Laminar flow about a rotating body of revolution in an axial airstream

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Title:
Laminar flow about a rotating body of revolution in an axial airstream
Series Title:
NACA TM
Physical Description:
43 p. : ill ; 27 cm.
Language:
English
Creator:
Schlichting, H
United States -- National Advisory Committee for Aeronautics
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NACA
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Washington, D.C
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Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
Rotational motion (Rigid dynamics)   ( lcsh )
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federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
A method is given for calculating the incompressible laminar boundary layer on a rotating body of revolution at zero angle of attack in an airstream. The method is based on two boundary-layer momentum equations, one for the meridional direction and one for the circumferential direction. A simultaneous solution of both momentum equations yields the boundary-layer parameters, the friction drag, and the torque. The boundary-layer flow about a rotating sphere, about a body with a blunt base, and about two streamline bodies is calculated. The separation point is found to be affected by the spin.
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by H. Schlichting.
General Note:
"Report date February 1956."
General Note:
"Translation of "Die laminare strömung um einen axial angeströmten rotierenden Drehkörper." Ingenieur-Archiv, vol. XXI, no. 4, 1953."

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University of Florida
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M & ThH9










NATIONAL ADjVISORY COM~I~TTEE FOR AERONAUTICS


TECHNICAL MEMORANSDUM 1415



LAMINAR FLOW ABOUT A ROTATINJG BODY OF REVOLUTION

IN AN AX.IAL AIR.STREAM*C

By H. Schlichting


1. INPR~ODUCTIONI


The flow about a body of revolution rotating about its axis and
simultaneously subjected to an airstream in the direction of the axis
of rotation is of importance for the ballistics of projectiles with
spin. In jet engines of all kinds, too, an important role is played
by the flow phenomena on a body which is situated in a flow and which
at the same time performs a rotary motion. Investigations of
C. Wieselsbergerl regarding the air drag of slender bodies of revolution
which rotate about their axis and are at the same time subjected to a
flow in the direction of the axis of rotation showed a considerable
increase of the drag with the ratio of the circumferential velocity to
the free-stream velocity increasing more and more, the slenderer the
body. Similar results were obtained by S. Luthander and A. Rydberg2
in tests on rotating spheres which are subjected to a flow in the direc-
tion of the axis of rotation. These authors observed, in particular, a
considerable shifting of the critical Reynolds number of the sphere
dependent on the ratio of the circurmferential velocity to the free-stream
velocity. The physical reason for these phenomena may be found in the
processes in the friction layer where, due to the rotary motion, the
fluid corotates in the neighborhood of the wall and, consequently, is
subjected to the influence of a strong centrifugal force. It is clear
that the process of separation and also the transition from lamilnar to
turbulent conditions are strongly affected thereby, and that, therefore,
the rotary motion must exert a strong influence on the dlrag of the body.




*"DiMe laminare Stro~mung um einen axial angestrojmten rotierenden
Drehkijrper." Ingenieur-Archiv, vol. XXI, no. 4, 1955, PP. 227i-244.- An
abstract from ths report was read on the VIII International Mechanics
Congress in Istanbul on August 27, 1952.
IC. Wieselsberger, Phys. Z. 28, 1927, p. 84.
2S. Luthander, A. Rydberg, Phys. Z. 36, 1935, p. 552.







NACA TM 14c15


In the flow processes in the corotating layer of the fluid, one deals
with complicated three-dimensional boundary-layer flows which so far have
been 1 ttle investigated, experimentally as well as theoretically. Th. v.
Karmin~ treated at an early date the special case of a disk rotating in a
stationary liquid, for laminar and turbulent flow, as a boundary-layer
problem, according to an approximation method. Later, W. G. Cochran4 also
solved this problem for the laminar case as an exact solution of the
Navier-Stokes equations. A generalization of this case, namely the flow
about a rotating disk in a flow approaching in the direction of the axis
of rotation, for laminar flow, has been treated recently by H. Schlichting
and E. Truckenbrodt The result most important for practical purposes
are the formulas for the torque of the rotating disk; it is highly depend-
ent on the ratio of the circumferential velocity to the free-stream veloc-
ity of the disk.

For the general case of a rotating body simultaneously subjected to
a flow, J. M. Burgers6 gave a few general formulations. We have set our-
selves the problem of calculating the laminar flow on a body of revolution
in an axial flow which simultaneously rotates about its axis7. The prob-
lem mentioned above, the flow about a rotating disk in a flow, which we
solved some time ago, represents the first step in the calculation of the
flow on the rotating body of revolution in a flow insofar as, in the case
of a round nose, a small region about the front stagnation point of the
body of revolution may be replaced by its tangential plane.

