The flow of a compressible fluid past a sphere fixed in a uniform stream is calculated to the third order of approximation by means of the Janzen-Rayleigh method. The velocity and the pressure distribution over the surface of the sphere are computed and the terms involving the fourth power of the Mach number, neglected in Rayleigh's calculation, are shown to be of considerable importance as the local velocity of sound is approached on the sphere. The critical Mach number, that is, the value of the Mach number at which the maximum velocity of the fluid past the sphere is just equal to the local velocity of sound, is calculated for both the second and the third approximation and is found to be, respectively, Mcr=0.587 and Mcr=0.573

Kaplan, Carl

English

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TECHNICAL NOTES
NATIONAL ADVTSORY _ ,r _- OLMI_*J_E_ FOR AERONAUTICS
No. 762
i
": " 1940
https://ntrs.nasa.gov/search.jsp?R=19930081579 2020-06-17T23:13:36+00:00Z
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL NOTE NO. 762
_ THE FLOW OF A COMPRESSIBLE FLUID PAST A SPHERE
By Carl Kaplan
SUMMARY _
i:__ The flow of a compressible fluid past a sphere fixed
in a uniform stream is calculated to the third order of •
• appro,,ximation by means of the Janzen-Rayleigh method.
The velocity and the pressure distributions over the.sur-
face of the sphere are computed and the terms invol_ing _.:
_ the fourth power of the _aeh number, neglected in Rayle_¢_h.S
i. calculation, are shown to be of considerable importance as
the local velocity of sound is approached on th_ sphere. _:_
. The critical Math number, that is, the value of the Mach
number at which the maximum velocit.v of the fluid past the
sphere is just equal to the local velocity of sound, is
calculated for both the second and the third approxima-
tions and is found to be, resoectively, Mcr = 0.587 and
Mcr = 0.573 ....
INT RODU CTI ON
i! • The irrotational flow of a compressible fluid pa_%.a!:i !__(cir6ular.cylinder and a s_here was first calculat.ed b
_bJanzen ( "re_erence l) and by Rayleigh (reference 2
.metEod consisted in obtainin_ _a correction term.t
compressible-fluid solution, but the results
to the terms involving only the sguare of t_he Mac!
:.,Recently, the author (reference 3)and Imai (refer_ence
exte.nded the calculations for the circular
eluding the terms involvin._ the fourth Dower o:
number. These higher-power terms, neglected in the ear-
lier calculations, were •found tO. be of considerable _mpor-:. •...._:.:
tance as the local velocity of sound is approached on the
surface of the cylinder. .It has therefore been thought
!_ • worth while to extend thu calculations in a similar manner
for the flow past a sphere.
.3:
" 2 N.A.C.A. Technical Note No. 762 . _ _:
i A!qALY S IS
_Pr._.!._%__nGr__&ev_elop_rn._en.ts.- The flow is assumed to be
" uniform _t a _reat distance from the sphere and. the mo ....
i; : tion to be everywhere irrotational and steady. Then, with
2 dp
o = -- (!7
-_i dp
the equations of motion reduce to _:
Ji
dp 1
C 2 _ --p _ d(v_) . (2)
where v is the fluid velocity; c, the local velocity i:,
of sound; p, the pressure; and p, the density. Then,
assuming the adiabatic relationship between p and p, . _.
it follows by integration of equation (2) that
i a _ P i U2 + 7 _P.9v + =- (S_
_ - ! p 2 7 _ ! po
and from equation (I) that
[ - (1 v_-] (4)cs= c _ I + _{ I M a ..... r::! o 2 U2 )
where U is the ve!ocity of the undisturbed stream; :Po,,
Pc' and Co, the corresponding quantities in the undis-
turbed stream; M (= U/co), the Mach number; and _, the - ,
ratio of the specific heats of the fluid.
Since the fluid motion is irrotational, there exi:sts
a velocity potential _ a.nd the eauation of continuity
: may be written as
a } i [av_ _ + +___+ _ _v_ 8_ 8v _ _
a=_ ay_ _ -_ % _ a_ a y ay a z _ (5)
Let r, 8, and q0 denote space polar coordinates and sup-
pose the origin of r to be at the center of a sphere of
radiu_ a and the initial line of 8 to be parallel to "
the direction of the stream° Then, designating by r the
..... + _
5
N.A,C.A. Technical Note No. 762 Z
ratio r/a, by _ the ratio _/Ua and, by v the ratio
v/U and taking into account the fact that the flows in
all meridian planes <P = constant are similar, equation(5) becomes
1 + _{ - 1 M e (I - v ) A{ = ! M_U 8_ 8v e
where
and
m= cos 8
It is now assumed that _ can be developed as a power
series in M s (reference 4) so that
-' /o / M4 + (8)-" # = + #IM 2 + _. ....
Since
v_: r___'__+r_(-_)_\3r] 8
then ::-
: v m = Voa + vl2M _. + _2M4 + .... (9)
where
=r_of ,-(_,_of "v°a vat / + g: a_-/ (ga):
v:_ = 2 .o3_,i_+. (9,b)8r 8r ra 38 38 /
4 N°A.C.A. Technical Note No, V62
_hen:'these expressions for _ and vs are inserted into
equation (6) and the coefficients of the same powers of M
oll both sides are equated,
a_o = o (loa)
z <a_o av° +-_ _ (_.ob)
_ = _ ar 8r r 8e 8e I:_
= --i-- (Vo_- i _r + 7 _- -_-)
B
l (8_ 8Vo_- + 1 8_ 0_8Voe+
.
• , • • I • • • • • • • • 6, • • • • • • • •
From these equations, any _iven approxim.s.tion _n clearly _'
depends only on the preceding approximations of which the
first one is the solution of Laplace's equation A_o = 0 J=for an incompressible fluid.
.... The first a_Drox_+_^_
...."-'......-_ ..............._ , Equation (lOa) is the dlf-
ferential equation __or tn.'_. velocity L.otential _o for the
flow of an incompressible, nonviscous fluid and may be
written as
t8 <r_. o + 8 1- I_m) =0 :.,;.::, -'
The solution of this equation is known to be
co / BD,_,_ .0 : n=O_i' <Anrn + rn_ Pn(b) "
where Pn(b) _s Legendre_s polynomial of order n and,
A n and Bn are arbitrary constants. In the case of a
sphere of radius a, supposec% fixed in a stream of uni-
form velocity U, the boundary conditions to be satlS-
fled are s.s follows:
_o
- -Sr - normal velocity = 0, at the surface of the
sphere

