The theory of Taylor and Maccoll (Ref,1) gives the surface pressure on an infinite cone in supersonic flow as a function of the cone vertex angle and the free stream Mach number and static pressure for a gas of vanishing viscosity. When a slender conical probe is used together with an impact pressure probe to determine the static pressure and Mach number in a low density gas stream, it is desirable to have some theoretical estimate of the effect of viscous boundary layer on the probe readings. Theoretical and experimental results with respect to impact probes have been presented in Refs. 5 and 6. A simple approximation for a conical probe based on linearized supersonic flow and compressible boundary layer theory is presented here

Lin, T. C.

Schaaf, S. A.

Sherman, F. S.

NASA Technical Reports Server

CASE FILE COPY REPORT NO. HE15C80
TECHNICAL REPORT 
UNIVERSIT
	
CALIFORNIA 
	
DEPARTM
	
GINEERING 
	
BERK
	
FORNIA 
INSTITUTE OF ENGINEERING RESEARCH 
BOUNDARY LAYER EFFECT ON THE SURFACE PRESSURE OF AN

INFINITE CONE IN SUPERSONIC FLOW 
By
T.C. LIN 
JHS UOCUMI$$1 O LWfROM 1H flL$ O	 S.A. SCHUF 
W11OML IWSOY COMITTT1 co	
., SHRM&N 
LANG.EY WmAhJT!AL tfMRAl0Ay 
IMGLEYFWA. Awplom. 
REflN TO ml AE AD[iRSS. 
AS WJV:
$1TT no Lr' 
TrLt,
SERIES NO .............................. .19 
ISSUE NO................................. 
MARCH 5 ., 1951. DATE........................... 
https://ntrs.nasa.gov/search.jsp?R=20150018653 2019-08-31T06:05:04+00:00Z
CONTRACT NAw6OO4 
REPORT NO. 1E=1500 
Series No. 194 
March 5, 1951
BOUNDARY LAYER EFFECT ON THE SURFACE PRESSURE OF AN

INFINITE CONE IN SUPERSONIC FLOW 
By 
T.C. LIN 
S.A. SCHAAF 
F.S. SHERMAN 
FACULTY INVESTIGATORS g 
R0G0 FOLSOM, PROFESSOR OF MECHANICAL ENGINEERING 
S.A. SCHAAF, ASSISTANT PROFESSOR OF ENGINEERING SCIENCE 
Approved by
NOMENCLATURE 
Cp = pressure coefficient (dimensionless) 
M = Mach number	 (dimensionless) 
p	 = static pressure	 (microns Hg in Sections 3.0 and 4O)')
lbft2 
= cone surface pressure (microns Hg in Sections 3,0 and 4oO) ?	 otherwise 
p1 = impact pressure
	 (microns Hg in Sections 3,0 and
r = radius of axisymmetric body (cone plus boundary layer)
	 (feet) 
Re = Reynolds number	 (dimensionless) 
S = cross-sectional area of axisymmetric body (cone plus boundary layer) (ft2) 
a = distance along flat plate from leading edge
	 (feet) 
T = gas temperature	 (°Rankine) 
U = free stream velocity (ft/sec) 
x = axial coordinate (inches in Sections 30 and 40)	 (feet in Red 
= ratio of specific heats = 
3* = displacement thickness of boundary layer (in in Sections 3.0 and 40) ft0 
otherwise 
fr4 = gas viscosity = lb/sec/ft2 
= kinematic viscosity = /A/O = ft2/sec 
= gas density = slug/ft3 
0 = cone smi-vertex angle	 (degrees) 
SUBSCRIPTS 
Flow properties with no subscript are those of the undisturbed free stream. 
Flow properties with subscript L are "local" properties of the potential flow 
at the seam ofthe boundary layer, e0g0
	 * 
U  
Re  - 
Report No. ]E1500 
BOUNDARY LAYER EFFECT ON THE SURFACE PRESSURE OF AN

