Theoretical analysis of stationary potential flows and boundary layers at high speed =


Material Information

Theoretical analysis of stationary potential flows and boundary layers at high speed = Theoretische untersuchungen über stationäre potentialströmungen und grenzschichten bei hohen geschwindigkeiten
Series Title:
Technical note / National Advisory Committee for Aeronautics ;
Parallel title:
Theoretisch untersuchungen über stationäre potentialströmungen und grenzschichten bei hohen geschwindigkeiten
Physical Description:
59 p. : ill. ; 28 cm.
Oswatitsch, K
Wieghardt, K
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Gas dynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: The present report consists of two parts. The first part deals with the two-dimensional stationary flow in the presence of local supersonic zones. A numerical method of integration of the equation of gas dynamics is developed. Proceeding from solutions at great distance from the body the flow pattern is calculated step by step. Accordingly the related body form is obtained at the end of the calculation. The second part treats the relationship between the displacement thickness of 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 behavior of subsonic and supersonic flows. Lastly, the decisive importance of the boundary layer for the pressure distribution, particularly for thin profiles, is demonstrated.
Statement of Responsibility:
K. Oswatitsch and K. Wieghardt.
General Note:
"Report date April 1958.."

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University of Florida
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Resource Identifier:
aleph - 003604221
oclc - 71056818
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Full Text
IACA- ~rv\~1\~;9





Br K. Oswatitach and K; Wieghardt

The present report consists of two parts. The first part
deals with the two-dimonsional stationer,. flow in the presence
of local supersonic zones. A numerical method of integration
of the equation of gas dynamics is developed. Proceeding from
solutions at great distance from the bod:, the flow pattern io
calculated step by step. Accordingly the related body form is
obtained at the end of th. calculation.

The second part treats the relationshipo between the dis-
placement thickness of laminar and turbulent boundary layers
and the pressure distribution at high speeds. Tha stability
of the boundary layer is investigated, rezulting in basic
differences in the behavior of subsonic an., supersonic flows.
Lastly, the decisive importance of the boundary layer for the
pressure distribution, particularly for thin profiles, is demon-



p pressure

P density

T absolute temperature

K ratio of specific heats


*"Theoretische Untarsuchun6sn 'ber a tationare Potentialstr'dmungen
und Grenzschichten bei hohen GeschwindicL.-iten." Lilienthal-
Gesellscheft fur L-ftfahrtforschung Bericht S 13/1.Teil, pp. 7-24.

NACA IM No 11U89




U, v



Ma = w/c

F = 1 Ma2

coefficient of friction

velocity vector

magnitude of velocity

velocity components

velocity potential

velocity of sound

critical velocity of sound

Mach number

0 = v/p stream density

Re* Reynolds number of the displacement thickness

6 boundary-layer thickness

5* displacement thickness

a momentum thickness

f stream filament section

r radius of curvature of the stream line

N normal to the stream line

H maximum bump elevation

R related radius of curvature

Subscript n refers to the conditions in the free-stream region,
subscript a to the outer flcw, subscript v to the wall. Subscript
m refers to the chamber or drum values in the phase quantities and in
the velocity to the highest obtainable value. The quantities of state
in part I are made dinensionless b-y the chamber quantities and the
velocities by the highest velocity obtainable.

TIACA TM No. 1189



The equation of gas dynamics is derived by means of the energy
equation rather than ti6 adiabatic equation js customary. A simple
formula is obtained rcr the stream density which is valid in a guide
range about the critical velocity of sound. By applying this formula
a simplified equation of gas dyiemics is derived which in the tran-
sition zone frcm subosonic to eupei'scnic, for sm-ll velocity compclients
v, describes the prccecses very acc.urateiy. Lastly, the problem of
flow around a crl.inidricsl bod:., rLrEtLi'j'cal in two diir-ctions, is
analyzed. It is found tnat, from 1 certain flow velocity on, located
above the critical -velocity, no nia.:maim velocity can occur at the
point of moxiimum bouy thicknoss.

In the de-:ription of a gas flow the mozt generall ca-e involves
six unrlnon functions, namelY', the three com.pon.snts of the velocity
and the three pihse quanti.tins of the gac, the preais-sure p, the
density p, and the absc'luto tanIp .-ra'ture T. The cluation of state
of the gas permits the teiippeiat.,-re to be exprezo'e. in terms of the
preessre and aensity, th s lcav'r:.3 five unk-inoxin f..nctlons fLr the
calculation of which the three Euler equations .nI tlhe ccntlnuit,:,
equation :re avai l.-blu, For ti'e mi.ss5ng eqtiat-'cn it is customary
to use the auiauatic curve to eliminate the pr'. ani density from
the equation ansi so arrive at an eyiration brt\ween the v;olocty com-
ponents, that is, the so-called eo'ut'tion cf gas .jnamics. IHowever,
it appears to be uaknmr.-.n that for th? d:lri-ation of the equ-'.tion of
gas dynamics tf.- asu:lpt'on of thie alabatic is not necessary at all,
but that the use of the c-ne-;,rgy th.3nrr.l itself i, sufficient. This
derivation Is brieily carried out in the, while havIng
recourse to the vector sa lthod.

Pressure, den..:ity, .ind tenpe'atu.ire ar-e iialo nconaimensicnal by
the corresponding "cn-mb-r qua,,tituie" p pm, Tm; that is, the
quantities of state at verloc'ty O, and all o.-curring -.'locitie.-
and velocity components by the maximrir: obtalnaile velocity, that
is, the velocity at pressure O. With iencting the ve -a-ity
vector, c the aseci'fic heat at constLnt .irS i.ce, ana r: the
ratio of the specific her;t:. this v.-locitly

S= 2,' Tm -1
pa r)

4 NACA TM No. 1189

The equation of energy of an ideal gas In stationary flow has
expressed norimenslcnally the following simple form

2 =1 t(1.1)

and the corresponding continuity condition reads

div w + grad 0 = 0 (1.2)

The oft repeated quantity
n = x 1

has a physical significance; it Indicates the degrees of freedom
of a molecule. For air n = 5 is very exact.

The Euler equation is then written as

ad rot xn = (1.3)
Grad w / (1.3)

The quantity xn enters the equation through the nondimensional

Pressure and density are eliminated by forming the gradient of
the energy equation,thus obtaining

0 = grad w2 + 1grad p + (w2 1) 1 grad o
0 0

Scalar multiplication by w and application of (1.2) and (1.3)
gives the equation of gas dynamics

(1 w2) div v + nw grid 0 (1. )

This equation is written here in a form where the velocity of sounf
is already eliminated and only the flow velocity itself is present.
We will use this equation in the next section.

For an insight into the Dotential flow properties in the speed
range of Ma = 1 the just derived equation is much too complicated.
So the processes in a flow filament are analyzed, unsteady variations

IJACA TM No. 1189

and friction processes excluded so that adiabatic changes of state
can be assumed. The enerr y equation (1.1) then affords a connection
between velocity arnd density, and the stream density wp can be
represented as a function of the velocity w:

0() = pw = w(l 2)n/2 (1.5)

It is known that this function reaches a maximum at the point where
the velocity is exactly equal to the velocity of sound. This
particular point is generally denoted as the critical speed of
sound c*. "ith f as the section of a flow filament the continuity
equation reads f9 = ccnst, The speed c* is there ore characterized
by the fact that a fl.o filament for Jhi. valuo w reaches a
smallest pcsciblc crosz section. For w> c* -- for w< c* the
flow filament section is .:reater.

Less familiar Is th.- ~rm ellnes. of the otr-am density changes
over a very subs-.tantial -peed ranse. To indicate it -,. is
represented for x = 1.40 In the ran6e of 0.5c* < w < 1.gc* in
figure 1. Q.ELuatity '"* denotes 'he vai.e of for w = :*, the
same applies to the derv:c'tive: ci .'. Trhi characteristic of the
stream density .s of decisive significance for th: effect of the
boundary layer -,n the flow, av3 vil]. be shl u- elsewhere.

