Moore, Franklin K

English

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.I
,
FOR AERONAUTICS
TECHNICAL NOTE 2722
DISPLACEMENT EFFECT OF A THREE-DIMENSIONAL
BOUNDARY LAYER
By Franklin K. Moore
Lewis Flight Propulsion Laboratory
Cleveland, Ohio
-
Washington
June 1952
https://ntrs.nasa.gov/search.jsp?R=19930083354 2020-06-17T19:26:26+00:00Z
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!DZCHIHCALl?OTE2722
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K. Moore
A method is described for determining the “displacement surface”
of a known three-dimensional compressible boundary-layer flow in terms
of the mass-flow defects associated tith the profiles of the two
velocity components parallel to the surface.
The height of the displacement surface above the body surface for
flow about a yawed infinite cylinder is shown to be equal to the height
characterizingthe mass-flaw defect of the chordwise velocity profile.
The displacement surface height is shown to differ, In general, from
that associated with the remil.tantmass-flow defect, even at stagnation
points of the secondsxy flow. Numerical vslues - found for the known
three-dimensionalboundary-layer flow about a cone at a small angle of
attack to a supersonic stream.
INTRODUCTION
The boundary kqer established in the flow of a slightly viscous
fluid about a body is normally considered an isolated region wherein
the effects of tiscosi~ predomimate= and outside of which the motion
of the fluid is governed by the laws ~’ JKuM.scous motion. For large
Reynolds number, the boundary layer is assumed to be so thin that the
nonviscous portion of the flow occurs as though there were KLOboundary
layer. This assumption is strictly correct in the limit of imfinite
ReynoMs -cr. l?orMge but finite Reynolds numbers, the growth of
the boundary layer causes the stream to be deflected away from the body
surface.
This displacement effect of the bound=y
_ on the nonviscous
flow my properly be deixm.ined from the behavior of the bom layer
as established
boundary-layer
either by experiment or by solution of tie
equations for Iam@ar flow.
.
“
.-- .. . ..—. . . ..— .-
.
—...— ._. ..- —.-.— ____ ..-. -.. . .__a ._
-.
2It
used to
Frandtl
NACA TN 2722
.
does not follow, however, that this revised outer flow may be
improve the solution for the boundary layer, still using the
equations. A new set of equations must be used for this pur-
pose tal&g into account the variation of pressure across the boundary
layer, which is neglected in the Prsndtl equations. (See Alden’s iter-
ative solution for incompressible flat-plate flow, reference 1.)
The customary definition of displacement thichess (reference 2)
is applicable
equation:
to k-dimensional fl& and is ~ressed ~ the following
Jh puo
where h is some location we13
at which pu = PIUV ~ beyond
viscous. (A ldst Of =yuibolsiS
asm.u@ion of an exkremely thin
W = py-q(h-~*) (1)
outside the boundary kyer (fig. 1),
which the flow may be considered non-
protided in the appendix.) Under the
boundary layer, h is so small that
plul may be tsken as @e evaluation at the body surface (y=O] of the
nonviscous flow obtained by neglecting the presence of the boundary
layer. Equation (1) equates the actual mass fl~nesr the surface with
the mass flow wMch wouldbe associated with a nonviscous fluw that
terminates at 5* rather than extending to the wall. Thus, the non-
viscous portion of the flow behaves as if it occurred h the presence
of a solid boundary given by the disp=cement thickness ~*(x)o Equa-
tion (1) maybe solved for 5*:
8*=~’(1-&~dy
Ordinarily, theoretical boundary-layer solutions for PU/PIUl am-
totically approach 1 for large values of Reynolds number based on y.
Therefore, displacement tht.@mess is often defined as follows:
n“
—— . . .——.—z ——— ..—.,= ... . ...—.—.——— _ –__— .+
NACA TN 2722 3
For three-dimensionalboundary-layer flows, two lengths character-
izing mass-flow defect may be defined in terms of the profiles of the
two velocity components tangential to the surface (fig. 2),
&
m and it is not cleaz which, if either, defines a displacement surface
m that properly describes the etient to’which the nonviscous flow is
deflected by the boundary layer.
Of course, it is expected that there does exist a displacement
surface for such flows. The analysis that follows shows that such a
surface may be described, using a definiqg equation more fundamental.
than equation (1).
The velocity and density profiles are assumed to be bow-n for the
three-dimensional flows under consideration. ~ases for which this is
true include: the lsminar boundary Layer on yawed infinite cy~nders,
treatedby Prandtl (reference 3), R. T. Jones (reference4), andby
Seers (reference5); the laminar boundary layer on a cone at a mall
angle of attack to a supersonic stream (reference 6). The dispkcement
effect of these flows wXEl be treated specifically.
This report may be considered as an addendum to references 6 and 7.
