THE EFFIDI'S OF NOSE AND CORNER RADII ON THE FWW

ABOUT BLUNT BODIES AT ANGLE OF ATrACK

By

James c. Ellison

Thesis submitted to the Graduate Faculty of the

Virginia Polytechnic Insti-:tute

in candidacy for the degree of

MASTER •.. OF SCIENCE

in

AEROSPACE ENGmEERlliG

August 1967

.APPROVED:

THE mmTS OF NOSE' AND. CORNER RADII ON THE FLOW

AB<>U'l1. Bt:.UNT BOm!S AT ANGLE OF ATTACK

By

.rames·c. Ellison

Thesi~ submitted to the Graduate Faculty of the

Virginia Poly'technic Institute

in candidacy tor. the degree of

MASTER or' SCIEE

in

AEROSPACE ENGIN.EEBING

k, J. B. Fades, Jr'?

Fred. R. DeJarn&!te f. J. Maher

August 1967 Blacksburg, Virginia

2 -

I. TABLE OF CONTENTS

I. TABLE OF CONTENTS·. . . . . . . . • • • • • • • . . . • •

II. LIST OF FIGURES • • • • • • • • • • • • • • • • • • • •

III. INTROilJCTION • • • • • • • • • • • • • . . . . . . . . IV. LIST OF SDtBOLS . . • • • • • • • • • • • • • . . . • •

PAGE

2

3

6

9

V. APPARATUS AND TESTS • • • • • • • • • • • • • • • • 12

VI.

Test Facility • • • • • • • • • • • • • • • • • • • • 12 Operating Procedure • • • • • • • • • • • • • • • • • 12 Test Conditions • • • • • • • • • • • • • • • • • • • 13 Mod.els • • • • • • • • • • • • • • • • • • • • • • • 13 Instrumentation • • • • • • • • • • • • • • • • • • • 14

DATA REWCTION • • • • • • • • • • • • • • Reduction of Pressure Data •• Reduction of Shock Data • • • •

• • • •

• • • • • •

• • • • • • • • • • • • • • • • • •

15 15 16

VII • THJ!X>REl'ICAL TECHNIQUES • • • • • • • • • • • • • • • • 17 Calculation of the.Pressure Distribution • • • • • • 17 Calculation of the Velocity Distribution • • • • • • 21 Calculation of the Velocity Gradient • • • • • • • • 22 Calculation of the Shock Detachment Distance • • • • 23

VIII. RESULTS AND DISCUSSION • • • • • • • • • . • • • • • • • 27 Can:parison of Experimental and Theoretical Results 27 Effects of Model Geometry on Pressure and Velocity 29 Effects of Model Geometry on Velocity Gradient • • • 30 Effects of An8J.e of Attack • • • • • • • • • • • • • 31

IX. CONCLUDING Rl!MARKS • • • • • • • • • • • • • • • •

X. ACKNOWLEDGMENTS . . . . . . . . . . . . . . . ,, . . . XI. REFERENCES . . . . . . . . . . . .. . . . . . . . . . .

XII. VITA . . . . . . . . . .. . . . . . . . . . . . . . . . 37

41

- 3 -

II. LIST OF FIGURES

FIGURE n~

1. Schematic of the Langley Mach 8 variable density tunnel • 42

2 • Model a.nd flow geometry • • • • • • . . . . . . . (a) Model geometry • . . . . . . . . . . . . (b) Coordinates a.nd flow geometry • • • • • • • • 43

3. Sketch a.nd dimensions of models. Dimensions are in inches

(a) K: 0.707

(b) K = 0.417

(c) K = 0

. . .

. . . .

. . . . . . . . .

. . . . . . . . . . . . .

. . . . . 4. Comparison of experimental. and theoretical. pressure and

44

44

45

46

velocity distributions • • • • • 47

(a) K = 0.707

(b) K = 0.417

(c) K = 0

. . . . .

5. Effects of model geometry on pressure ••

(a) RC = 0 .8 .

(b) RC = 0 .4 .

(c) R = 0 • c . . . . . . . .

6. Effects of model geometry on velocity •

(a.) RC = 0.8 . . (b) RC = o.4 . . ( c.) R = 0 c . . . . . . • . . .

. . . . . . . . . .

. . . • •

. . ~ . . . . . .

• • • • • • •

47

48

49

50

50

50

50

51 51

• • 51

• • 51

7. Effects of bluntness para.meter on velocity gradient • 52

8. Effects of corner radius on velocity gradient • • • 53

- 4 -

FIGURE

9. Effect of angle of attack on pressure •

(a) Model 2-0 . . • . . . . . • . . • . . . (b) Model 2-4 . . . . . . . . . . (c) ·Model 2-8 . . . . • • • . . . . . (d) Model 4-0 . • . • . . . . . . • . (e) Model 4 .. 4 . . . . . . (r) Model 4-8 . . • . . . . • . (g) Model F-0 . . . . . • . . . (h) Model F .. 4 • . . . . . • . • . . • . . . (i) Model F .. 8 . . . . . . . . . •

10. Effects of nose bluntness and corner radius at angles of attack on pressure . . . . . . . . . . (a) Effect of nose bluntness, Re = 0 • •

(b) Effect of corner radius, K = 0.707 •

11. Location of stagnation point at angles of attack

. .

(a) K = 0.707 and K = o.417 (flagged symbols)

(b) K = 0 . . . . . . . . . . . . . . . 12. Detachment distance measured in vertical plane of

. .

.

.

.

. •

.

PAGE

54

54

54

55

55

56

56

57

57

58

59

59

59

6o

60

60

SyIDDletry- • • • • • • • • • • • .. • • • • • • 61

(a) Model 2-0 . . . . (b) Model 2-4 . . • . • . ( c) Model 2-8 . . • . . . . . (d) Model 4-o

( e) Model 4-4 . . . . . . (f) Model 4-8 . . .

.

. . . . . .

. . . .

. . .

• . . • .

. . .

. . .

. . •

. .

.

.

61

61

FIGURE

(g) Model F-0

(h) Model F-4

( i) Model F-8

- 5 -

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . TypicaJ. schlieren photographs . • • • • • • . . . . .

(a) Model 2-0 • . • • • • • • • • • • • • • • • • • •

(b) Model 4-o

(c)

(d)

(e)

Model F-0

Model 2-4

Model 4-4

(f) Model F-4 {g)

(h)

Model 2-8

Model 4-8

(·1) Model F-8

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

PAGE

64

64

65

66

66

67

68

69

69

70

71

72

73

- 6 -

III. lNTRODUCTION

At present, vehicles which are capable of performing missions

requiring reentry into the atmosphere at hypersonic speeds are of pri-

ma.ry interest in aerospace research. Various configurations of the

li~ing capsule type are being considered as reentry vehicles; however,

due to their bluntness, problems related to the large aerodynamic forces

and extreme heating rates acting on the vehicle's surface must be

evaluated, in addition to payload and mission requirements, in order to

establish design c~iteria •. As 8Jl. aid to the designer, it becomes most

desirable, if not necessary, to detemine the effects of vehicle geome-

try on these forces and heating rates.

During the last decade, considerable interest has been focused on

the study of flow over blunt bodies in an attelicpt to aileviate the

problems met with in flight by reentry and hypersonic cruise vehicles.

A selected list of such studies is presented,., here to am;plify the impor-

tance of this investigation.