Consider the fiction C in thr- vicinity of the maximum
developed and signify it-i d--rivative vith e*W, *, ., etc. Nov
it is found that the parabola

S- c2 -i(1.6

is already sufficientl seccu.rate for a wide seed ran.-e. This
approximation is indicated by dashes in figure 1. The calculation
for the coefficient of the quadrstic term livess the simp e result

1 i+2 1w + ] n+1 (1.
2 -= -..

The equation (1.6) serves in good stead for the derivation of a
simplified dynamic ges equeti.on for twc-dime-si.nal on the
limiting assumptions that the y component of veloc-.t- w,
signified by 7, is small comLpared tc the vel.ocit, of sound i'. that
u, the x component of the velocity, does not differ too much
from the velocity of esund. The rtrcar1 up can be

NACA TM No. 1189

replaced by up = e(u) and the equation of continuity (1.2) on
applying the same omissions as effected with respect to the terms
with the factor v for the derivation of the Prandtl law, can be
written as

(up) + 'U + = 0

The coefficient of depends only on u; it is simply a dif-
ferent method of expressing the wel..-known quantity 1 -.Ma-.
Assuming, aside from the smallness of v, that -Pu can be
regarded as constant results in the Prandtl--GL.u"'-t 'vnaeloy.
If this coefficient were plotted against u in the vicinity of
the sonic velocity,it would show that it can a-sume 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 u changes signs at
sonic velocity, as Jeen from figure 1. The variations of u- 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 law u- can very well be
put equal to this quantity in the free-stream region, but not for
eu. This quantity is computed by (1.6) and gives

-. ei. + .. --) --- i =O
So c- ^ ox e' / y ck

The subscript o denotes the quantities in the free-stream region.
Using the notation

U V (e + 1) V y 6 (o 9 V) (1.7)
c i) c -)' v 0 c

gives for 1 and -1 the simplified equation of gas dynamics

Iu ,v
U -= 0 (1.7)
ix Cy '

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), because it combines the simplifying assumption of small v

NACA T' No. 1109

with a very accurate description of the processes in the critical
sonic speed range. This wa- the reason foi the brief 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 bodies, as will be demonstrated
for the case of two-dimensional flow pa;t a body that is symmetrical
about two mutually perpendicular axes. The flow direction is to
be a-ong one body axis, that is, the aigle of attack equal to zero.

The flow is to be adiabatic and iriotational, the latter
characteristic being expr?3s2d by

w w w
7N =-7

where N is the no-mal to the streamlinee and r its radiuz of
curvature. The sign for N is so cho- .n that it is positive when
the normal points out from the radius cf curvature. Equation (i.8)
holds exactly for a.l two-dimenrional potential flow. By the
continuity condition in the form,

f = Ccnstnnt

and the freedom from rotation (1.e) the flow is completely defined.
The origin of the coordirnte system x and y i: placed in the
center of the body, axis x iL made ccincident with the flow
direction (fig. 2). and the r' or -C ts't~v ;-, 7vaue naL:'rzed.
The cylindrical body is visualized as exposed to a flow
velocity which Jeads to the formation oL a supersonic zone near
the thickest part of the body and it 1 as.oumed tnat in every
stream filament the maximum velocity 1i reached at the point
x = 0, an assumption which certainly should be fulfilled for
subsor-ic flows. A point on the y--axi. with zupers-onic opened
must have a maximum btroam filyment width, a point with su.bsonic
speed, a minimum of stream filament width. In the sLper.ornic
region the curvature cf the streamlines on the positive portion
of the y- axis must deciense leso rapia.i' than on concentric
circles, in the subsonic zone the stream line curvature must
decrease more rapidly than for; concentric circle.. Hence no
great error is introduced when in the vicinit; of the point on
the y- axis where donic velocity is reached, the etreamlineu are
replaced by concentric circles, and it will not lead us far astray
when this is assumed up to a value of y equal to twice the
distance from the cylinder of the point with the critical sonic

NACA T4 No. 1189

velocity c*. After the streamline curvatures are approximately
known the velocity distribution on the --axis in this zone is
completely defined by (1.8). If the piece which the body cuts off
from the y-axis is denoted by H and the radius of curvature of
the profile on the y-axis by R, its velocity distribution is

u R
UY__ -F- -H+ Y (1.9)

According to (1.6) It may be stated that 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:t; 'ul ~' oint the inequality

u < u < 2c* uo

is fulfilled. Since the velocity distribution for x = 0 is
defined by (1.9) up to the constant uy_, it also is the
difference in through flow v-nlurne for > H T.1, the free-stream
reaon and on the y-axis. It may now be asked at what value of the
constants the absolute amount of this through "ow d..'rferencL reaches
its highest possible value and the answer is found in the fairly
accurate equation

u=g = 2c* u (1.9a)

that is, that the stream density on the .---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) is applicable up to the attain-
ment of speed uo and that from this y value on, the constant
flow velocity prSvails. This break may occur at the value y = y
for which the equation reads

u = u for o R -

As near the body more can flow by than on a strip of equal width
in the free-stream region, because of th3 increase in density. we
must proceed from the cylinder only as tar as the free stream is
displaced. The result is therefore a highest possible value of H,
denoted by Hmax, which is given by the equation

oHmax = (e 0) dy (1.10)

NACA TM IHo. 1189

The integrand is given by (1.6), (1.9), (l.9a), H is to be replaced
by Hmax, since u is equal to w on the y--ax-.s. The evaluation
of (1.10) gives the following relation between flow velocity and


- 0.70 0.75 0.80 0.85 0.90 O.Q5 1.00

-e 0.053 0.026 0.013 0.009 0.0020 0.0004 0

The extension of the speed by pieces at y y unquestionably
introduces an error; but it can only callse a shift in Hmax, while
not changing the existence of such a va't-e. In the subsonic zone a
streamline may be re.7?rded a.s bnimp and the residual rise in
through flow volume iup to inrreass in .-.locity corputqd by in approxi-
mation process that applies in the subcs-nic range. The result then
is a finite variation o' the inte&-rl in (1.10) and a corre.spcndi'ingly
different Hmx. The possibility cf a a3red increase in ; direction
in the subsonic region must be rejected, as it would invalidate the
present considerations. Hence it is seen that thhe assumption cannot
be applied to all bodies and therefore .he fo'.lo'-in,-. principle:

To each flow velocity u,, there corresponds a definite ratio
H /I. If the ratio H/'R exceeds this l:iit for a body symmet-
rical in two mutually perpendicular directions and lying along the
flow direction, there is no flow for whlic: velocity ma: im'.ums can be
reached on the entire y-axis.

It must be expected that. the maxyim-n, speeds on the y-axis
disappear or-y in chA supersonic rnnge. But since it cannot be
assumed that velocity maximums in the sicnrsoni' range disaspear
on a part of the y--axis while a velocity maximum appears on the
body, we are led to the following principle.

From a definite value of H/P. on, for bodies and flow directions
of the described type, there is no flow at which a speed maxim'n with
local supersonic zone is reached at the noint of maximum thickness
of the body.

A boundary point for these specific values of H/R is given
in table 1.

In the subsonic zone this principle has no analogy.

NACA TM No. 1189



A numerical 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 rr peculairities.

