferent method of expressing the welA-known quantity 1 — .M.-^.-.
Assuming, aside from the smallne^^s of v, that ^ 9,, can he
regarded as constant results in the Prandtl-Glauert analo{ry.
If this coefficient were plotted against u in the vicinity of
the sonic velocity, it would show that it can aosume negative as
well as positive values and at u = c* is equal to zero. So
the premise of constancy of this quantity can no longer be main-
tained, especially since the derivative 0^ changes signs at
sonic velocity, as seen from figure 1. The variations of — on
the other hand are no more weighty than the variations of the
entire coefficient anywhere in the range of not too, high subsonic
speeds. Thus in support of Prandtl's jaw — can very well be
put equal to this quantity in the free-stream region, but not for
6^. This equantity is computed by (I.6) and gives
80 c ^^ ^ \ c' ^j ox Ic" / by c'-
The subscript o denotes the quantities in the free-atr«am region.
Using the notation
T (.-c ■'- I) (^•■7)
the simplified equation of gas dynamics
(1.7)
U can be positive and negative. Here also the introduction of
a velocity potential is accompanied, although in simplified form,
by the undersirable change of the equation from the elliptical
to the hyperbolic type. To secure solutions which have supersonic
zones by an analytical method it is advisable to find solutions of
(1.7) J because it combines the simplifying asaumption of small v
u
c-^
©o
(..
+
1) \-
-)'
^^o
9-
9
gives
for
V
c*
«
1
and -^
~1 1
the Sim]
dv
?1
WACA TM IIo. 1169 '
with a very accvrate descrij-ition of the processes in the critical
sonic speed ranee. This wac the reason for the oilef derivation
of the equation.
The fact that the flow filament section has a minimum at the
critical speed may, under certain circumstances, have very
characteristic consequences for the velocity distribution at the
appearance of supersonic zones on todieG, as will he demonstrated
for the case of two-dimensional flow patt a hody that is symmetrical
about two mutually perpendicular axes. The flow direction is to
be aj.ong one body axis, that is, the axii^le of attack equal to zero.
The flow is to be adlabatic end iriotational, the latter
characteristic being expressed by
^w _ w
tl ~ ~ r
(1.8)
where TI is the noimal to the streamline and r its radius of
curvature. The sign for K is so chocsn that it is positive when
the normal points out from the radius of curvature. Equation (I.8)
holds exactly for all tvo-dimenf:ional potential flow::. By the
continuity condition in the form,
0f = CorLstn.nt
and the freedom from rotation (l.S) the flow is completely defined.
The origin of the coordiriate system x and y i:j placed in the
center of the body, axis x is made coincident with the flow
direction (fig- 2) , and tbo are-", of •DC3:^1.1vf- ;, value f.^naiyzed.
The c;/lindricai body is visualized as b-^-ing exposed to a flow
velocity which Deads to the formation of a supersonic zone near
the thickest part of the body and it is assumed that in every
stream filament the maximum velocity is reached at the point
X = 0, an assruaption which certainly should be fulfilled for
subsonic i'lowd . A point on the 3''_^axis with supersonic speed
must have a ma:>:imian stroam filpjnent width, a point with subsonic
speed, a minimum of stream filament width. In the 3>.ipereonic
region the curvature of the streamlines on the positive portion
of the y— axis must decrease leso rapiaiy than on concentric
circles, in the subsonic zone the streoii line curvature must
decrease more lapidly than for concsntnc circle:>. hence no
great error is introduced when in the vicinity of the point on
the y— axis where sonic velocity is reached, the sti-eamlineo are
replaced by concentric circles, and it will not lead us far astray
when this is assumed up to a value of 5" equal to twice the
distance from the cylinder of the point with the critical sonic
8 NACA TI'l No. 1189
velocity c*. Aftar the streamline curvatures are approximately
kno-vm the velocity distrihution on the y— axis in this zone is
completely dLefinod "by (I.8). If the piece which the body cuts off
from the y— axis is denoted by H aiid the radius of curvature of
the profile on the y— axis by R^ its velocity distribution is
u R
^y=H R - H + y
(1.9)
According to (1.6) it may be stated thah the volume of flow through
a section of the y_ axis is then greater than on an identical
section of the free stream, if at a ,.a:ti ula:' ^olnt the ineq.ual.ity
u < u < 2c* — u
is fulfilled, fiince the vej.ocity distribution for x = is
defined by (1.9) up to the constant ^y-E' ^^ also is bhe
difference in through, -flow Tolume for >> H iu the rree-streani
region and on the y^axis. It may now be asked at what value of the
const.anta the absolute amount of this through ^^ow dic'ference reaches
its hlg-hest possible value and the answer is found in the fairly
accurate equation
Vh " 2c* - u^ (1.9a)
that Is, that the stream density on the y— axis must nowhere be
less than in the free stream. For a simple picture it is imagined
that (1.9) with the constant (l.9a) ia applicable up to the attain-
ment of speed u and that from this y value on, the constant
flow velocity prevails. This break may occur at the value y = y^
for which the eq.uation reads
As near the body more cexi flow by than en a strip of equal width
in the free-stre-sm region., because of th3 increase in density , we
must proceed from the cj^'lindei' only as lar as the free stream is
displaced. The result is therefore a highest possible value of H,
denoted by H^ ., which is p;iven by the equation
^oHnax =7 (^ " ^o) ^y (l-^O)
y~nsia.x
MCA m No. 1189
The Integrsnd is given "by (1.6), (1.9).^ (1.9?) j H is to be repla.ced
by -^Wx^ sinG3 u is equal to w on blie y — axis. The evaluation
of (l.io) gives the foxl owing relation "between flow volocity and
E
TABLE I
^0
0.70
"i
j 0.75
0.80
0.85
0.90
0.95
1.00
H
E
0.053
1
0.026
0.013
0.0059
0.0020
o.oooi;
0.
The extension of the sneed "by pieces at y = y unquestionably
introduces an error; but it can only caiise a shift in H ,, while
not changln3 the existence of such a value. In the subsonic zone a
etreamllne loay be rej^arded as s. bimip and the residual rise in
through flow voluae lue to increase in velocity computed by an approxi-
mation process that applies in the subsonic range. The result then
is a finite variation of the integral in (l.lO) and a correspondingly
different H^^g^. The possibility of a opeed increase in y direction
in the subsonic region must be rejected, as it would invalidate the
present considerations. Hence it is seen that the assumption cannot
be applied to all bodies and therefore ^-he folio-ring principle:
To each flow velocity u^, there corresponds a definite ratio
H „ /E, If the ratio H/'E exceeds this limit for a body s;^/iimet—
rical in two mutually perpendicular directions and lying along the
flow direction^ there is no flow for whjcli velocity maximums can be
reached on the entire y — axis.
It must be expected that the maximiEn speeds on the y — axis
disappear only in th« supersonic rr.nge . But since it cann.ot be
ass^'jraed that velocitj'- ms.ximum3 in the svporsonic range disappear
on a part of the y— axis while a velocity maximum appeal's on the
body^ we are led to the following principle.
From a definite value of H/E on, for bodies and flow directions
of the described type, there is no flow at which a speed maxim'.an with
locgj- supersonic zone is reached at the ^ooint of maximum thickness
of the body.
A boundary point for these specific values of H/E is given
in table 1.
In the subsonic zone this principle has no analogy.
10
MCA TM No. 1189
2. METHOD FOR THE NUMERICAL INTEGF;ATION OF THE EQUATION
OF GAS DYNAMICS
A nianerical - graphical method is indicated for finding
solutions of the equation of gas dynamics with supersonic zones,
by progressive calculation of the entire flow, starting from an
exact solution at great distances from the body. The exact body
form follows at the end from the shape of the streamlines . Exceeding
the sonic velocity causes no special difficulties r^r oeculairities.
Limited to two-dimensional, irrotational flows with w = grad
past a cylindrical body, equation (l.-i) raves for the velocity
potential cp a nonlinear differential eq^.iation of the second order
$
D :$
] =
1 -• (n + 1)
Sx
^2
a2$
-(f -(-KD
ox
(2.1)
^2$
2n — ~
Sx dy dx dy
0; n
- 1
The zero point of the coordinate sj'stem is placed in the body, its
dimensions are of the order of magnitude of unity, and the flow
strikes the body along the positive x--a::is . The boundary conditions
for $ then read:
^ = at the body itself, N
for z s w'x^ + y- ->!» ^r — ^0
z H^x^ + y--^»^
denoting the normal, and at infinity
and '^: >u = (dimensionless) flow
o
ox
velocity.