In our problem regarding the rotating: body of revolution in a flow,
for luminar flow, one of the limiting cases is known: that of the body
which is in an axial approach flow but does not rotate. The solution of

XTh. v. Karmin, Z. angew. Math. Mech. 1, 1921, p. 255.
W. G. Cochran, Proc. Cambridge Philos. Soc. 3O, 1934, p. 365.
H. Schlichting, E. Truckenbrodt, Z. angew. Math. Mech. 32, 1952,
p. 97; abstract in Journal Aeron. Sciences 18, 1951, p. 658.
J5. M. Burgers, Kon. Akad. van Wetenschappen, Amsterdaml 45, 1941,
p. 13.
I1t is pointed out that the turbulent case, for the rotating disk in
a flow as well as for the rotating body of revolution in a flow, mean-
while has been solved, in continuation of the present investigations, by
E. Truckeabrodt. Publication will take place later.- E. Truckenbrodt,
"Die Strojmung an einer angeblasenen rotierenden Scheibe bel turbulenter
Strajmung," will be published in Z. angew. Math. Mech.- E. Truckenbrodt
"Ein Quadraturverfahren zur Berechnung der Reibungsschicht an axial
angestrtjmten rotierenden Drkehkijrpern." Report 52/20 of the Institut fur
Strojmungsmechanik der T. H. Bra~unschweig, 1952.






NACA TIM 1415


this case was given by S. Tomotikai, by means of transfer of the well
known approximation method of K. Pohlasen9g to the rotationally symmt-
rical. case. The other limiting case, namely the flow in the neighborhood
of a body which rotates but is not subjected to a flow is known only for
the rotating circular cylinderlO, aside~ from t~he rotating disk. In, the
case of the cylinder one deals with, a distribution of the circuimferential
velocity according to the law v = ER2/r where R signi fies th ecyli nder
radius, r th~e distance from. the center, and a the angular velocity of
the rotation. The velocity distribution, as it is produced here by the
friction effect is therefore the same as in the neighborhood of a poten-
tial vortex. In contrast to the first limiting case (nonrotating body
subjected to a flow), the flow in, the case of slender bodies which rotate
about their longitudinal axis in a stationary fluid does not hav
"boundary-layer character," that is, the friction effect is not limited
to a thin layer in the proximnityr of the wall but takes effect in the
entire environment of the rotating body.

Very recently, L. Howarthll also made an attempt at solution for a
sphere rotating in a stationaary fluid. This flow is of such a type that
in the friction layer the fluid is transported by the centrifugal forces
from the poles to the equator, and in the equator plane flows off toward
the outside.

When we treat, in what follows, the general case of the rotating: body
of revolution in a flow according to the calculation methods of Prandtl's
boundary-layer theory, we must keep in mind that this solution cannot con-
tain the limiting case of the body of revolution which only rotates but
is not subjected to a flow. However, this is no essential limitation
since this case is not of particular importance for practical purposes.

The dominant dimensionless quantity for our problem is the ratio


Circumferential velocity _V, Rmu
Free-stream velocity U, U,


where R, is to denote the radius of the maximum cross section of the
body of revolution. The calculations must aim at determining for a
prescribed body of revolution the torque, the drag, and beyond that,
the entire boundary-layer variation as a function of V,/Um. The

8S. Tomotika, "Laminar Boundary Layer on the Surface of a Sphere
in a Uniform Stream." ARC Rep. 1678, 135).
4K. Pohlhausen, Z. angew. Math. Mech. 1, 1921, p. 253.
10H. Schlichting, "Grenzschicht-Theorie," p. 63. Karlsruhe 1951.
lL. Howarth, Philos. Mag. VII Ser. 42, 1951, p. 1,j08.






NACA TM 14c15


particular case V,/Um = O is already known from the boundary-layer
theory established so far. Considering what has been said above, we
must not expect our solution to be valid for arbitrarily large Vjm*um
The upper limit of the value of V,/Um for which our calculations
holds true, still remains to be determined. Presumably, it will lie
considerably above Vz U, = .


2. THE FUNDjAEDWEAL EQUATIIONS


We take the coordinate system indicated in figure I as a basis for
the calculation of the flow. Let (x,y,z) be a rectangular curvilinear
fixed coordinate system. Let the x.-axis be measured along a meridional
section, and the y-axis along a circular cross section so that the
xy-plane is the tangential plane. The z-axis is at right angles to the
tangential plane. Let ut, v, w be the velocity components in the direc-
tion of these three coordinate axes. Furthermore, let R(x) be the
radius of the circular cross section, a, the angular velocity of rota-
tion, U(x) the potential-theoretical velocity distribution, and
v =ppthe kinematic viscosity.

The equations of motion simplified according to the calculation
methods of the boundary-layer theory a-re for this coordinate system


+u u dR aw- (continuity) (1)
ox R dx ae



u-- --, --- w- U--+ v--- (momentum, meridional) (2)
Ax R dx ae dx iz2



u21 uv dR avy & 1 (momentum, azimuthal) (5)
ax R dx az az2

The cioundary conditions are


z = 0: u = 0, v = v0 = Rmu, w = 0; z = m: u = U(x), v = 0 (4)






NACA TIM 1415


A solution of this system of differential equations for an arbitrarily
prescribed body shape R(x) with the pertaining potential. theoretical
velocity distribution U(x) leads to insurmountable mathematical. diffi-
culties. We use therefore the more convenient approximation method which
makes use of the momentum theorem. We obtain the two momentum equations
for the meridional and the azimuthal direction by Lntegration of the
corresponding equations of motion over z from the wall a = O to a.
dis tance z = h > 6 which lies outside of the friction layer.