Limited to two-dimensional, irrotational flows with w = grad
past a cylindrical body, equation (1.4) L:iv3s for the velocity
potential cp a nonlinear differential equation of the second order

D = -(n +1) -'

S- (n (2.1)

x y ox y 1)%

The zero joint of the coordinate system is placed in the body, its
dimensions are of the order of magnitude of unity, and the flow
strikes the bod, along the positive x-a:is. The boundary conditions
for then read:
= 0 at the body itself, IN denoting the normal, and at infinity
for z x + y -ja r----O and --- u = dimensionlesss) flow
yx ox 0

On passing thriourh the local velocity of sound ,
n +
equation (2.1) changes from the elliptic to the hyperbolical type.
For this case the -xact integration has b.-on successfully secured
for single specific examples only. For the subsonic range several
general approximate solutions are available, the sim-lest and best
known of which is the solution I. obtained by the Prandtl rule.
This satisfies equation (2.1) better as the body becomes more slender
and the distance from the body becomes greater.

NACA 1T No. 1189

The following method is therefore indicated. Compute the
Prandtl solution 0p, for the entire flow and attempt to secure
the correction cp in arch a way that ) + c = 0 becomes a
solution of the complete equation (2.1). As the analytical
calculation of q is too complicated,a numerical method is
advisable, starting from the outside (z > 1) where = z 0,
and progressively continuing Inwardly toward the body. The
exact body shape follows at the end of the calculation from
the streamline distribution; however it is to be suspected that
it ossontially remains similar to the form of the Prandti solution.

For this purpose the differential equation for p = : _
is set up; Q is an exact solution of the considerably simplified
equation (2.1:

o 2 2
!1 -Mno) &2 + 0

Ma, = = n uo (2.2)
o 1 Uo

The subscript c refers to the condition at Infinity, (p
fulfills the complete equation (2.1) up to an error ep which
can be computed by metns of (2.2):

D[J "=2(Ma2 +n
L n)

\ ^y/
+ (n + 1) Ma2n (2 Ma2

p 2,-
S bx (2.3)
ox Cvy Bx ,V

Putting = 0 + 9c in (2.1) and regarding it as a differential
equation for cp, p follows as term of zero degree in p. As
<'P ip in the entire range, it is assumed that it applies to the
derivatives as well. Merely the terms of the zero and first degree

NACA TM No. 1189

and the greatest term of the second degree in p need to be Included;
There results

- (S

-(n + 1) -) -.
0y) 6y2

- (n + 1) ()2


ox- dX

1) Maoa -



- n '---
ox -,y

n -7-p ---
- d)7- cl

+ n 32
cIx cy

3x Idy

= -P

+ 2 KMea2 + n)
0 /

S2 (n

+ 2n

"y ex 6y

+ 2 (a + )

6p ?cp


The better satisfies the equation of gas dynamics, the
easier is the determination of 'p. Cc, at great distances from the
body the equation can be substantially simplified. For z >> 1,

especially on slender bodies, --~ --; hence we can put
cy x

and = 0 in 2.4 but in contrast to the Prandtl rule

consider the variation of Since p 0 for z->*m, the term
of the second degree is emitted also. And equation (2.4) is simplified




- Fo + F'



1 --(n 1) --

,,1-- ( i
,^ .l-l^. ---- ---- Z
w -1

y 0

c .p .-
1 ).) Zp

NACA TM Ho. 1189

Fd ----

This equation could also be
u .
vlation for -u

2n I ,

derived from (1.7);

F is an abbre-

The boundary condition for z -> is p = 0 where the Prandtl
rule applies exactly. On the other hand, however, the disturbance
of the flow by the body is very extended when the flow velocity
approaches sonic velocity. It is thercfo.'a necessary to determine
an initial approximation for p analyti. flyr so as not to be com-
pelled to start at unduly great distance'. For this purpose (2.5)
is transformed further. While Ma = Mao, by the Prandtl rule, hence

F = Fo, the more exact term F = Fo + F'. u so that

Fo 2? 32?
F -
SX- o*-

= F'

I 4-


- u + -1
/ CX .AJ


There --- can be ignored with respect to and it is assuim
-c x2
that x- x u at great distances; for example, in the cal-
culated case: For a parallel flow and dIpole, uo dies dowi

1/z2, but e as 1/z4. Hence finally:


4) (F 1. F) --
6x 1 /


i as

(2. 10)

All the equations for the process are nc: available. The general
process of calculation is as follows: First determine the entire
field of flow of the incompressible fluid numerically, and then the
compressible approximate solution by the Prandtl rule for a fixed
Ma number of the flow velocity. The correction cp on a strip for
great y follows from (2.10). Frcm here on q' is computed
numerically, step by stop. At great dictz'i.ce from the body we

NACA TM No. 1189

therefore first use equation (2.5) and later in the neighborhood
of the body the more complete equation (2.4). Having thus
determined cp for the entire field, the stream lines and there-
fore the body contour itself, as well as pressure and velocity
distribution,are obtained from $ = p + p. Naturally the process
can be built on another approximate solution; however, the formulas
probably become simplest when the Prandtl rule is used.

For the present the range of application of this method is
confined to the flows where the Prandtl rule affords a good
approximate solution and the fundamental assumption p << pt is
actually fulfilled. p

Excluded are accordingly flows around not sufficiently slender
bodies, as well as areas in which the velocity of sound is sub-
stantially exceeded. This also manifested itself in the calcu-
lated example. Flow around a cylinder (circular in the incompressible
case) at Ma = 0.7454. The calculation was perfectly smooth into
the supersonic range, where it had to be broken off especially
since -c and -- quickly rose to the order of magnitude of --
ox 8x 6y ox
and but exceeding the sonic velocity itself involved no
8x 6y

In principle we can also free ourselves from the approximation
that p << -p, when in the formulation of equation 2.4 we consider
terms of the third degree in qj; the length of the calculation,
however, becomes disproportionately large. In another more appro-
priate method the assumption p << OP is emitted and the tedious
calculation of Sp in the entire field of flow is eliminated. The
previously described p method is utilized only for computing the
initial values for large 5. The new method is as follows: 0p is
evaluated at large distances from the body, for y >> yl; where yl
is chosen so large that the error of the Prandtl solution is suf-
ficiently small; p is evaluated from equation 2.10. This affords
the exact solution of the dynamic gas equation D = p + p in an
initial strip. From here on $ itself ic calculated step by step.
The width of the initial strip from -xl to + x21 must extend
upstream and downstream from the body so that for all y at
x X1 and x > x2 the Prandtl rule is applicable with sufficient
accuracy. From yl on, where $ is then known,
620 1
-- is graphically extrapolated to yl y for certain
6y2 2

'For bodies which are symmetrical relative to the y-axis
also, xl is naturally = x2.

NACA 24 No. 1189

fixed abscissae x (Ly is the length of the step; it can be
assumed quite large at first, and reduced again later in proximity
of the body). Next

"71-y 1 : l'

is plotted against x and graphically differentiated, which gives
-2- Plotting I- against y and integrating gives

yC -Oy
6 -Y'1
y-1 y1ly
1 = --~+ y
ox) ox J ,- oO:
Y 1. -AY Yl Y1

Lastly the variation of f over x fieldsds by graphical
Yxl ox
differentiation. Uith it

,- 6 62 and
x' Oy ox' x2
are known for y = y Ay.

From the equation of gas dynamics (-.l), in which the simpli-
fication 0 can be made so long as it is valid that
<< 6 -). is calculated for the required x values and plotted

against y. Then the calculation is reported, -2 extrapolated
for yl y and so forth.

If the values of computed for ; > yl, the value
extrapolated at 21 2Ay and that comoited for yl Ly 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

NACA TM No. 1189

of the curves x and is uncertain at the
y=const. y=const.
boundary points x = xl and x = x2, it is advisable to compute
D = -p + q also in two vertical strips x <- xI and x > x2 and
to join the progressively defined points to these edgp strips.

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 alon3 x. is sure to be
greater than almost everywhere, especially in the supersonic
zone. Thus at entry in a supersonic zone the coefficient of -
in (2.1) goes through zero. This does rot interfere in the above
method since (2.1) is used for computing -y.

