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REPORT 1147

THE SIMILARITY LAW FOR HYPERSONIC FLOW AND

REQUIREMENTS FOR DYNAMIC SIMILARITY OF RELATED BODIES IN FREE FLIGHT

By FRANK M. HAMAKER, STANFORD E. NEICE, and THOMAS J. WONG

Ames Aeronautical Laboratory Moffett Field, Calif.

National Advisory Committee for Aeronautics Headquarters, 1724 F Street NW, Washington 26, D. C.

Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific study of the problems of flight (U. S. Code, title 50, sec. 151). Its membership was increased from 12 to 15 by act approved March 2,1929, and to 17 by act approved May 25,1948. The members are appointed by the President, and serve as such without compensation.

JEROME C. HUNSAKER, SC. D., Massachusetts Institute of Technology, Chairman

DETI.KV \V. KKONK, I'll. I)., President, Rockefeller Institute for Medical Research, Viic Chairman

HON. JOSEPH P. ADAMS, member, Civil Aeronautics Board. ALLEN V. ASTIN, PH. D., Director, National Bureau of Standards. LEONARD CARMICHAEL, PH. D., Secretary, Smithsonian Institu-

tion. LAURENCE C. CRAIOIE, Lieutenant General, United States Air

Force, Deputy Chief of Staff (Developmant). JAMES II. DOOLITTLE, SC. D., Vice President, Shell Oil Co. LLOYD UARRIMON, Rear Admiral, United States Navy, Deputy

and Assistant Chief of the Burenu of Aeronautics. R. M. HAZEN, B. 8., Director of Engineering, Allison Division,

General Motors Corp. WILLIAM LITTI.EWOOD, M. E., Vice President—Engineering,

American Airlines, Inc.

HON. ROBERT B. MURRAY, JR., Under Secretary of Commerce for Transportation.

RALPH A. OFSTIE, Vice Admiral, United States Navy, Deputy Chief of Naval Operations (Air).

DONALD L. PUTT, Lieutenant General, United States Air Force, Commander, Air Research and Development Command.

ARTHUR E. RAYMOND, SC. D., Vice President—Engineering, Douglas Aircraft Co., Inc.

FRANCIS W. REICHELDERFER,

Weather Bureau. THEODORE P. WHIOHT, SC. D.

Cornell University.

Sc. D.; Chief. United States

Vice President for Research,

lluoii L,. UKTUEN, in. D, DirecUu

JOBN W. CROWLEY, JR., B. 8., Associate Director for Rtuarch

JOHN F. VICTORY, LL. D., Executive Secretary

EDWARD H. CHAMBFRI.IN, Executive Officer

HENRY J. E. REID, !). Eng., Director, I.angley Aeronautical Laboratory, Langley Field, Va.

SMITH J. DEFRANCE, I). Eng., Director, Ames Aeronautical Laboratory, Moffett Field, Calif.

EDWARD R. SHARP, Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland Airport, Cleveland, Ohio

L/NOLEI AERONAUTICAL LABORATORY,

Laugley Field, Va. AMES AERONAUTICAL LABORATORY,

Moffett Field, Calif. LEWIS FLIGHT PROPULSION LABORATORY,

Cleveland Airport, Cleveland, Ohio

Conduct, under unified control, for all agencies, of scientific r'.tearch on the fundamental problems of flight

II

REPORT 1147

THE SIMILARITY LAW FOR HYPERSONIC FLOW AND REQUIREMENTS FOR DYNAMIC SIMILARITY OF RELATED BODIES IN FREE FLIGHT l

By FHANK M. HAMAKKR, STANFORD E. NKICK, and THOMAS J. WONT

SUMMARY

The similarity law for nonsteady, in viscid, hypersonic flow about slender three-dimensional shapes in derived in term* of customary aerodynamic parameters. The conclusions drawn from the potential analysis used in the development of the law are shown to be valid for rotational flow. A direct consequence of the hypersonic similarity law is that the ratio of the local static pressure to the free-stream static pressure is the same at corresponding points in similar flow fields.

Requirements for dynamic similarity of related shapes in free flight, including the correlation of their flight paths, are obtained using the aerodynamic forces and moments as correlated by the hypersonic similarity law. In addition to the conditions of hypersonic similarity, dynamic similarity depends upon con- ditions derived from the inertial properties of the bodies and the immersing fluids. In order to have dynamic similarity, how- ever, rolling motions must not occur in combination with other motions.

The law is examined for steady flow about nlahd three- dimensional shapes. The results of a computational investiga- tion showed that the similarity law as applied to nonlifting cones and ogives is applicable over a wide range of Much num- bers and fineness ratios. In the special case of inclined bodies oj recvtudun, the law is txttnde<i i>i include twine significant effects of the viscous cross force. Results of a limited experi- mental investigation of the pressures acting on two inclined

y.o l*i ^

revolution.

«J fn „h„„h lhc no it n-nrt/ifiQ In ftrWoVe of

INTRODUCTION

The 113-personie similarity law for steady potential ilows about thin airfoil sections and slender nonlifting bodies of revolution was first developed by Tsien in reference 1. Hayes (ref. 2) investigated this law from the standpoint of analogous nonsteady flows and concluded that it would also apply to nonpotential flows containing shock waves and vorticity, provided the local Mach number was everywhere large with respect to 1. He also reasoned that similitude could be obtained in hypersonic flows about slender three- dimensional bodies of arbitrary shape; however, the form of the similarity law in terms of customary aerodynamic parameters was not determined. Oswatitsch (ref. li) investi- gated the law for two-dimensional steady flow in the. limiting case where the Mach number lends toward infinity and, hence, ceases to be a flow parameter. His formulation of the

law, therefore, involves only thickness ratio and angle of attack. Goldswcrthy (ref. 4) investigated the effects of rotation on the hypersonic similarity law for two-dimensional steady How. His results corroborated, in part, the previous findings of Hayes and showed the potential analysis of Tsien to be valid.