For the meridional direction there results by integration of (2)
over z, with consideration of the continuity equation (1) and after
introduction of the wall-shear stress for the x-direction


(bul~l


the momentum equation for the meridional direction


7O


U2, ,, x + 2,+


+ ~U 4x+ v '24 )


Therein, as is well known,


Bx 1 i


- uds


is the displacement thickness whereas


9, 8
x Oul


-u dz


may be denoted as momentum-loss thickcnesses for the x- or y-direction.


9 1O6~62






NACA TM 1415


In an analogous manner there results for the animuthal direction by
integration of (3) over z with consideration of the continuity equa-
tion (1) and introduction of the wall-shear stress for the y-direction


ato = 1 (10)


as the momentum theorem for the circumferential direction


m=-d _{URd 39Y= -R2 70 (ll)
dx\ xyP

Therein


6,y =hU1 u ZOd (12)


has been introduced as the "momentum loss thickness due to spin."


3. APPROXDIATION6 METHOjDS

(a) The Velocity Distributions


According to the approximation method of the boundary-layer theory
as given first by Th. v. Karmain and K,. Pohlhausen, the momentum equa-
tions (6) and (ll) are satisfied by setting up suitable formulations for
the velocity distributions u and v which satisfy the most important
boundary conditions. For the present case, two parameters may still be
left undetermined in these equations for the determination of which the
two momentum equations are then available. As expressions for the veloc-
ity distribution, polynomials in the distance from the wall have proved
to be suitable, with the property that the boundary layer joins at a
finite wall distance z = 8 the frictionless outer flow. The boundary-
layer thickness may be different for the meridional and the azimuthal
velocity component. Let these boundary-layer thicknesses be by and
6 respectively; we introduce the dimensionless wall distances formed
with them


z t and z- t' (15)







NACA TM 1415


For the velocity distributions u and v, we select polynomials of
the fourth degree in t and t', respectively. These contain five coeffi-
clents each so that, for determination of these coefficients, we can
satisfy five boundary conditions each for u and v. We choose the
following 10 boundary conditions:


VO2 dR
R ax



z22


2u
z2

dz


dU


t = 0: u = 0



t= 1: u =U


(lka,b,c,d,e)


2

az2

' -


t' = 0: v =v0 Ru


t' = 1: v = 0


(15a,b,c,d,e)


The boundary conditions (14a, b, c) and (10a, b, c) result immediately
from the fundamental equations (2) and (3) with (4) for z = O and
z 8 or S The remaining boundary conditions provide a gentle
transition of the boundary layer into the outer flow. Taking these
boundary conditions into consideration, one obtains the following poly-
nomials as expressions for the velocity distributions


u= 2t -2ti + t + K1 t -t2 + 3tS t
U6


1 2t' + 2t'3 t'







8x2 dU'01U dRI
Sdx \U R dxJ


(16)



(17)







(18)


Therein







NACA TIM 1415


is a form parameter of the u-velocity profile, which is analogous to the
form paramter h of the Poh~hausen meth~odl2. The velocity distributions
ulU and .vlvO are represented in figure 2.

Let t;he point of separation be given by the beginning of the return
flow of the meridional velocity component u(z) in the proximity of the



\a z=0

This yields


K = -12 (separation) (19)

The expression for the u-component is the same as in the Pohlhausen
method for the plane and rotationally symm~etrical. case. This guarantees
that our solution in the case without rotation, w = O, will be trans-
formd inrto the solution of S. Tomotika and F. W. Scholkemeyerli for the
nourotating body of revolution. Introduction of the expressions (16)
and (1'1) into the momentum equations (6;) and (ll) yields two differential
equations for the still unknown boundary-layer thicknesses by(x) and
6 (x) or the quantities derived from them.


(b) The Momentum Equation for the Circumferential Direction

We present first the further calculation for the momentum equation
of the3 circumferential direction. With

TY0 lv = -2-'-Rm.v (20)
p dqJ s


there results from (ll), after division by w,


1- ~R5Us = 2-'R3 (21)


120f. H. Schlichting, "Grenzschicht-Theorie ," p. 195-
13F. W. Scholkemeyer, "Die laminare Reibungsschicht an rotations-
syrmmetrischen Kojrpern." Dissertation Braunschweig 1945, Of. H.
Schlichting, Grenzschicht-Theorie, p. 204.






NACA TlM 1415


With introduction of the further parameters


gO = and n = (2


as well as

8_xyr (23)

and


vx R dxl

one obtains from (21_) the following differential equation for O(x)

=~E G(K,AL) (5
ax U

Therein is

G(K,a) = gO 2a (26)

a universal function of the two parameters K and C.

This function has been determined already by W. Dienemanal4 in the
calculation of the temperature boundary layer on a cylinder (two-
dimensional problem).1) For the temperature distribution in the boundary
layer there we chose the same polynomial of the fourth degree as we did
for the azimuthal velocity distribution according to (17). According to
(12) we have

9 1l
x~Y u v~t
a, J U YO

Because of


x, y x,

1.Dienemann, "Berechnung des Warme~berganges nliarutote
Korpern mit konstanter und ortsveranderlicher Wandtemperatur." Disserta-
tion Braunschweig, 1951, Z. angew. Math. Mech. 55, 1955, p. 89.
15with the symbols according to W. Dienemann there apply the
identities Ht a gO and A K.