If, however, we integrate in the x-direction and solve
equation (2.1) for L- then difficulties will result. The
coefficient of --2 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 with
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.


A flow symmetrical in x and y is computed for a Mach number
of flow of 0.7454. 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

The described method is tried out on a very simple example;
we start from the incompressible flow subscriptt i) past a circular

i = xi + (3.1)
xi2 + y2

NACA TM Io. 1189

the free-stream velocity and the radius are taken as unity. Pr.ndtl's
rule is applied to a fixed Mach number of flow Mao to which the
dimensionless flow velocity

uo i/

I 2
Ma + n
= 1,13.0 + n

corresponds. The abscissas for this transformation are contracted

by \ Mao2

x x= 1 a o2 Xi


The ordinateh remain the same: y = y = yri so that


-a0 i
. .TMac ,i

To compute p by equation (2.10) the coordinates xi
are used, so that

xc2 y2 dx'2
X y d x


and y = yi


Uo ox
M1 2 xi 2

1 + Ma 2J

Development of the right-hand side for

using equation
S=- and

1 (n + 1) uo2 (,i/3xi)2

u i 2 i)2


la..-ge zi xi- + Y- by

(3.1) gives for the first ;?poroximation, when

= 2--=- 1 -

c =
3x = uo


(1 Ma2)

NACA vT No. 1189

= Mao 4 ij
nuo 1 Mao2 Z

12Mao Xi

nuo %1 Mao2 Z1

- 6i + 892)

10,o 8 2\
-3 s+ b,

A particular integral of this Poisson equation is obtained

with the help of the separation formula
general solution is written

S=-- -f ( ).

-12Ma4 Xi
nu 1 -- -
nuo l Ma2 Zi

with tApot = 0 and c arbitrary.

As boundary condition for p the sile requirement is that
it shall be small compared to Op, that Is,decrease more rapidly
than q. But for the rest ePot end c can be chosen at random.
The physical meaning of this ambiguity is as follows: Owing to the
disregarded terms of higher order in ths formulation of (3.5) only
the effects of the first order of the bod;. at great distances are
taken into account. But these are the saze for different body
forms. So the calculation yields different section forms, depending
upon the choice of c and ppot. The manner in which c and
Q ot affect the body form cannot be evaluated until several examples
h6e been worked out. Up to now only one such example has been
worked out, owing to lack of time.

The Mach number of flow Mao = V5/9 = 0.7454 had been
specifically chosen. The dimensionless velocity is then
Uo = 7/10 = 0.3162 and x -E p = xi.
In (3.6) only half 9 ot 0 and c = 1/8 were assumed for
simplicity, thus eliminatiRg the linear term in ;

Lp(xi, y)


Thus the


c -- + Qpot

NACA TM No. 1189

Sr (3.7)


with this p the velocities (and their derivatives) of the exact
solution 6 = Op + rp for y = 10 and 0< x< 6, as well as to
x = 6, 0 < y< 10 were computed and 0 was determined for
y < 10 and x <6 by the described step-by-step method. In view
of the symmetry of flow relative to x and y the calculation
in one quadrant was sufficient. The step length 4y up to y = 1.5
was ay = 0.5; from there on 0.2.

While the exceeding of ths sonic velocity (first at y = 1.35)
caused no difficulties, the calculation could not be carried out to
the body because of another reason but hl.d to be broken at y = 0.6.
For at x 0 0.6 the horizontal components of the velocity
Changes so rapidly for smaller ordinates y that the graphical
differentiations became too uncertain tD compute the next step.
As is seen from the contour of constant velocity (':i. 3) a
steady but still very sudden rarefect.on occurs and on a point
symmetrically situated with reference t- x = 0 a compression
occurs. This phenomencn would of course not be pl&in at a lower
flow velocity, but it is certainly characteristic of the flow in
proximity of the stagnation point where the speed increases quickly
from subsonic to supersonic. For this point of the flow field
another method must therefore be developed.

So while unable to obtain the flow around. a finite body
with a stagnation point, the data obtained thus far are nevertheless
very informative for subsonic flows with supersonic zones. The
calculated streamlines and lines of constant velocity are shown in
figure 3. Visualizing, the lowest streamline in figure 3 as rigid
wall, we get the flow along a smooth buip with a supersonic zone
near the highest point. Since this streamline is already very
steep for x" 0.55, it can be assumed that the velocity distribution
around the finite body (with s;tmmetry axes x = 0 and y = 0.5)
indicated in figure 5 is fairly accurately reproduced by the dotted
line. Incidentally,it is notad th&t even Prandtl's rule yields
considerable errors near the stagnation zint.

Figure 4 shows several streamlines maznified five times in
elevation, along with the respective velocity distributions. Not-
withstanding the similarity of the individual peaks the velocities
differ considerably at various places. The velocity and with
it the pressure distribution of thin bodies is therefore at high
flow velocities markedly dependent upon the exact shape of the body.

NACA TM No. 1189

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 set up in section 1 according to which the highest
speeds under certain assumptions do not occur at the point o?
maximum thickness of the body. Even the equation (1.9) 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 curve w(x = 0)/c* at
w/c* = 1. From the far-reaching agreement of the curves it follows
that in the vicinity of w = c* the expression u = w(x = 0) = a
b +y
13 a good approximation. With this example the accuracy of table I
can be checked. In view of the flow velocity of uc/c* = 0.7746
Hp. /R = 0.019 would have to be expected according to this table,
bu by the calculated example it is proved that from H,,a/R = 0.031
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 hy,/R. The difference is attributable to the
fact that the hyperbola used for the approximation gives too low
speeds in the subsonic range.



Studies of the behavior of supersonic flosv in parallel
channels disclose that in the supersonic zone,principal flow
and boundary flow are in unstable equilibrium in certain circum-
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 stationary flow around
a circular cylinder at a high but still subcritical Reynolds
number. Computing the pressure distribution at the body with the

NACA TM No. 1189

aid of the potential theory on the assumption that the cylinder has
no dead-air region behind it and then calculating on the basis 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 corresponding to reality when the
dead-air region is so assumed that the related pressure distribution
yields separation exactly at the starting point of the free stream-
line. Thic example shows that potential flow and boundary-layer
flow usually depend upon each other. In general, we can say that
the potential flow determines the bounda-.y-layer flow, also that
the boundary-layer flow deter-ines the potential flow. The former
can be stated with great approximation in flow without pressure rise.

It is a knonm 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 pressure rice, which in turn favors
a stronger growth in boundary layer. In contrast to subsonic flow,
an unstable equilibrium of boundary layer and principal floor is
involved in this instance and a very considerable boundary layer
growth must be reckoned with in certain circumstances. It may,
in a straight channel result in a sudden ctrcng pressure rise at
the flat wall and so in the formation of a dead-air region (fig. 7(a)).
(Compare reference 11.) If the pressure rise is so great that the
flow becomes subsonic, the relation of main flow and boundary-layer
flow is stable again, the dead-air space cannot 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 hava to revert into the
subsonic range. It is therefore to be a-suxmed that at an oblique
compressibility shock, as met with in fiLvre 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 formation of dead-air regions even at
a flat wall.

NACA TM No. 1189

A disturbance of the unstable stato of equilibrium of principal
and boundary-layer flow in the supersonic range is favored by the
fact that any minor disturbance in a supersonic flow is propagated
undamped along Mach lines. Thus, a pressure rise in a supersonic
tunnel can be dispersed by a small disturbance far upstream; on the
other hand, the pressure rise sets in again some distance downstream
as we can also infer from our example.