6 N.A.C.A. Technical Note No. 762
A
v
rm+s pn(_)
= (m + 2) (m + 3) - n(n + i_
, Hence, the solution is given by
_ = Z An rn + __nr-(n+i) +
n=0 L (m+2) (m+3) -n(n+!/
except when (13) <
m = n - 2 or m = - (n + 3)
Accordingly, the solution of equation (1"2) is
2
_i = (Air + _lr- )P!(_) + (£3r3 + _3r'4)P_ (_)
- y_ r +--- r-_ (_)176
Since _o already satisfies the necessary boundary condi-
tions of the problem, the higher approximations _z,
_. , •.. must satisfy the conditions
3_ _ o --- = e _
8r ' 8r ' """
for both r = 1 and r = c_. Hence, after a simple calcu-lation,
\
! 27
Ai = A3 = 0 and Bi = _, B3 = _
Therefore,
._ _i -- r 2 -_ r- +-- r-8 _i(_)24
+(..3 _ 27 Z 3 ) ...._ r- +-- r-4 ___ r-_ +--- r-_ _(_) (14)55 i0 176
i
The third a_Droximation.- Substituting from equations
(ga), (9b), (ll), and (12) into the right-hand side of
equation (10c), it follows after a strai_hoforward__ calcu-lation that