On passing through the local velocity of sound
equation (2.l) changes from the elliptic to the hyperbolical type.
For this case the exact integration has been successfully secured
for single specific examples only. For the subsonic range several
general app-^oxiniate solutions are available, the simplest and best
known of which is the solution obtained by the Prandtl rule.
This satisfies equation (2.I) better as the body becomes more slender
and the distance from the body becomes gi-eater.
WACA TM Wo. 1189 11
The followlngr method Is therefore Indicated. Compute the
Prandtl solution 'i^ for the entire f3-0w and attempt to secure
the correction cp in ai^ch a vay that 5,, + cp = hecomes a
solution of the complete equation (2.1). As the analytical
calculation of cp is too complicated, a numerical method is
adviBahle, starting from the outside (z ;:§> 1) where cp ~ 0,
and progressively continuing inwardly toward the hody. The
exact hody shape follows at the end of the calculation from
the streamline distribution; however it is to he suspected that
it ossontially remains similar to the form of the Prandtl solution.
For thits purpose the differential equation for cp = 5 —
is set up; is an exact solution of the considerahly simplified
equation (2.1;:
A2r ^.
^■^^r. -i^A
'° Sx2 oy?
with
o il -^ ^o
The suhscript c refers to the conditlona at Infinity, cp
fulfills the complete equation (2.1) up to an error e^ which
can he computed hy me?ins of (2.2):
nM
= €, = -', (Ma/ + n) ' ^'^'
p 1^^ o v^ ex
•^o
J Sx2
_2l, _2 _P __P (2.3)
ox dy Sx 5y
Putting = + cp in (2.1) and regarding it as a differential
equation for q), e fo.llov8 as tern of zero de.^ree in 9. As
cp« Op in the entire range, it is assumed that it applies to the
derivatives as well. Merely the terms of the zero and first degree
12
MCA TM No. 1189
and tho greatest term of the se^.ond dsgree in cp nesd to be included.
Thex'e results
ax
(a + 1)1^)
S^cp
oj'
.2
ep -!1- (n + 1) U-^
■^.fT- \ 2
+ 2
-J- + n -; ■ -"s-
S% Bf„
Sep
i
+ a ■ |(n ■ 1) m^- - aj ^£ T^^ + n -
[
0^5p Oil
§x^ oy
•^^ii) S5p[
:Z£s
SCD
dx Sy dx I dy
S"
+ 2n
% ^^p ^2fp 3$p S2cp $cp
^'
. ., + 2 (a + 1) ^.
dy ex dy ^-'^ -^^d
3x
{2.k)
The better 0^ satisfies the eq.uation of gas dynamics, the
easier is the determination of cp. Co, at great distances from the
body the equation can be substantially simplified. For z » 1,
especially on slender bodies, Tr-^« ^r-^: hence we can put
oy ^^ '
Sx
d®ri riCP ,
- — ±- = and -— - in 2.4- but in contrast to the Prandtl rule
oy
consider the variation of
3$
Since cp -> for z-^co, the term
of the second degree is emitted also. And equation (2.4) is simplified
to
^.-(F-F, .F
^'^ . a^cp
— F T— oJ
dx'^
with
dO.
ox
1^^ s 1-Ma2
VOX /
(2.5)
1 -
'%
.ax
MCA TM No. 1189
13
and
F« 5
cLF
2n
r%)
^SX /■
dx y J
This equation conld also be derived from (l.T)j F is an abbre—
u
viation for — 0„ .
o u
Tiie "boundary condition for z -> oo is cp = where the Prandtl
rule applies exactly. On the other hand^ however j the disturbance
of the flow by the body is very extended v/hen the flow velocity
approaches sonic velocity. It is therefore necessary to determine
an initial approximation for cp analytic -i-lly so as not to be com-
pelled to start at unduly great distance' . For this purpose (2.5)
is transformed further. While Ma = MaQ^ by the Prandtl rule, hence
F = F^,
the more exact term F = F + F'
^
P
Sx
— u
so that
F_^ + ^._F'
o -- 2
dx^
There
d'-cp
Sx'^
Vdx2
'Sfp
Ur
VX
can be ignored with respect to
c
x By
difficulties.
In principle we can also free ourselves from the approximation
that cp « $ , when in the formulation of equation 2.4 we consider
terms of the third degree in cp; the length of the calculation,
however, becomo'S disproportionately large. In another more appro-
priate method the assumption cp « ^n ^"^' omitted and the tedious
calculation of <|p in the entire field of flow Is eliminated. The
previously described cp method is utilizod only for computing the
initial values for large Z. The new method is as follows: fp is
evaluated at large distances from the body for y » y-j^; where y^,
is chosen so large that the error of the Prandtl solution is suf-
ficiently small; cp is evaluated from equation 2,10. This affords
the exact tjolutlon of the dynamic gas equ-'.tion "I -"^ri + 'P in an
initial strip. From here on $ ittjelf ifj calculated step by step.
The width of the initial strip from — xi to + xg^ must extend
upstream and downstream from the body so that for all y at
X < - xi and x > X2 the Prandtl rule is applicable with sufficient
accuracy. From y-j_ on, where $ is then laiown,
S^^ . , . . . . , . , , 1
Sy2
is graphically extrapolated to y± - ~ Ay for certain
^For bodies which are symmetrical relative to the y-axis
also, x-[_ is naturally = Xo.
NACA Wi No. 1189 15
fixed abscissae x (Ay is the length of the step; it can "be
assumed quite large at first, and reduced again later in proximity
of the body ) . Next
S$ ,' _ Si) I So
Sy ^y| ay
yi~^y ^1 i'l-^y
Ay
1/
is plotted againot x and graphically differentiated, which gives
^ ■ ^ . Plotting . ^^ . againot y and integrating gives
oy ax I dy ox
Sx| (3x| J 07 hx^"^
7^-^j yi yi
Lastly the variation of 2z-\ over x yields - — !. by graphical
dx' >. 2
yi ox
differentiation. With it
ox' ciy' dy ^x' dx^
are knovm for y = y, - Ay.
From the equation of gas dynamics (p.l), in -which the simpll-
Sy
f Ication ~ = can be made so long as it is valid that
•?— « •?— , ^-^ is calculated for the required x values and plotted
against y. Then the calculation is repeated, ^^—-^ extrai-iolated
for J 2, " pAy and so forth. ^
S2 J-\, the value
extrapolated at y-, - iAy and that computed for y, - Ay do not
form a smooth curve, the step must be repeated with a differently
extrapolated value for the particular abscissae . In this event it
is better to reduce the length of the step. Since the differentiation
Ig MCA m No. 1189
of the curves ^ and g-/ 13 uncertain at the
7=const. y=const.
■bo'ondary points x = — x-^ and x = Xp, it is advisatle to compute
§ = Q + cp also in two vertical strips x < — x-|_ and x > Xp and
to Join the progressivelv dofined points to these edpr' stripe.
The direction of integration for this step method must "be
chosen at right angles to the flow for the following reason. At
flow around a "body exposed to a flow alon.Q; x, t— - ie sure to be
c)6 ' ox
greater than r— almost everywhere, esi^ocially in the supersonic
zone. Thus at entry in a supersonic zona the coefficient of ■— -
Sx2
in (2.1) goes through zero. This does rot interfere in the above
method since (2.1) is used for computinfj, ^r-^.
oy-
If, however, we Integrate In the x -direction and solve
^2ff)
equation (s.l) for ^-~, then difficulties will result. The
coefficient of ■^-~ can, on the other hand, disappear only far
^^ ?,^
above the speed of sound where ^^ is not important at the point
under consideration. We can also prove this state of affairs v^ith
the help of characteristics. When discontinuities in the velocity
or their derivatives appear we cannot integrate across a character-
istic. On the other hand the characteristics of our flow become
nearly vertical so that again we can not calculate in the x direction
in this supersonic region.
3. ILLUSTRATIVE EXAMPLE
A flow sjHBmetrical in x and y is computed for a Mach number
of flow of 0.7)+5U. The flows on smooth bumps with supersonic zones
are obtained exactly, but on the other hand the flow past a closed
body is obtained only with errors in the region of the stagnation
point .
The described method is tried out on a very simple example;
we start from the incompressible flow (subscript i) past a circiLLar
cylinder
^1=^1+ P (3.1)
^1 + ^i^
MCA TM IJo. 1189
17
the free-stream velocity and the radiua are taken an unity. Pr.?ndtl'3
rule is applied tc a fixed Mach number of flow Map to which the
dimensionles3 flow velocity
3$ j
Ma.