The unstable behavior of the boundary layer in the supersonic
zone must disappear when the principal flow approaches sonic
velocity. In the critical speed range w = c*, which is of particu-
lar interest in flows past bodies with high speed, the fact stands
out that this is the range of maximum flow density. But the proce-
dure in computing the incompressible flow past an airfoil is such
that the pressure distribution is obtained from the potential flow
without consideration of the displacement effect of the boundary layer,
and then the boundary layer is computed with the aid of this pressure
distribution. This is not permissible however in the region of the
velocity of sound, because a minor variation in stream density e
exerts a very substantial effect on the speed. This is readily
apparent in figure 4, where peaks with comparatively minor form
changes produce very unlike pressure distributions. 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 simple example, which,
although it involves no flow problem, is nevertheless informative
for the appraisal of the displacement effect of boundary layers
at high speeds. The velocity distribution in the nozzle used by
Stanton (reference 1) for his experiments was :omputed by appli-
cation of the simple flrw filament theory, once without 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 section. The initial
value of momentum and displacement thickness at x = 0.20 was
estimated. The dimensions of the nozzle are so small that it can
be assumed that no turbulent transition takes place. Stanton's
test series C is illustrated in figure 7(b). The velocities were
determined 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 filament solution with and
without nbundary layer allowed for are included for comparison.
The former was computed only as far as the separation point. It is
seen that the aPymmetry 'is reproduced qualitatively correct by the
flow filament solution with boundary layer taken into account.
The displacement thickness at the narrowest point of the nozzle is
not quite 2 percent -f the nozzle radius. Computing the velocity

NACA TM No. 1189

distribution for the same nozzle in incompressible flow with and
without consideration of the boundary layer, the results in both
cases are essentially even lines. Even aL speeds about 15 percent
below 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 distribution of the displace-
ment thickness from the stream filament solution and then a new
stream filament solution taking into account the calculated displace-
ment thickness is proof in itself that ouch an iterative procedure
is prrmissable at very high subsonic speeds. Displacement thickness
and stream filament solution are obtained step-wise at the same time
in the downstream direction.

The influence of the boundary layer on a submerged body will be
handled in section 7. Our example, however, shows that we cannot
hope to obtain results that 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.



For more accurate calculations on boundary-layer effect in
flows at high speed, formulas for the variation of the displacement
thickness are necessary. 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 cci:pressible 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 temperature variations.
Thus, if sonic velocity prevails in the outer flow the relative density
variation inside the boundary 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.

NACA TM No. 1189

So as not to exclude the possibility of compressibility shocks
beforehand, the flow outside of the boundary layer is called outer-
or-principal flow instead of potential flow.

Being primarily interested in the behavior of the displacement
thickness, the behavior of other quantities is studied only to the
extent that it appears in the result witliout 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.
.ess dezivrd. from it.

The process is based upon an improved Pohlhausen method in
conjunction with the reports by Bohlen referencee 3) and Walz
(reference 4).

Boundary-layer equation and continuity condition for the
stationary case are written as follows

u u dp 8 / u
pu F + pv y = d+ 7y
x (5.1)

(pu) + T (pv) = 0

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 ply Jj i 'artesian coordinates in the following;
A is the friction coefficient dependent on temperature. The
quantities p, p, u, and v are not made dimensionless. It further
is assumed that the boundary-layer thickness 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 Karman momentum
equation which is written in the form

d- (au + PaU 5jd ( (5.2)

(-.2) s. s Lc iLeTTr an G M'PF 'S
A,j TM..11, "t (10.3.) s" a Sp>ii cs. E: (CCA

NACA TM No. 1189

The subscript w indicates the values at the wall, subscript a
those in the outer flow. In the momentum equation displ.coment
thickness and momentum thickness are defined, by

*5 i( -s 7;u \ u u

where 5 is the so-called boundary-layer thickness which is now
chosen so great that 5' and :* can be regarded as independent of
5. The displacement thickness has the hiysicsll significance that
the through flow volume in the bouindiry la:er is reduced by an
amount that, on the assumption of pure yp.tentisla flow. is equivalent
to a shift of the wall by piece 5' in positi'r3 y direction.

Equation (5.2) .can also be given the form

u 2 2 du + -2 2 /a

Aside from the usual assunrtions with the aid of which the boundary-
layer equations are derived, no restrictions of anR kind have been
made so far. Only the problem without heat transfer cn the wall,
the so-called thermometer problem, is tre-ted in the ollciwing..
Further we will make one .approximation, b: which we will specify
the form of the bomudary-layer profile b-; only one parameter in
addition to the Mach number of Tne floe. Next It is
necessary to make an assumption concernin- the configuration of the
density profile. Having se3rn that in the vicinity of the critical
velocity the velocity profile especially ,ioht be decisive, while
the density variation is unimportant, the case of laminar boundary
layers is limited Go Lhe assuimption that the tempsrotur.) at the
wall always attains the tank temperature Tm and eas.sfies the
energy theorem within the boundary layer. This ties in .lso the
assumption that temperature and velocity boundary layers are
of equal thickness. Accurs-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 temperature variation as

NACA TM No. 1189

1 -/
0 Ta Q'Imi
P T--
pt, -c 2


The parameter for the boundary-layer characterization is derived from
the known boundary conditions which for u = v = 0 in the boundary-
layer equation leads to

+ P -

= P aUa

Since the internal friction is only dependent on T and no heat
transfer takes place, = 0, hence with application of 4
t s a, Y

* =
L /'^ "

O'- dua
Pw d4

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 4). The parameter k* differs from the
conventional Pohlhauson parameter by the factor -2 the density
refers to that outside, the internal friction to that at the wall.
For a class of velocity profiles, such a- for M = 0, for instance,
the individual profiles which are represented by the magnitude of
parameter ~* the quantities and can be taken as
S t
function of the parameter %* from the class of profiles, hence

d Pae2
obtain us by (5.2a) simply as function
Ix II,

of %*. Choosing

the Pohlhausen profiles as profile class 3Ives the curve of Bohlen-
Holstein (reference 3) in figure 1, vhile the Hartree or Howarth pro-
files result in Walze' curves of figure 1. Moreover, it is not
necessary 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



OL ) ;

6_y (Ry~

NACA T1 No. 109

of nur ignorance in the sphere of comcressible velocity prcfiles
two known facts are tak.n advantage of: Firzt, we Jnow that the
velocity profiles on the plate aL constant pressure are not verya
closely related with the 'Mach n'rmbrs of the outer flo'r (reference 5).
Therefore, this is assumed to be the case in retarded or acce-lerated
flow also; secondly, we know that the single pa-ametric riathed in
the incompressible, which utilizes the Pohlhausen profiles, leads to
fairly practical results, althcugJ the ?ohlhausen profiles themselves
do not represent the actual profiless very well. An e:-xct represen-
tation of the velocity profiles thernel-ves is not needed, the main
point is the displacement thickness for which the integration over
the profile form is already accomplished.

On the basis cf these arcuirLents we therefore select, independent
of the Mach number, the Hartroe profiles f'nr the velocity distribution,
which, as e rule, p:-obably r.irzsent the in7om-iressible boundary
layer, best of all. The density variation is then (iven by (5.!,) as
function of the -.elocity distribution and owing to th' presence of
quantity -- as function of the Mruch nLrnuber.
By a numerical m sthod th3 quantities -- and L are
u -17 u./
i j /w
then obtained as functiorn.i of and Ika, as exemplified for the
'ollowinr Ma which correspond to the -- values:

si 0 1 I 1 l. .O0

\- o.,'. 0 .hTB O.57 07.'5 07

L I I -

We obtain the c:-rve system. cf fi-ur'e S.

The curves are shown at the left as tar as the separation
point, at the riCht they proceed to the point .p to which the
Hartree profiles are calculated. Several ci.rvs 'ere extra .clatd
beyond it, and indicated by dashes. The CLoav? la = 0 is identical
with the Hartree curve from Walz.