10 MACA TM 1415


one obtains after calculation of the integral with the velocity distri-
butions (16) and (17) the quantity gO as a function of K and n.
According to W. Dienemann, there results


g0(K,6) = gl~a + 2(LA)


15 140 180


90 84 5b0 1,080


(27)


(28)


A 2 : gl( )= + 3
10 10 G 15 02


)
140


1 1 1
S+ -- -
4 180 5


g2(n)120 a 180 a2 840 ,4 3,024 p


g0(a) as a function of a for various values of


The function


K is


represented in figure 3. Table I gives a few numerical values of the
functions gl(A) and g2 A)*


AND g2 ()


TABLE I.- THIE UN~IVERSAL, FUNCTIONS gl(n)


ACCORDING TO EQUATION (28)






















































16Cf. H. Schiliehting, Grentschicht-Theorie, p. 195-


NIACA TIM 141j 11


(c) The Momentum Equation for the Meridional Direction

FrL~ther transformation of the mometum equaion for the meridjional
direction yields, if one introduces, according to Iiolstein-Bohlenl6 and
analogous to (25)


v


dx


(29)


the following differential equation for Z(x)


dZ= F(K n
dx U




F(K~,.d = 2 f (2Xi + Y:)- g gRI h
R dx Uf0~-


( 50)


Therein


(3l)


G(K,C) in (26j) a universal. function of the two
Individually, the following relationships apply:


exactly as
K and 12.


parameters


f0(K) 9x K_ K2
8x 315 945 9,072


fl(K)- x_ K
6K 10 120


8x* fl(K)
2(K)
4, f0(K)


($2)


6x


s
x


s
x
6x


f0(K)


(35)


f3(K) =_0


9Ux~~ a~l~


O )f(K)


(36)











6, 126

The above functions of K are already known from the calculation of the
boundary layer of the two-dimensional case.17

The connection between Z and. K results from (18), with considera-
tion of (29) and (32), and is

Kf02(K) =~ Z 6 + '0 dR g


Taking X = ZU', from (29), into consideration, one may write this because
of (32) also in the form


xE + -- P' =X* =K7 K_~ ~\g0 (39)
U R U' Sl 945 9,072~

In figure 4 the universal functions f0> f2> f, and K a-re
represented as functions of X+. At the point of separation, for arbi-
trary rotational velocity, one will have, because of K = -LB, the
parameter X* = -0.1567. At the stagnation point, without rotation,
K = 4.716 and :Ir* = 0.05708, whereas with rotation the values at the
stagnation point are dependent on the spin parameter v0 U (cf. the
following section). From (58) the form parameter K can be determined
when Z is given. Furthermore, for the later calculation a connection
between the parameters 6,gO, 13 Z, and K is needed. There results
according to (22), (25), (29), and (32) as follows


OgO(K,n) = f0g(K) (40)

The two differential equations (25) for 93(x) and (30) for Z(x)
are two simultaneous differential equations coupled by the universal
functions G(k,a) and F(K,n). In the case of the nonrotating body,
VO = O, the coupling is eliminated since then, according to (31), the
function F becomes independent of C and remains dependent only on K.


17Cf. Hi. Schlichting, Grenzschicht-The~orie, Chapter XII.


NACA TJM 1415



































- O20" L 0 2 =


__


NACA TIM 141'j


In this case, one can first determine Z(x) from (50), and sulbsequently
S~x) from (25). This solution for 8 has it is true no physical
significance. It serves merely for giving the: limiting value for
vanishing speed of rotation.,


(d) The Initial Values at thez Stagnaion Point

At the stagnation point where~ U = 0, the two differential equa-
tions (25) and (50) have a singularr vaue since in. both equations on
the right side thei denominator vanshes. In order to obtain at the
stagnation point initial slopes of finite magnitude, d6/ldx and dZ /dx
finite, the numerators also mut disappear iLn these two equations for
the stagnation point. This requirement yields the initial values of
the parameters KO and 60O at the stagnation point. For the~ potential
flow there applies at the stagnation point


R= 1
dx


x--4: U(x) = U0'R = aR


(41)


The initial values of the meridional equation are
F = 0 according to (31)


obtained from


f30 2XD


With


according to (38)J with


KO 002 D 2c


.


according to (36), and
brief calculation



KO .
1 /,2
1 +


hO according to ( 7), there results after a




2 + l gK


(42)


157 29 1 2
210 ,2 ,4


-25p






NACA TM 1415


For a given speed of rotation m/a, this is the first equation between
the initial values KO, and 60. For the case without rotation, a = ,
the boundary-layer thickness ratio 13C drops out from this equation,
and an1 equation for the initial value KO only remains which reads


2 + -Kg_ + 29 -K2+ __ 24O = 0


(45)


The physically useful solution of this equation is


=~ KOO= .716


(44)


as known according to S. Tomotika.

For the initial values of the azimuthal equation, one obtains from
G = 0 according to (26)

280 O,60 ~- a KO*0g =


Because of


kaO2 0~2 2


a KO,60= 4;ae0


according to (24) and


according; to (18) and because of (27)
mediate calculation

KO

1 +o 2 gl 0)


one obtains after a short inter-


1
+ K(OE2 0 02


(4S)


For a given speed of rotation m/a, this is the second equation between
the initial values KO and nO*


(KO m=0


,fi 2 1-
Kg aB, 1+I-
v \ai]






M~ACA ITM 1415


For the case without rotation, a = O, one obtains from (45) with
KO = KOO = 4.716 according to (44) for the initial value of

6~0 m 0= 00 the equation

1 9.432C002 Pcl i OO + 4.716g2 00 (46)


Bence results with gl(n) and g2(a) according to (27) and (28)


n00 = 0.915 (47)

The ratio of the boundary-layer thicknesses A = by/6x for the azimuthal
and meridional velocity distribution therefore lies near I which is
physically plausible.

