Investigations on the supersonic flow around bodies

INVESTIGATIONS ON

THE SUPERSONIC FLOW

AROUND BODIES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAp· AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 24 OKTOBER 1962 DES NAMIDDAGS TE 2 UUR

door

PIETER JACOBUS ZANDBERGEN Vliegtuigbouwkundig Ingenieur Geboren te Schiedam. Bibliotheek TU Delft P 8251 1302

\\\\\\ \\\\\ \\\\\ \\\\ \\\\\\\ \ \\\\\1

c 24312

Dit proefschrift is goedgekeurd door de promotor Prof. Dr. E. van Spiegel

De schrijver van dit proefschrift betlligt hiermede gaarne zijn dank aan het bestuur van het lJederlands Insti tuut voór Vl1egtuigontwikkeling voor ~e toestemming resultaten van onderzoekingen, wèlke in opdracht van dit Instituut zijn uitgeVoerd, in dit proefschrift op te nemen. de directie van het Nationaal Lucht- en Ruimtevaartlaboratorium voor de aan de schrijver toégestane vrijheid, hem opgedragen onderzoekingen de voor dit proefschrift wenselijk geachte vorm en afwerking te geven, lalsmede voor het geven van de mogelijkheid van dezelfde reproductie~

middelen gebruik te maken als voor de gelijktijdig met dit proef-sohrift verschijnende laboratoriumpublicati~ over dit werk.

Summary.

1 General introduction. 2 The basie equations.

Con t e n t s.

2.1 'rlhe field equations.

2.2 The equations for shook transition.

, ' ~ The equa tions for linearized potential flow.

i

Studies on supersonie flow around axial1~symmetrie

3

8 8 18 21 bodies. 22

~ On the validity of linearized theory for

axially-symmetrie flow. 23

3.1.1 Integral expressions for body area and drag. 24 3.1.2 Tha 1inearized flow around a eone. 28 3.1.3 The 1inaarized flow around a body. 32 3.1.4 The exaet flow around a eone. 36 3.1.5 The exaet flow around a body. 38 3.1.6 Comparison of the prassure distributions. 42

3.1.1 Comparison ofthe floW fie1ds. 45

-3.1.8 Coneluding ramarks. 46

~ On the detarmination of optimum axially-symmetrió shapes.

3.2.1 The raquiramants for minimum drag. 3.2.2 App1ieation of variational theory. 3.2.3 Detarmination of the optimum velooity

distribution.

3.2.4 Determination of tha optimum shape.

!

Studies on quasi axially-symmetric flow.

~ On the validity of linearized theory for quasi axia11y-symm~trie flow.

4.1.1 Integral expression for the lift as faund by using linearizedtheory.

Linearizad theory foraninelined eone. Linearizad theory for the flow around an inelined body. /' 41 49

50

55

51

59

60

61 63 68

li

4.1.4 The ~irst order theory of Stone ~or the ~low

around an ino1ined cone. 69

4.1.5 The li~t on a oone aooording to Stone'e ~iret

order theory. 72

4.1.6 Comparison of the pressure distribution as obtained by different theories for an

ine1ined eene. 79

4.1.7- Comparison of the flow fielde for inc1ined cones.

80

4.1.8 Conc1uding remarks on the flow over inc1ined

bodies. 82

~ A first order perturbation theory for the oa1cu1ation of the invisoidsupersonic flow around

axia11y-symme-trio oon~igurations with arbitrary axis inclinations. 82

4.2.1 Out1ine of the methode 86

4.2.2 The ca1cu1ation of the transformed ~low field. 90 4.2.3 The boundary condition en the body. 101 4.2.4 ~e boundary conditions at the shock wave. 110

4.2.5 The ca1ou1ation prooedure. 117

4.2.6 On the ca1cu1ation o~ the rea1 ~low field

and the 1i~t. 119

4.2.7 Summary of ths investigation of ths flow around a deformed axial1y-symmetrio

oonfi-guration. 127

Referencea. 129

Appendix : On certain geometrio re1ations of an

arbi trary sur~ace. 132

Summary in Dutoh. 135

6

tables 38 figures

Summary;.