Thus with the velocity distributica. the variation of the Mach

number of the outer flow, and an initial value of the ouraieter

A* can be formed; with it and observirng Mi the quantity

NACA TH No. 1189

d apg2 Da P
u d a can be taken from figure 8 and the variation of --a
a CL 1w Pw
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 b* is wanted, it was preferred to compute La-

because the equation for this quantity is very much simpler. Know-
ing the outer flow, -32 can now be specified. To determine 5*
thus further requires which is a function of X* and Ma,

represented in figure 9.

Since the quantity in the
be defined as accurately as possible

1 di* B ,i 5* di*
b: dx 6* ,"* d

boundary-layer equation is to

the following formula is of

.0 6
S* v

( 5 d Ua

d Pa
dx pw


This formula contains only quantities dependent either on Ma and
*, or which can be taken from the previous boundary-layer calcu-
lation. The derivative with respect to u which is a function
of Ma, was preferred over that with respect to Ma for reasons of
simplicity. The first term at the richt-hand side in (5.6) is
generally the principle term.

Ii *t
If it is desired to eliminate V.

so as to secure

then because of

AS! merely in relation

and a

to i d- ua
dx dx2

in (5.6)


--* enters in the equation ) and coefficients

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 bsing suitably referred to the out-
side density, the coefficient of friction at the wall and the
displacement thickness:

Re* = UaPa*

2 a2
a (


JACA TM No. 1189

The formula reads

xd6* 1 R1 *\ 2 a
dx Re* -3 (4 W

+ *dua i
u yix fu

a o 2* 8* "
2 + Ma2
* 3S1?* -

ua + I R5 "
+- _- I-- i
m u ,

f a.
- (2 -( 4 ivia)
(ua dx

l 56* dua &*2 d ua /E dua\
. _- a. aR* aRs*
Re* ua dx ct3 u c d.x \aa /x


It contains the first and second derivative of ua made dimension-
less with the displacc-ment thickness and th3 outside rlorcity, and
also Re* and coefficients that are dependent c-n A* and Ma.

The coefficients al, a-, and a? for 0* = 0 are given in the
following table:



N:a 0

a1 1.4 |

a0 5.31

rL 21,T

-I V~~r



o..? u




0.- 2

'.* = 0




J, .

2 -C.



r. 1.1

Later on the equation is to be applied to the case where the
velocity distribution differs little and monotonically from
Uat d a 5*2 d2ua
us = Const, so that and are also regarded as
Ua dx ua d:2.

0.21-y U...)(+|
.. I

30 NACA TM No. 1189

being small. Restricted to the linear terms in the derivatives of
u1a, the term with na can be stecs: out and the coefficients a2
and a3 taken at the point X* = 0 for the specified Mach number.
The dependence of al on the derivatives of ua proves to be so
small that the sane can be done for this quantity too.

So on the assumption of smaall derivatives of ua simply
(5.8) is taken vith the constants of table II for the corresponding
Ma, for of a laminar boundary layer.
To obtain a formula for the variation of the displacement
thickness of a turbulent boundary layer a different procedure is
required. On analyzing the cause of the variation of an incom-
pressible turbulent velocity profile at a specific -pressure
variation it is found that the pressure forces are primarily
responsible. The shearing stresses introduces by the turbulent
intermingling play, however, a subordinate part. It is true that
the difference of the two effects is not so far reaching that a
second profile could be computed accurately eroiug from the
specified velocity profile when the ehearirng tr-.sses are discounted,
because the shearing stresses are able to substantially modify the
character of the profile; but for the calculation of the variation
of displacement thickness, which essentially involves an integral
over the velocity variation in the incompressible case, the shearing
stresses can be disregarded.

The result at Ma = 0 is the following ap'-ro:'imation formula
for the turbulent boundary layer:

d. a / u (5.9)
da- ua dx u a

As the intoerand is always positive, it can be tarzon 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 outside velocity the displace-
ment thickness ra.malns constant, according to (5.9). This result
is naturally wrong, 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 formula could be improved
by the addition of the conventional formula for the variation of

NACA TM No. 1189

the displacement thickness, but it would serve no useful purpose, as
will be seen. It is of greater significance that in contrast to
equation (5.3) for the laminar boundary layer the second derivative
of ua is lacking in (5.9). Since (5,9) was obtained by several
rougher omissions its practicability is illustrated in figure 10.
The experimental values of dua an3i are shown plotted
ua dx dx
against the arc length x of Gruschwit-'s (reference 6) test
series 3, along with the variation in displacement thickness calcu-
lated by (5 9), the integral being forced at = 0. It is found
that the formula reproduces the actual conditions s adequately, as far
as the area of greater accelerations, whore errors begin to be intro-
duced. This is, of course, due to the fact that 8* = 0 imposes a
limit on the decrease in displacement t.lcihness.

These experiences in the incompressible zone can now be
interpreted to the effect that the turbil-nt shearing stresses for

the calculation of i* can also be carcelled in the compressible
zone. But even this assum-tion is insufficient to develop a law
for the variation in displacement thicknes; 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. i
probably-twice as great as the velocity-boundary layer; 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 6* as
the variation in speed within the boundary layer. Unfortunately
only one measurement of a turbulent supersonic profile is available,
and naturally there is little sense in developing a theory without
further basis. However, in order to reach a tolerably correct
numerical value, the part of the boundary layer in which the density
alone varies is disregarded for the present, since it involves
only about 10 percent of the displacement thicknsss, and, in the
remaining portion, putting the stream de.isity as a unction of the
velocity as follows:

pu- '- (5.10)
Paa u.

H to be taken from the experiment. Now the derivatives of p can
be expressed by derivatives of u, u., and pa with the aid of
(5.10). This enables us to derive a for-nila for the variation of
displacement thickness, neglecting the turbulent shearing stresses.

NACA TM No. 1189

d* = du I Ma2
dx iu dx

Jo P


- 1) e --*
/ &*

5* dua
= a dx
2t 6-id


E' is the derivative of the function H according to the argument
u-u and 5 the place where -- can be put equal to unity
Ua Ua
- = 1), while 6* represents the corrnc-.t displacement thickness

hence, integrate up to a point where -2- itself is equal to unity
L = j). This means, we state that the 10 percent of the dis-
placement thickness between the point =1 and = 1
u Pa
contributes to the variation of the displacement thickness an amount
which corresponds to its portion of the disolacement thickness.
For Ma = 0, equation ( 11l) 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 -d = 0, and for the
velocity and density profile represented in figure 11; thus we obtain
the constant a2 for two values of the Ilach number.



!Ma 0 1.7


5.1 2.2 i

The close agreement of coefficient a2 i'or the turbulent and the
laminar velocity profile is noteworthy.


A study of the equilibrium of boundary layer and supersonic flow
on the flat plate indicates that an unstable state is involved. The
growth of a small disturbance in a laminar boundary layer differs
somewhat from that in a turbulent layer and is, especially in the last

I.ACA I M No. 1189

case, very rapid. In incompressible flow a stable equilibrium
exists between principal flow and boundary layer.

Having secured the variation of displacement thickness 8*
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, naiely, in the flow at the plate
without specified pressure distribution.

Since the effect of small disturbances is to be involved, the
Mach number of the outer flow Ha is regarded as constant and the
v component of the velocity considered small relative to the
velocity of sound. Pfter introductionn of a velocity potential the
simple equation (2.2) is.involved, and written in the form

S ( 1) (6.1)

The x-axis is to be in plate direction, the :y-Ixis normal to it.

Now it is necessary to represent the effect of the boundary
layer on the potential flow in form of a boundary condition. The
boundary layer is thersfcre 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 Doundary layer .,t the particular place for the
prevailing velocity distribution. That is, the equation

v = u1 (6.2)
must be satisfied for y = 5*.