The shock detachment distance and shock shape, which can be

employed to calculate the flow field behind the shock, have been inves-

tigated by Ambrosio and Wortman (ref. 1) who presented an expression for

the stagnation point shock detachment distance and compared it with the

result of Serbin (ref. 2). Kaattari (ref. 3) has devised a technique to

calculate the shock shape for very blunt bodies, having various corner

radii, at a.ngJ.es of attack. Boison and Curtiss (ref. 4) experimentally

investigated the stagnation point velocity gradient on blunt bodies in

supersonic flow, and Zoby and Sullivan (ref. 5) more recently presented

- 7 -

the results of a study of the effects of corner radius on the

stagnation point velocity gradien;t·. In addftion, Griffith and Lewis

(ref. 6) investigated laminar heat transfer on blunted cones and

hemisphere cylinders, while Cleary (ref. 7) investigated the pressure

distribution and flow field on blunted cones. Other studies on blunt

bodies include those of Reshotko and Cohen (ref. 8) on stagnation

point heat transfer and Cooper and Ma.yo (ref. 9) on local pressure and

heat transfer. Although these, and other investigations (refs. 10-20),

have analyzed the flow over blunt bodies, there is insufficient data on

very blunt (blunter than a hemisphere} axisymm.etric bodies - especially

at angles of attack - to provide correlations between body geometry

and flow pare.meters or to allow for comparisons with theoretical

results.

The-present investigation incorporates a compilation of theoreti-

cal results and experimental data for a family of blunt, axisymmetric

bodies which included a range of ratios of body radius to nose radius

from 0 (flat-face} to 0.707;and ratios of corner radius to body radius

from 0 (sharp corner) to o.4. The tests were made at a f'ree stream

Mach number and unit Reyno+ds number (per foot) of approximately 8.0

and 4.10 x 106, respectively, for angles of attack varied f'rom o0 to

29°. Also, included in this ir.ivestigation are theoretical predictions,

experimental pressure and velocity distributions, and the shock-

detachment distances fof' blunter-than-hemisphere and flat faced bodies. ; . ~

The experim,ental pressure an4 velocity dihributions are compared with : ,, .,

theoretical predictions, at z~ro angle of attack; in addition,

pressures, velocity, and shock-detac.fuaent distance data obtained at

. ' ~·

- 8 -

angles of attack are presented. Where possible, correlations between

model geometry and flow ~roperties are presented.

A

a

b

c

H

h

k

M

n

p

p

r

- 9 -

IV. LIST OF SYMBOLS

constant in equation (39)

defined in Figure 2

defined in Figure 2

( ) -1 velocity gradient due/ds , sec

pressure coefficient, (P - P00) /~

diameter of model

total enthalpy, ft2/sec2

2/. 2 static enthalpy, :ft sec

incremental height of manometer fluid, in.

bluntness parameter, ~/~

free stream density divided by density behind bow shock at

body axis

Ma.ch number

exponent in equation (39)

manometer reference pressure, psia

pressure (local surface pressure when used without a

subscript) 1 2 :free stream dynamic pressure, 2 P00 V00

body radius·

corner radius, in.

nose radius

shock radius

radia.l qoordinate

model base radius

rmax r*

s

T

TP

U'

u

v

x

y

a.

7

/:::.

p

Subscripts

- 10 -

equivalent to body radius

radial distance to sonic point

coordinate along surface, measured from stagnation point 0 temperature, R

distance along surface from x-axis to the point of tangency

velocity gradient defined by equation (27)

velocity component in the s-direction

free-stream velocity

velocity component in the y-direction

coordinate along body axis of symmetry

coordinate normal to Surface

angle of attack

ratio of specific heats

shock detac~ent distance, incremental change

angle between the x~a.xis and a vector normal to the surface

at the stagnation point

density:

density of ~meter fluid

angle between the f'ree stream velocity vector and a vector

normal to the B\ll'.'face

defined in Figure 2

e at the edge of the boundary layer

stag stagnation point on bow shock

t stagnation point value

2 conditions behind normal shock

- 11 -

00 free-stream conditions

The symbols 2, 4, and F have been used to denote the models

having a 2.83 inch nose radius, a 4.8o inch nose radius, and a flat

face, respectively; the numbers O, 4, and 8 have been used to denote

models having a sharp corner, a o.4 inch corner radius, and an o.8 inch

corner radius, respectively. Models are generally ieSignated by a

hyphenated combination of these symbols; for exa,m:ple., 2-8 desj.gnates

a model having a 2.83 inch nose radius and an o.8 inch corner radius.

- 12 -

V. APPARATUS AND TESTS

Test Facility

The experimental program was conducted in the Langley Mach 8

variable-density hypersonic tunnel at the Langley Research Center

(see Fig. 1). This is an intermittent, blowdown tunnel which

is equipped with a quick injection mechanism and schlieren windows.

Basically, the tunnel consists of a heat exchanger, a contoured,

axisymmetric nozzle, an 18 inch diameter test section, a dif:f'user

section, and a vacuum system. The facility is equipped to accommodate

pressure, force, heat transfer and oil flow tests.

Operating Procedure

The following procedure is generally used in conducting tests in

the above mentioned facility: Initially, the tunnel pressure is

reduced below that required for starting the tunnel by the vacuum

system. Next, air from a high pressure storage field is heated to the

desired stagnation temperature. Once the desired stagnation tempera-

ture and pressure are obtained a quick opening valve allows the air

to expand down the nozzle into the test section. After passing

through the test section, the flow is decelerated. through a diffuser

and exhausted into the vacuum system.

An injection mechanism is used to place the experimental model in

the :free stream once the desired stream conditions are established.

This mechanism is located in a chamber beneath the test section. All

models are strut mounted to a plate which is fastened to the injection

mechanism. The models are injected into the flow after the desired

- 13 -

flow has been established and retracted after obtaining the data. This

is accomplished by means of a pneumatic piston which operates the

injection mechanism.

The models are positioned in the test section by use of a

coordinate cathetometer; a graduated inclinometer can be used to set

the angle of attack. The accuracy for measuring angle of attack is

±0.05°.

For the pressure tests, at angles of attack, a test run is made

and the manometer levels observed. When subsequent runs are ma.de

the angle of attack is adjusted until the pressures at the orifices -

immediately adjacent to (approximately 0.150 inch) the desired

stagnation point - are equal. When this balanced condition has been

attained the procedure is repeated in order to obtain additional

stagnation point locations. On the average, four runs are required

to define the correct angle of attack for a desired stagnation point.

Test Conditions

All tests for this investigation were conducted in air at an

average stagnation temperature of 146o0 R, and for a stagnation

pressure of 1000 psig. The corresponding free stream Mach number and

unit Reynolds number (per foot) were 8.0 and 4.10. x 106, respectively.

Run times of approximately 9() seconds were obtained in the tunnel;

this allowed the coll,Ullns of ·Jna.llom.eter f'luid to reach equilibrium.

?-f_Odels

During the investigation a family of blunt, axisymmetric bodies

having arbitrary nose and corner radii, any one of these might be

- 14 -

considered as a reentry vehicle,· were studied. Shown in Figure 2

is the model geometry, and in Figure 3 .sketches and dimensions of the

models are shown. In this program a total of nine models were tested,

representing nine configurations. All models were 4.o inches in

diameter.

The models were constructed from 347 stainless steel and had

0.100-inch walls. Pt"essure orifices were formed as the inside diameter

(o.o4o inch) of a stainless steel tube which ha.d been soldered into

holes in the model and cut off flush.with the model's outer surface.

Orifices were located along the axes of symmetry, approximately

0.150 inch apart.

Instrumentation

A dial gage and a chromel alum.el total temperature probe were

used to measure the stagnation pressure and temperature, respectively.