2
Ma^ ■ + n
Z->±oo
corresponds. The abscissas for this trajisfoiination are contracted
by \/l - 1%^
X = X
(1 - Ma^^ x^
The ordinats^ remain the same: y e^ y - y • sc that
(3.2)
33^ = uo ^ and -^ = -=—^
Uq a^^i
-gax^-
(3.3)
To compute cp by equation (2.10) the coordinates x^ and y = y^
are used, so that
2\ ^^cp cl'^cp B-^cp 5^cp
y"^ dx^^ oy"
(1 - Ma 2\ Zcp a:
ax'^ ■ a:-2
1 - Ma,
- 1 + Ma 2
1 - (n + 1) u^2 (S5i/axi) =
1 - Ur
Pj
i3'^)
Development of the right-hand side for large z. = V-'^-i'' "•" ^ "^^
using equation (3.I) gives for the first approx5.mation, when
I =
z^
-3 and T) = ^-g = 1 - I
3_8 TIACA n-1 No. 1189
^(xi, y) = ^ ^^'^ ^ (1 - 6t, + 8^2)
nuo l/l - Meo2 ^ i
___^__^1 3^+3 gy (3.5)
A particiilar Integral cf this Pclason eq^uation la obtained
with the help of the separation formula cp = — r- f (g) , Thus the
general solution is written z±
-la-^^o H i A IV 1 ^c
nuo Jl - Mao2 Zi L ^
+ Vt (3.6)
with ^-pot = emd c arbitrary.
As boundary condition for cp the sole req^uirenent is that
it shall be small compared to Q~, that Is^ decrease more raDidly
1
than — • ^^"t ^'^r "the rest O-^Q-t £nd c can be chosen at random.
The physical meaning of this ambiguity is as follows: Owing to the
disregarded terms of hif-her order in the formulation of (3.5) only
the effects of the first order of the tod;; at great distances are
taken into accour.t. But these are the same for different body
forms. So the calculation yields different section forms, depending
upon the choice of c and ^ot • '^^° mcjiner in which c and
©4. affect the body form cannot be evaluated until several examples
h&ve been worked out . Up to now only one such example has been
worked out, owing to lack cf time.
The Msch number of flow M?.^ = \/5/9 = 0.75j.5l4- had been
specifically chosen. The dimensionless velocity is then
vlq = i/T/lO = 0.3162 and x h Xp = ^ x^.
In (3.6) only half Cp o^ = end c = 1/3 were assumed for
simplicity, thus eliminating the linear cerm in c,
MCA TM No. 1189 19
9
with this cp the velocities (and their derivatives) of the exact
solution = $p + 9 for y = 10 and <_ x< 6, as well as to
X = 6. <^y<_ 10 were computed and $ was deterMned for
y < 10 and x <6 "by the descrihed step—hy-step nisthod. In view
of the symmetry of flow relative to x and y the calculation
in one c]_tiadrant was sufficient. The step length /^^ up to y = 1.5
was Ay = 0.5; from there on 0.2.
While the exceeding of the sonic velocity (first at y = 1.35)
caused no difficulties, the calculation could not he carried out to
the hody hecaiise of another reason hut h'.d to he brc>:en at y = 0.6.
For at X '^ 0.6 the horizontal co!riponen':s of the velocity
^^ chajages so ra,pidJ-y for smaller crdinates y that the gr3.phical
differentiations hecame too uncertain to compute the next step.
As is seen from the contour of constant velocity (^'i'''. '^) a
steady hut still very sudden rarefaction occurs and on a point
syratnetricaJ-ly situated with reference to x = a compression
occurs. This phenomenon would of course not he plain at a lower
flow velocity, hut it is certainly characteristic of the flow in
proximity of the stagnation point where the speed increases ci^iickly
from, suhsonic to supersonic. For this point of the flow field
another method must therefore he developed.
So while unahle to ohtain the flo\v around a finite body
with a stagnation pointy the data ohtained thus far are nevertheless
very informative for suhsonic flows with supersonic zones. The
calculated streamlines and lines of const.ant velocity are shoT^m in
figure 3. 'Visualizing, the lowest stresmline in figl.^re 3 3.s rigid
wall, we get the flow along a smooth huiiip with a supersonic zone
near the highest point. Since this streamline is already very
steep for x~ 0.55? it can he assumed that the velocity distrihutlon
aroujid the finite hody (with sjTnmetr^'- axes x = and y = O.5)
indicated in figure 5 is fairly accurately reproduced hy the dotted
line. Incidentally, it is noted that even Prandtl's rule yields
conslderahle errors neao" the stagnation p.-)int.
Figure k shows several streamlines magnified five times in
elevation, along with the respective velocity distrihtitions. Not-
withstanding the similarity of the individual peaks the velocitios
differ considerahly at various places. The velocity — and with
it the pressure distrihutlon of thin holies ^ is therefore at high
flow velocities markedly dependent upon the exact shape of the hody.
20 MCA TK Nc . U89
Noteworthy also is the steep velocity increase at a point where
the streamlines themselves are still comparatively flat.
The contours of equal velocity in the supersonic region prove
the principle sot up in section 1 according to which the highest
speeds under certain assumptions do not occur at the point of
maximum thickness of the body. Even the equation (I.9) applied
for the derivation of this principle is satisfactorily confirmed
in figure 6, where the velocities on the y— axip are plotted along
with the hyperbola (dotted) that touches the ourre w(x = 0)/c* at
w/c* = 1. From the far—reaching agreement of the curves it follows
a
that in the vicinity of w = c* the exnression u = w(x = 0) = ,
b + y
is a good approximation. With this exanple the accuracy of table I
can be checked. In view of the flow velocity of u^/c* = 0.77^6
Hjj^^/R = 0.019 would have to be expected according to this table,
but by the calcixLated example it is proved that from H^ /R = O.O3I
on, the speed maximum is no longer situated at the greatest ordinate.
Thus, it is seen that table I is a good representation of the order
of magnitude of H y/^^- The difference is attributable to the
fact that the hyperbola used for the approximation gives too low
speeds in the subsonic range.
PART II
nrTRODUCTORY N0T3S On BOUlTOARy LAYERS AT HIGH SPEEDS
Studies of the behavior of supersonic flows in parallel
channels disclose that in the supersonic zone, princit)£l. flow
and boundary flow are in unstable equilibriim in certain circjua-
stances. An effect of the boundary layer on the principal flow
In the zone of the critical speed is to be expected for the reason
that here small variations in stream density cause considerable
changes in speed. This is particularly plain in the calculation
of the flow through a Laval nozzle at high subsonic speed with
observance of the boundary layer.
In order to gain an insight into the condition of the
boundary-layer flow at high speeds, which we will study in the
following, consider an example from the sphere of incompressible
flows, where the conditions are better controlled. We consider
the circulation—free, incompressible, and stetionary flow around
a circular cylinder at a high but still subcriticaJ. Reynolds
number. Computing the pressure distribution at the body with the
WACA TM No. 1189 21
aid of the potential theory on the assumption that the cylinder has
no dead-air region "behind it and then calculating on the haais of
this the boundary "layer conditions, say, with the aid of a refined
Pohlhausen method, we find a separation point in the zone of rising
pressure. It is found that the omission of the dead-air region was
wrong. The pressure distribution on the body must therefore be
computed with due allowance for the dead-air region and then, it can
be hoped to attain a result correspondinr, to reality when the
dead-air region is so assmied that the related pressiore distribution
yields separation exactly at the starting point of the free stream-
line. This ezample shows that potential flow and boundary -layer
flow usually depend upon each other. In general, we can say that
the potential flow determines the boundary -layer flow, also that
the boundary -layer flow deterjiines the potential flow. The former
can be stated with great approximation in flow without pressure rise.