This condition is incc-.venient to the *xt-:nt th-t the boundary
for which it is to be fulfilled 'i not specified beforehand. But,
inasmuch as the disturbances are to be small, hence the outerflow
is to differ very little from a flow ua = Const., the boundary
condition for displacement thickness 5* in -undisturted flow is
assumed. By assumption the departure of 3* from the valie of
the displacement thiclneas for the undisturbed flow must be small.
Hence it seems immaterial whether v is specified at y = 5* cr
at y = 5* + dS* in the lincarized problem. Besides, the study is
to be restricted to such a small area that -- itself can be
regarded as constant at ua = Const.

NACA TM No. 1189

Furthermore the boundary condition (6.2) has the property of
giving the same v component of the velocity at y = 5* as a
boundary layer with equal -, on the assumption of potential
flow in the entire space.

In (6.2), v and u are none other than the components of
the outer velocity, hence in the notation of the proceeding section
equal to va and ua.

Applying (5.8) or (5.11) to dX gives then as boundary
condition of the problem,linearized in the derivatives of u, the
following equation for

y = 8*: v = u al a 25. aRe* -2 5*2 (6.2a)
Re- -x x2

If a laminar flow is involved the corresponding constants
must be taken from table II; if, turbulent flow, table III; in the
latter case, a3 must be put = 0. In view of the linearization
5* 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

Now it is attempted to find the solution for the cabe that
the plate is exposed to a flow with the velocity u(x,y) = u = Const
and at a point x = 0 at the plate the velocity is artificially varied
by an amount U <, uo. The coordinate system is turned through a
small angle so that the x-axis in point x = 0 is exactly in flow
direction and the y*-axis 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

dB' J
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 so 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*.

NACA TM No. 1189

These new coordinates are denoted with

5 x =y- 8*

and after introducing the velocity potential in (6.2a) give the
following differential equation with the respective boundary

at 'y'y = "x'x' (Ma' 1) (
y' = 0; y' = 2t'x, a RetO'x'xy (6.2b)

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

0= uo*X' 4 u* f(x' -4:.Ia2 )

+ ( + \W2 1 y')

f and g are arbitrary functions of which it is merely required
that their sum at x' = y' = 0 be equal to unity. It is seen
that g gives Mech 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. Introduction of' the thus obtained solution in the
boundary condition gives the functional form of f. Denoting the
derivative with respect to the argument

T = x JMa2 1 y'

with subscript Tr, we get

x' = u 0* + U6*fT ; xx'W = ~8*f.i

x'x'x' = b6*f ri; 1 y' = lb* Ma2 If

which inserted in (6.2b) gives an ordinary differential equation of
the form

Re*f + f Ma2 = (6.3)
3 1Tfrj 71 1

30 NACA 4TM Io. 1189

This equation is easily solved. First postulatine a laminar boundary
lawyer, hence a3 / 0. Then from the renuireLent for x' = y' = 0:
f = 1, a requirement for the second derivative f can be satisfied,
because the upper equation can be regarded as differential equation
of the second order of f,. Since at this point only the consequences
of small velocity disturbances, not the consequences of disturbances
of the velocity difference are to be studied, the aaded requirement
for x' = y' = 0 is fq = 0, which gives the solution

t t
f 1 e t 2 e- t tl
f ti t tl t2

and t2 are abbreviations for the expressions

t1,2 2

Ma3e* -

1 t 2 a2 -_1
i/ 41 2 1e
q e, --/,Re*
3 -.


As Re* in general
term under the root
appraisals at high

has the order of nmanitude of 103, the last
is a term of greatest influence. Thus for
Be* we can put


t l+2 Z VNa
1 aRe*
V1 J

It is to be noted however that the critical Re* which corresponds
to a value of abnut 1.4 x 103 must not be exceeded as will be
shown later.

By use of (6.4) the velocity distribution in a laminar boundary
layer on the plate is obtained as:

1 x' =

+ t2- t2let2 x- Ma -ly t2e tl '- a 2ly (6.5)
tl t2 I'C\

which by (6.4a) is reduced to the simple form

u + u coh\(x a 1 y

where tl


NACA TM 1o. 1189

If a turbulent boundary layer were involved, hence a. = 0.
the first summand in (6.3) cancels out and only one boundary condition
can be satisfied. Again requiring f = 1 for x' = y' = 0 gives

fMa2.-. 1
ft =e cu?

The velocity field under the assumption of a turbulent boundary layer
at the plate is

u = u + ue a2 (6.6)

From (6.5) and (6.6) it is seen that the boundary disturbance along
Mach lines is propagated into the flow. The interference velocity
U is always accompanied by a function which grows considerably
with rising value of the argument, while in the case of the laminar
boundary layers the coefficient a3 plays the principal part. In
turbulent boundary layers the coefilcient a2 is essentially
involved. Thus the boundary layer of a flat plate in flow with
constant 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 sm.ll disturbance grow
infinitely. The type of growth is, of course, quite dissimilar
on the two boundary layers. To secure a measure for the instability
of the state, we may ask for which value of x' = -g at y' = 0
the disturbance has grown to twice the amount and call this quantity
the length of growth A. It is not made dimensionless 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 1.3 to the amount 2. Accordingly

a Re*
A 1.3 ,* (6.7)
1. Ma2 1

The length of growth of a turbulent boundary layer At is

At = 0.70 ,2 5* (6.8)
MaS 1

NACA TM No. 1189

Postulating a laminar boundary layer at Re* = 1000, tables II
and III give the following length of growth


Noteworthy is the unusually short length of growth in the turbulent
boundary layer; but even that in the lam'n-Tr l-yer 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 5* = 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 became evident in a pressure rise,
probably an oblique compressibility shock. It would not be sur-
prisirg 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, perhaps cven 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 turbulent.

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
b* would have to be analyzed first.

One unusual fact is that in the measured pressure distribution
on a wing, such as those by Gbthert (rofer-ence 7), for instance,
pressure increases were almost over observed in the supersonic
zone, except in form of compressibility shock or occasionally at
small Reynolds numbers, where laminar boundary layers must be assumed.

Ma 1.2 1.5 1.7 2.0

S 25 18 12

At 1.1
& ,

IIACA TM No. 1189

It appears entirely possible that this fact might 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 distances from the wall, that is, for great values of y,
must disappear. On these premises, (, (6.2b) by the same method
of calculation give

u = uo + ue -B -'-2'cos (02x' Ply) (6.9)

with the abbreviations

1 C 2)+ ( 3Re ")2 +
0l am3BRe*j g + -3

2 3Re*][( (ct ) (

The decisive term at high Re* numbers is again a Re*. For
Re* = 500 it approximately is

Il 2 m 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-xtremely stable according to it. This method of
analyzing offers the further possibility of exploring the stability
of laminar subsonic boundary layer relative to nonstatic.nary 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

Incidentally, the requirement of damping of the disturbance
for great y is not fulfillable in subsonic flow on the assumption
of a turbulent boundary layer at the plate. This result may have
its cause in the fact that (5.9) does not meet all requirements.

NACA TM No. 1189



Appraisals indicate that the flow in the critical range of
sonic velocity is very substantially affected by the boundary
layer. Without its inclusion a correct calculation of the pressure
distribution therefore seems, in general, not very promising. In
many instances the behavior of the boundary layer actually governs
the pressure distribution.