The surface pressures on the models were measured on one mercury and

two butyl phyhalate manometers. Two Wallace and Tiernan (dial) gages

were used with a vacuum pump to regulate the reference pressures.on the

butyl phthalate manometers; a barometer recorded the atmospheric

pressure to which the mercury manometer was referenced. From photo-

graphs of the manometer the pressure levels were read to within

±0.05 inch of fluid.

Photographs of the flow field about each model were taken during

each run using a single pass schlieren system. A wire was located

across one of the schlieren windows, parallel to the tunnel centerline,

for reference purposes.

- 15 -

VI. DATA REDUCTION

Reduction of Pressure Data

From photographs of the manometer tubes the fluid levels were

read to an accuracy of ±0.05 inch. All columns on the mercury

manometer were referenced to the atmosphere; and, the absolute

pressures were obtained by the simple expression

(1)

where p is the pressure, hf is the difference in height between a

given column and the reference column, and pf is the density of

mercury. Each butyl phthalate manometer had two reference columns;

these pressures were measured on the mercury manometer. To obtain the

pressure levels recorded on the butyl phthalate manometers an

expression similar to equation (1) was used; precisely,

where p and hf have the same definitions as before; P is the

manometer reference pressure, and pf is the density of butyl

phthalate.

The pressure:ratio P/Pt. (local pressure divided by the pressure

at the stagnation point) has.been used to present the pressure

distributions. From the pressure distributions the velocity distribu-< ,,,

tions were calculated using the isentropic relation

u e -=

- 16 -

r - 1 r

1 - (~t) , which is later derived as equation (25).

Reduction of Shock Detachment Distance

(3)

Schlieren photographs were ta.ken during each test run after flow

had been established over the model. From the finished prints (Fig. 13)

values of r;r were determined; at these points, the distance from. max .

the body surface .to the shock, along a line parallel to the x-axis, '

was measured. Approp_ri'ate scale .factors, based on model diameter,

were used to determine the actual values of the shock detachment

distance. Using'this techniq~e the complete shock shapes, about each

model at ~es of attack,_ were obtained. These data are presented

as plots of the dimensionless quantity A/d versus r/rmax..

- 17 -

VII. THEORETIC.AL TECHNIQUES

Calculation of the Pressure Distribution

Three methods for predicting the pressure distribution over a

blunt body are presented for comparison with the experimental results:

two of these methods are for bodies with a finite nose radius, and one

method is for bodies with an infinite nose radius (flat face).

Newtonian impact theory expresses the local pressure coefficient

in terms of the angle between the free stream direction and a vector

normal to the local surface of the body. In eq\;lS,tion form, the

pressure coefficient is given by (ref. 21)

2 c = 2 cos e . p

In general, the Q.efiniti.on of the pressure coefficient is

where

p - Poo c ' = p ··~

(4)

(5)

(6)

Equation (4) overpredicts the pressure coefficient at the stagnation

point but can be modified to ·yield the conect value. This is

accomplished, as suggested by Lees, by replacing the constant in

equation (4) with the maximum (stagnation) value of C , defined as p

(7)

- 18 ...

thus,

c 2 __R._ = cos a • cp,t

(8)

From these previous expressions the ratio of the local to maximum

pressure coefficient is related as

c p - p __.i.,_ : OD •

cp,t Pt - Poo (9)

Equating the last two equations, and manipulating the terms, yields

n Pt 2 2 ....._ = -- cos a + (1 - cos a) , Pco Pco

(10)

which may be written as

(11)

The modified Newtonian pressure distribution over part of the exposed

surface of. .. ~ b~unt body is given by equation (li); however, it is not

applicable to a body of infinite nose radius (flat face) since it

would suggest'thai~ the pressure is cp~stanton the face.

Anotli~r·modificat~qn of' Newtonian im,pact t,heory is concerned with

the corr~tiott for centrifugal force ·e:ff'ects. In reference 21 the

Newtoni~-~us-centritugal-fo~ce relatio:n p~edicts the pressure !" ' coef'f'icient op a .sphere by means of·. the following equation:

. "

(12)

As before, it is desirable to obtain the correct value at the stagna-

tion point; therefore, equation (12) is modified to yield

- 19 -

c 2 1 2 ~=cos e - 3 sin e • p,t

(13)

Equating equations (9) and (13) and rearranging terms will give an

expression for the pressure distribution as

(14)

By manipulating the terms in the last equation, one finds that

I, 2 pex> 2 ~ [pco ( 2 2 2 ) l 2 l ~t = ~OS a + Pt sin ej + p; 1 - cos a - 3 sin a - 3 sin 9j •

(15) The first quantity in brackets is identical to the right hand side of

equation (11); consequently, the second quantity in brackets can be

attributed to centrifugal force effects. While equation (lll predicts

a pressure equal to the :free stream pressure at 9 = 90°, it can be

shown that equation (15)" predicts the :free stream pressure occurring

at 9 = 6o0 for a sphere. Even though equation (15) was derived on

the assumption o:f' a sphere, it is vf,1.1.id (in this investigation) since

a majority of the models test~d had' aphe~ically blunted forward

surfaces.

Another techniqu~ used~to predict the pressure.dfatribution over

· a blunt body has been develop"¢" 'Py Li and. Geiger. The analyai s begins ' .

with the equations of motion aiidconservation of ass for an inviscid ~ ·.· ' . . ~ .

fiuid at hypersoilic :free stream Mach numbers. Several assumpti·ons are t ..

ma.de to simpli:f}r" the govel"ll.ing' differential equa tlons ( see ref. 22) ' '

. ' .!.

- 20 -

which permit their solution. These assumptions include: (1) r ~ s;

(2) y /~ = O [A/~] and can be neglected since A/~ << 1 f'or the

hypersonic case; (3) the flow near the stagnation point is considered

incompressible, theref'ore, p ~ p2 = p00 /k; (4) k << 1, and ue/V00

and v/Voo = o[k] so that higher order terms involving these terms can

be neglected; (5) the analysis is applicable to the nose region (on

the basis of' the f'irst assumption); and (6) ~~Rs. Application of'

these assumptions to the governing equations results in the f'ollowing

expression for the pressure coefficient:

(16)

By rearranging terms and making substitutions, as in the previous

analysis, the last expression becomes

(17)

Equation (17) is applicable to either a two dimensional or an

axisymmetric flow; however, it is limited to the stagnation region of'

a blunt body.

The last prediction of pressure distributions considered here is

for a body with an infinite nose radius (flat face), developed by

Probstein (ref. 23). This solution has been obtained by assuming

incompressible irrotationaJ. flow over a circular disk, and by matching

the results to a curved shock wave displaced from the body. It is

also assumed that the afterbody has no effect on the. flow in the stag-

nation region. The resulting equation for the pressure coefficient is

- 21 -

(18)

where r* is the re.dial distance to the sonic point. Although this

equation was developed for a disk with a cylindrical afterbody it may

be applicable to a flat-faced body with curved or sharp corners by

using the appropriate value of r*. Once again making substitutions

and rearranging terms, equation (18) may be written to express the

pressure distribution as

(19)

Calculation of the Velocity Distribution

For a body in hypersonic :f'low the adiabatic energy equation is

u2 h + 2e = H (20) e e

Solving for the velocity and writing the equation in dimensionless

form., one finds that UN e . e -......-.. 1--. y2He He (21)

For a perfect gas

(22)

and, for a perfect gas and isentropic :f'low,

- 22 -

(23)

Substituting the last expressions into equation (21) results in the

f'ollowing:

u e y2He =

"! - l

l -(:: r (24)

It is generally assumed that the pressure through the boundary layer

along a normal to the surf'ace is constant; thus P = p. Therefore, e the velocity distribution over the body surface may be expressed in

terms of' the local surface pressure by the expression

u e -=

"! - l

l - ~;) 7 (25)

Equation (25) is dependent on the body geometry, since the surf'ace

pressure depends on body geometry, and is used exculusively in this

investigation to calculate the velocity distribution over each model

tested. This is accomplished by utilizing either the experimental or

a theoretical pressure distribution.