An investigation of the law as H applies in nonsteady flow was made by Lin, Reissner, and Tsien (ref. 5). In particular, the necessary conditions for similarity of hypersonic flow about oscillating two-dimensional bodies were determined. The analysis for more arbitrary motion of two- or three- dimensional bodies is appurently not available.

Khret, Rossow, and Stevens (ref. 6) investigated the hyper- sonic similarity law for steady flow about nonlifting bodies of re-, oiutiun by comparing pressure distributions calculated by means of the method of characteristics. They found the law to be applicable over a wide range of Mach numbers and thickness ratio0.. Their investigation did not, however, include the effects of vorticity arising from the curvature of the nose shock wave. Rossow (ref. 7) continued this in- vestigation and found that the law was equally valid when the effects of vorticity were included in the calculations. These findings corroborated, in part, the observations of Hayes and indicated that the law may be used with con- fidence to investigate the. aerodynamic characteristics Un- steady flow about nonlifting bodies of revolution at hyper- sonic speeds.

it appears desirable, therefore, to attempt to unify tlio different treatments of the similarity law into a single formu- lation. The primary purpose of this report is, then, to determine the form of the hypersonic similarity law for non- steady flow about slender three-dimensional bodies of arbitrary shape and to present the results in terms of custom- ary aerodynamic parameters. It is further undertaken to examine the hypersonic similarity law in some detail as it applies to steady flow.

The possibility of obtaining a hypersonic similarity law for correlating the aerodynamic forces and moments on related shapes in free flight suggests a more general dynamic problem, that of correlating their motions with the aid of this law. Hence, it is also undertaken in this report to determine the requirements on the inertial properties of related bodies and the immersing fluids in order that such bodies may have similarity.

similar free-flight paths, that is, dynamic

1 Supersedes NACA. TN 2U3, "The similarity Law fur llyiwramle Flow At t Kfe-ruti-r Three Dimensional SIIHIHS,"by Flunk M. IIIUIMW, Stanford K. N.'ltv, and A. J. Eiders, Jr., 1951, and NACA TN 2031, "The Similarity Law tor Nonsteady Hypersonic Plows and Ueiiu'nments for the Dynamical Similarity of KeteiUti Bodies in Free Flight," by Frank M. Haniaker and Thomas J. Wong, 1952.

1

(I

A b

<'r

Cc

<"„ V„ C

r, (',.

r, rm

6m

(\

£, <", c. c

<\

<\

It

it F f in a

1:3

REPORT 1 1 -17

SYMBOLS

NATIONS! VIA 1HOHV COMMITTEE KOK AERONAUTICS

speed of sound eliaracleristic reference urea of body, A — bl characteristic width of body

side force ' IPOWA

aide-force coefficient

side-force function

drag coefficient, ,- .,. ; ipo v o -'1

drug function

rolling-moment coefficient,

rolling-moment function lift.

•oiling moment ip„WAb

lift coefficient, IPOVO'A

pitching moment JpolV-le

lift function

pitching-moment coefficient,

pitehing-moment function a, • , vnwing moment

yawmg-moment coefficient,' . .,.2 ., 3 Po I I) •

yawing-moment function specilic heat at constant pressure specific heat at constant volume characteristic length of body section drag coefficient of circular cylinder

with axis perpendicular to the (low mean vd for a body of revolution

displaced-fluid-mass factor, Q

length of flight path viscous force or moment function (limensionless perturbation potential function general functional designation body-shape function (limensionless body-shape function vector from the origin of the coordinate system

to any pomi on the body

unit vectors along coordinate axes /,.y,r, re- spectively

moments of inertia of body about the I,IJ,Z

axes, respectively

6 v c

Ka = M,a, K, .U,0

Kt = &.K„- U,('(?)

JC-i/.(f>*-A/.(fJ

•I)

• hypersonic similarity para- meters

. , A ._, , , K, I r-r 'r -1

7/

K.z-i

i,m,n

M

, A' I)

dynamic similarity parameters

direction cosines of the unit outer normal vector to the body surface

Mach number

X r /'.<7.''

It

v A.'/.:

<5 t

f.i.r

e M

P

#,Q

0 /• l,2,n

moments acting on body about r,y,: axes, respectively

unit outer normal vector to surface of body static pressure rolling, pitching, and yawing velocities, re-

spectively radius of curvature of flight path cross Reynolds number based on maximum

body diameter and the component of the free-stream velocity normai to the body axis

radius of body of revolution at any station x, cross force per unit length characteristic depth of body components of body velocity along the i,y,~

axes, respectively resultant velocity Cartesian coordinates fixed relative1 to the

body forces on body along r,;/,r axes, respectively angle of attack angle of sideslip

(' ratio of the specilic heals, 7=yr

angle of roll orifice location on the test cones (limensionless coordinates corresponding to

x,y,z, respectively time coordinate mass of body density of the fluid

(limensionless time coordinate, — °- c

perturbation potential function j>r\*eT1.*ml function

alternate time variables angular velocity of the body

.>^«^*. Ut4 .

free-st ream condit ions viscous cross-force effects different functions F, (',„

noted or ('„, except as

SUPERSCRIPT

- vector quantities

Except for symbols noted above, all variables used as subscripts indicate partial differentiation with respect to the subscript variable1,

THE SIMILARITY LAW FOR NONSTEADY THREE-DIMENSIONAL FLOW

DEVELOPMENT OF THE L*W

The hypersonic similarity law is derived from the equations of motion and energy and from the boundary conditions. In deriving the law. the following assumptions are made: (I) The Mach number of the uniform stream is large com- pared to 1; (2) the disturbance velocities are small compared

THE SIMILARITY LAW KOH HYPEK8UNIC FLOW

to the free-stream velocities; ami (3) the How is of the potential type. These assumptions imply that the analysis is restricted to hypersonic (low over slender bodies at small angles of attack and to irrotational flows, respectively. As was indicated in the introduction, the law has been extended to rotational flows by both Hayes and Goldsworthy. An analysis is presented in Appendix A to show that the rota- tional effects in a three-dimensional nonsteady flow obey the hypersonic similarity law as formulated by the potential analysis. Hence, the conclusions derived from the analysis hased upon potential flow will also be valid for rotational flow. The purpose of making the assumption of potential flow is merely to simplify the analysis.