INFINITE CONE IN SUPERSONIC FLOW 
1 0 0 INTRODUCTION
The theory of Taylor and Maccoil (Ref,1) gives the surface pres-
sure on an infinite cone in supersonic flow as a function of the cone vertex 
angle and the free stream Mach number and static pressure for a gas of vanish-
ing viscosity 0
 when a slender conical probe is used together with an impact 
pressure probe to determine the static pressure and Mach number in a low density 
gas stream, it is desirable to have some theoretical estimate of the effect of 
viscous boundary layer on the probe readings. Theoretical and experimental re-
sults with respect to impact probes have been presented in Refs. 5 and 6 A 
simple approximation for a conical probe based on linearized supersonic flow 
and compressible boundary layer theory is presented here., 
20 ANALYSIS
The cone is oriented parallel to the undisturbed free stream, its 
axis of symmetry in the x'-direction and its vertex at the origin. The semi-
vertex angle is 0 
One considers the linearized supersonic flow to take place around 
a body consisting of the cone plus a displacement thickness of the boundary 
layer. The radius of this body is (see HYD 2653) 
r = x tan e +	 * sec 0	 - - - - - - - - - - - 	 ( i) 
Ref.2 shows that the integration of the laminar boundary layer equations for a 
cone in supersonic flow can be based on that of a flat plate, and that in 
particular
* (cone) =	 *(plate)	
- 	
- - - - - -	 (2) 
where 8 * is calculated from the local" properties of the potential flow at 
the cone surface. This calculation, under the additional assumptions that/. 
is proportional to T. that Prandtl's number = 100 and that 1' 
= 
Cp C = 1400, 
may be obtained from Ref030 The result is 
S*(plate) = 1.73 (1 + 0.277 ML2) (1)L8 ) 2 	
- - - () 
where the subscript L refers to the aforementioned "local e' conditions, and s 
is the distance along the plate from the leading edge. For the cone s = x 
sec , so that the displacement thickness on the cone 
S*= (1+ 00277ML2) '	 0 (L UL
	
)2 - - - -  " (4)
Substituting Eq04 into Eq0l,
rax+bx2	 -	 (5)
where
a = tan 0 
b = sec3/'20 (1 + 0.277 ML2) ( AL) * 
UL 
Thus the cross-sectional area of the "effective body" is 
S11 (ax +bx2 )	 ---------------( 6) 
Let p2 be the pressure at the surface of the cone and define the surface pressure 
coefficient C as
P2 p 
p
 j/0 U2 
where quantities without subscript are those of the undisturbed free stream. Ref. 
4 relates this pressure coeffiôlent to the body geometry-and the free stream Mach 
number M, by a first-order solution of the linearized equations Of supersonic 
flow about uniformly continuous bodies of revolution. The result is 
	
1 [El 
	 ( 1 +	
r\M2_l) 5,v
	 s''' +	 s	 +_2	 (6) +]
Cp =Wx 	 480 	 000 
(r')2 + 0(r3) -----------------------------(8) 
where primes and superscripts in parentheses indicate differentiation with res-
pect to x0 It is pointed out further in Ref04 that for the case of an Infinite 
cone, the rigorous first order linearized solution is
-------(9) 
P	 Tr	 x 
Since all applications in the wind tunnel facilities of this project are for 
M> \fl this expression for CwIll be used. Substituting Eqs05 and 6 into Eq09 
C,	 2 18
2
	 x4] in (a + bx4) ----- ( 10) 
For small-angled cones we may make the approximations tan 	 & , sec&
	
1, 
whereby
9 , b	 (1 + 0.277 ML2 ) () 	 -  - - - - (ii) 
Then
-22 + Q (1 + 0.277 ML2)Jin (0 + + 0.277 ML2)	 (12) 4 
where
ULX 
ReL = 
This expression may be simplified by expansion of the logarithm, and the 
dropping of all terms containing powers >-I- in 1/eJ , since terms of these 
higher orders are neglected in all boundary layer theory 0
 One obtains 
(1 + 0.277 ML 2)
	
-	
- 
2e 2 31)	
+ ( 0 
in	 20	 (13) )	
\ ReL' 
To a consistent degree of approximation In the linearized non-viscous flow, ML 
Is related to M,&, by 
ML 	 M2l+ 02(1+2 inO)] 	 - - - - - - - - - - -  ( 14) 
Note that for vanishing viscous action, (Re L— 00), 
cp	 i.) 
--------------- (15) 
Ideal	 e 
3O R E S U L T S
- For wind tunnel applications the ratio P2/P2ideal
	