Calculation of the Velocity Gradient

The experimental velocity gradient (rate of change in velocity

with distance) was approximated f'rom the expression

(26)

- 23 -

where,

U' = 6(ue[ \f2He) ~(s/~) '

and du

c e = ds

In the absence of an experimental velocity distribution, an

estimate of the velocity gradient may be obtained from several

expressions. For instance, Newtonian flow gives (ref. 9)

1 c = 11i

(27)

(28)

(29)

and, according to reference 5, the velocity gradient for a blunt body

is given by

c l ~ Reff

1Ft. vp; (30)

The quantity Reff is an effective nose radius defined (ref. 5) as

the radius of a hemisphere which will produce the same velocity

gradient as that computed for the blunt body.

Calculation of the Shock-Detachment Distance

Several equations are presented as a means of predicting the

stagnation point shock detachment distance; all of these express the

distance in terms of the density ratio across the shock. In their

analysis of flow over a blunt body, Li and Geiger obtained an equation

- 24 -

which predicts the stagnation point shock detachment distance. The

analysis is for an inviscid fluid at hypersonic Mach numbers and

begins with the equations of motion and conservation of mass. Again,

simplifications are made which limit the analysis to the stagnation

region (see page 20) and the following equation results:

In addition to the assumptions of the previous analysis, that

L~/Rs << l and ~ ~ R8, Reshotko (ref. 24) assumed: a Newtonian

pressure distribution; the shock layer was of uniform (constant)

thickness near the stagnation point; and, the velocity gradient was

linear. Af'ter applying the conservation of mase for axisymmetric

flow, on~ obtains

where the velocity is given by

and, the velocity gradient, c, can be defined as

voo \~ c=rVJ+k · s '

(32)

(33)

(34)

Combining the last two equations with equation (32), Reshotko obtained

the following expression to predict the stagnationpoint shock

detachment distance:

- 25 -

(35)

In an article by Ambrosio and Wortman (ref. 1), two expressions

are presented for the prediction of the stagnation point shock

detachment distance for a sphere. The first expression,

6 stag 2k ~ = 3(1 - k) (36)

was taken from reference 2; when this is compared to experimental data

at low Mach numbers (M < 7.0), it is found to be less accurate than

the expression,

Astag (. k )0.861 ~ = 0.52 ~ - k ' (37)

which was derived by Ambrosio and Wortman in reference 25.

In Probstein's analysis (ref. 23) of the flow over a flat faced

body the stagnation point shock detachment distance was predicted by

A stag 8 _;,:: r* = O. 33 vk , (38)

which is valid for small values of k, that is, hypersonic Mach

numbers.

The relation

(39)

- 26 -

has been presented in references 26-28 for predicting the shock wave

shape. VaJ.ues of the coefficient A, and the exponent n, have been

obtained f'rom. experiment&l data for several shape~ with various nose

radii by plotting the experimental shock wave coordinat~~l on logari th-

mic paper (see ref. 26). A . ·more thorough method of predicting the

shock wave shape and other pertinent properties of the shock wave has

been developed by Kaattari in references 3, 29, and 30.

- 27 -

VIII. RESULTS AND DISCUSSION

Comparison of Experimental and Theoretical Results

The experimental pressure and velocity distributions,, at zero

angle of attack, are presented as a function of sfRB and compared to

theoretica.l results in Figure 4. The data ar¢ presented as the

pressure ratio P/Pt and the velocity parameter ue j J2He, ~hich was

calculated from the corresponding pressure distribution by using

equation (25), plotted as a function of the parameter s/RB.

In Figure 4(a) the exper1menta.l distributions for the !llOdels

with K ~ 0.707 are presented and compared to distribtuions ca.lculated

by Newtonian impact theory and the theory of Li and Geiger for a sharp

cornered body. It is apparent that the modified Newtonian theory,

given by equation (11), closely predicts the pressure in a region near

the stagnation point while slightly overpredicting the pressure toward

the corner of the model; consequently, this method underpredicts the

velocity distribution. On the other side of the data curve is the

modified Newtonian-plus-centrifuga.1-force theory; given by equation

(14), which underpredicts the pressure and overpredicts the velocity

by an amount comparable to that of the modified Newtonian theory. The

Li and Geiger theory, given by equation (17), predicted the pressure

and velocity distributions best; in fact, equation (17) only varied

slightly from the data near the stagnation point and predicted the

magnitude of the pressure ratio at various locations along the

periphery of the body. However, the Li and Geiger theory does not

- 28 -

give p /Pt = 1.0 and ue/ y2He = 0 at s/RB = 0 as Newtonian

theories do; this slight deviation is due to the quantity (2 - k),

in equation (16), which is not identically equal to C t. p,

Experimental date. for the models having K = 0.417 are presented

in Figure 4(b) in a form identical to Figure 4(a); however, the results

are not the same. As can be seen from the figure, none of the theories

provided very accurate predictions. It should be noted that the

modified Newtonian-plus-centrifugal-force theory most closely predicts

the distributions away from the stagnation point; this should be

expected since this theory employs a correction for increasing

curvature (decreasing values of K), even though it assumes the shock

is on the body.

The experimental data for the flat faced models are compared to

the theory of Probstein, given by equation (19), in Figure 4(c).

The theory inaccurately predicted the distributions cal.cu.lated for the

flat faced, sharp cornered body; instead, it very closely predicted

the distributions for the bodies having a corner radius. This result

can most likely be attributed to the dependence of equation (19) on r*.

From the comparisons shown on Figure 4, it appears that the

modified Newtonian theory may be applied with accuracy to bodies having

a value of K > 0.7 (where the body approaches the shape of a sphere);

and the modified Newtonian-plus-centrifugal-force theory is reasonably

accurate for the range of values 0 .40 < K < 0. 7. The Li and Geiger

theory appears to be applicable for a range of K which overlaps

both of the Newtonian theories considered, and the range 0.5 < K < 0.75

seems a reasonable approximation. In applying these theories to bodies

- 29 -

having a corner radius, proper geometric relations beyond the point

of tangency must be used; also, these predicted limits on K may be

extended easily, provided the application is limited to a region near

the stagnation point. For a prediction of the pressure and velocity

distributions over flat-faced bodies, the theory of Probstein, at

first glance, appears inaccurate; however, if one realizes that a

sharp corner forces the sonic point to occur at a point having nearly

the same value of r* as a body with a small corner radius, the theory

has merit.

Effects of Model Geometry on Pressure and Velocity

To show the influence of model geometry on the pressure

distribution, at zero angle of attack, the pressure ratio P/Pt has

been plotted against s/'fl:s in Figure 5. Considering bodies with the

same corner radius (Figs. 5(a), 5(b), and 5(c)), it is evident that the

nose radius significantly influences the pressure distribution,

especially outside the stagnation point region. It is also apparent

from the figure that the nose radius has the greatest effect when the

corner of the body is sharpest {Re is smallest). The effects of

corner radius on the pressure distributions can be seen from Figures 4

and 5. First, for a given no·se radius, the pressure gradient (the

change in P/Pt with s/'fl:s) decreases as the corner radius decreases;

secondly, the effect of increasing the corner radius is more pronounced

on the blunter bodies.