The two equations (42) and (4;) now represent, for prescribed
angular velocity m/~a, two equations for the initial values KO and GO*
A solution was obtained by determining from both equations the values of

KO T1 + 2 as a function of 60 for various fixed values KO'
Rence, the initial values indicated in table 2 result. These values are
presented in figure 5 as a function of m/a. It was found that for
values of m/a > 0'.815, no usable initial values of KO and SO exist;
that is, our method fails for these larger values of o/a. The limit
beyond which our calculation method fails coincides with the value K = 12
of the form parameterl8 The initial values XO, O*, and g00 deter-
mined fromn the initial values KOj and 60, are represented in figure 5
and table 2, as a function of oj/a.


18For K( > 12, because of the effect of the centrifugal forces, it
is entirely possible in the present case to obtain velocity profiles
with ufU > 1.













a


TABLE 2.- ~INITIAL VALUES AT THE STAGNATION POINT


0 4.716 0.915 5.71 5.71 0.0629
.221 5 .908 5.71 5.99 .0652
.454 6 .882 B.T 6.89 .0640
.679 8 .858 5.69 8.52 .0651
.785 10 .781 5.69 9.19 .0661
.815 12 .726 5.69 9.49 .0664


19See footnote 5 on page 2.


NTACA T1M 1415


2


100 XO+


100 XO


Finally we obtain the initial
manner with U0' = a


value for Z


simply in the following


zo = 5
a


( 8)


The initial value for 8 results with GO = 2gOO according to (24) as


S=1 g00
26= a


(49)


The expression for the velocity distribution used here (parabola of
the fourth degree for u and v) is different from that of our former
calculationl9 for the rotating disk in a flow. It must be expected,
however, that the boundary-layer parameters of the rotating; disk in a
flow should agree approximately with those at the stagnation point of
the rotating body of revolution if both methods are to yield usable
results. We give this comparison for the mcomentum-loss thickness in
x-direction (8) at the stagnation point and for the meridional component
of the wall shear stress at the stagnation point. The dimensionless
momentum-loss thickness at the stagnation point is according to (29)
with Ux=0' = a


4x0I~ 0 ~


(5O)







NACA TM 141',


The meridional component of the wall-shear stress at the stagnation
point iTO x=0 = TPO is according to (5), (16), and (j2)


"u;o-I2 +Y~i (51)


The values calculated accordingly are compared with those of the rotating
disk in figure 6.20 The agreement up to the validity limit of our cal-
culation (w/a = 0.815) is quite satisfactory.

Hence we conclude that our present calculation yields satisfactory
results in the entire range 0 I m/a 5 0.815.


4. TORQUE AN~D FRICTION~AL D1RAG

(a) Torque


The entire torque of the body of revolution may be easily ascer-
tained from the results of the boundary-layer calculation in the
following manner: The contribution of an element of the btody of revolution
with the radius R.(x) and the arc length d~x is (fig. 7)


E = -2nR2 iygdX

and thus the total torque


M =-2xS A YR2dx (52)


where xA signifies the arc length from the stagnation point to the
point of separation. Taking the momentum theorem for the circumferential
direction (ll) into consideration, one obtains



20Whereas the values for the wall-shear stress could be taken
directly from the report referred to in footnote 5 (p. 227, table 2),
the values for the momentum-loss thickness were calculated subsequently
with application of equation (8) with the velocity distributions indi-
cated there.







NACA TM 1415


M =2non [R ,UG A = 2npRA IUAB ? xyA


where the subscript A denotes the values at the separation point.
From the boundary-layer calculation, one knlows the value of the momentum
thickness due to spin at the separation point in the dimensionless form



R, v

where E, is assumed to denote the radius of the maximum cross-sectional
area.

If one Lntroduces in the same manner as for the rotating disk -
a dimensionless spin coefficient by






one obtains






where Vm = Rmm is the circumferential velocity of the maximum cross-
sectional area. Since, as the completely calculated examples show, the
dimensionless momentum thickness due to spin B varies at the separa-
tion point only a little with V /Um, eg is in first approximation
proportional to Um V, and inversely proportional to the Reynolds

number UmR d~v.

For the case of the rotating disk in a flow, with the radius R, = R,
one obtains because of RA = R, UA = aR from (56) in combination
with (54)






NACA T1M 1415 19


and with the numerical value



=~ m/Ve~2I~a=0 = 0.2$1


according to (49)





in very good agreement with the former investigation where the num~eri-
cal value is 3.17.