The coordinate system is fixed with respect to the body, as shown in figure 1. Also shown are the possible angular velocities of the body and the direction of the velocity vector of the free stream. The angles have the conventional positive sense of angles of attack and sideslip. Under assumption (2), these angles must be small.

KICHRE I.—Schematic diagram of orientation of body in How.

The development of the law involves, first, derivation of a simplified potential conation describing the flow second, the statement of the boundary conditions, and third, the trans- formation of these equations into nondimensional coordinates.

The simplified potential equation is obtained from the nonsteady equation of motion and the energy equation which nre written in the following potential form:

2(*„*,*,+*„*„*,+*„*,<f\.) ,

2(*r*-.+ *„*„.+ <f>;-l>,8) <>

*,+], (*/+<IV+<JV)+ ";, - 2*+£",

(In)

(lb)

These equations are expanded by expressing the potential- function derivatives in the following perturbation form:

1'ior Vn/3"', +*= 1 o— ,j ~ •> ~l~'fi'

*„= — V0& •*••/>„

$i = <pi

(2)

The local speed of sound a can be eliminated by combining the expanded forms of equations (lb) and (la). The result- ing equation can be simplified by neglecting higher order terms keeping in mind that for hypersonic (lows about slender shapes tpx, ipt, <pI} and a0 are small compared to \'0

and that tpx is small compared to </>„ and <p,. The simplified potential equation then assumes the following form:

^+.u„wi-^„ r.'iw-i (7-1) 'V" **-(-H o if" 0*„+ ".! L "" "•»

(7_l) „„ a*- + 2 „v + 2 «„-' + (7-,)«7~1J+

fe f"a/,VH (7- 1) t *>,-(y- i)"!/" fo,+(y + I) '!,/n «* +

7 — i <fit',y

a„ <lu

«,.-

•Pli. / , , , Mn&f: .\f,,aip„ <pu(fi.\. ,r ( ., ,

''o'J "V «u <>n a02 r a0 o0- /

The shape of the body can be expressed by the functional relation

tfCr,?/.-) = <> (4)

The unit outer normal at a point on the body surface is given by the vector

A - ••/; I w] ]• nk (5)

mid the requirement tluit the body be slender is satislied by the condition

/«1 (6)

There are two boundary conditions which must be satis- fied. The first of these is that the perturbation velocity, imposed by the presence of the body, must vanish at targe distances ahead of the body. Consequently,

-'I ill r — — re (7)

The other boundary condition is given by the fact that the How is tangent to the body nt the surface, that is, for no angular velocity

XV = 0 (8)

The angular velocity of a body will cause an apparent dis- tortion of the velocity vector at the surface of the body. By expressing the angular velocity in the form,

«=/» I gj+rk m the velocity of each point on the surface of the body is then given by the vector cross product

5XA=-=(23— ry)i-t (rx—p:)j+(.py—gt)k (10)

The boundary condition on the surface ui the body then becomes

(T-ZyJn-X^i) (ii)

JPORT 1147—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

After equation (10) is expanded and combined with equation (11), and higher order terms neglected in accordance with equation (6), the second boundary condition assumes the following form:

VeQt-(Vt0-i>,+rx~pz)Gl+ (I\a+v.+q*-/'?/)G, = 0 at G=0 (12)

In obtaining the similarity law for flow about related bodies, the equations of motion and boundary conditions are expressed in a nondimensional form. A nondimcnsional coordinate system is introduced by the following affine transformation:

t f V y 2 6(i0M0

COt C (13)

and a nondimensional perturbation ])otential function is defined by the relation

/(fitj.r. r)—— p-.-r, a.il/„f (|)

(14)

where c, b, and < are a characteristic length, width, and depth of the body, respectively. Under the coordinate transfor- mation given above, equation (4) takes the form

0(«.i».r) = O (15)

By substitution of equations (13) and (14), equations (3), (7), and (12) become, respectively,

A?(./„ -f /«)+(0V,, [AV -I (T - i) A7/< -

(7+D @ A,A^,+ (7-D ff,/C./r+

^rY^y^'V+^WV+fY- n A77,- il+^rAV+

(7-l)A7.A-(7-l) (j|) A'.AV,-)-(7 I l)K,KaJt+

72 ' (©S KW+y 2 ' A,7rM(7- l)A,a/,- l]+

2Kf)^[(l)^-^^>(f)^[-A'-^- K.Kfft + K, (|) A (A'. I• A7r)]+ A',/t( (/u + /v',/t) J +

»-[(t)-(g)V.+(t)<-K;i)'0-+[(t)+

*+®KK')']*-° <l8)

where A-, = Af„ (10)

AV Mn" c

A'„ = Me a

Kt=M„p

*-=M®

(20)

(21)

(22)

(23)

(24)

(25)

It is seen then that, if two related bodies are flying with given motions and attitudes so that the parameters, equations (19) through (25), are the same for both bodies, the flows arc characterized by the same function/({,ij,f,r) and are there- fore similar. The requirements expressed by the non- dimensional form of the body-shape function, equation (15), and the similarity parameters, equations (19) through (25), therefore constitute the similarity law of hypersonic flow.

A closer examination of the parameters A",, Kb, Ka, and A'o reveals that an essential property of similarity is that the lateral dimensions and the slopes of a body with respect to the (low direction are in inverse proportion to the flight Mach number. In fact, the remaining parameters Kp, K„ and Kr, which relate to nonsteady motion of the body, can be interpreted by means of the same property. In rolling, for example, points on the body surface , '-m helical motions, and the quantity pb/]~0 in equat is simply proportional to the slope of the helix with rtt (i> to the flow direction. This slope must be inversely proportional to the flight Mach number. Similar arguments may be applied to A, ami Kr.