In Figure 6, the dimensionless velocity parameter ue / y2He

(calculated from the pressure distributions of Figure 5) has been

- 30 -

plotted against s/~· Figure 6 shows that at a given location -

measured from the stagnation point - the magnitude of the velocity

increases if: (1) the corn~r radius is increased for a given nose

radius, and (2) the bluntness parameter is increased for a given

corner radius. It is important to note the region where the velocity

changes linearly with distance from the stagnation point, since it

has been sUggested (ref. 21) that the magnitude of the inviscid

velocity gradient determin'es the local heating rate. The figure

indicates that the region of a linear change in velocity with distance

from the stagnation point decreases if the corner radius or the nose

radius (a decrease in K) is increased.

Effects of Model Geometry on Velocity Gradient

The slopes, at the stagnation point, of the curves in Figure 6

have been determined and the velocity gradients calculated using

equation (26). These velocity gradients have been plotted against

the bluntness parameter, K, and the dimensionless quantity Re/~

in Figures 7 and 8, respectively.

The results in Figure 7 imply that the stagnation-point velocity

gradient varies linearly (except when Re/~ approaches zero on a

flat-faced body) with the bluntness parameter and increases as K

increases. Also, the value of C for a sphere is predicted if the

curves are extrapolated to a value of K equal to unity. Furthermore,

the value of C obtained by this procedure differs from the value

predicted by simple Newtonian theory, equation (29), by less than

7 percent.

- 31 -

As shown in Figure 8, an increase in corner radius causes an

increase in the value of C. The rate of change in C with respect

to Re/~ is linear; however, this rate of change is considerably less

than the change in C with respect to K. Figure 8 also shows that

the corner radius affects the velocity gradient more as the bluntness

parameter decreases.

Effects of Angle of Attack

Pressure distributions.- Shown in Figure 9 are pressure distribu-

tions which have been measured on the windward side (t = 180°) of

each model at several angles of attack. 0 The distributions for a = 0

are shown in Figures 4 and 5, and have been discussed previously. As

Figure 9 shows, angle of attack significantly affects the pressure

distribution and consequently the velocity distribution. (See

equation (25).) It is quite apparent that as the nose or corner

radius increases, the pressure gradient (the change in P/Pt with

distance along the surface) increases with increasing angle of attack.

Several additional observations may be made by recalling the effect

of decreasing pressure on the velocity and heating rates. Realizing

that as the stagnation point approaches the corner of a body

(increasing a) the pressure gradient becomes much larger; thus,

there is an increase in the velocity and the heating rate. A second

interesting observation, which can be concluded f'rom Figure 9, is

that in many cases the f'low over certain portions of the body is no

- 32 -

longer subsonic; and, consequently, many of the assumptions made for . . 0

the a.naJ.ysis of the flow over the body at a. = 0 can no longer be

applied.

Presented in Figure 10 are several of the pressure distributions

taken from Figure 9 to show more vividly the effects of nose and

corner radii on the flow when the body is at angle of attack. Figure

lO(a) shows the trend in pressure distributions for a given corner

radius (sharp corner) as the bluntness of the nose varies. It is

evident that the nose radius (or bluntness parameter) greatly

influences the pressure distribution; and, as shown in the figure,

·an increase in nose radius (a decrease in K) tends to yield a

smaller varying pressure. In the case or·a :flat faced bod.ya nearly

constant pressure exists, even at fairly large angles of' attack. From

Figure lO(b) it may be concluded that the. ef'f.ect ·on pr~ssure due to

corner radius is less si'gnificant than the effectdueto nose radius.

However, here an unusual trend is indicated. By comparing

Figures 9(a) - 9( c), it can be seen that at <li = ·o0 the magnitude

of the pressure decrea~es over the surface if the t:orr.i.er r~dius is ,,:,·· '

increased; while Figure_.lO(b) shows that th~ magnituge o:t the pressure,

at a given location from the.stagnationpoi?lt, increa~esif the corner

radius is increased (from 0.4 inch to 0.8 inch). Be:f't>re attempting to . . .

draw any definite conclusions, it must be pointed out that the angle

of attack is not identical in both tests.

Stagnation point location.- On Figure 11 the stagnation point

location, in terms of the angle (or s/~) measured from the

stagnation point to the x-axis, has been plotted against the angle of

- 33 -

attack. Due to the manner in which this data were obtained the

accuracy may not be precise; however, the trend is indicated. Figure

11 shows that the stagnation point location varies linearly with angle

of attack, over most of the range of ~. In general, the change in

stagnation point location with angle of attack was less on models

having the larger nose and corner radii.

Shock detachment distance.- Shown in Figure 12 are the shock

shapes for all of the models at angle of attack, presented as the

nondimensionalized shock detachment distance (A/d) plotted against

r/rmax. The figure indicates that the detachment distance is generally

smaller on the windward side and larger on the leeward side when the

body is at angle of at~ck. At zero angle of attack, Figures 12(a) -

12(f) show that the value of A/d (at r/rmax = 0) · for the models

with a finite nose radius decreases o:i:lly slightly if the corner

radius is increased, while Figures 12(g) - 12(i) shQw that the value

decreases significantly if the corner radius is increased on a flat

faced model. A significant change in D./d occurs when the ~ose

radius is increased; as shown in Figure 12, the value of A/d (at

r/rmax = o) changes from 0.10 to 0.25, approximateiy, as the bluntness

parameter decreases from 0.707 to O.

- 34 -

IX. CONCLUDING REMARKS

An investigation was conducted to determine the.effects of the

nose and corner radii on the flow about blunt bodies at &ngle of

attack. Experimental pressure distributions were measbredon a family 0. . 0 of blunt, axisymmetric bodies at aneJ.es qf' attack from 0 to 29 • The

experimental data were obtained at a free stream Mach number and unit 6 Reynolds number (per foot) of approximately 8.o and 4.10 x 10 ,

respectively.

The experimental pressure distributions {at ~ = o0 ), and velocity

distributions calculated from them, were compared to several existing

theories. The Li and Geiger theory and Newtonian theories used

predicted the distributions very closely on the models with K = 0.707;

however, these theories were less accurate for the models with

K = 0.417. Probstein's theory, used for predicting the distributions

on the flat faced, sharp cornered model, was found to be inadequate;

although, it is expected that the theory would be applicable to flat

faced models with a corner radius.

Results showed that the pressure and velocity distributions were

significantly affected by the nose bluntness; and, while there was an

effect due to corner radius, 'it was less noticeable than the effect

of nose radius. Also, a decrease in the bluntness parameter (K) or

the corner radius increased the magnitude of' the pressure ratio P/Pt

over the surface of the model.

Stagnation point velocity gradients obtained from the experimental

data showed a dependence on nose and corner radii; in fact, the value

- 35 -

of C (the stagnation point velocity gradient) on the model with a

sharp corner and K = 0.707 was about two and one-half times the value

for the flat faced sharp cornered model. Results turther indicated

that for a given corner radius, the value of C decreased when the

bluntness parameter {K) decreased; and, for a given nose radius, the

value of C increased when the corner radius increased.

Data obtained at angles of attack showed that the nose and corner

radii affected the pressure distribution and stagnation point location.

The magnitude of P/Pt increased on all models tested as the angle of

attack was increased, and this increase generally became more pro-

nounced when the nose or corner radius was increased. Also, the change

in stagnation point location with angle of attack (d~/da.) increased,

and approached a value of unity, as the bluntness parameter (K)

increased·.

Results obtained f'rom the schlieren photographs revealed that the

stagnation· point shock detachment distance increased with decreasing

corner radius or increasing nose radius.