TMs investigationwas conducted at.the NACA Lewis laboratory.
l
.
THEORY
Defining Relation for Displacement Surface
The boundsry-layer solution (assumed known) yields a ceqtain dis-
tribution of velocity ~bl(xjz) normal to the body surface at-the
outer edge h(x,z) of the boundary layer, where p, u, and w may be
taken essentially equal to pl, ~, and wl (see fig. 2). Under the
assumption that the nonviscous flow is altered only slightly by dis-
placement, the most direct way tjocompute this effect would be to sup-
pose that the nonviscous equations hold for yzh(x,z) and to impose
the follting boundary condition on the normal.velocity vof in the
outer flow:
Vof = ~bz(x,z) at y = h(x,z) (3)
,
--—— ----— -. -—-— ... -—-—.. ., _. ______ _. ---- ._ -._.. _
4However, since the
viscous flow specify an
M.CA TN 2722
boundary conditions usually encountered in non-
permeable surface, it is convenient to recast
the boundary condition given in equation (3) in answer to the question:
What impermeable surface y = A(x,z) would deflect a nonviscous fluid
in such a Way as to produce a normal velocity satisfying condition (3)?
This fictitious surface may be called the displacement surface.
Because A(x,z) is imagined to be a stream surface, it is neces-
sary to specify that at y=A the resultant velocity vector (u,v,w)
be tangent to the surface y=A. Thus, at y~ the ratio of v to the
_tude of q, wlich is defined as the vector (u,w), must be set
equal to the slope of the surface y=Aj measured in the direction of
the vector g, or, eqylvalently, equal to the component in the direction
of g of the vector grad A. In vector notation, therefore, the nor-
msl velocity v which would be produced in a nonviscous fluid at an
impermeable surface y~ is q- grad A. The vector g maybe
obtained by evaluating the velocity vector of the unrevised nonviscous
flow at the body surface y=Oj under the related assumptions that the
velocity vector varies only slightly over distances of the order of
the actual boundary-layer tbiclmess, and that the revision required to
take account of displacement is slight. Thus, at y=A (see fig. 3),
‘of ‘91 *grad A
The increment in vof between A ‘and h is approximately (h-A)hof/ti,
again assuming a thin boundary layer and using only the first term in a
Taylor series.
.
To the order of approximation contemplated in this analysis,
&of/ay
at y=O.
at y=h,
may be obtained frmn the unrevised nonviscous flow evaluated
Thus, the fictitious impermeable surface A would produce,
a normal velocity (see fig. 3)
v
of = % “
Introduchg this result into
tion (3) yields the defining
( ).avofgrad A+ (h-A) ~ y+
the boundsry
relation for
The boundary-layer sOIUtiOn yields vbl.
condition given in eqya-
A(x,Z) :
+cJl”gradA (4)
—.. —— ———-———— .——.—.—....— ——— ..--—. — —. —.-—.—.—. .—. —.. - ..- J
N
CJl
CN
0-)
NACA TN 2722
Expression of A in Terms of llass-FlowDefects
The displacement surface A may be related to the mass-flaw
defects (equations (2)] which characterize the bounhx’y layer as
fOllows: in a Cartesian coordinate system (fig. 2), the equation of
continuity for both the boundary-layer and nonviscous flow is
$=-g-$ . (5)’
lIuderthe Prandtl boundary-layer assumptions, the Cartesian equations
of motion, and hence equation (5), may be applied in an orthogonal
curvilinear coordinate system in which the surface of the body is
givenby Y=O, provided the radius of curvatuxe of the body is every-
where large as compared with the boudary-layer thiclmess. Integrating
equation (5) across the boundary layer yields
or, inasmuch as h is outside
(3W= Pl~Tlj
the boundary layer where pu = plul and
where 5X and bz are the mass-flow defects defined
For the nonviscous flow, vof = O at the body surface
tion (5) becomes
,
d%y=o=-(~+w
in equations (2)0
(y=O), and equa-
(7)
.. . ..- ------ ..--. .—.. -.--— .-. .————.. . . ..—. ..4-.. —. .—. ._—. . . ..— — —.——- .—. —-
6 NACA TM 2722
Introducing equations (6] and (7) into equation (4) yields
%&”l’A-5x~ ‘: [’lW1(A-’Z!I = 0
(8a)
In a Csrtesian coordinate system, the displacement surface is related
to the mass-flow defectsby equation (8a}. When cases arise for which
other coordinate systems must be used, the fo~owihg generalization in
vector form of eq&tion (8a)
[
div pfilA
—
may be used: 2
.
-Jh(’fi@E=‘8b’
where y is the distance normal to the body surface and where the div-
ergence operator involves differentiation only with respect to the two
coordinates parallel to the body surface.
E&mples
Plane flow. - Equation (8a) maybe integrated to yield; for plane
flow (a/az = o):
where K is a constant of inte@xrbion.