Because of the complexities of algebra involved, the effects of angle of roll were not included in the previous equations. II<itl they been includcdj however, the result would h» the same as above with the additional requirement that the angle of roll must be the same for the related bodies. Hence, the additional hypersonic similarity parameter is

Kt (26)

CORRELATION OF AERODYNAMIC FORCES AND MOMENTS

The correlation of aerodynamic forces and moments on related bodies in unsteady hypersonic flows can be developed by consideration of the pressure distribution over the bodies. The pressure relation is obtained from the energy equation, equation (2), and is given in the following form:

/' , ,7-1 r,

i+72""!(rM 20.)

(27)

When this expression is simplified (in a manner paralleling the development of the preceding section) to include only higher order terms and put into nondimensional form, i(

THE SIMILARITY LAW FOR HYPERSONIC FLOW

reduces to a function only of the nondimensional coordinates nnd tiic similarity parameters (for a constant y).

p p i> = r> (£> i. £, T; Kt, Kt, Ka, K$,Kt, A,, K„ A,) (28) l o * 0

It is clear from this relation that for similar flows, the ratio of the local to the free-stream static pressure is the same at corresponding points in the flow fields.3

The correlation of the aerodynamic, forces and moments is then obtained with the aid of equation (28) by integration of the appropriate components of the pressure forces over the related shapes. This correlation can be given in the following forms:

••A/ L= (\— ft(A|, A&, Ka.'Kii, Aj. Kp, A,, A,)

•WrD=rD= (~D(K„ K„ Ka. K,, A,. K„ A'f. AM

M0CC= rc= CC(K„ Kb, Ka, KB, A8. K„ K„ Kr)

1 r. MaCm=Cm = Cni{K, . . . Kr)+^Vm2(K, h\)

<\=<\ = ('mi{K, . . . K)+jfj (\t(K, . . . Kt)

M0e,^C,*=C,(K, . Kr)

• (29)

it appears, from the equations for the pitchiiig-moment and yawing-moment functions of equation (29), that these two functions cannot be correlated for related but otherwise arbitrary body shapes. However, a careful examination of the order of magnitude involved in the analysis indicates that the second term on the right in these expressions becomes negligible in magnitude in all but two very special cases. In tho case of the pitching-moment function, both terms on the nghi side become ol Hie same order of magnitude when the / and n components of the unit normal to the body surface are very small. This condition corresponds to an r..v.i i .. ..*: .i /^.. . . I -. . ft r» / % Tf 1 ..iiriuuu >utnui 1111 «!.-> anutlil 111 H^uu' & til). 11, IIOIVCVIT,

the vertical fin is mounted on a body, or used in combination with a body equipped with horizontal wings, the contribu- tion of the vertical tin to the total pitching moment will be very small indeed. The contribution of the second term in the pitching-moment function for the. entire body will, of

(a) (b) (a) Pitching moment. (b) Yawing moment.

Kir;rRE 2.—Bodies excluded from similarity considerations as applied to pitching end yawing moments.

course, he correspondingly small. An analogous situation exists in the yawing-moment function for an isolated wing (fig. 2 ()))) in which the / and m components of the unit normal vector are both small. For most practical aero- dynamic shapes, therefore, the offending terms can be neglected, and correlation of the aerodynamic coefficients can be achieved as shown in the following relations:

•W',. (', - I ',(K„ A\, Ka, Kt, K>, K„, K„ Kr)

•W<',, ("',, -- (\{K„ Kh, Ka, K«, Ks, KP, A„ A,)

.!/,/',.= Cc= CC{K„ K„, K„ K„, Aj, A„, K„ A,)

Mat'„ = (\ = CJK„ A„, A„, K,, A,, A„ A„ A,)

('„ — ('„-- ('„(A,, A», ha, Ka, Aj, Kp, A,, A,)

.»/.''• (\{K„ A„, Ka. Kt, AJt A',. K„ K,)

Y (30)

DETERMINATION OF REQUIREMENTS FOR DYNAMIC SIM- ILARITY OF RELATED BODIES IN FREE FLIGHT

The requirements for dynamic similarity of related bodies in free flight are developed on the assumption that the forces and moments on such bodies are correlated by the law of hypersonic similarity. In order to determine the conditions for dynamic similarity to be coexistent with hypersonic similarity, the dynamic equations of motion should be, transformed to the same dimensionless coordinate system that was used in developing the requirements for hypersonic similarity. In addition, the velocity and force quantities should be expressed in terms of hypersonic similarity param- eters.

In this dynamic system, only those forces are considered which correspond to the "power-off" conditions in free flight. The coordinate axes are taken to be principal axes of the body so that the products of inertia vanish. The dynamic equations of motion of the body are given by the relations

I/J- re +(/>r

rs — /nr— ru

»•« — IJU rl>v

A

Y

Z M J

GiO

•J/J 052)

The translalional and rotational velocities may be expressed in terms of hypersonic similarity parameters, the Mach number, and the speed of sound of the free stream by the relations

u = a0^[(l, V-— aaKa, w=aBK,\

K, K, A„ (**) V ~aa JT» 7 •'"'"u—- r=o0- la f> 7

1 Analogous statements can be made for the ratios of local to fre&*streani values of temperature, 'lenity, and Math rjumber.

6 REPORT 1147 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

Similarly, the aerodynamic forces and momenta are given in terms of the correlation functions by the relations

(34)

,tf.V 2 )

v * /aB2Mnbtpa\

}-Cc\ 2 )

y^p /a„2M0btp„\

By substituting equations (33) and (.14) into equations (HI) and (32), and by treating only that length of flight path over which M0 ean be considered constant, the following set of equations is obtained:

'/A 3 . ,, Kpha

(IT A* KJ'C

dK„ A.pAjj >lr n« A, KJ\

(35)

(38)

(37)

1 dKp / 1 1 \ KfK, _ A^ /-, ,„fiv ,_XA6 rfr VAV, A,_ J :W 37? ' {M)

J_rfA,_/ 1 _ 1 \A,A„=f, A,,_„ rfr \ff,., Ki-Z/ Kb

A',_, </T \A',_T A',_,/ A'(,

(39)

where A',, is given by

and where

A' = /> M

C2/> A - J = =:/;^

A, -|f =

1 y-y

f clD A, — * ~ /,-.