X. ACKNOWLEDGMENTS

The author wishes to express his appreciation to the Nationa.l

Aeronautics and Space .Administration for the pel'Jlission to use the

data and material. contained in this thesis w~ch wa.s obtained from a

research project conducted at the Langley ,Research Center.

For their assistance and adYice du.r1ilg th~,experimental. phase of

this thesis, the author wishes to thank, and

The author wishes to express hi~ thanks tO,Dr. Fred DeJa.rnette ' • ' .. T .

:f'or his assistance in preparing this thesis.

.. - 37 -

XI. REFERENcES

'~·

..•. •, '. "'·) .·•· · l. Ambrosio, Alphonso; ~'and• .. ~or~, Andrzej: · StagnatiQ,n .. P<:il<nt Shock-

. . ··~~ . . " • '• .i \:,' ;: '<~ ·.·~ •w • .::1>

Detachment Distance for El.aw Aro\lnd Spheres and CYl'inders in

Air. JAS, vol. 29, np. 1, 19El2, 1H 875., ' ' ~ , .

2. Serbin, H.: Supersonic Flow Ai-oudd' Blunt BOdies.' t~ · ,A~z:on. Sci.,

vol. 25, no. 1, Jatl. 1958. ·,

' .· . ; " ' :' ' ... '·" ::;, .. 3. Kaa ttari, George E. : ·fredieted Shock Envelopes About, Two Types of

. . ~ . ·. Vehicles at Large AngJ..es' ot,Attack.. ~··TN n.:.86o(i96i. " ., ·'

4. Boisen, J. Christopher; and CU:rti~; Howard A.:. kn. ~·~erimental - . ~·

Investigation of Blunt Body Stagnation Point Velocity Gradient.

AAS Journal, vol. 29, no. 2, Feb. 1958, pp. 130-135. ' '

5. Zoby, Ernest V.; and SUl.livan, Edward M.: Ettects of Corner

Radius on Stagnation-Point Velocity Gradients on Blunt

.Axisymmetric Bodies. NASA TM X-lo67, 1964. 6. · Griff'i th, B. J.; and Lewis, Clerk H.: A Study of Laminar Heat

Transfer to Spherically Blunted Cones and Hemisphere Cylinders

at Hypersonic Conditions. AEDC~TDR-63-102, 1963.

7. Cleary, Joseph W.: An Experimental and Theoretical Investigation >' •

of the Pressure Distribution and Flow Fields of Blunted pones at

Hypersonic Mach Numbers'. NASA TN ~2969, 1965.

8. Reshotko, Eli; Cohen, Clarence B.:. Heat Transfer at the FOrwe.rd

Stagnation Point of Blunt Bodies. NA.CA TB 3512, 1955.

- 38 -9. · Cooper, Morton; and Mayo, Edward E.: Measurements of Local Heat

Transfer and Pressure on Six 2-Inch-Diameter Blunt Bodies at a

Mach N\.U!iber of 4.95 and at Reynolds N\.U!ibers Per Foot up to 6 81 x 10 . NASA MEMO l-3-59L, 1959·

10. Maslen, s. H.; Moeckel, W. E.: Inviscid Hypersonic Flow Past

Blunt Bodies. J. Aeron. Sci., vol. 24, no. 9, Sept. 1957,

pp. 683-693.

11. Lees, Lester: Recent DeveloP111ent in Hypersonic Flow. Jet Propul-

sion, vol. 27, no. 11, Nov. 1957, pp. 1162-1176.

12. Kemp, Nelson H.; Rose, Peter H.; and Detra, Ralph W.: Laminar

Heat Transfer Around Blunt Bodies in Dissociated Air. J. Aero/

Space Sci., vol. 26, no. 7, July 1959, pp. 421-430.

13. Oliver, Robert Earl: An Experimental Investigation of Flow About

Simple Blunt Bodies at a Nominal Mach Number of 5 .8. J. Aeron.

Sci., vol. 23, no. 2, Feb. 1956, pp. 177-179·

14. Laurin, Roberto Vaglio: Laminar Heat Transfer on Blunt-Nosed

Bodies in Three-Dimensional ·Hypersonic Flow. WAOO Technical

Note 58-147, 1958.

15. Van Dyke, Milton D.; and Gordon, Helen D.: Su::r;>ersonic Flow Past

a Family of Blunt AXis~et:tic Bodies~ ·· NASA TR R .. 1, 1959.

16. Wisniewski, Richard J.:. Methods of Pr~dicting Laminar Heat Rates

on Hypersonic Vehicles. NASA TN D-201, l.959·

17. Katzen, Elliott D.; and Kaattari, George E.: .Flow Around Blunt

Bodies Including Effects of High Angles of Attack,

- 39 -

Nonequilibrium Flow, and Vapor Injection. Proceedings o:f the

AIAA Entry Technology Conference, Williamsburg, Va. , Oct • 1964;

pp. 106-117.

18. Lomax, Harvard; and Inouye, Mam.oru: Numerical Analysis of Flow

Properties About Blunt Bodies Moving at Supersonic Speeds in

an Equilibrium Gas. NASA TR R-2o4, 1964.

19. Inouye, Ma.moru; Rakich, John V.; and Lomax, Harvard: A

Description of Numerical Methods and Computer Programs for Two-

Dimensional and Axisymm.etric Supersonic Flow Over Blunt-Nosed

and Flared Bodies. NASA TN D-2970, 1965.

20. Fey, J. A.; and Riddell, F. R.: Theory of Stagnation Point Heat

Transfer in Dissociated Air. J. Aeron. Sci., vol. 25, no. 2,

Feb. 1958, PP• 73-85, 121.

21. Truitt, Robert W.: Hypersonic Aerodynamics. New York: The

Ronald Press Company, 1959·

22. Li, Ting-Yi; Geiger, Richard E.: Stagnation Point of a Blunt

Body in Hypersonic Flow. J. Aeron. Sci., vol. 24, no. 1,

Jan. 1957, pp. 25-32.

23. Probstein, Ronald F.: Inviscid Flow in the Stagnation Point

Region of Very Blunt-Nosed Bodies at Hypersonic Flight Speeds.

WAIX: TN 56-395, 1956.

24. Reshotko, Eli: Estimate of Shock Standoff Distance Ahead of a

General Stagnation Point. NASA TN D-1050, 1961.

25. Ambrosio, A.; and Wortman, A.: Stagnation Point Shock Detachment

Distance for Flow Around Spheres and Cylinders. .ARS Journal,

vol. 32, no. ?, Feb. 1962.

- 40 -

26. Seiff, .Alvin; and Whiting, Ellis E.: A Correlation Stud.Y: of the

Bow-Wave Profiles of. Blunt Bodies. NASA TN D-1148, 1962.

27. Seiff, .Alvin; and Whiting, Ellis E. : The Effect of the Bow

Shock Wave on.the Stabilityo:f' Blunt-Nosed Slender B9d,1es.

NA.SA TM X· 377, 196<>. '

28. Van Hise, Vernon: Analytic Study of Induced Pressure on Long

Bodies of Revolution with Varying Nose Bluntness at Hypersonic

Speeds. N4SA TlfR-78, 1960.

29. Kaattari, George E. : Predicted Gas Propeties in the Shock Layer ,~ ;

D-1423, 1962.

30. Kaattar1, George E.: slio.ck E:lweiopes of.:si$t '.Bo_d{es 'a~ Large

Angles of Attack. XASATN D-198o, 1963.