(b) Frictional Drag

The frictional drag of the rotating body of revolution may be deter-
mined by integration of the wall-shear stress components 7 0 A sur-
face ring element of the body of revolution with the radius R,(x) and
the are length dx (fig. 7) yields th~e drag


dW = 2nRTxd (~b)


Therein TI is the coordinate measured along the body axis. Integration
from the stagnation point x = 0 to the separation point xA, where

TXO = 0, yields

w =2n7 ,RaX (59)

We shall refer the drag to the maximum cross-sectional area nR 2 and
define the drag coefficient


0, (60)

2


21Cf. footnote 5 on page 2, equation (49a).






NACA TM 14151


Since we obtain the wall-shear stress in the dimensionless form


pU= T (61)



we ma~y write for tlhe drag coefficient


0 = 4 (62)



5. EXAMPLES

(a) Sphere


As the first example, the friction layer on the rotating sphere was
calculated. When, R, signifies the sphere radius, x the are length,
and xlR = Q, the center angle measured starting from the stagnation
point, the radius distribution is


R(x) = Rm sin 9 65

and the theoretical potential velocity distribution


U(x) = IU, sin rp (6 )


Thfe velocity gradient at the stagnation point is


dx =0~ 2 R, U

and thus










Since, according to the explanations in section ), the calculation can
be carried out only for m/a 5 0.815, we must limit ourselves to

V, U- <3 0.815 = 1.22.


The solutions are obtained by numerical integration of the two sim~ul-
taneous differential equations (25) and (30) for the two cases V,IUm = 0
and 1. The calculation scheme is given in table 3. The results for
further values of V /Um could hence be obtained conveniently by inter-
polation. The case Vm U, = 0 (nonrotating sphere) agrees with the case
of Scholkemeyer22. The results of the calculation are represented in
table 4 and figures 8 to 12.

TABLE :.- CALCULATIONS SCHEME FOR THE SOLittOI O THE TWO
SIMULTANEOUS CIIIFERE21IAL SQUATIONS (25,) AND (J))

a prrecribea
ollt.) dU K f
9 aR~) =R' U(x) -=U' v0 = J
dz dr


To be calculatePd line by line


TABLE 4.- POSITION OF SEPARATION POINT AND OF THE TO~RQUE TN


spin Separation
Torque,
parameter, point,





o 108.2 9.15
.25 108.0 9.14
.50 107.5 9.06
.75 106.2 9.03
1.00 104.9 8.95
1.22 los.) 8.85


22Footnote 13 on page 8.


NACA TIM 1415


Initial
givn
(eq. (48))


Eq. (59)


(table 2)


Fig. 4 | Fig. 4 |Fig. 4


Given
P--- body fonn and potential flow -


To be caClculted
( LinL~e by Lne )


FOR THE ROTATING SPHERE IN A FLOW


DEPENDENCE ON V, Um






NACA TIM 1415


Figure 8 gives the variation of the form parameter K of the merid-
ional component of the velocity distribution in the boundary layer. The
initial values KO at the stagnation point are Lanediately given in
table 2 with equation (0)). At maximum velocity, 9 = 900, K is,
according to (18), equal to zero for all VIUm, because in a sphere at
the point where dU/dx. = O, also dR/ldx = The value K = -12 gives
the position of the separation point A. In figure 8 the variation of
the boundary-layer thickness ratio a = B 8,x is also plotted; it always
lies close to I and also changes only little with V~JUm. Figure 9 shows
the variation of the momentum thickness due to spin 4gy. The curves for
various V, Um almost coincide. The same is true for the momentum-loss
thickness 9, and the friction-layer thiicknesses by and by. Figure 10
shows the variation of the meridional and aziauthal component of the
wall-shear stress. The meridional component TXO increases with the
spin coefficient V, Um only a little whereas the azimuthal. compo-
nent TYO in first approximation is proportional to the spin coeffi-
clent V, Um. The position of the separation point as a function of
the spin coefficient VUml" is given in table 4. For the nonrotating
sphere cpA = 108?.20, and for V@Um = *22 the separation point shifts
forward to VA = 103.5o. This displacement of the separation point
because of the rotation is due to the effect of the centrifugal forces
and is, clearly, immediately plausible. For the velocity profiles
behind the equatorial plane ((9 > 900), the centrifugal forces have the
effect of an additional pressure increase in flow direction and there-
fore cause the separation point to shift forward. In figure 11 the
dimensionless torque coefficient formed according to equation (56) is
represented as a function of the spin coefficient Vm/U,. (Cf. table 4.)
One sees that the proportionality with V /Um is fulfilled with very
good approximation. Finally, figure 12 shows several velocity profiles
in photographic reproduction.

A sphere is rather unsuitable for the comparison of the theoretical
calculation with test results, because of the large dead-water zone which
has the effect that even in the case of the nonrotat~ng sphere the posi-
tions of the separation point according to theory and to measurement do
not agree when the boundary-layer calculation is based on the potential-
theoretical pressure distribution as we have done here. A valid compari-
son regarding the influence of the rotation on the behavior of the fric-
tion layer can be made only for a slender body where no noteworthy
dead-water zone develops. Nevertheless we mention here the measured
results of S. Luthander and A. Rydberg 23. In figure 13 the drag coef-
ficient of the sphere in dependence on the Reynolds number Re for
various values of V, Um is givell according to these measurements. For
the nonrotating sphere, VmlUm =0, and up to values of V,/U, to about
3, the curve c, against t Re shows the characteristic variation with
2}Footnote 2 on page 1.