D: Cbtp„ 2

(40)

(41)

(42)

(43)

(44)

(45)

all the terms involved may be of comparable order of magni- tude. Consequently, since equation (38) is the relation for rolling effects, it is indicated that flight paths which include rolling cannot be correlated by this method for obtaining dynamic similarity. For motions that do not involve roll, it is seen that dynamic similarity will exist for related shapes if the hypersonic similarity parameters and the dynamic similarity parameters given in equations (41) through (44) remain invariant. These dynamic similarity parameters re- late the masses of the bodies and the immersing fluids, as well as the distribution of the mass in the body.

For rolling motions only, correlation can again be achieved but with a slightly different set of parameters. In this case, only equation (38) remains and can be rewritten as

xk^-*' (46) where, now

AV^r (47)

The initial conditions to this set of equations are the initial values of the hypersonic similarity parameters,

If both hypersonic similarity and dynamic similarity are to be achieved, it is required that equations (35) through (40) be independent of the. Mach number as a separate variable. The elimination of .\f0' from equation (38) is impossible in the general case, even approximately, because

3 The parameter A", Is equivalent to a familiar stability-analysis term known as the relative mass factor.

so that correlation for pure rolling motions is now given by the hypersonic similarity parameters and the parameter A",.,:

A familiar example of motions where rolling effects would be absent is the case of motions confined to the plane of symmetry of the body, the so-called longitudinal motions. To extend the application of this law to the more general case where there are lateral motions as well as longitudinal ones, but no roll, it is necessary to have a suitable symmetry of shape and to have the inertial properties satisfy the relation

Ky„y=Kt.t (48)

When these conditions are fulfilled, the flight paths of related bodies ean be correlated. As an illustrative example, the disturbed motions of related missile shapes can be ex- amined. The lengths of corresponding portion? of related flight paths would be proportional to the corresponding lengths of the shapes. This property can be used to relate the amount i>f damping in the disturbed flight paths. As shown in Appendix B, the radii of curvature at corresponding points of the flight paths would be proportional to the product of the body length and the flight Mach number. Some of these points are illustrated in the example given in figure 3.

APPLICATION OF THE LAW TO PARTICULAR SHAPES IN STEADY FLOW

In steady How, the three similarity parameters Kp, A',, and K, are zero and equations (30) reduce to the following form:

•IA/ i— (\— ' t(A,,A»,A«,A'fl,Ai)'

•W '„=P»= <\(K„Kb,K.,KM

•I/B/'C— f C— t 'r(A,,A„,A"a,Ajj,Aa)

•M(/ m "" I m— ( •(A,,A(,,A«,A,i,Aa) (49)

THE SIMILARITY LAW FOB HYPERSONIC FLOW

^•4.0 '/t«.300 */c-.300

M.-Q.0 '/t-.l50 fct.JSO <W-2eA

FIOI'BE 3.—Ro!atod*ving-bo<Iy combinations at hypersonic sjM'cds.

It is important to note that the correlation of the aerodynamic coefficients given by equations (30) was obtained oil the basis of two restrictions as to allowable body shapes. (See section Correlation of Aerodynamic Forces and Moments and also fig. 2). These restrictions applv equally well to equations (49).

BODIES OF REVOLUTION

For bodies of revolution, equations (49) reduce to '

.U«/\='', = '',. (AT,, A\n

(.50)

where Kb is eliminated as it is identical to K,.b It is apparent from these relations that the corresponding force and moment parameters have identical values for related bodies of revo- lution provided the corresponding similarity parameters have identical values. It will now he shown that this conclusion can be generalized to include significant effects of the viscous cross forces on related inclined bodies.

The viscous cross force arises from the boundary-layer flow transverse to the body axis. A method of estimating this force along with the lift, drag, and pitching-moment co- efficients associated with it has been suggested by Allen in reference 8 a:-.;! is presented in Appendix V. The resulting expressions for these coefficients (sec eqs. (C*«i) in Appendix C) are transformed to the nondimensional form, and the following relations are obtained:

J\(K,J<a) y

M<,('.M=cdcF3(K„Ka)

(M)

For slender bodies of revolution of the type under considera- tion, &i is primarily a function of the Maeh number and Reynolds number of the flow component normal to the body

axis. Consequently, t-bexe expressions can be reduced to the form

MAt= ctr= <\{KJCM

M„Cm>=rm=(\(K„Ka, Re) J

(52)

where Re is the cross Reynolds number. For small angles of attack, the cross Maeh number is identical to Ka. It is clear, when comparing these relations with those, of equation (.50), that the latter relations apply with equal validity when viscous cross-flow effects are considered, provided that /?c

is included as a similarity parameter.8

Nonlifting cones and ogives.—In reference 6 an analysis was performed to determine the limits of applicability of the hypersonic similarity law for nonlifting cones and ogives.7 To determine this limit for cones, surface pressures were calculated using reference 9 and were plotted as a function of the similarity parameter K, as shown in figure 4. A single curve favoring the slender cones was faired through the calculated points. It is apparent that the similarity in pressure holds for a wide range of values of K, for slender cones. If it is assumed that a pressure deviation of 5 percent from the faired curve can be tolerated in using the similarity law, then limits of similarity can be determined as a function

0-0°

*.»

4.4 ft 1

4.0 / ! f

3« j f 3.2

Seml-vei'ex angle

o 5* D 10* O 13" " 20* * 30'

h , /

2.8 / /

2.4 t»

2.0 t>

1.6 'a u

1.2 J 4

.8 ,A f iJ r

A P f j 1 fr-

-i P0

.8 3.2 3.6 1.2 1.6 20 2.4 2.8 Similarity parameter, K,

KIIMHK I. -Variation of pressure ratio, /'/''o, with similarity parameter, Ki, for nonlifting cones.