On examining the press'ire distribution at a bump computed in
section 3, (Cig. U), a symmetrical 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. 12).
where symmetrical peaks were Invariably accompanied by asymmetrical
velocity distributions. Naturally the question ia whether there
is only one solution for each bump but it will be shown that, owing
to the boundary-layer effect, symmetrical 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

5* = 2a1
S1 u Pa
what is the possible extent of the bump in order that the boundary
layer remain laminar? Figuring with tests in a low-pressure tunnel,
the values at critical velocity are

ua = 3 x l04cm/sec; na = 0.8 x 103g/cm3; 1. = 1.8 x 10-

It is to be presumed that the critical Reynolds number at sonic
velocity does not differ substantially from that In incompressible
flow. Taking the critical Reynolds number formed with the plate
length at

Rcrit. = 5 x 105

NACA TI1 No. 1189

gives the critical Reynolds number (5.7) formed with the displacement
thickness at

Re*crit = 1.4 x 103

with the previous values of ua, Oa, and Pw the critical.values of
plate length and displacement thickness are

Xcrit. = 3.8 cm; B*crit. = 1.1 x l0-2 cm

So in order to prevent transition from laminar to turbulent flow in
the boundary-layer model, lengths of only a few centimeters may be
permitted in the usual test arrangements, provided that no strong
accelerations are involved.

Conversely, the critical length indicated here gives a measure
for when the transition point is to be expected on a plate flow in
an exhaustion tunnel at sonic velocity. In a free-air test this
length is reduced by about half because of the higher density.

In the schlieren photograph of an infantry shell in flight at
around sonic velocity (fig. 13) (references 9 and 10) the oblique
compressibility shock is evidently released by transition, its
effect being probably amplified by the unstable behavior.of the
boundary layer. The fact that a missile at small supersonic speed
is involved is immaterial; since a straight compressibility shock
prevails in front of nose of the missile, it actually flies as if
in a subsonic flow.

Analyzing the bump in figure 4, 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 thickness at the peak Is certainly greater than that of a plate
1 centimeter in length in flow at sonic velocity. Therefore

5*,=0 > 0.56 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

NACA TM No. 1189

from point X* = 0 to the separation point from value 3.2 to 4.7.
Considering the fact that the momentum thickness itself increases
up to the separation point, an empirical rule can be established
according to which the displacement thickness is doubled between
k* = 0 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

6* =o 0.56 x 10-2 cm
separ. x=o

The difference in height of the highest point and at point of
separation hsepar is (compare fit. 4)


While the variation in h due to the boundary-layer effect
amounts to a mere 20 percent, the illustration shows that a change
in height of bump by this amount must be followed by an extra-
ordinarily great chance in velocity distribution, so that there
can be no question of attaining symmetrical results in the

The conditions in the presence of a turbulent boundary layer
are considerably worse. A little calculation on Grushwttz's 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 displacement 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 bump. In turbulent bound-ry layer
and thin profiles or low bumps this is possible only to the extent
that a compressibility shock occurs at the point of creetly reduced
profile thickness; furthermore a compressibility shock would have
to occur so much farther downstream as the bump or tha profile is
flatter. It also is feasible that the effect of the increase in
displacement thickness is raised by stron, return flow behind the
point of separation. Th.-e qualitative re-sults can be checked
against the work cf GCithert (reference .).

NACA TM No. 1189

The fact that a compressibility shock can occur when there is
enough space available for the increased displacement thickness
caused by it is to be regarded as reason for the fact that the
separation computed by stream filament theory in figure 7(b) is almost
exactly coincident with the start of the compressibility shock in the

It may be asked how the streamline pattern in a flow problem
must look, in order that the compressibility shock be possible. This
can be answered to the effect that the compressibility shock on slender
bodies is to be expected near the point of vanishing streamline curva-
ture. Since the 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 acco-ding to.(1.8). In a flow
that differs little from the critical so-ic 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 the outer velocity must be
counted on. The marked velocity variations in figure 4 coincide with
the streamline inflection points. On flat profiles a point of separa-
tion can be regarded as starting point of 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
shock. This argument is therefore not suitable for finding the location
of a compressibility shock.


The proceeding work shows that in a calculation conforming 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 permits 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 displacement effect that is found on an
experimentally investigated body including its deadwater region and

4 IIACA TM No. 1189

the displacement offoct of its boundary layer. We can also anticipate
from our calculation a strong velocity increase at the body nose and
a strong velocity decrease at the point where the curvature of the
substitute body disappears.

Translated by J. Vanier
National Advisory Committee
for Aeronautics

NACA TM No. 1189


1. Stanton, T. E.: Velocity in a Wind Channel Throat. Aeron.
Res. Comm. R. & M., No. 1388.

2. Oswatitsch, K1., and Rothstein, W.: Das Strbmungsfeld in einer
LavaldUoe. Vorabdr. d. Jahrb. 1942 d. deutsch.
Luftfahrtforsch. in den Techn. Ber. Heft 5, Oct. 15, 1942.

3. Holstein, H., and Bohlen, T.: Verfahrcn zur Berechnung laminarer
Grenzschichten. Lilienthal-GCs. f. Luftfahrtforschung
Bericht S 10 (Preisausschreiben 19'0.), p. 5.

4. Walz, A.: Ein neuer Ansatz fUr das Gezchwindigkeitsprofil
der laminaren Reibungsschicht. Lilienthal-Ges. f.
Luftfahrtforschung Bericht 141.

5. Hantzsche,W., and Wendt, H.: Zum Kompressibilit&tseinfluss bei
der laminaren Grenaschicht der ebenen Platte. Jahrb. 1940
d. deutsch. Luftfahrtforsch. I, p. 517.

6. Gruschwitz, E.: Die turbulente Reiburigeschicht in ebener
Strtmung bei Druckabfall und Druckanstieg. Ing.-Arch.,
II. Bd., 1931, p. 321.

7. Gothert, B., and Richter, G.: Messung am Frofil NACA 0015-64 im
Hochgeschwindigkeitskanal der DVL, FB 1-47.

GCthert, B.: DruclckerteilunGs- und Impulsverlustschaubilder
fur die Profile NACA 0006-1, 130, usw. bei hohen
Unterschallgeschwindigkeiten. FB 1505/1-5.

8. Tollmien, W.: Ein allgemeines Kriterium der Instabilit&t
laminarer Geschwindigkeitsverteiluneen. Nachr. d. Ges. d.
Wiss. zu GOttingen, Math.-ohys. Kl., Fachgr. I, Math.,
Neue Folge, Bd. I, Nr. 5, 1935, p. 79.

9. Cranz, C.: Lehrbuch der Ballistik, Bd. II, p. 452, Fig. 22.

10. Ackeret, J.: Gasdynamik. Handb. d. Physik, Bd. VII, p. 338.

11. Busemann, A.: Das Abreissen der Gren.schicht bei Annaherung
an die Schallgeschwindigkeit. Jahi'b. 1940 d. deutsch.
Luftfahrtforschung I, p. 539.

12. Fr6ssel, W.: Experimentelle Untersuchung der kompressiblen
Strboung an und in der Nahe oiner Eewblbten Wand, 1. Tell.
UM 6608 (Abb. 12 entstammt einem nicht
verbffentlichten Vorversuch).

46 NACA TM No. 1189



u .s

-I O

a) 0


/ ir^ ^
/ S3
/~~ P7 <" "

\ ^

NACA TM No. 1189


/ I
y s,



48 NACA TM No. 1189

S ii


>? a)








)3 F-)

/ i
0 M


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 t 1.5

0 1 2 3 4 X 5

Figure 4.- Velocity distributions over various bumps.


i .
-- --

NACA TM No. 1189



ni -4

0 '-
C a D
0 o


~ ,-


NACA TM No. 1189





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-^ S 1-i
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52 NACA TM No. 1189




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NACA TM No. 1189 53



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54 NACA TM No. 1189



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NACA TM No. 1189



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56 NACA TM No. 1189



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NACA TM No. 1189 57




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NACA TM No. 1189 59

o'i :4


Figure 13.- Infantry bullet at slight supersonic speed (reference 9 ).


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