NACA T1M 1415


the familiar sudden drop at the so-called critical Reynnolds n~umber. It
is known that for Reynojlds numbers below the critical Reynolds number
the friction layer undergoes laminar separation, and for numbers above
the critical Reynolds number, in contrast, a turbulent one. In the case
without rotation, the laminar separation point lies at about (O = 81",
the turbulent one, in contrast, at about (p = 1100 to 1200. The meas-
urements with rotation show for Vm/U, = 0 to 1.4 a shifting of the
critical Beynolds numbers toward higher values of Re. This shifting
of the critical Reynolds number to higher values for small V,/Um is
probably brought about by the fact that for Vm/Um, = O the laminar
separation point is shifted from (p = 810 to higher (p-values, with the
separation still remaining laminar, however. Only for higher values of
V,IU,, the rotation causes the friction layer to become prematurely tur-
bulent, and it then has the effect of a trip wire whereby a shifting of
the critical Reynolds number to lower Reynolds numbers takes place.

Whereas in our theoretical calculations a forward displacement of
the separation point occurs, due to the influence of the rotation, the
measurements for small values of V /U, indicate a shifting of the
separation point toward the rear. On the basis of the effect of the
centrifugal forces, this must be expected, if one takes into considera-
tion that in the case without rotation the laminar separation point lies,
according to theory, behind the equator, according to measurement, how-
ever, ahead of the equator. In both cases, the separation point is
shifted toward the equator by the effect of the centrifugal forces as
is to be expected, at least for small Vm Um, as long as no premature
laminar/turbulent transition has been produced by the rotation.


(b) Bodies With a Base (Half-Bodies)

As a second example we shall now treat the so-called half-body
(body of revolution I) which originates by superposition of a transla-
tional flow on a three-dimnensional source flow. If one denotes by Rm
the largest radius at infinity, the following parametric representation
for the geormetrical data of the body24 is


-= sin (o
Rm 2




24For these relationships as well as for the numerical calculations
of section Sa, I am indebted to Dr. E. Truckenbrodt. The example calcu-
lations of sections 5b and c are taken from the thesis of K. B. Gronau,
1952.







NACA TM 1415


S= tan sel 1 sin2 9E + F ,(67)25


Here is the angle measured from the forward stagnation point.
F and E 3-re the incomplete elliptical integrals of the first and
second kind for the modulus a = OO

The velocity distribution is


U 2 sin 2E ,1- sin2! 2 (6g)
Um 2V 2

The form of the body anrd the velocity distribution are represented in
figure 14. Thiis figure shows, for various values of the spin param-
eter V IU,, the variation of the form parameter K with the distance
along the bodly. One sees that already for VU, Um l .j only positive
values of K result. This means that due to the rotation the laminar
friction layer has become more stable because in the present case the
centrifugal forces accelerate in the direction of the flow and thus
have the effect of an additional pressure drop. We shall forego dis-
cussing here all the results. In figure 15 we have represented the

torque coefficient rE eg against the length L R, of the half

body. Moreover, the asymptotic solution was drawn in for comparison;
one can derive for it the relationship





Aside from the torque, the frictional drag also was determined.
Figure 16 presents a compilation of the torque coefficient and of the
drag coefficient in dependence on the spin parameter Vm Um for vari-
ous body lengths L R It should be emphasized that the drag coeffi-
cient is increasing about quadratically with the spin parameter which
is in qualitative agreement with the test results that have become
known so far.


25E and F signify


FG~I) = 09 1 d sin2 6 and E~ ~ = i24d






NACA TM 1415


(c) Streamline Bodies

As further examples we also calculated two streamline bodies of the
thickness ratio D/L = 0.2 (bodlies of revolution II: and III). The body
shapes and the pertaining velocity distributions were taken from the
report of A. D. Young and E. Young2i (fig. 17). The body of revolution II
has as a meridional section a normal profile; the body of revolution III,
in contrast, has a laminar profile withi the velocity maximum lying rela-
tively far downstream. Of the results, figure 18 shows the torque coef-
ficient and the frictional drag coefficient as a function of the spin
parameter V JU,. In both cases, there are not large differences between
the bodies. For the rest, the variation is similar to that in the case
of the body with a base. In figure 119, the position of the separation
points is shown as a function of the spin parameter V ,U,. In agreement
with the values for the rotating sphere (cf. table 4), the separation
point shifts forward with increasing rotational speed. This displacement
is larger for the body of revolution II than for the body of revolu-
tion III which is made understandable by the position of the velocity
maximum. Finally, we gave for the body of revolution II a graphic repre-
sentation of the velocity distributions in the friction layer for the spin
parameters V /Um = and VG/U, =1 (fig. 20). Fr~m it one sees that
ahead of the pressure minimum the meridional velocity component does not
vary noticeably due to the influence of the rotation whereas between the
pressure minimum and the separation point the influence of the rotation
is considerable.


a. SUMMZARY


A calculation method is given by which the flow about a rotating
body of revolution in a flow which approaches in the direction of the
axis of rotation may be determined on the basis of boundary-layer theory.
The investigations yield a contribution to the aerodynamics of a pro-
jectile with spin. The calculation is carried out for the lamrinar
boundary layer with the aid of the momentum theorem which is stated for
the meridional and for the circumferential. direction. The performance
of the calculation requires the solution of two ordinary staultaneous
differential equations of the first order. It yields, in addition to
the boundary-layer parameters, the frictional drag and the torque as a
function of the dimensionless spin coefficient Vm/Um = circumferential
velocity free-stream velocity. The displacement of the separation point



2A. D. Young, E. Young, "A family of streamline bodies of revolu-
tion suitable for high-speed and low-drag requirements." ARC Report 2204,
1951.