!8 N.A.C.A. Technical Note No. ?62
m
where
BI = 0.03566(?-1) + 0.32614
Bs = 0.02268(7-1) + 0.74107
B5 = -0.18261(?-1) + 0.212].6
From equations (Ii), (14), and (16), it may be easily
calculated that the velocity of the fluid at the surface
o{ the p'
- s. nere is given by
- ! _8_ = Z sin 8 + 1 (989 sin @ - 1215 sin 38)M er 2 V57.t.5
t
4- (0.I0572 sin8- 0.16008 sin Z@+ 0.0643,_ sin 5@)M _
* (7-i)(0.01168 sin6- 0.02475 sin 38+ 0.02582 sin 5e)M4 _:_
The _ppropriate value of _/ "_'or air bein_ _/ = 1,408, this
equation can be written
. _z_8£ "_ _in e + _z__ (989 _in e - z_15 sin se)M _ ,'r 3e : _ 7o_o
+ (0.11048 sin8- 0.17018 sin 58+0.0?489 sin 5e)M 4 + .,.
_(ZV)
The critical value o£ the Zach number for th 9
.... -_ _s]_h__eLre.-When the velocity of the fluid equals the velocity
of sound at any point in the field, a change in the ty]ae
of f:low occurs and _otential flow may no lon_er exist.,:. It _.-
will, therefore, be of interest to determine the value of
..:_ U/c o at which the velocity of the fluid just equals _._.he .
local velocity of sound in the field of flow past a re."
This velocity is first attained on the sphere at the nt °,
, of minimum pressure or of maximum velocity. According :%0
equa'_ion (17), with 8 = w/2, the variation of the malx__-
-- mum velocity with the Mach number M(= U/co) is give[n!:;by
Vmax= 1.5 + O.31ZOVM 2 + 0.Z5555 M4 -,-
... (i8)
Now, desi_natin@ by c _ the velocity of sound at a
point where the velocity of the fluid is equal to the lo-
cal velocity of sound; that is, at a ooint where
V = C = C _
<'?;i::,
N,A.C.A. Technical Note _o_ 762
_ • it follows from equation (4) that
c'2 = _+ 1 2
It is to be noted that this oquation is essentially
_ernoulli's equation and does not depend on the shape of
the body immersed in the fluid.
In conformity with the usage in this paper, c • is
replaced By _c_*(= v*) and the equation (19) becomesi _ U
/
.. v ._- _ 2 1 7 - 1
7 + 1 MR + (20)Y + 1
The so-called critical value of U/c o (= Mcr) for the
flo_ past a sphere is then obtained by putting = v*Vmax
TabLes I and II show, respectively, the values of Vma x
and v* calculated from equations (1B) and (20) for vari-
_ ous values of the Mach number These values are alsoJ_shown
in figure l, and the points at which the curves intersect
give the corresponding values of the critical Mach number
Mcr. The critical values are, respectively, for the sec-
ond and the third approximations_ Mcr = 0.587 and Mcr =
0.57_. The corresponding critical values of M for the
case of an infinitely long circular cylinder are Mcr =
0.420 and Mcr = 0.409 (reference 3).
-T-he___velqc!tyand the_]_ressure distributions.- l_f_ Po_ 9 _
Pc, and co are the pressure, the density, and the velec ....
ity of sound in the undisturbed stream, then the density
P of the fluid at a point where the velocity is v is
gi yen by
1
[ yP 1 + 7-1 _ 7-Ii_ Pc : --_ M2(1 - v )
and the pressure p at this point is
P P _ _ _ 1 __7_
or
1 x .... ! + M2(I - v2) -1 (2S)
. _o u SM_ 2
I0 N.A.C.A. Technical Note N9. 762 .
For an incompressible fluid, M = 0 and expression (23)
reduces to
P - Po
/_ _1 Po U'_ = 1 - v a (24a) ,.2 ii !i
.
and, for a compressible fluid with 7 = 1.408 and M = _:_;_0.57 ,,....
-- , - (24b)
-o ..a = 4.372 l + 0 06628(1-v a) s,45!2
The values of v to be used in equations (24) are ob-
tainel from equation (17). Thus, for the incompressiblefluid
v = _ sin 8 (25a)
,_ and, for the second and the third approximations to the•.-{!_}i,
;.i_}j_ compressible.fluid with M = 0.57
v = I,'54564 sin 8 - 0.05607 sin 38 (25b)
v = 1.55731 sin G - 0.07404 sin se
+ 0.00791 sin 5,e -(25c)
Table III lists the values of v computed from these
formulas and tabl& IV _ives the corresponding values of
P - P._o obtain@d _rom equations (24) Figures 2 and 8;
-i PoU2 ,,
show, respectively, the _raphs of the velocity andthe
pressure _istributions of tables III and IV.
Langley Memoriai Aeronautica.1 Laboratory,
National Advisory Committee for Aeronautics,
Langley Field, Va., May 4, 1940.
y• N.A.C.A Technical Note No. 762 Ii ....• _!_
[ii,
" REFEREIT CE S i
i. Janzen,T 0.: Beitr_g zu einer Theorie der stationaren
Stromung komDresslbler Fl_{_si_keiten. Ph_,s. i_
_i Zeitschr. , 14. J,ohr_., Nr. 14, !S. July 19!Z S. _
2. Raylei_h, Lord: On the Flow of Compressible Fluid
:_st an 0b_tacle. Phil. Mag., ser. 6, vol. 32,
no. 187, July 1916, pp. 1-,6.
• Kanlan, C_rl: Two-Dimensional "Subsonic Compressible
Flow past Elliptic Cylinders. T.R. No. 624,
}[.A.C.A., 19_8.
_! 4. Imai, !sac: On the Flow of a Comoressible Fluid past
a Circular Cylinder. Proc. Phys.-Math. Soc. of i-:
Japan, ser. ,$, vol. 20, no. 8, Aug. 19_8.
:_ 12 N,_,.C.A. Technical Note No. 76? ,
TABLE I
M Vma x
Second Third
approximatfoh approxTmation
q
0 1.5 1.5
• ,I 1.50051 i.50317
.2 1.51252 1.51Z09
p "
.8 1".52818 1.5ZI05
i .4 1.55009 1.55919
.5 1.57827 1.60049
}!
.6 1.61271 1,65878 _
.7 1.65Z4 1.73877
TABLE II
o._ , o._ 1o._ 1o._ f-o.L--
- _--- 4...........-q ..........r..............F F--',-a-
v_qg.1228I4.57ssis.0651Ia.sz52i1.86s6II.57s7Iz.s655;: [ 3_ _J...............L ...... t.......... L ..... .--:-
N.A,C.A. Technical Note No. V62 13
. • "!

..... ,i_;,::


The flow of a compressible fluid past a sphere

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