Te a

ir·

supp

ly

Qui

ck o

peni

ng v

al.v

e C

onto

ured

noz

zle

Tes

t se

ctio

n A

f'ter

cool

er

Dif

fuse

r

0 ex

chan

ger

Set

tlin

g ch

ambe

r

Tes

t ca

bin

To v

acuu

m

Figu

re l

.-Sc

hem

a.tic

of'

the

La.l'.

lgiey

Mac

h 8

vari

able

den

sity

tun

nel.

+:""

I\)

,I

y

- 43 -

(a) Model geometry.

(b) Coordinates and flow geometry.

Figure 2.- Model and f'low geometry.

Model 2-0 Model2•4

~- = R = c

b = rb =

2.83 0

1.648 1.618

Model 2-8 RN = 2.83 -

· R = 0.8 c a = 1.19.3 b = 2.010

rb = 1. 706 TP = 1. 770

(a) K = 0.707.

RN -R. = c

a = b

rb = TP =

Figure 3 .... ·Sketch and dimensions of models • Dimensions are in inches.

2.8.3 0.4 0.991 1.715 l.704 2.010

.. 45 -

Model 4-.:0 }iodel 4-4 RN = 4.80 ~ - .. 4.80 R = 0 R· = 0.4

c c b = 1 • .312 a = 0.701

rb l.597 b = 1.558 rb = 1~642

TP ::: 1.77~

Model 4-8 RN = 4.80

\ R = o.8

\ c a = 0.984 b - .l.858

:rb ::: 1.680 TP = 1.454

(b) K = 0•417.

Figure 3 .• Continued.

Model·F-0 Flat face R = 0 c

b = 1.263 rb = 1.496

.-·46·-

Model F-8 Flat face. R = 0.80 c

a = .0.80 ·• b = 1.673 = 1.676

1.200

(c) K = O.

Figlire 3.- Concluded.

Model F-4 Flat face RC = 0.40 .a = 0.40 b - 1.344

r· = 1.600 b TP = 1.600

- 47 -

o.4

u __!,__ .2

~

• ·; , ..

.8 Sym R c 0 0 a o.4 E... 0 o.8

Pt Mod. Newtonian .6 --- Mod. Newtoniah

plus centritugal force

-- Li and Geiger {ref. 22)

<> .4 0 .2 .4 .6 .8 1.0

(a) K = 0 .• 707.

Figure 4.- Comparison of experimental and theoretica1 pressure and velocity distributions.

u e

~

.8

0

0 0

0 o.4

- 48 ...

<> o.8~~1...-....-.1.--....-L-~.L-~_,.._--w-....-~~ ----- Mo • Newtonian - - - Mod. Newtonian

plus centrifugal force---1--.::1io1r.;.._1--__.,..__~

- ~ . Li and Geiger (ref'. 22) Cl

.2 .4 .6 .8 1.0

(b) K = o.417.

FiSure 4.- Continued.

u e

R..... Pt

LO

.9

.8 0

- 49 -

CJ

0 0 0

0 0 0 0.4 0 0 0.8 - Probstein (ref. 23) <>

.2 .4 .6 .8 1.0

(c) K = O.

Figure 4. - Concluded.

... . ... v 18 1.0

.8

.6

1.0 ...... . 'V ~ ;~

.6

LO ~ A .A .. ti:.I \9 l~

.8

.6 0 .2

- 50 -

J ~ <> <> () Cl <: -0 ....

D c• "'

'

(a) ' Rc = o.8 .

q;

~-(

<> <> 0 <> <I> 0 0 r1

0 0 0

\II

(b) RC = 0.4 .

0 <> D 0 0

0

.4

( c)

< ~ <> a r.

0 (

R = O. c

.6

I I

Sym K

0 0.707 <> D 0.4i7 n 0 0

0 °"1 . <>.

(

<> Cl <>

r1

0 0 . <> -

v .. Oo

'

<> ~ (

a Q

0 Ir:" <>

0 0 I":'

[J 0

c: .8

Figure 5.- Effects of model geometry on pressure.

l.O

o.4

.2

0

o.4

u e .2 {2H;

0

o.4

.2

0

c:;

~ < ~

i

8 ~

.. (b)

( -

0 v I.; 0 .e 8 <> ¢

0 .2

- 51 -

0 ·) " 0

0 0 < 0 0 [J A v

R = (a) c 0,.8".

R = c

0 0 [J ,.,,

0 <>

.4

( c)

o'.4 .

( •t:\

0 r;

~

~ > v

.6

R = O. c

,i.;i 0

":'"Q ( 0

D ~ -v Sym K

0 0.707 0 o.417.,.. 0 0

I I

OIJ

0

~

b € - r. CJ

v 0 .· D <> - [J "'-' <> 0

0 A

.8 1.0

Figure 6.- E:f:fects of model geometry on velocity.

-1 c, sec

12

10

8

6

4

2

0

_/' ~ / v ·v

I~

0

- 52 -

h ~ ~

/ ~

~ ~ .y ~

~ /

.2 .4

K

I I I Newtonian theory

(for a sphere) 'I)

d/-;,u

~ . ,,

ri Sym R c

0 0 D 0.4 <> 0.8

.6 .8 1.0

Figure 7.- Effects of' bluntness parameter on velocity gradient.

12 x 103

-1 c, sec

10

8

6

4

2

0

- 53 -

\

•I ... 1,,

~ --L---~~

0 .1

,, ;

.J '\

"

• i1 ·~

i---~ l----'~

Sym K

0 0.707 Cl o.417 <> 0

.2 .4

Figure 8.- Effects of corner radius on velocity gradient.

.5

Sym a., deg

0 0 o· 8.17 <> 16.25' A d28.o

0 c 0 a 0

.6 0:

{a) Model 2-0.

.8

- y 9 e Sym deg ~ ~

a., •a .

v 8 0 0 0 8.o

8 <> .19.0

<> II

8

1.0

.

v , c 0

.6 ~ J

... ~)

J! ' ' 0 ~· . '

' . ; ~

.. ... ~ ' ;' .4 . / 0 .2: .4 .6 .8 LO

s/~ \

{b) .Model 2-4. '

Figur~ '9 •• E:rte9t of angle of attack on pressure.

- 55 -

1.0

Sym a., deg

0 0 0 10

.8 ·¢ 20 A 28.5

L \A Pt

.6

<> 0 .4

( c) Model 2-8 •

1.0 .. ~ -• . ¥ l!f Q (t

A ~ 8 Q r ~,. L 0

deg 0 0 Sym a.,

("\

w ~

0 0 0 6.o ¢ 15°5 r

A 21.5 . • 6

0 .2 .4 .6 .8 1.0

s/~

(d) Model 4-o. Figure 9.- Continued.

,• ·'('

' - 56 -

1.0 -" - '

" ¥ ¥ ' r;> ~ ,, 0

6 Q n L 0 ,,

0 a 6 n

Sym a., deg ,,

0 0 a 0 D 5.5 <> 13-75 0

.6

( e) Model 4-4.

1.0 -~ ¥ ¥ e u ll. D

<> e ,... -ll. D

(1

<> 0 .8

~II. ~ [ I n v

.6 Sym a., deg ll.

0 0 <> a 0 D 9.0 <> 16.o ll. 25.0

n .4 0 .2 .4 .6 .8 1.0

s/RB

(f) Model 4-8.

Figure 9·- Continued.

. - 57 -

... ...... ...... -J..0 ". ~ ¥ ~ ~ 0 • g 8 0 :> A <> 0 0

u 0

Sym a., deg 0

0 0 0 5.0 <> 12.0 A 14.5

.6

(g) Model F-0 •

.. --1.0 \ ... y 9 ~~ Q 8 A 0 0 a 0 0 0

.8 A r I

0

<> .6 Sym a., deg

0 0 .!~ 0 0 5_.5 <>. 12.5 A 18.5 0

.4 o• .2 .4 .6 .8 1.0

(h) Model F-4.