It is a known experimental fact that for large expansions a
flow simply does not follow the boundaries of the region; but it
should be remembered that for the development of a dead-air region
not the expansion of the stream filament but the fact of a pressure
rise is decisive, which only in subsonic flows goes hand in hand
with an increase in stream filament section. In supersonic flow
on the other hand a contraction of the stream filament results in a
pressure rise. Thus visualizing a parallel channel with a flow of
Ma > 1 a too strong growth in boundary layer caused by some
disturbance is followed by a pressui-e rise, which in turn favors
a stronger growth in boundary layer. In contrast to subsonic flow,
an unstable equilibrium of boundary layer and principal flow is
involved in this instance and a very considerable boundary layer
growth must be reckoned with in certain circumstances. It m.ay,
in a straight channel result in a sudden strong pressure rise at
the flat wall and so in the formation of a dead -air region (fig. 7(a))
(Compare reference 11.) If the presrm^e rise is so great that the
flow becomes subsonic, the relation of ma.in flow and boundary -lay er
flow is stable again, the dead-air space cai:inot remain in this part
of the channel. If a small pressure rise is involved of, say, a
weaker oblique compressibility shock, the principal flow experiences
a directional change in the sense of a channel contraction. The
dead-air space must increase wedge-like, but this holds only over
a short distance, otherwise the flow would have to revert into the
subsonic range. It is therefore to be a::^sumed, that at an oblique
compressibility shock, as met with in figure 7(a), the turbulent
intermingling imposes a limit on the growth of the dead-air space.
These qualitative reflections lead to the conclusion that in the
range Ma > 1 an unstable state of equilibrium must be reckoned
with in certain circumstances between principal - and boundary -layer
flow, which may promote the fom^ation of dead-air regions even at
a flat wall.
22 MCA m No. 1189
A disturliance of the unstatle stato of equllilDi iim of principal
and iDoiandary-layej:' flow in the supersonic range is favored, ty the
fact that any minor disturlDance in a supersonic flow is propagated
undamped along Mach lines. Thus^ a pressure rise in a supersonic
-^ tunnel can te dispersed Isy a small disturbance far upstreamj on the
^ other hand^ the pressui'e i-iae sets in again some distsxice downstream
^ as we can also infer frcm our example.
The unstahle "behavior of the "bcundary layer in the supersonic
zone must disappear when the principal flow approaches sonic
velocity. In the critical speed range w = c*^ which ia of particu-
lar interest in flows past todies with high speedy the fact stajids
out that this is the range of maximum flow density. But the proce-
dure in computing the incompressihle flo^r past an airfoil is such
that the pr-ossure distribution is ottainefl from the potential, flow
without consideration of the displacement effect of the "boundary layer,
and then the toundary layer is computed with the aid of this pressuz^e
distribution. This is not permissible however in the region of the
velocity of sound^ because a minor variation in stream, density 6
exerts a very substantial effect on the speed. This is readily
apparent in figure k, where peaks with comparatively minor form
changes pi'oduce very unlike pressure distiibuticns. This effect
increases with increasing flow velocity*
The effect of the boundary layer on the flrw in the vicinity of
the velocity of sound is illustrated by a aimple example , which,
although it involves no flow pioblem. is nevertheless informative
for the appraisal of the displacement effect of bcundary layers
at high speeds. The velocity distribution in the nozzle used by
Stanton (reference l) for his erperinents was ocmputed by appli-
cati'-n of the simple flrw filament theoiy_, once vri.thout boundary
layer, and once on the assumption of a laminar boundary layer.
The boundary -layer calculation is made with the help cf a process
which will be explained in the following secti^'n. The Initial
value of momentum and displacement thicltness at x = - 0.20 waa
estimated. The dimensions of the nozzle are so small that it can
bo assumed that no turbulent transition takes place. Stanton's
test series C is illustrated in figure 7(b). The velocities were
deteimined by measuring the static pressure on the axis of the
axially symmetrical nozzle (lower test points) and adjacent to the
wall (upper test points). The theoretical curves by Oswatitsch and
Rothstein (reference 2) and the flow filaiaent solution with and
without boiondary layer allowed foi' are included for comparison.
The former waa computed only as far as the separation point. It is
seen that the aBymmatry 'is reproduced qualitatively coi-rect by the
flow filament solution with boundary layer taken into accc-unt.
The displacement thickness at the narrowest point of the nozzle is
not quite 2 percent "^f the nozzle radius. Computing the velocity
NACA TM Wc. 1189 23
distrlliution for the sane nozzle in incompressi"ble flow with and
without consideration of the toimdary Isyer, the results in "both
cases are essentially even lines. Even ab speeds atout 15 percent
helow those of test C, any boundary-layer effect is quite insignificant.
This may be taken as proof that the asymmetry in nozzle flows which
at the most, manifest local supersonic zones, are caused by boundary-
layer effect. As to making the computation, only the following is
mentioned. That one gets at first the d'^stribution of the displace-
ment thickness from the strenia fileanent solution and then a new
stream filament solution taking into account the calc\'.lated displace-
ment thickness is proof in itself that euch an iterative procedure
is pprmissable at very high subsonic speeds. Displacement thickness
and stream filament solution are obtained step— i/ise at the same time
in the downstream direction.
The influence of the boundary layer on a submerged body will be
handled in section 7. Owe example, however, shows that we cannot
hope to obtain results tha.t correspond to the real process in some
degree, for the flow problem with high velocity, without examining
the boundary layer. We then have to remove, in practice or experiment,
the influence of the boundary layer, perhaps by suction.
5. CALCUIATI'CN OF DISPLACEJffiNT THICKNESS OF LAKENAJR AND TURBULENT
COI-IPRESSIBLE BOUNDARY LAYERS
For more accurate calculations on boundary -layer effect in
flows at high speed, formulas for the variation of the displacement
thickness are necessaxy. This is done in this section, first for the
laminar and then for the turbulent boundary layers. In the derivation
of the formulas for laminar boundary layer a refined Pohlhausen method
is given for computing laminar compressible boundary lasers for given
pressure distribution.
Inasmuch as the pressures in the coi-ipressible zone transverse
to the flow direction in the boundary layer is also to be regarded
as constant, the relative density variations within the boundary
layer are in amount equal to the relative temperatux-e variations.
Thus, if sonic velocity prevails in the outer flow the relative density
variation inside the boxmdary layer amounts to about 20 percent,
since stagnation point temperature can be approximately assumed at the
wall. So, at not too high supersonic speeds a qualitatively identical
behavior in the boundary layer and in the incompressible range is to
be expected.
2k
NACA m No. 1189
So as not to exclude the possltility of compressibility shocks
■beforehand, the flow outside of the boundary layer is called outer-
or— pi'incipal flov; instead of potential flow.
Being primarily interested in the bsliavior of the displacement
thickness, the behavior of other quantities is studied only to the
extent that it appears in the result without loss of time. This is
the case in the ctudy of laminar boundary layers, where the momentum
thickness is comparatively easily obtained', and the displacement thick-
ness derived from it.
The process is based upon an Improved Pohlhausen method in
conjunction with the reports by Bohlen (reference 3) and Walz
(reference k) .
Boundary— layer equation and continuity condition for the
stationary case are written as follows
pu
Su
+ pv
5u
dp d
dx ^
Su'
(5.1)
Ie (p^) ■' h ^^^^ '- °
The coordinate system is chosen in the ucual manner so that x is
tangential and y normal to the contour of the body. Thus, in the
general case, x, y ' 'imply jae Cartesian coordinates in the followingj
\i is the friction coefficient dependent on temperature. The
quantities p, pj
u.
and V are not made dimensionless. It further
is assujned that the boundary— layer thickneas is small relative to
the radius of curvature of the wall, so that curvature effects can
be disregarded. The second Navier— Stokes equation gives then
exactly as for incompressible flow the result that p is merely
dependent on x, but not on y, which in the boundary— layer
equation was already evidenced by formation of the ordinary
derivative of p.
After integration of the boundary— layer equation over y, the
application of the continuity condition gives the von Karma'n momentum
equation which is written in the fonn
d / 2 \ "^^a
■K2 ^ Pa^a ^) + Pa% IT ^*
(I)
— t
*v
(
(5.2)
MCA TM ETo. II89 25
The subscript w Indicates the values at the wall^ siabscript a
those in the outer flovr. In the moment'u;.i equation displacocient
thickness and momentum thickness are defined by
nS ft^
*i(^-^)"^*us(-i)- ■<">
where o is the so-called boundary-lajer thickness , which is nox>r
chosen 30 great that 5"^ and -5- can be regarded as independent of
5. The displacement thickness has the physical significance that
the through flow volume In the boundary layer is reduced by an
amount that^ on the assumption of piure potential flow, is equivalent
to a shift of the wall by piece 5'* in positive y direction.
Equation (5.2) .can also be given the fori
■m
s^-^^%^¥^^-4-^^^ <'•-'
Aside from the usual assumptions with tho aid of which the boundary-
layer equations are derived^ no restrictions of any kind have been
made so far. Only the problem without heat transfer on the wall,
the so-called thermomoter problem is treated in the following.