The supersonic flow around symmetrio and quasi axially-symmetrie bodies is investigated with a twofold purpose. One purpose

is to ·determine whether or not the linearized potential flow theory

can give an adequate description of the flow-field around such bodies.

The other purpose is to forward more reliable methods of computation for those cases where the results of the investigations lead to the conclusion that thia theory is inadequate.

A consideration of the mass- and momentum flow through conveniently chosen control surfa?es, proves that one can obtain a quantitative

measure for the error made by using linearized theory. The usefulness

of this concept is emphasized by making a direct comparison between

the results of linearized theory and those of more exact theories.

For axially-symmetrie bodies such a comparison can ba obtained by using an exact method of cparacteristics. The results show that the linearized theory is of only limited value, particularly so when

an interferenee between various parts of a configuration occurs. This leads to thè inv'estigation of optimum· shapes

ofaxially-syrr~etrio bodies with a given base area by uBing the non-linear

differential equations of isentropic flow. The same mass- and momentum flow equations are used here as for the determinatian of thc adequate- . nesa of linearized theory.

For the quasi-axially-aymmetrio badies a comparison can only be obtained for the flow around an inclined cone, since it is

2

the only case ~hich has been studied by using more advanced methode. Once more it is found that in most cases th~ linearized theory does not give reliable rssults. TherefOre a .method is 'presented tor the calculation of ths flow field around axially-symmetrl0 bodies with axis-inclinations. This method oonsists, analogous to that tor the cone, of superposing a perturbation on the purely axially-symmetric flow field. It is 'given in such a form that it is possible to perform the calculations by using a method of oharacteristics based on the characteristics of the axially-symmetric flow field. The analysis ls . restrioted to terms which depend on the first order of a small

1 General introduction.

The study of supersonic flow has a history of about one century. It wae init1ated by inveetigating the wave phenomena related to the propa-gation of sound. A now classical paper was written in 1860 by Riema~

(ref.l) on the theory of waves of finite amplitude paving the way for the development of the mathematical theory of hyperbolic equations.

Although the possibility of d1scontinuous solutiQne was recognized rather early, it was not until the publication of the works of Rankine

(ref.2) and Hugoniot (ref.3) that the equations for shock-waves were established as they are known today. At about the same time the first practical application of supersonic flow was made by the Swedish engineer Gustave de Laval, the discoverer of the nozzle named af ter him. Thie type of duct is and has been of fundamental importance for the development of supereonic aerodynamics, eince it plays an essential role in the operation of wind tunnels.

In the beg1nning of this century progress into the study of plane supersonic flow was made through the important work of Prandtl and his co-workers. They discovered and elaborated the so called simple wave flow,

thus °mak1ng it possible to design two-dimensional de Laval nozzles that are perfect.

°However, it may be stated that the great impetus to the 1nvestigation of supersonic aerodynamics was not made until about 1930. Two d1stinct lines of approach were then init1ated.

The first approach relies on the assumption that the disturbance veloeities, caused by bodies moving faster than the speed of sound, are small compared to the undisturbed veloc1ty. It 1s evident that such a otheory is rest~ioted in its range of applicability,i.e. the bodies have

to be slender and the Mach number not too high. On the other hand, the sim-plificat10n reached by linearizing the governing differential equations opens ths possibility to obtain results, which otherwise canonot °be found. The researohes of Ackeret oon plans flow (ref.4) and of von Kármán and Moore (ref.5) on axially-symmetric flow were the starting point for nume-rous applications of theee perturbation methods.