NACA T1M 141'5


with the spin coefficient also is obtained. As examples, the flow about
a rotating sphere, about a body with a base, and about two streamline
bodies is treated.


Translated by Mary L. Mahler
National Advisory Cormmittee
for Aeronautics





NACA TIM 1415 27




U (x)

a Lu-x,z)



''R ( I x


o'









b c U(x)



V (x,z)


Rc, R -?x)


Figure 1,- Explanat~ory sketch.



















































8 10 0 20 0 22 C


Figure 3.- Tie universal function L = s7" x as a function of g~ = xy /x
according to (2r7) and (28).


K=-19 -to8 -4-2 O 2 4 ~6 8 10 2


I ~ _


. .


NACA 'IN 1415


Z.
t=-


s Z
t


Figir~e 2~.- Velocity distributions in meridional and in azimuthal direction.


04


0.e


00cnn /


on r~


016~ 'nf






NACA TM 14c15


Figure 4.- The universal functions fil = 9,5 x, f2 = Sx*Sx, f3 = rx0 x/9>,U

and K as a function of :l* according to (32) to (39).






NACA TM 141)


O 0.2 0.4 0.6 0.8 1.0


Figure 5.- The initial values at the stagnation point h0, KO' 0O, and x 0

















2


NACA TM 1415


UR- xo ~


1.6 E


T---
4UR
4V


-~--


RI


O.8 ~


0~4t---


According t

H. Schlichting and E. Truckenbrodt
0.4 0.6 0.8 1.0


Figure 6.- Momentum-10ss thickn~ess a, and wall-shear stress r,0 at the
stagnation point. Comparison with the values of the rotating disk in a
flow according to H. Schlichting and E. Truckenbrodt.


Figure 7.- Sketch for calculation of the torque and of the frictional drag.






NACA TM 14.15


Figure 8.- Form parameter K: a~nd boundary-layer thickness ratio
6 y/SX for the rotating sphere in a flow for different spin
parameters Vm/TJ.






NACA 'IN 1415


Figure 9.- Iv10menturn-10ss thickness due to spin 29 x for the rotating
sphere in a flow. Re = U. Rm/ *




NACA TIM 1115


-O~ pUm


O8 = LO
'I UmD


O 6 0.75






O.25





O 200 40* 600 80" 100.

Figure 10.- The wall-shear stress components in meridional and
azimuthal direction -rz0 and -ry0 for the rotating sphere in a flow.
Re = U.R m'






NACA TM 1415


vm
U00


Figure 11.- Torque


coefficient cy, for the rotating sphere in a flow.
Re = UmRm; -.






NACA 'lM 141'5


1 ?:
'1 \
I:s u!,
'~~


Figure 12.- Ph-otographic representation of the velocityr distributions
in the boundary lawyer of the rotatin~g sphere in a flow, Vm .u = 1,
A = separation line.


Figure 13.- Drag coefficients of the rotating spihere in a flow as a
function of the Reynolds number, according to measurements of
S. Luthander and A. Rydberg.






NACA TM~ 1415


x y


Figure 14.- Form parameter K for the rotating body with a base in a
flow (body of revolution I).






NACA I1 1415


Figure 15.-
a flow as


Torque coefficient cpJ for the rotating body with a base in
a function of the length of the body. Re = U Rm/v .

































r


~t~t~_


......---


NACA 'IN 16 15


VcM


vm
U,


I-ReCw


L
-6
R

4


- 2


Vm
Um,


1.0 1.2


O 0.2 0.4 0.6 0.8


Figure 16.- Torque coefficient cMn and drag coefficient cw for the
rotating body with a base in a flow (body of revolution I) as a function
of the spin parameter Vm/U,. H~e = U,Rm/V -






NACA '1M 1415


O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 17.- Body shape and potential flow for tw~o streamline bodies
(bodies of revolution L1 and Il); D/L = 0.2.










IVm ~ec,
Um,


----~-~--t


V,
Um,
0.2 0.4 0.6 0.8 1.0


4




Vm
Um


NACA TM 12C1'


20











8



4



C


Re c sp


0.2


1.0


Figure 18.- Torque coefficient eM and drag coefficient ew for the
two streamline bodies according to figure 17 as a function of the
spin parameter Vm/U,. Re = U,R,/V.






NACA TM 1415


Body of revolution II


Body of revolution U.


O 0.2 0.4 0.6 0.8 1.0


Figure 19.,- Position of the separation point A for the two streamline
bodies according to figure 17 as a function of the spin parameter
Vm/IU,. Position of the separation point: o for Vm/: U. = 0; for







N~ACA 1I 141+ 1 54






o 0o





Or


O OO




rD~ IS


oa o
od

o9
Ic I



a~ a o



O so O
.J 8 11- i I.. O L --:


NACA Langley FLEld. Va.














































































































































































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