Further we will make one approximation, by which we will specify
the fon:n of the boundary— layer profile by cn3.y one parameter in
addition to the Mach member of the outside flow. Next it is
necessary to make an assi-Huption concerning the configuration of the
density profile. Having seen that in the vicinity of the critical
velocity the velocity profile especially might be decisive, while
the density variation is unimportant, the case of laminar boundary
layers is limited to the assi.miption that the temperature at the
wall always attains the tank temperature T^^^ and satisfies the
energy theorem within the boundary layer. This ties in .also the
assumption that temperature — and velocity — boundary layers are
of equal thickness. Accur-s-te calculations on flat plates indicate
that this assumption also holds in a considerable supersonic range
(reference 5) .
The pressure in the y direction being constant, the density
variation follows from the tem.^erature variation as
26
MCA TM No. 1189
m
J'
p_
Ta
T
u
w.
ni.
(5.M
The parameter for the boundary— layer characterization is derived from
the known boundary conditions vjhich for u = v = in the boundary-
layer equation leads to
£^i\ /du\ f>^u^
dUg
Pa^a dF
Since the internal friction is only dependent on T and no heat
transfer takes place, (t^ ) = 0, hence with application of •^
X* =
5^
cV^
^^
1
""iT"
w
2
w
du,
a
dx
(5.5)
We have avoided the introduction of the boundary— layer thickness
itself in the equations other than at the unimportant place as upper
limit of the Integral, The version of (5.5) was largely taken from
Walz's report (reference k) . The parameter X,* differs from the
/S*\2
conventional Pohlhausen parameter by the factor ( — I , the density
refers to that outside, the internal friction to that at the wall.
For a class of velocity profiles, such an for M = 0, for instance,
the individual profiles which are represented by the magnitude of
— and / — rr— \ can be taken as
\ ^ /w
from the class of profiles, hence
parameter X* the quantities
\*
function of the parameter
d Pa^"
by (5«2a) simply as function of \* .
obtain Uo -—
Choosing
dx [x^
the Pohlhausen profiles as profile class ^ives the curve of Bohlen—
Holstein (reference 3) in figure 1, while the Hartroe or Howarth pro-
files result in Walz's curves of figure 1. Moreover, it is not
necessai-'y at all tc have an analytical representation of the profile
class, it can equally be given as family of experimental curves.
If the outer flow is dependent on a Mach number, one profile class
is used for each Mach number. In order not to come to grief because
MCA TI4 So. 1189
27
of o-ur Ignorance in the sphere of compres^ihle velocity profiles
two known facts aie takon e.dvantag-e of: Firat, we Icnow that the
velocity profiles on the plate at constant pressure are not very
closely related with the Mach nijiiihere of the outer floT.' (reference 5) •
Therefore, this is ass-jmed to he the case in retarded or accelerated
flow also; secondly, ve knew that the single parametric raethod in
the incompressible, which utilizes the Pohlhausen profiles, leads to
fairly practical results, althcugii the Pohlhausen profiles themselves
do not represent tlie actual pi'ofiles very well. An exact represen-
tation of the velocity profiles themselves is not needed, the main
point is the displacement thickness for which the integration over
the profile form is already accomplished.
On the tasis cf these arguments we therefore select, independent
of the Mach number, the Hartree profiles for the velocity distribution,
which, as a r\LLe, probably represent the incompressible boundary
layer, best of all. The density variation is then £iven by (5.'0 s,s
fivnction of the velocity distribu.tion and owing to the presence of
quantity
"■a
w.
as function of the Mach number.
m
By a niimerical method the Quantities -
&*
and I .^ JL-] art
a / w
then obtained as f unctioix^ of
and ha, as exemplified for the
following, Ma which corre3t)ond to the — values :
w„.
m
Ka
1
:
1
1.2
1.5
2.0 .:
1 i
L
0.U30 '
0.!i73
0.557
0.667
We obtain the c-.u-ve system of flgui-e ci.
The curves are showr. at the left as far as the sepai'ation
point, at the right they proceed to the point u.p to which the
Hartree profiles are calculated. Several cvjtvos were extrapolated
beyonl it, and indicated by dashes. The cuj've Iia =0 is identical
with the Hartree curve from 'Walz .
Thus with the velocity distribution, the variation of the Mach
number cf the outer flow, and an initial value of — - — the loarameter
^\'
A* can be formed; with it and observing Ma the quantity
28 MCA m No. 1189
Ua T can be taken from figure 8 and the variation of
computed. From it we obtain again this quantity at a point shifted
by one step and the calculation can then be repeated (reference 3)'
Although o* is wanted, it was preferred to compute -S —
because the equation for this quantity is -^qty much simpler. Know-
ing the outer flov:, -3^ can now be specified. To determine 6*
represented in figxire 9*
thus further requires —• which is a function of A* and Ma.
(3
Since the quantity ~- in the boundary -layer equation is to
dx
be defined as accurately as xjossible the following formula is of
advantage
6«^ dx o* d?^* i3 dx &* u„ U J dx w„
a— ^
^m
+ 1 /^_1_ JL Pa£ ^Ha2 ^\ (^^.5)
2 (Pg^TSS dx n^ Ua Ix
This formula contains only quantities dependent either on Ma and
A*, 01' which can be taken from the previous boundary -layer calcu-
Uq
lation. The derivative with respect to -— ^ which is a fianction
of Ma, was prererred over that with repect to Ma for reasons of
simplicity. The first term at the right' hand side in (5-") is
generally the principle term.
,-s „ , p .^ 2
If it is desired to eliminate — r~ and -? in (5.b)
djc ax |i
so as to secure Sii_ merely in relation to — — , — i^ (which
dx dx' ^^2
then because of entera in the eauation ) and coefficients
dx
solely dependent on A*, Ma, and Re, the expression becomes
fairly long, but since this relationship is used later it is
given here, the Re number being suitably referred to the out-
side density, the coefficient of friction at the wall and the
displacement thiclcness:
Re* = ^^£ML (5.7)
MCA TM Eo. 1189
29
The formula reads
d6*
1
Ee* \i3 /
^*f
S -8 /w
Ua _^ _a__ S*
/:,
A
Vi3
&*
Uz u,;^^. 5* BA- .
r
2 d'^ri
*a
5* p
2 + Ma'^
i3
duA 2
5*
^a ^
.f..
Ma''
Ee*
a^
6* ^
6*2 art. It is true that
the difference of the ta-ro effects is not 3o far reaching that a
second profile could be computed accurately enougli from ttie
specified velocity profile when the shearin,? ctresses are discount©!,
because the shearing stresses are able to substantially modify the
character of the profile; but for the calculation of the varjaticn
of displacement thicVaiess, which essentially involves an integral
over the velocity variation in the Incompressible cc-se, the shearing
stresses can be disregarded.
Tiie result at Ma = is the following appro;:imation formula
for the ttirbulent boundary layer :
d6* _ _ 5^^ / 1/^^^ ii
■ ^a
u„ dr:/ K uy
dx ~ Ua dr.^.
&l^ (5.9)
6-^
As the integrand is alvays positive, it can be tai^en from this
formula that a speed increase is accompanied by a decrease in
displacement thickness and a speed decrease by an increase in
displacement thickness. At constant outsido velocity the displace-
ment thickness reraains constant, according to (5-9) • This result
is naturally -vnrong, as Indicated by experiments on the plate at
constant pressure. For In this case the variation in displacement
thickness is contingent upon the turbulent shearing stresses, so no
correct result is to be expected. The forxJiula conld be improved
by the addition of the conventional formula for tlie variation of
MCA TM No. 1189 31
the displacement thicloiess, but it would serve no useful purpose^ as
will be seen. It is of greatei' significance that in contrast to
equation (5-8) '-^or the laminar boundary layer the second derivative
of u^ is lacking in (5 •9)- Since (5 --9) was obta,ined by several
rougher omissions its practicability is illustrated in figure 10.
The experimental va.lues of — 5; and -I—- are shown platted
Ua dx dx
against the arc length x of Gruschwit'; ' s (reference 6) test
series 3; along with the variation in d;;.sTjlac3ment thickness calcu-
, . ■" dUg^
lated by (5-9), the integral being forned at ~ — = 0. It is found
that the formula reproduces the actual conditions adequately, as far
as the area of greater accelerations, vrtiere errors begin to be intro-
duced. This is, of course, due to the fact that 5* = imposes a
limit on the decrease in displacement tiiicicness .