Fi6ure 9.- Continued.

- 58 ..

1.0 ~ ~ "J y ~ u. .;; I

~ A <>' .· b b Sym a., deg . <: n r ~ 0 0

0 0 7.5 n <> 11.75

A A 19.0 <>

.8

0 0

.6 /J. <> 0 0

6. <)

.4 0 .2 .4 .6 .8 1.0

s/RB

(i) Model F-8.

Figure 9 ... Concluded.

1.0

.9 .....

.8

1.0

.9

Sym

.8 _o D <>

0

A

""

Sym

0 D <>

(a)

'"a

Re

0 o.4 o.8

.1

- 59 -

~ A

"' Vo ""' ::> 0 c 0

n K a, deg c

0.707 16.25 o.417 15.5 \;)

0 12.0

Effect of nose bluntness, R = o. c

~ J 0 <2i (>

~] 0 IJ,l.J

<> r. a, deg

16.25 19.0 20.0

.2 .4

0

c

(b) Effect of corner radius, K = 0.707.

<>

0

r\

.

r\

0 .;/

.5

Figure 10.- Effects of nose bluntness and corner radius at angles of attack on pressure.

0

.6

- 60 -

CJ . 0

24

l) o<> - .N - V' ~I deg 16

O' c ~ 8 11 ~

~ HC

rfJ <!} 0 0 -a o.4 ,o o.a· .. "' I • .. 0

(a) IC= 0.707 and K = o.417 (flagged symbols).

o.a 0 ~

·~ n

(j] (

0 . 0 8 24 32

a., deg

(b) K • O. Figure 11.- Location of stagnation point at a.Dgles of att$ck.

~ d

.20

0

.16 ti

0

.12

• os

• 20 ()

cl

.16 <>

.12

.08 -1.0

Sym

0 a <> A

., [ J

·o 4 •

Sym

0 a <>

'~ [ J

<~

- 61 -

a, deg 0 8.17

16.25 28.0

I n 0 H a .

(a.) Model 2-0 •

a, deg 0 8.0

19.0

H ~ .. I l ..

0 ,·; ..... ·o .....

., 0

tb) Model . 2-'*~

4 • 11

0 <> ~~

I J , . ,

u 41). 0

0 II w

ll (~

(>

,,

11

<>

Il " ; <> ;

l I Cl <>

~r J,/lf~ p , .. ~

»

.,

.' " -.: .. "

1.0

Figure 12.- Detachment distende mea.sUred in vertical plane o:f'; symmetry.

. \

~ d

~ d

.20

.16

.12

.08

• 22

.18 (~

n .14

<>

.I :i.

.10 -l.O

()

[J <>

~~

0

u <> .o.

Sym

0 D <> A

IJ ,, .::~

Sym

0 D <> A

C I

.:::.

... 62 -

·~ ~

a, deg 0

10 <> 20 ·;

28.5· .,

'

0 I J

~ r\

.::~ 0 <> [) ... Cl 0

() oli). 1 • ()

~ . 11 s ,_ r '

(c} Model 2-8 •

0 a, deg <>

0 I

6.o <> 15.5 .::~ []

·~ 21.5 ~ > H n () (~

r11 0

H 0 1t (~

H J.

"~

0 1.0

(d) Model 4-o. Figure 12.- Continued.

~ d

_JL d

.22

.18

.14

.10

• 20

.16

.12

.08 -1.0

u

r lot .

<~

( I

HI

(~

J~

Sym

0 a ¢

0 11

<>

Sym

0 0 ¢ Is

(I)

[~

<~ . ~

- 63 -

et, deg 0 5.5

13.75 < I)

< I> l ~

g H c ~ (I) u ~ ')

11 p I l

' ~>

(e) · Model 4-4 .

Ct' deg L~

0 9.0 0

16 •. 0. -·~~ ~ 25.0

< > \.I . ~ . CJ tJ I l

o· I ) H ~ t () ( I ()

I~ d' l~

~ 11

~ II>

0 1.0

r

(f') Model 4 .. 8.

Figure 12.- Continued.

-A_ d

~ d

.26

·.22

.18 ID

II

.14

• 24

.20

.16

.12 -1.0

I ]

I J

4 ~

I~ ~ ~

41' . ~ • ti.

0

0 [~

Cb ~ I> '

• I>

- 64 -

u 0 ~ . 0

0 ~ u 0 a

• 11

·~

(g) Model F-0 •

g L. . ll

u <) . ~ 0 ·h

o _r_ rmax.

(h) Model F-4.

Figure 12.- Continued.

0 h 4 r. ltl < )

-i~

0 Cb " ID

41> I tJ

Sym a, .. deg 1 D

0 o a 5.0 <> 12.0 A 14.5

a A~ A~ ~ [)

< I> I J 0

• i.

0 Sym a, deg

0 o 0 5.5 <> 12.5 A 18.5

1.0

~ d

.22

.18

.14

.10

.06 -1.0

',

()

[ J

< >

~ ~

', n • I

() <> if '

4 ~

J \.

.a

- 65 -

'

,H

I I 4~

' ' 4 .. ,, ~~ "'""' I~ I J " ll 0

~ > 0, [ ) ' \ A ~

'•"

0 ()

Sy1n a, deg

0 0 0 '7. 5 <> 11.75 A 19.0

0 1.0

(1) Model F-8.

Figure 12.- Concluded.

- 66 -

8 .17°

a, = 16.25°

(a) Model 2-0.

Figure 13.- Typical schlieren photographs.

0 a, = 0

- 67 -

(b) Model 4-o.

Figure 13.- Continued.

a, = 60

a, 21.50

- 68 -

a a

a a

(c) Model F-0.

Figure 13.- Continued.

- 69 -

a =- o0

(d) Model 2-4.

a =- o0

(e) Model 4-4.

Figure 13 .- Continued.

a =- 8°

0 a= 5.5

- 70 -

a, 00

(f) Model F-4.

Figure 13.- Continued.

0 a,= 5 .5

a, 18.50

- 71 -

a,

a, = 20°

(g) Model 2-8.

Figure 13.- Continued.

a = o0

a = 16°

- 72 -

(h) Model 4-8.

Figure 13.- Continued.

0 a = 25

a =- o0

a =- 11. 750

- 73 -

(i) Model F-8.

Figure 13.- Concluded.

0 a =- 19

THE EJ!'FllVrS OF NOSE AND CORNER RADII ON THE FLOW

ABOUT BLUNT :BODIES AT ANGLE OF A!r'l'ACK

By

Jam.es c. Ellison

ABSTRACT

An investigation has been conducted to determine the effects of

nose and corner radii on the pressure distribution and shock geometry

for blunt bodies at a.ng:Le of atta-ck. EX,Perimental pressure distribu-

tions were measured on a family of' blunt,. axisymmetric bodies which

included a range of' ratios of' body radius to nose radius from o.

(flat face) to 0.707 and ratios of corner radius to body radius from 0

(sharp corner) to o.4. The tests were made at a free stream Ma.ch num-

ber and a unit Reynolds number (per foot) of approximately 8.0 and

4.10 x 106, respectively. The range of angle of attack w.s o0 to 29°.

Included in this investigation are pressure and velocity distribu-

tions, shock detachment distw:ices measured from schlierenphotographs,

and a comparison of the experimental pressure and velocity distributions

(at zero a.ng:Le of attack) with theoretical predictions.