Interest. 
From the perfect gas law, 
p u2 = .7M2p 
and we obtain from the pressure coefficients 
l+(M2C 
2 =
	 2	 P	 (16) 
P2	 l+'1M2c ideal
	 2	 Pideal 
Substituting from Eqs013, 34 9 15 
P2 -
	 ____ 
2ideal	 !4 L
where
f(M,9) = . 
M2(3 in	 4&) [ + 0.277 M2 [1	 92(2 ].n(--)
(18) 
+ 22 
in(*) 
f(M,O) is plotted versus M in HYD 2651+, for 0 = 5 0 9 7..5 0 9 1000 
A somewhat less general result is more convenient for use in 
work with the No.3 Wind Tunnel (Ref07).. Assuming adiabatic flow from a stagna-
tion reservoir temperature of 5400 Rankine, and employing the Sutherland 
formula for the air viscosity, one may write 
ReL 
= 9 N --------------(19) 
P2 x 
This function is plotted for p2 in microns Hg and x in inches (HYD 2655) To 
maintain the degree of approximation expressed in Eq.,17, p 2 may be replaced by 
ideal	 ideal 
in Eq019.. Then, since p2	 /p and	 both functions of M and9, 
we modify Eq019 to read
ReL
(2o) 
or, substituting Eq..20 in q;17,
	
P2	
Ce 
+	 (M&) , - - 
	
- - 	 (21) 
P21dea1 
where 4(M, 0) has the dimensions nècssary to render the equation dimension-
less. (p (M,9. ) in (microns Hg in.)V is shown in HYD 2656.. 
400 NUMERICAL EXAMPLES NO., 3 WIND TUNNEL 
Two examples characteristió of the smallest and largest boun-
dary layer effects on conical probe pressure measurements in the No-3 Wind 
Tunnel may be àalculated from Eq021.. Both examples refer to the No015 probe, 
9 = 5°, x = 0.700 in.. 
(1) No.,4 Nozzle, 21 lbs/lw, M = 275, p = 120 microns (nominal values of M 
and p, ignoring viscous effect)., From HYD 2656 2 = 0..7 From 
Eq..21,
P2 0 
__
	
	
p	
.	 • 
 
-J_+ 
P2ldeal -	 (]20)(07) .	 -. 
The effect of this - ratio- on M and p may be . computed roughly if the 
Rayleigh (ideal fluid) impact pressure,Pji.d... , is known. As explained 
eal 
in Ref., 8, M may be found as a function of the ratio (pi ideal/'P21d1)0
Our noininalM = 275 was computed from the ratio(Piidea1/P2)' which had 
the value 8.84, whereas the true free stream Mach number corresponds to the 
ratio,
P1	 Pi 
ideal	 ideal _______ 
	
=	 x -	 = (884) (108) = 9055 
P21deal	 P2	 P2idl 
and has the value M = 2.88. The true static pressure p follows from the 
relationship between M and P2 
ea1
/Po The measured value of p2 for this ex 
ample was 139 microns0 Then,  
P2	
ideal 139 
x	 =	 129 microns 
	