These experiences in the incompressible zone can now be
Interpreted to the effect that the turbTid.3nt shearing stresses for
the calculation of -— - can also be cancelled in the compressible
dx
zone. But even this assumption is insufficient to develop a law
for the variation in displacement thickness," additional data on
the density distribution in the boundary layer are needed. In the
case of turbulent boundary layers the energy theorem is not
directly applicable, because the density -boundary layer- is
probably twice as great as the velocity -boundary layerj hence, the
density varies in an area in which the velocity is already practically
considered constant (fig. 11) • The result of it is that the varia-
tion in density plays the same role in the calculation of 5* as
the variation in speed within the boundary layer. Unfortunately
only one meas-arement of a tuL-'bulent supersonic pi'ofile is available,
and naturally there is little sense in developing a theory without
further basis. However, in order to reach a tolerably correct
mnierical value, the part of the boundary layer in which the density
alone varies is disregarded for the present, since it involves
only about 10 peixent of the displacement thiclaisss, and, in the
remaining portion, putting the stream density as a function of the
velocity as follows:
Pa^
.E(^j (5.10)
H to be taken from the oxperiment. Wow the derivatives of p can
be expressed by derivatives of u, Ug_, and Pg^ with the aid of
(5-10) . This enables us to derive a for-inila for the variation of
displacement thickness, neglecting the torbulent shearing stresses.
32 NACA m No. 1189
tto 7- (5 •11)
H' is the derivative of the xunction H according to the argument
— and & the ulace where -— can be vnt equal to 'onity
'^ " %
1 — = 1), while b* represents the corrj(-.t diSDlacement thickness
hence, integrate up to a point where -^ itself is equal to unity
Pa
(-2- = 1). This means, we state that the 10 7:)ercent of the dis-
VPa /
placement thickness between the point — = 1 ' and -^ = 1
% Pa
contributes to the variation of the displacement thickness an amount
which corresponds to its portion of tho displacement thickness.
For Ma = 0, equation (5-11) naturally changes to (5-9) • If the
density and speed in the boundary layer are specified, the integral
can be evaluated also. We have calculated the expression in paren-
theses for a profile by Gruschwitz, for which — r: = 0, and for the
velocity and density profile represented in f igvxe 11; thus we obtain
the constant aj for two values of the Kach nijmber.
TABLE III
TUKBULEira BOUim\RY MYER
I m 1.0 1.7 !
' i • !
I ttg i 5.1 ' 2.P I ■
The close agreement of coefficient a^ ^'or the turbulent and the
laminar velocity pi-ofile is noteworthy.
ST.'VBILITY STUDY ON THii: FLAT PLATE
A study of the equilibrium of boimdary layer and supersonic flar
on the flat plate indicates that an unstable state is involved. The
growth of a small disturbance in a laminar boimdary layer djffers
somewhat from that in a turbulent layer and is, especially in the last
MCA Wi No. 1189 33
case, very rapid. In incompressible flov a stable equillbriuni
exists between principal flow and boundary layer.
Having secured the var:^ation of displ ".cement thickness 6*
in relation to the velocity variation of the outer flow, the
reciprocal effect of principal -- and boundary layer flow is now
analyzed in the simplest case, naiiiely, in the flow at the plato,
vrithout specified pressure distribution.
Since the effect of small disturbances is to be involved, the
Mach number of the outer flow Ma is regarded as constant aiid the
V coD^ponent of the velocity considered small relative to the
velocity of sound. After introduction of a velocity potential the
simple equation (2.2) is. involved, and written in the form
5yy = 0,,^(Ma2-l) (6.1)
The X-axis is to be in plate direction, the y— axis normal to it.
Now it is necessary to represent the effect of the boundary
layer on the potential flow in form of a bo-ondary condition. The
boundary layer is therefore visualized as being replaced by an
elastic layer superimposed on the plate, which has the property
of always attaining the thickness equivalent to the displacement
thickness of the boimdary layer t.t the particular place for the
prevailing velocity distribution. That is, the equation
I
V = u ^ (6.2)
must be satisfied for y = 8*.
This condition is inconvenient to the extont that the boundary
for which jt is to be fulfilled is not specified beforehand. But,
inasmuch as the disturbances are to be small, hence the outerflow
is to differ verji little from a flow u^ - Const., the boundary
condition for displacement thickness 5* in undisturbed flow is
assumed. By assimption the departure of o* from the value of
the displacement thicioieas for the undisturbed flow must be squall.
Hence it seems immaterial whether v is specified at y = S* or
at y = 5* + d5* in the linearized problem. Besides, the study is
d5*
to be restricted to such a small area that —3-- Itself can be
dx
regarded as conctant at u^ = Const.
Ik MCA TM No. 1189
Furthermore the boimdary condition (6.?) has the property of
givinc the same v component of the velocity at y = 6* as a
d6*
boundary layer with equal -■?-, on the assumption of potential
flow in the entire space.
In (6.2), V and u aro none other than the components of
the outer velocity, hence in the notation of the preceedlng section
equal to Vg and u^.
Applying (5v8) or (5.3-1) to —, — gives then as boimdary
condition of the problem, linearized in the derivatives of u, the
following equation for
y = 5*: V = u -^ - a„&-*|li- a-.Re*^ 5*2 (5,2a)
Se^ 2 5x ^ -
If a laminar flow is involved the corresponding constants
must be taken from table II j if, turbulent flow, table III; in the
latter case, a-, must be put = 0. In view of the linearization
S* and Re"* must also be regarded as constant, although the
variation of u in the first term is not Important, it is considered
nevertheless, because the solution then is reduced to the treatment
of a homogeneous linear differential equation, which means some
simplification.
Now it is attempted to find the solution for the ca^e that
the plate Is exposed to a flow with the velocity u(x,y) = u = Const
and at a point x = at the plats the velocity is artificially varied
by an amount il « u^. The coordinate system is turned through a
small angle so that the x*-axis in point x = is exactly in flow
direction and the y*-axi3 normal to it. The tangent of the angle
of rotation is defined by the variation of the displacement thickness
at X = y = 0, which is equal to
d5* ^ _^
dx Re*
Strictly speaking a transformation of the coordinates in the
equations themselves should be effected. But since the boundary
layer itself makes no difference between these two directions,
and 80 a rotation merely involves more paper work without any
physical significance, it is disregarded and the equations applied
to the new coordinates. The coordinates are in addition visualized
as being made dimensionless by the displacement thickness and the
origin shifted to the point x = 0, y = 5*.
KACA TM Ko. II89 35
These new coordinates are denoted with
, X . y - 5*
^ 5*^ / - 5*
and after introducing the velocity potential in (6.2a) give the
following differential equation with the respective "boundary'
condition
Oy,y, =5j.txi (Ma^ - 1) (6.1a)
at
y» = 0; 5y' = - ajOj-i.-t - a^Re^S^f x'x' (6.2^)
Assuming a very general solution of (6,3a), and writing the
potential as sum of a potential of a principal flow u and a
small disturbance
Uq5*x' + u&'*^
f(x' ~-^j'A&^ -1 y'
g(x» + ^Ma2 - 1 y') |
f and g are arbitrary functions of which it is merely required
that their sum at x' = y' = be equal to unity. It is seen
that g gives Mach lines which point toward the boundary layer,
hence stem from a disturbance from the outside. This function is
thus put identically zero since such disturbances are to be
disregarded. Introdvction of the thus obtained solution in the
boundary condition gives the f^onctional form of f . Denoting the
derivative with respect to the argument
1] = x' - y'Ma" - 1 y'
with subscript ri, we get
^x« = u 5* + u5*f^ ; $x'x' = ilg-^ . •
^x'x'x' = uo% ; %' = - u5* jMa2 _ If^
which inserted in (6.2b) gives an ordinary differential equation of
the form
a-Ee*f + a„f -JlAa^- -If =0 (6.3)
3 THTi 2 1]'. V T)
3o
NACA TM Ko. II89
This eqiiation Is easily solved. First poBtuJ.atine a laminar bouncLary
layer, hence a^ ■/ 0. Then from the raquireijient for x' = y' =0:
f = 1, a requirement for the second dei-ivative f^_^ can be satisfied,
because the upper equation can be regarded as differential equation
of the second order of f^. Since at thio point (^nly the consequences
of small velocity distur-bances, not the consequences of disturbances
of the velocity difference are to be studied, the added requirement
for x' = y' = is f = 0, which gives the solution
f_ =
T) t-, - tr
jtgn
t-L - tg
,\^
where t-, and to are abbreviations for the expressions
a_
^1,2
2 a^Ee*
a.