The appearance of this constant means,
(9}
in general, that the
revised boundary condition on we nonviscous flow near the wild may be
applied at any surface in the boundary-layer region; for example,
along the wall itself. Of course, if there is a stagnationpoint on
the body where U1 vanishes, then K must be taken equal to zero and
the revised boundary condition must be imposed at the location A= ~,
at least near the stagnationpoint.
Stagnation point of secondary flow. -~ the vertical plane of sym-
metry of the flow about a body of revolution at au angle of attack, the
circumferentialvelocity component vanishes in the boundsry layer as
well.as in the nonviscous outer region. Such stagnation of the second-
ary motion would occur in a variety of cases, tiparticulsr, wherever
an essentially three-dimensional flow has a plane of symmetry. Vhen
WI is designated as the component of secondary flow (circumferential
velocity for a body of revolution} at the outer edge of the boundary
layer, and equation (8a) is written in the form
——_ —_———— — -- .._ .—_ _ . _
_—___
—.— -.—. . .. ..= .——._ _ —A
INACA TN 2722
it is clear that A cannot equal ,5X,in gener~j even if both W1 and
the boundary-layer profile of w vanish, unless h~~z also vanishes.
Flow about yawed infinite cylinder.
- W this case, there is a
spamise boundary-layer velocity profile and an associated spsmwise
mass-flow defect. (See references 3 to 5.) If x is taken in%he
chordwise direction (fig. 4}, the entire flow depends only on x. Thus,
derivativeswith respect to the spanwise coordinate z vanish; and,
hence, from equation (8a), the plane-flow result (equation (9)) applies.
Accordingly, the spanwise mass-flow defect representedby 52 does not
enter into the determination of the displacement surface.
Supersonic flow about cone at small an@ e of attack. - When the
coordinate system shown in figure 5 is used, equation (8b) becomes
(l-l)
where
Because the outer flow is conical, PI, Ul, and W1 are functions only
of q; and equation (Xl) may be written
(13}
In the case of conical outer flow, the associated boundary-layer
profiles show similsri~ of the Blasius type in meridional pQnes (see
reference 7 or 8). Thus, in a meridional plane, A, 5X, and 5 are
proportional to ~ Incorporation of this information into eqya-
tion (13) gives
(14)
.. . . .- . .. . .-— _ .. -,_ _. ___~,, .- ____ ----
----—— .- ..-.. -—.. .. .. . ... . .. . __ _ . -. . . ,
8For a cone at small
the veloci~ on the cone
ties PI, A, ~j a ~
whereas
angle of attack
surface at zero
a, U1 is nearly
angle of attack.
WI =& A2Sin Q
where A2 depends only on the cone vertex angle and
ber and is defined in reference 6 as follows:
A2 =-&__ &
u u sin @
The quantities z, x, rind ~ are in the notation of
wherein they are tshulated.
IWCA TN 2722
equal to u,
The quanti-
attack,
the flow Mach
reference 9,
(15)
nu3n-
To the first order in a, the substitution of equation (15) into
. .
equation (14) yields the resuit -
(16)
The smalysis Qf reference 6 yields the values of ~ and 5W
Clearly, A dHfers from 5X in the plane of symmetry q = 0, x,
where the chcum@rential veloci~ w vantshes. It might, however, be
fi 311
noted that in the plane Q = ~, ~, A = 5X. Figure 6 shows the pro-
portional rate of increase of displacement tMclmess A with angle of
attack in the plane of symnetry Q = z at zero angle of attack. The
corresponding rate of increase of the mass-fluw defect ~ is shown
for comparison. These curves are-obtained from equation (16) and the
results of reference 6. me she of the s-vertex angle is introduced
as a factor primrily to permit presentation of the case @ = O as the
Mm&t of an indeterminate form. For a stream Mach nuuiberof 2, the
change in displacement thiclmess appears to be of the order of 50 per-
cent larger than the change in the meridional mass-flow defect.
coIWIXIDIIW’REMARKS
The forego~ analysis deals only with the displacaent effect of
a lmown boundary l~er on the nonviscous outer flow, and hence can be
applied only if the boundary-lsy= behavior has been determined either
theoretics32y or experimentally. The latter approach might possibly
L
——-——— ———.— .__. .-. —. 2
2M NACA TN 2722
find application in the correction of nozzle contours for
9
boundary-layer
development. In this connection,perhaps,it should be noted that-we-
, analysis is not restricted to laminax flows.
Lewis FMght Ihmpulsion laboratory ~
National Advisory (!ommitteefor Aeronautics
Cleveland, Ohio, Msxch 6, 1952
*
---- . ...__ ._. ..__. .__..- . . . .. . ___ -_ _.a..