• Bemuse of the axial symmetry of bodies cf revolution, only angles of attack arc considered. This latin consideration obviates a discussion of force ami moment characteristics at angles of side. or combined inplc.i «f attack and sideslip, while roll, of course, itas no incanine. It is clear, then, thai the similarity parameters A'.iand A** are eliminated from I his analysis

8 If the angle of attack Is wro, i.« Is also zero, and the expression for tho drag parameter reduces t" a form equivalent to that obtained iiy 'IV iti refeicmv I. • It is assumed that the viscous flo'v considered here does not significantly Influence the potcniial. in viscid flow discussed previously. Hence, the force and moment coefficients result im:

from these flows may bo superimposed. ' It should be notod thet oelv»j v.u act exactly a related set of bodies; nevertheless, they wen' chosen in this study since the configuration is of Interest, and the ile\ latimi i,t (hickness

distribution Is not significant for slender bodies.

REPORT 1117 -NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

of Much number ami fineness ratio c/t of the cone. The limits determined in this way tire illustrated in figure 5 (a). The shaded area indicates the regions of Much number and fineness ratio where the similarity law as applied to pressure on the cone will be in error 5 percent or more.

Since the surfnee slope of an ogive is largest at the vertex, the pressures at this point should provide a critical test for similarity of pressures. Accordingly, the limits of applic- ability of the law for ogives were determined in reference 0 from consideration of the pressures over a cone tangent to the ogive at the vertex. Figure 5 (b) presents the limits of applicability for ogives as obtained by this method. These results illustrate the conclusions of reference 4 that the law, as applied to nonlifting cones and ogives, is applicable over a wide range of Mach numbers and fineness ratios in spite of the simplifying assumptions made in the derivation.

A check of the applicability of the hypersonic similarity law in a rotational flow field was performed in reference 7 by comparing the pressure distributions, obtained by the method of characteristics, over ogive cylinders at several values of K,. The pressure distributions for two ogive cylinders at a value of K, of 2.0 are presented in figure fi and serve to illustrate the general results obtained in refer- ence 7. The high degree of correlation of pressures in figure 0 indicates that the hypersonic; similarity law applies in a rotational flow field and verifies the analysis presented by Hayes in reference 2.

Lifting cones.—For bodies of revolution al angles of at- tack, a limited experimental check was made in the Ames 10- by 14-inch supersonic wind tunnel. Two cones having fineness ratios of 3.0 and 4.9 were tested at Mach numbers of 2.75 and 4.46, respect iv-.-Iy; thus, the value of A', >vas 0.91. Overlapping values of K, up to 14° were obtained. Pressure measurements were made at the locations shown in figure 7 fur angles up to ri° The results are shown in figure 8 as a function of K„. Agreement with the prediction of the similarity law is generally observed, in that ihe values of />//>0-for corresponding points on the two bodies lie essentially along the same curve. The exception to this agreement i* on the lee sides of the cones (e=180°) where it is noted that significantly different curves are defined. This difference

af

0 2 4 6 Fineness ratio, c/t

? 4 6 8 10 Fineness ratio, c/t

(a) Cones. (b) Ogives. KiuniK 5.— liange of applicability of similarity law for nonlifting eone

and ogives.

is believed to be the result of the dissimilar flow separation from the two cones, caused by the fact that identical values of Rc could not be obtained for the two cones at the same value of K„. This difference in Re should not affect the pressures appreciably where separation does not occur.

8 M0c/t

. •

o 6 3* <=^_,

L oi26<^ \ ^ — *v

\< 2 4

\ r including rotation m

J

3 \ t 3

\ Neglecting rotation *

A * \ i N

* ._' Hi »

20 40 60 80 KK> 120 140 160 ISO 200 Longitudinal coordinate, percent note length

Kim HK <>.—Variation of pressure ratio, I'/Pn, along nonlifting ogive cylinders for a value of the similarity parameter, K,, of 2.0.

(a)

(C)

(a) r/(^3.0 (I)) r/( = 4.9

(r) Orifice location, «, in transverse plane, A—A.

Phil-Re 7-—Location of orifices on two cones tested at r-.". = 0.'.)l.

THE SIMILARITY LAW FOR HYPERSONIC FLOW 9

1.8

Flogged: M0 • 4.46, c/t • 4.9 Unf logged: M0 -2.75, c/t • 3.0

1.6

o" o 2 1.4

1.2

0 4 8 12 Similority porometer, Ka, degrees

FIIIIRK 8.—Variation of pressure ratio, l'/l\ with fV«, for two rones tented at K, 0.91.

WINGS AND WINC-BODY COMBINATIONS

If, for spanwiso symmetric wings, only angle of iittack is considered, the similarity parameters A'jj and K& vanish from equations (49) and only three of the aerodynamir coefficients remain. The corresponding force and moment functions arc reduced to the following form: '

M0<'„= ('•„= ('m(A'„A'»,A'„) *

(53)

These relations also apply, of course, to wing sections. In this case, b and therefore A"6 arc infinite and it is seen from equations (16) through (18) that the terms involving A'j, vanish and the equations reduce to the two-dimensional equations for hypersonic flow. The similarity parameter Kb is thus eliminated from equation (33). This result is equivalent to that presented in reference 1 *

Of practical importance is the conclusion to be drawn from application ot the dimetisionless equation of motion (eq. (16)) and the dimensionless boundary condition (eq. (18)), to steady flow about thin wings at zero angle of yaw. It is noticed in the equations that the parameter, Kb, always appears in the form

If 6 is of the same order of magnitude as c, then, consistent with the other approximations made in developing this equation, the terms involving (K,/Kb)2 are to be neglected. Performing this operation, however, yields the equation of motion for two-dimensional flow. Thus, it is indicated that, if the aspect ratio is of the order of magnitude of one or greater, hypersonic flow about wings may be treated ap- proximately as a two-dimensional-flow problem. The latter problem is, of course, relatively simple to solve.

From a physical point of view, this conclusion stems from the fae.1 that, in supersonic, flow, the effect of a disturbance at a poi;;t is confined to the conical zone formed by the

1 Parameters equivalent 10 these were obtained l>y Tsien and, although tint iiuhlisaed, we completion of this Investigation.