1/ ._
^^. +
VMa'^ ~ 1
a^Re*
(6.1+)
As Re* in general has the order of marnitude of 10^, the last
tei-m under the root is a term of greatest influence . Thus for
appraisals at high He* we can put
1;2 \\ ttoEe*
(6.1ia)
It is to be noted however that the critical Re* which corresponds
to a value of ab^ut 1.1+ X lo3 must net be exceeded as will be
shown later .
By use of (6.U) the velocity distribution in a Laminar boundary
layer on the plate is obtained as :
u
5* 'y-
't' V ' = U„ +
u
*1 " ^2
tie*2fx'-^/Ma?-ly') - tpe'*^lUx ' -/Ma^-ly •;
(6.5)
which by (o.Ua) is reduced to the simple form
Mar- - 1 /
u a u^ + u coah\/-~jp— (x' - \/Ma2 - 1 j^)
\! 3 ^
(6.5a)
MCA TtA No .1189 ^'^
If a turbulent boundarj"- layer were 5nvolved, hence an =
the first aimimand in (6.3) cancels out and only one boundary condition
can be satisfied. Again requiring f = 1 for x' = y' = gives
\
The Telocity field luider the assumption of a turbulent boundary layer
at the plate is
u = Uj^ + ue
(6.6)
From (6.5) and {^:>,G) it is seen that the boundary disturbance along
Mach lines is propagated into the flow. The interference velocity
u is always accompanied by a fijinction which grows considerably
with rising value of the argument, while in the case of the laminar
boundary layers the coefficient a^ plays the principal part. In
turbulent boundai'y layers the coefiicient ap is essentially
involved. Thus the boundary layer of a flat plate in flow with
r,onstant velocity is in both instances in an unstable state of
equilibrium with the principal flow, which with observance of the
terms of the first order only, lets a small disturbance grow
infinitely. The type of grov/th is, of course, quite dissimilar
on the two boundary layers. To secure a measure for the inctability
of the state, we may ask for which value of x' =-"cir at y' =
the disturbance has grown to twice the amount and call this quantity
the length of growth A. It is not made dimeneionleas by the
displacement thickness.
The length of growth in a laminar- boundary layer A^ is assessed
by (6.5a). The hyperbolic cosine grows for a value of the argument
of around I.3 to the amount 2. Accordingly
:2"
A^ ~ l.'il- .^"^' : b* (6.7)
jl Ma'=^ - 1
The length of growth of a turbulent boundary layer A^ is
At = 0.70 i.M ^ - 6* (6.8)
^Ma- - 1
38
MCA m No. 1189
Postulating a laminar "boundary layer at Re*
and III give the rollowing length of growth
1000, tables II
TABLE
IV
Ma
1.2
1.5
l.T
2.0
5*
25
18
12
At
1.1
Noteworthy is the unusually short length of growth in the turbulent
boundary layer; but even that in the lamVnur layer is still very
small when bearing in mind that the displacement thickness in
supersonic flows is of the order of magnitude of 10"*3 to
10~2 centimeters.
The investigation was restricted to small disturbances. The
extent of growth once they have reached greater amounts remains to
be proved. One thing is certain that the outerflow cannot increase
to great velocities, because the boundary layer cannot drop below
the amount &* = 0. Thus no limit in velocity decrease is imposed.
It may be presumed that the velocity decreases until the boundary
layer breaks away. In general, the instability of the discussed
equilibrium condition will become evident in a pressure rise,
probably an oblique compressibility shock. It would not be sur-
prising if oblique compressibility shock occurred in the center on
a flat wall (fig. 7(a)). The example cited here could be multiplied
by many others, pei'haps even by flow around conical tips. It should
be kept in mind that a pressure rise can cause transition of the
boundary layer. In the example adduced here the boundary layer is
already certainly tuirbulent.
This study of plate flow can be regarded as first result in
this sphere of instability of supersonic boundary layers. It would
be desirable to get away from the assumption of small disturbances
and constant flow velocity. This seems altogether possible by a
combination of characteristics method and boundary layer computation.
For the turbulent boundary layer, of course, the laws of variation
6* wouIq have to be aiialj-'zed first.
One unusual fact is that in the measured pressure distribution
on a wing, such as those by Gothert (reference 7), for instance,
pressure increases were almost never observed in the supersonic
zone, except in form of compressibility shock or occasionally^ at
small Reynolds numbers, where laminar boundary layers must be assumed.
KACA m No. 1189
39
It appears entirely possible that this fact mipht be explainable
by the cited properties of the supersonic boundary layer.
The corresponding behavior of a laminar boundary layer in
incompressible flow (Ma = 0) is briefly indicated. The disturbance
at great distaxices from the wall, that is^ for great values of y,
must disappear. On these premise s, (6.1a), (6,2b) by the same method
of calculation give
u = Uq + ue
-p^x'-p^y'
cos (P2X' - Piy')
(6.9)
with the abbreviations
1 J'^ I 1
liV^
-^Y . i.^.^)2 . (^f
Po =-' M C-l + (a,Be*)2 -f^f
The decisive term at high Ee* numbers is again a-J^e^
He* = 500 it approximately is
For
Pt ^ Po
J 2a Re*
0.053
that is, a strongly damped oscillation is involved. The analyzed
equilibrium of laminar boundary layer and outer flow in the sub-
sonic zone is e:±remely stable according to it. This method of
analyzing offers the further possibility of exploring the stability
of lami.nar subsonic boundary layer relative to nonstationary dis-
turbances and comparing the results with Tollmien's calculations
(reference 8). For nonstationary velocity variations Pohlhausen's
method is, of course, not practical in general, in the form given
here.
Incidentally, the requirement of damping of the disturbance
for great y is not fulfillable in subsonic flow on the assumption
of a turbulent boimdary layer at the plate. This result may have
its cause in the fact that (5-9) does not meet all requirements.
1^0 KACA TM No. 11 39
7. SIGMFICANCE OF BOUTTDAEY lAYEE IN THE PRESSURE
DISTRIBUTION ON A BODY
Appraisals Indicate that the flow in the critical range of
sonic velocity is very suhstantip.lly affected by the houndary
layer. Without its inclusion a coi'rect calculation of the pressure
distrihution therefore seems, in general, not very promising. In
many Instances the behavior of the boundary layer actually governs
the pressure distribution.
On examining the pressiire distribution at a bump computed In
B'^'ctlon 3, (fig. ^4-), a s:,Tnmetrlcal velocity distribution is also
found on a body symmetrical about the y-axis. This is, however,
in great contrast to the experience in tests (compare, fig; IP) ,
where symmetricai peaks were invariably accompanied by asymmetrical
velocity distributions. Naturally the question ±3 whether there
is only one solution for each bump but it will be shown that, owing
to the boundary— layer effect, s;>Tmnetrical solutions can be expected
as little as in the example of the velocity distribution in a nozzle
{fig. 7(b)).
By (5.8) the displacement thickness of a laminar boundary layer
for constant outer speed is
S^ = 1 ^ai
^^w
^oPa^
What is the possible extent of the bump in order that the boundar;-
layer remain laminar? Figuring v/ith tests in a low-pressure tunnel,
the values at critical velocity are
Ua = 3 X lO^cm/sec; n^^ = 0.8 X 10~3g/cn3; ^^ = 1.8 x 10"^
CG3E
It is to be presumed that the critical Ee;>Tiolds number at sonic
velocity does not differ substantially from that in incompressible
flow. Taking the critical Reynolds number formed with the plate
length at
^^cTlt. = 5 X lo5
NACA TO No. 1189 ^1
gives tha critical Eeynolds number (5.7) formed with the displacement
thickness at
^«*crit. = 1-^ >; 10^
with the previous values of 'J,^, Pa? and [i-j^ the critical .values of
plate length and displacement thickness are
^crit. = 3.S cmj &^crit, = ^'^ - 1°"^ <^™
So in order to prevent transition from laminar to turbulent flow in
the boundary-layer models lengths of only a few centimeters may be
permitted in the usual test arrangements, provided that no strong
accelerations are involved.
Couverselyj the critical length indicated here gives a measvure
for when the transition point is to be expected on a plate flow in
an exhaustion tunnel at sonic velocity. In a free— a.ir test this
length is reduced by about half because of the higher density.
In the schlleren photograph of an infantry shell in flight at
around sonic velocity (fig. I3) (referencos 9 and 10) the oblique
compressibility shock is evidently released by transition^ its
effect being probably amplified by the unstable behavior. of the
boundarj"- layer. The fact that a missile at small supersonic speed
is involved is Ijmmaterialj since a straight compressibility shock
prevails in front of nose of the missile^ it actually flies as if
in a subsonic flow.