--------—..._.._. _________ . ......_______..+-r~-.___
.,
10
AZ
h
-v
%1
Vof
x
z }
Y
a
A
AmENmx - SYMBOLS
NACA TN 2722
The following symbols are used in this report:
function of cone angle and Mach nuniber(equation (15))
height above wall at which p, u, w = Plj ulj W1~ ~ beyond winch
nonviscous equations aypl.y(equation (3))
s,tresmMach number
.
velocitj vector co~osed of components parallel to body surface
u,v
velocity component in x-direction
meridional velocity component at surface of cone at zero angle of
attack
velocity component in y-direction
boundary-layer solution for velocity normal to surface, evaluated
at outer edge of boundary layer
nonviscous solution for veloci~ normal to suxface, evaluated near
the surface
velocity component in z-direction
coordinates in body surface
coordinate normal to surface
angle of attack
. .
height above body of displacement surface
displacement thickness in plane flow
length characterizingmass-flow defect of u-profile (equa-
tions (2), (X2})
—— - —.— ______ ._ _____ ._—____
———— . ——-.—-. .- . .. .—
l!M.CATN 2722 , U
5Z length characterizingmass-flow defect of w-profile
(equation (2)]
q$ length characterizingmass-flow defect of w-profile .
(equation (12))
e semi+ertex angle of cone
P density
9 angular,coordinate(fig. 5]
Subscript 1 denotes evacuation of nonviscous flow at body surface,
taken equivalent to conditions at outer edge of a boundary layer of
infinitesimal thichess.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Alden, Henry Leonard: Second Approximation to the Laminax Boundary
Layer Flow over a Flat Plate. Jour. Math. and ph.yS., VO1. -I,
no. 2, Jul.y1948, pp. 91-104,
F!randtl,L.: The Mechanics of Viscous Fluids. Vol. III of
Aerodynamic Theory, div. G, sec. 14, W. F. Durand, cd., Julius
Springer ~Ber13n], 1943, pp. 80-82.
Prandtl, L.: On Boundary Layers in Three Dimensional.Flow, Rep. &
Trans. No. 64, British R.A.E., my 1, 1946.
Jones, Robert T.: Effects of Sweepback on Boundary Layer and
Separation. I@CA TN 1402, 1947.
Sears, W. R.: The Emndary Layer of Yawed Cylinders. Jo&. Aero.
Sci., vol. 15, no. 1, Jan. 1948, pp. 49-52.
Moore, Frsmklin K.: Iarhar Boundary Iayer on a Circular Cone in
Supersonic Flawat a Small An@.e of Attack. NACATN 2521, 1951.
Moore, Franklin K.: Three-Dtiensional Compressible Lsminar
Boundary-Layer Flow: NACA TN 2279, 1951.
Hayes, Wallace D.: The Three-DimensionalBoa Layer. NAVORD
Rep. 1313, N(Y!X384, U.S. Naval Ordnance Test Station (Inyokern),
May 9, 1951. (Bur. Ordnance Task Assignment NOTS-36-Re3d-441-3.)
Anon.: Tables of Supersonic Flow Around Yawing Cones. Tech. Rep. .
No. 3, Dept. Elec. Eng., M.I.T., 1947.
-— —..——... ___ . __— -.. ..—-_. ._..__ _
-..—— ——-—-.—.-..— .-. . . .. .
hm“’(x)
‘4 ,WE57”L Body surface
Figure 1. - Plane boundary-layer flow.
——— .---- —--
—.—— ._.. _—.————
___. __— ..——_ —. ----- ------
–.
NACA TN 2722
/
13
‘b%
...............
.................
t
Figure 2. - !lbree-dimensional”bcmndary-l.ayerflow (Cartesiancoordinates).
1
. . . ..~.:::::y. :::.
Surface
= A(x,z)Y
R@ surface
.
Figure 3. - Displawment surfacein three-dimensionalflow
(Cartesiancoordinates).
. .. —-..—. - . . . .._. _ ____. . . . _________ ___ _+___..-
...-. .—..- ....—---- .-
—------ .— ---
J!IACATN 2722
14
\
Figure 4. - Coordinate system for yawed infinite cylinder..
Coordinate
pi!!=
system for cone at angle of attack.
.
.1
——..——- —
___— .—. —.. ——. —- -—– --- -
-.
_——
.—— —
,,1
NACA TN 2722
3.0
‘%
In
N
2.5
1.5
1.0 I I I I I I I I I I I I1 2 3 4 5 .6
Figure 6. - Proportionalrate of Increase of displacementthichess with
angle of attack on a cone.
X5
NACA-I.a@eY -6-10-59- 10+0
.. ...= ... . ___ ..._ ___ ------ __-.J_ ----
. ----- ...— .—..-— _.._.._.. ... . .. . . .


Displacement effect of a three-dimensional boundary layer

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