• The exponent? of -W0 obtained here are dillerent from those obtained in reference i. becails

Much lines from that point. For very high Mach numbers, this zone of influence is a narrow region behind tho dis- turbance. Consequently, conditions along a streamline are, for the most part, independent of the conditions along adjacent streamlines.1* For thin wings in hypersonic flow, therefore, it can readily be seen that the zone of influence of disturbances caused by wing tips will, for example, be small compared to the wing urea if the aspect ratio is greater than one. The effect of the lip disturbances on the aerodynamic characteristics of the wing will, of course, be correspondingly small.

Wing-body combinations may bo thought of merely as irregular-shaped bodies. As such, the aerodynamic coeffi- cients are correlated by equations (49) with the restrictions discussed in relation to these equations. The illustrative example, given in figure 3 in connection with the free-flight motion of a wing-body configuration, can be re-examined on the basis of steady flow. It is seen that in going from a Mach number of 4 to a Mach number of 8, the wing and body lengths are doubled, the angle of attack is decreased by one-half, while the body thickness and wing spans remain the same. The changes in some of the aerodynamic coeffi- cients are also shown in the figure.

CONCLUDING REMARKS

The similarity law for nonstcady, inviscid hypersonic (low about slender three-dimensional shapes has been derived in terms of customary aerodynamic parameters. The conclu- sions drawn from the potential analysis used to derive the law were found to apply also to rotational flows. As a direct consequence of this law, it was found that the ratio of the local static pressure to the free-stream static pressure is the same at corresponding points in similar flow fields. With the aid of this law, expressions were obtained for correlating the forces and moments acting on related shapes in hyper sonic flows.

It was found that the motions :>f related bodies in free flight could be correlated usinc the hypersonic similarity parameters and additional parameters relating the incrtial properties of the bodies and the air densities. The dynamic similarity of the free (light of related bodies can be obtained for motions which include pitching and yawing but no rolling. For pure rolling mottous, similarity can again be achieved.

In the case of steady (low a be it inclined bodies of revolu- tion, the correlations of forces and moments derived from the similarity law can be generalized to include the significant effects of the viscous cross force.

The results of a computational analysis, using the method of characteristics, showed that the similarity Saw as applied to nonlifting cones and ogives is applicable over a wider range of Much numbers and fineness ratios than might be expected from the assumptions made in the derivation.

AMES AERONAUTICAL LABORATORY,

XATIOXAI. ADVISORY COMMITTEE FOB AERONAUTICS,

MOFFKTT FIELD, CALIF., June ">, t95t. .resented in the form of lectoru notes which wore brought to the attention of the authors after

t is used as a reference area, rather ' an c-6. '• This res lit holds, In fact lor nonstoady as well as steady hy,wrsonic flow about thin wings, as pointed out by Kggcrs In reference 10.

APPENDIX A EXTENSION OF POTENTIAL FLOW ANAPTSIS TO ROTATIONAL FLOW

The hypersonic similarity law ran he extended to rotation ill flows by (he method of Hayes (ref. 2). This extension is in fact demonstrated by Hayes' results. However, to under- stand fully the reasoning involved, it is instructive to elabo- rate on his analysis. Hayes showed that the hypersonic potential equation for steady flow about slender shapes was identical to the nonsteady potential equation in one less spatial coordinate under the transformation

x=--a0.\f„e (Al) In the case of two-dimensional How, the transformation,

equation (Al), allows, for example, the upper surface of (he body profile to be replaced by the upper surface of a moving piston as shown in figure 0. The piston motion must be such that a given piston displacement >h al time 0, will be the same as the ordinate on the body profile at the coordinate J-I given by the relation /i - «0.\/t,fl,.

Pision

/,(*>

(a)

(a) Su-ildv flow.

Ki«;i KK (I.—Two-dimensional steady flow and analogous onc-dinii'ii sional nonstcadv How.

(b)

(li) AnaloKoni nonstendv flow.

In investigating tiie physical significance of this trans- formation, Hayes pointed out (hat its existence resulted from the basic assumptions of slender bodies and large Much numbers. Since, as a result of these assumptions, the x component of the fluid velocity does not change appreciably and is always much greater than the local speed of sound, there is essentially no chance for disturbances to propagate in the x direction. This is the essential feature (hat permits the. replacement of x bv the time variable 8 and, hence, lite existence of an analogous nonsteadv flow.

Hayes further showed that in hypersonic flow about slender shapes the local Much number remains large com- pared to one, even in the presence of strong shock waves caused by small surface inclinations. Consequently, the consideration of the, hypersonic flow about a slender body as a nonstationary problem in one less spatial dimension re- mains valid when shock waves and the resultant entropy gradients are present.

One further feature of Hayes' analysis, which is not explicitly stated in reference 2, is (hat similarity follows directly from (he existence of the analogous nonsteady flow.

This feature is illustrated for two-dimensional flows as follows: The motion of the nonsteady boundary (in this case, tHo piston face) can be expressed in the following dimen- siomloss form:

Upon transforming to the two-dimensional steady flow sys- tem, by the substitution of equation (Al) into the functional relationship on the right side of equation (A2), we obtain

'•(? ,.;,„)•'»(;,,s)-" a K) ««>

i-„(j> K, = const ant (A4)

Equation (A4) expresses the conditions for which the non- ste ady flow system can replace a steady flow system; namely, tliut the body profile must be expressible in a specific non- dimcnsiomil form and that the parameter, K,. must be constant for all profiles given by this form. These are, of eo«irse, the conditions of hypersonic similitude in two- dimrnsioniil steady flow. The extension of these considera- tions to three-dimensional steady flow is straightforward

To extend these concepts and results to three-dimensional, nonsteady flow, the nonsieady part of the flow may be co nsitlered, in the analogous nonsteady flow, as a nonsteady :.w-ntnnn t on the already iious'.cady boundary. This can be (|r»nioiislrated with reference to the potential analysis as follows: If the transformation, i — a0.\f04>, is used on the rojnation for steadv-statc hypersonic flow in perturbation fo»rm ''

JU .,--[ 1 (7 - 1) .U„ •V f ft

(In

[ , Mr, •fit—

7 -! 7 ! i

-' "+ an *' -) (1

llirtv is obtained the 1 '(III! ion

"n" L (7-

_7 1 )

a> «»"

V\it~

/ M *,<?.•

(A">)

'M.-J <Pm'

I -(7-1) [ .> iVtfi " \ «()•

",l"

"ii" a02 /

7 T

'I (A6)

In nil thi- criuattnnsof this »rtlim, iln-1

10 i'I ;L\CS air HKKIO to coincitlr « ith Jic hwl} a\ft

I liy applying the same transformation to the nonsteady flow I ( Hjiiation

"lt-r not to obM'iiiv ihr areumrat.