Analyzing the bujap in figure k, which at the point of its
greatest height has nearly constant sonic velocity for some distance,
and supposing the points of strong velocity rise and velocity
decrease (x = 0.6) to be about 2 centimeters apart, the displace-
ment thlcknevss at the peak is certainly greater than that of a plate
1 centimeter in length izi flow at sonic velocity. Therefore
5*_Q > 0.50 X 10-2
cm
At the point of substantial speed decrease, separation must be
definitely expected. A calculation by the expanded Pohlhausen
method shows that the momentum thickness grows with increasing arc
length. Much greater is the rise in the ratio of displacement
thickness to momentum thickness (fig. 9) which for Ma = 1 increases
lj.2 WACA TM Ko . II89
from point X* = to the separation point frcm value 3.2 to U,7.
Considering the fact that the momentum thickness itself increases
up to the separation point, an empirlca3 rule can bo established
according to which the displacement thickness is doubled between
X,* = and the point of separation.
The difference between the displacement thickness at the
separation point and at the highest point of the peak is in the
example; therefore
5* - 6* = O.-^Jo X 10-2 cm
separ. x=o
The difference In height of the highest point and at point of
separation h ,^ is (compaie fig. k)
h =: 3' X 10-2 cm
separ.
While the variation in h j„ duo to the boundary-layer effect
amoi-mts to a mere 20 percent', -Ehe illustration shows that a change
in height of bump by this amount must be followed by an extra-
ordinarily great change in velocity distribution, so that there
can be no question of attaining symmetrical results in the
experiment.
The conditions in the presence of a turbulent boundary layer
are considerably worse. A little calculation on Grushwitz'e test
series 3 (reference 6) discloses that the displacement thickness
multiplies from the point of transition to the point of turbulent
separation by about 25 times. Assuming turbulent separation at
the point of severe velocity drop the greatest dlsplacaroont effect
(height of bump + displacement thickness) would also exist on a
bump of considerably greater absolute dimensions at the po:'nt of
separation due to boundary— layer growth. It is supposed that the
displacement effect of the body, increased by the displacement
effect of the boundary layer, undergoes no substantial increase
behind the highest point of the bimp. In txirbulent boundary la.7er
and thin profiles or low bumps this is possible only to the extent
that a compressibility shock occurs at the point of rreetly reduced
profile thickness; furthermore, a compressibility shock would have
to occur so much farther downstream as the bvmip or the profile is
flatter. It also is feasible that the effect of the increase in
displacement thiclaieas is raised by strcn;^ return flow behind the
point of separation. Th'^oe qualitative results cen be checked
agair-st the work of G<'thert (reference 1).
MCA TO No. 1189 h3
Tho fact that a compressitility shock can occur when there Is
enough space avallahle for the increaced displacement thickness
caused hy it is to he regarded as reason for the fact that the
separation computed hy stream filament theory in figurG TCt) is almost
exactly coincident with the start of the compressihility shock in the
test.
It may he asked how the streamline pattern in a flow protlem
must look, in order that the compressibility shock he possible. This
can he answered to the effect that the compressibility shock on slender
bodies is to be expected near the point of vanishing streamline curva-
ture. Since tho streamlines in the zone of critical sonic velocity
are approximately parallel, the points of vanishing streamline curvature
must lie near a common orthogonal trajectory, hence, a potential line.
Along it the velocity changes little according to (l;8). In a flow
that differs little from the critical sonic velocity, the free -stream
velocity is therefore to be expected in the vicinity of points with
zero streamline curvature. If the curve decreases rapidly at a place
with supersonic velocity a decrease to tJie outer velocity must be
counted on. The marked velocity variations in figure k coincide with
the streamline inflection points. On flat profiles a point of separa-
tion can be regarded as starting point o-f^ a free streamline with very
little curvature. The streamline curvature must thus decrease very
substantially in the separation point and it is seen that a strong
compressibility shock produces through the separation connected with
it a streamline pattern that favors the appearance of the compressibility
ehock. This argument is therefore not suitable for finding the location
of a compressibility shock.
8. CONCLUDING REMARKS
The proceeding work shows that in a calculation con^forming to
reality the pressure distribution of a body in a flow at supercritical
free-stroam velocity may not be given by the potential flow, that the
boundary layer plays a decisive role here. In general, the potential
flow around the body penaiits not even an approximate calculation of
the boundary layer. This means that in contrast to incompressible flow
the pressure distribution on flat bodies can also be much different.
It is therefore Intended to first improve the process of calcula-
tion of the potential flow with a supersonic region. With the process
we will ascertain the flow around a substitute body. This will have
approximately the same displacc^ment effect that is found on an
experimentally investigated body including its deadwater region and •
1+1+ MCA TM No. 1189
the displacement of foot of its iDoundary layer. \Je can also anticipate
from our calcolation a strong velocity increase at the "body nose and
a strong velocity decrease at the point whore the curvature of the
substitute body disappears.
Translated by J. Tanier
National Advisory Connnittee
for Aeronautics
NACA TM No. II89 ^5
REFERENCES
1. Stanton, T. E.: Velocity in a Wind Channel Throat. Aeron.
Res. Comm. E. & M., No. 1388.
2. Osvatitsch, Kl., and Rothstein, W.: Das Stromungsfeld in einer
Lavaldu3e . Voratdr. d. Jahrls . 19^2 d. deutsch.
Luf tf ahrtf orsch . in den Techn. Ber. Heft 5, Oct. 15, 19^2-
3. Holstein, H., and Bohlen, T.: A'"erfahren zur Berechnung laminarer
Gren^schichten. Lilienthal-G')s . f . Luftfahrtforschung
Bericht S 10 (Preisausschreiben 19'+0. ) , p. 5-
k. Walz, A.: Ein neuer Ansatz flir das Ge3chwindiglceit3prof il
der laminaren Reibungsschicht. Lilisnthal-Ges . f.
Luftfahrtforschung Bericht iHl.
5. Hantzsche, v., and Wondt, E.: Ziim Kompressibilitatseinfluss hei
der laminaren Grenzschicht der ebeiien Platte. Jahrb. 19^0
d. deutsch. Luf ti ahrtf orsch. I, p. 51? •
6. GruBchwitz, E.: Die tiirbulente Reihungsschicht in ehensr
Stromung hei Druckabfal.l und Druckanstieg. Ing.-Arch.,
II. Bd., 1931, p. 321.
7. Gathert, B., and Richter, G.: Mcssujig am Frofil NACA C015-6^ im
Hochgeschwindigkeitskanal der BYL., FB 12^7 .
Gothert, B.: Druclr/erteilungs- und Impulsverlnstschaubilder
fur die Profile NACA OOO6-I, I30, asw. bei hohen
Unterschallgeschvindigkeiten. FB 1505/l~5'
8. Tollmien, W.: Ein allgemeines Kriterium der Instabilitat
laminarer Geschwindigkeitsverteilungen. Nachr. d. Ges = d.
Wiss. zu Gottingen, Math.-phys. Kl., Fachgi-. 1, Math.,
Neue Folge, Bd. I, Nr. 5, 1935, P- 79-
9. Cranz, C: Lehrbuch der Ballistik, Bd. II, p. U52, Fig. 22-
10. Ackeret, J.: Gasdynamik. Handb . d. Physik, Bd- VII, p. 338-
11. Busemann, A.: Das Abreissen der Grenzschicht bei Annaherung
an die Schallgeschwindigkeit. Jahrb. 19^0 d. deutsch.
Luftfahrtforschung I, p. 539-
12. Frosael, W.; Expsrimentolle Untersuchung der kompressiblen
Strom^ong an und in der Nahe einor gew51bten Wand, 1. Teil.
UM 6608 (Abb. 12 entstammt einem nicht
veroffentlichten Vorversuch) .
46
NACA TM No. 1189
*
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NACA TM No. 1189
47
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48
NACA TM No. 1189
I
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I*
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NACA TM No. 1189
49
Streamline with the asymptote y = 0.6
Streamline with the asymptote y = 0.8
Streamline with the asymptote y = 1.0
Streamline with the asymptote y = 1,2
Streamline with the asymptote y = 1,5
/ 7. 3 L^ X
Figure 4.- Velocity distributions over various bumps.
50
NACA TM No. 1189
-
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NACA TM No. 1189
51
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52
NACA TM No. 1189
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