ideal =	 p2	 1008 = 
For M = 2.88YP21dealIP = 1167, and p = 129/1167 = 111 microns( is the 
true static pressure. According, then, to this highly approximate calcu-
lation, the use of the No015 probe without regard to the effect of viscous 
boundary layer causes an underestimate of M by 5 percent, and an ovèresti 
mate of p by 8 percent, in this case. The effect on the free stream 
Reynolds number may be read from HYD 2655, and is an underestimate by 5 
percent. 
(2) Similar computations may be performed for the example M = 278 9
 p = 38 microns 
(3 lbs/br, No03 nozzle). Here the neglect of boundary layer in the No015 
• probe causes an indicated underestimate of M by 8 percent, an overestimate 
of p by 17 percent, and an underestimate of Re by 7 percent. 
500 DISCUSSION NATURE AND LIMITATIONS OF RESULTS 
- Eqs.17 or 21 show that the viscous boundary layer on the sur-
face of an infinite cone at zero angle of attack in a uniform supersonic air 
stream causes an increase of the surface pressure on the cone, and makes this 
pressure dependent on the distance from the vertex of the cone. 
Roughly speaking, the real surface pressure will differ appre-
ciably (say by 1 percent) from the ideal pressure when the local Reyiolds 
number of the flow at the seam of the boundary layer is less than 10, the dif-
ference increasing in proportion to 1/4 Re as ReL decreases. For free stream 
Mach numbers greater than 25, the , theoretically calculated ratio P2/P2 
is sensitive to variation in M and in ê , the cone semi-vertex angle. ideal 
The limitations of this theoretical analysis are severe and 
should be recognized in some detail. By the nature of its basic assumptions 
boundary layer theory is applicable only when viscous layers are reasonably 
thin, (i.e. when l/ReL is fairly small). Consequently the results of the 
present analysis should be numerically significant only in the range of 
Reynolds numbers 104 < ReL < 105 (say) and serve only to indicate a trend of 
events for lower ReLO Similarly, linearized supersonic flow theory is best 
restricted to very slender bodies and moderately low Mach numbers, although the 
form of theory employed here does not become progressively worse for increas-
ing M.
Such physical approximations as Prandtl number = 1 and,/A T 
are not serious as respects theoretical trends, but will have considerable 
effect on numerical results. 
60 CONCLUSION
A simple approximate analysis based on compressible boundary 
layer theory and first order linearized supersonic flow theory indicates that 
the pressure on the surface of an infinite cone in a viscous fluid is greater 
than that in an ideal fluid. The difference between real and ideal pressures 
is proportional to the inverse square root of the local Reynolds number for 
given M and
	 For given ReL and. , this difference increases with increasing 
M. for Re and M fixed, it decreases with increasing 9
	
For 5°& &< 1009 
M) 15, the difference is not negligible for ReL< lO 
An experimental program is being developed for the investigation 
of this matter in the No03 Low Density Wind Tunnel. 
700
1) G.I. Taylor and J.W. Maccoil "The Air Pressure on a Cone Moving at High 
Speeds", Proc0Roy 0Soc0,
 Ser0A, Vol-139, pp 27298, 1933 
2) W0 Hantzsche and H. Wendt "Die laminare Grenzschicht bel einem mit 
TJéberscha llgeschwindigkeit añgestroemten nichtangestellten 
Kreiskegel", der Deutachen Luftfahrtforschung, Jahrbuch 1941 
3) L. Howarth "Concerning the Effect of Compressibility on Laminar Boundary 
Layers and Their Separation", Proc0Roy0Soc., Ser0A, No.1036, 
Vo10194, pp 16=42, 194 
4) E.V. Laitone "The Linearized Subsonic and Supersonic Flow About Inclined 
Slender Bodies of Revolution", Jour0Aero0Sci., Vo1014, No.11, 
pp 63 =643 9 1947 
5) P.L. Chambr	 "The Theory of the Impact Tube in a Viscous Compressible Gas" 
Univ. of Calif Eng. Projects Report	 15050, November 1948, 
6) E0D0 Kane and G.J. Maslach "Impact Pressure Interpretation in a Rarefied 
Gas at Supersonic Speeds", NOA OCOAO TN 2210, October 1950 
7) S.A. Schaaf, D.O. Horning and E.D. Kane "Design and Initial Operation of a 
Low Density Supersonic Wind Tunnel", presented at Heat Transfer 
and Fluid Mechanics Institute, Berkeley, June 1949 and pub-
lished by AOSOMOEO 
8) E0D0 Kane "Drag Forces on Spheres in Low Density Supersonic Gas Flow", 
Univ. of Calif Eng. Projects Report 1E.=150.65, February 1950, 
T. C. LIN 
S. L SCHAAF 
MARCH 591951
	
F. S. SHERMAN
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Boundary Layer Effect on the Surface Pressure of an Infinite Cone in Supersonic Flow

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