T1IK HIMlbAKITV UW HI It H Vl'KHHONIC VUOVt

•ft)» A^v^-h- . n.U„ 7 [-1 v%2 7-1 *'

«ii J L "o « "n - "II

(.7-1) - ii {•fin I -' I „ <P,<fin-\- Mu+„t-*i')+ "II J \ "o "II "ll /

2U, *"+i?lW+5?^"a (A7) with an additional variable change of

n-=9H + (AS)

the same equation (AC) is obtained with ^ replaced by S2. Hence, Haves' conclusions concerning steady-state, three- dimensional flow should apply equally well to uonsteady, three-dimensional flows.

APPENDIX B CORRELATION OF THE FLIGHT-PATH CURVATURE

Consider related bodies moving through properly related fluids in paths of finite radii of curvature, Equating the centrifugal force to the side force, the following relation is obtained:

M-/-rcip.r.M (BO

After rearranging in terms of similarity parameters, equation (Bl) becomes

"-A =rcKj, r;=eonstant (B2)

The parameter A/0c//? correlates the radii of curvature at corresponding points of similar flight paths,

This conclusion is also true for curved flight in the vertical plane.

APPENDIX C FORCES AND MOMENTS DUE TO VISCOUS CROSSFLOWS ON

BODIES Or REVOLUTION

In reference 11, Prandtl demonstrated that laminar viscous flows over infinitely long inclined cylinders may be treated by considering, independently, the components of the flow normal and parallel to the axis of the cylinder. Jones, in reference 12, applied this concept to the study of boundary- layer flows over yawed cylinders. The work of Prandtl and Jones suggests, as indicated by Allen in reference 8, that the cross force on slender inclined bodies of revolution may be estimated in the following manner: Each cross section of the body is treated as an element of an infinite cylinder of the same radius. The cross force per unit length on such a cylinder is given by the following equation:

,s-p-- /r,;f/3„\ *,,-' sin2 a (Cl)

The incremental lift, drag, and moment produced by this cross force are then given by the relations

lift irJcpn\'u2 sin2 « cos or

drag —rc,,^,,!',,2 sin1 c

moment---rjT^poV,,2 sin2 a

1!

(02)

Retaining leading terms in a and integrating over the body, where. r=r(x), the aerodynamic coefficients are given by the equations

Ci.

c„

~A J„"/J

A lr'lr

2r\,r«2 f,

Ac J„

(03)

where the reference area is proportional to the maximum cross-sectional area of the body, and the reference length is the body length. The coefficient c,ic is the mean cdc for the body of revolution, and has therefore been taken outside the integral.

REFERENCES

1. Tsit'n, Hsuc-shen: Similarity I.aws "f Hypersonic Flows. Jour. Math, anil 1'h.vs., vol. 25, au. .(, Oct. 1940, pp. 247-251.

2. Hayes, Wallace 1).: On Hypersonic Similitude. Quart. Appl. Math, vol. V, no. I, Apr. 19-17, pp. 105 IOC.

:(. Osuatilseli, Klaus: Similarity Laws for Hypersonic Flow. KTI1 Aero T.\ Hi, Royal lust, of Tech., Division of Aeronautics, Stockholm, Sweden, 1950.

•I. (ioldsworthy, F. A.: Two-Dimensional Rotational Flow at High Mach Number Past Thin Aerofoils. Quart. Jour. Mech. and Appl. Math., vol. V, pt. 1, Mar 1952, pp. 54-63.

S, fin, C. C, Reissner, Eric, and Tsien, II. S.. On Two-Dimensional Non-Steady Motion of a Slender Hotly in u Compressible Fluid. Jour. Math, and Phys., vol. 27, no. Ii, Oct. 1948 pp. 220-231.

(i. Khret, Dorris M., Rosso", Vernon J., and Stevens, Victor I.: An Analysis of the Applicability of the Hypersonic Similarity IAW to the Study of Flow About Bodies of Revolution at Zero Angle of Attack. XACA TX 2250, 1950.

7. Itossow, Vernon .1.: Applicability of the Hypersonic Similarity Rule to Pressure Distributions Which Include the F.ffeets of Rotation for Bodies of Revolution at Zero Angle of Attack. XACA TX 2399, 1951.

8. Allen, II. Julian: Pressure Distribution and Somo Effects of Viscosity on Slender Inclined Bodies of Revolution. NACA TX 2041, 1950.

9. Mass. lust, of Tech., Dept. of Klec. Kugr., Center of Analysis: Tables of Bupersonic Flow Around Cones, by the Staff of the Computing Section, Center of Analysis, under the direction of Zdenek Kopal. Tech. Rep. So. 1, Cambridge, 1947.

10. Kggcrs, A. J., Jr.: On the Calculation of Flow About Objects Traveling at High Supersonic Kpcerla. XACA TX 2811, 1952.

11. Prandtl, I..: On Boundary Layers in Three-Dimensional Flow. Ministry of Aircraft Production, Volkenrode VC5 84. (Reports and Translations Xo. (14) May 1, 1940.

12. Jones, R. T.: Effects of Sweep-back on Boundary Layer anil Separation. XACA Hep. 884, 1947.

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