The second approach tries to f1nd phys1cally acceptable solutions of the non-linear d1fferential equations, governing supersonic flow. For plane flow several exact solutione ware known. The first exact solution for an

4

axially-aymmetric superaonic flow was given by Taylor and Maoooll (ref.6). Their work on the flow around a eone, oan still be oonsidered the starting point for later investigations of more general flow fields, by the method

.

of oharaoteristics.

It is interesting to see how these two approaches have developed sinae their initiation.

Especially during the second world war and thereafter the number of problems studied and solved by using parturbation methods, leading to linearized equations, are uncountable. Attention may be drawn to examples , such as ,the ,supersonic flow around inolined bodies, and the study of the

optimum shape ofaxially-symmetrio configurations with respeot to wave drag. To account for suoh braad applications of in fact only approximate methods, various reasons may be given.

One of the most important reasons in the opinion of the author , is the taot that there was already a well-developed mathematical theory for linear partial differential equations, whioh together with the principle of super-position could be used to reduce many very complicated problems to a few simpl erones •

The study of exaot flow fields around axially-symmetrio configurations was etimulated by the publication of a comprehensive table of the flow around a oone by Kopal et.al. in 1947 (ref.7). Thie table was obtained by numerioal integrations of the equations of Taylor and Maccoll. It is

interesting to note that these oomputations were performed with the aid of ordinary desk computers.

A further step forward in this field was made by Stone, who determined the flow around an inolined cone,correot up to the first order in the angle of inoidenoe(ref.8}.The seoond order term has subsequently been determined. Extenaive tables ot the data obtained have also been given by Kopal (ref.9

and 10).

'In the mean time several papers had appeared, exploring the applioabi-lity of, the theory of oharacteristio surfaoes and oharaoteristio equationa pertinent to hTPerbolio equations, for the numerioal ealculation ot ths

flow field around axially-symmetr10 bodies. Tbe researohes of Ferri may be mentioned here, 8speoially sinoe he tried to generaliz8 the metho,<l of Stone'

tor bodies at an angle of attaok (ref .11)', by usinga method of oharaoter-ietios. Ferri (ref.12) was also ths first to point out an inconsistenoy

in the theory of Stone who ignored tha singular behaviour of the entropy at the surface of aninclined c.one. This criticism, laading to the conoept of a vortioal layer, d.oes not influence, however, the preasure distribu-tion 'obtained' by the first or.der theory. of Stone.

A natural and important question is:"How do the rasults obtained by the linearized and the exact theory compare?"A direot oomparison, however, is only possible if there are bodies for which the flow oan be oalculated by using both methods. As .is evident from the foregoing disoussion, this is the case for the eone. Already in 1947 this comparison was made by Kopal (ref.13). Although only valid for a cone, this work oonstitutes a sharp oritioism against the use of the linearized theory. Already at that time it was remarked: "if we wish to progress with quantitative in-vestigations .of supersonio flow around eolid bOdies ••••• , we oannot avoid

the non-linear character of these problems". It is quite astoniahing that this serious warning against the use of linearized theory seems to have had no effect, for since that time a tramendous number of papers on linearized methods have appeared.

However, though it is very easy to say that problems should be solved by more exact meth~ds, such aremark has little signifioance when such more exaot methods are not available, or if time and money are prohibitive to their applioation, whioh was certainly the oase at the tima they were propossd. On ths other hand, quite a number of papers have appeared which have attempted to define the range of validity of the linearized theory. Aa an example of suoh a paper, the one by Miles may be mention~d (ref.14). However, all the reeults of these researches have the drawback that they lead to rather vague requirements, not giving a quantitative meaaure for the error whioh is made by using linearized theory.

Moreover a variety of methods have been proposed to improve the re- .

suIte of the first-order linearized theory. Suoh a prooedure for instanoe is given by van Dyke (ref.15). This eeoond-order theory, however, does not extend the range of validity very much~ so that its practical useful "-ness is only limited. A oomparison of the results obtained by using these improved methods has been given bi Ehret (ref.16). Tbe oonolusions reached are that the range of body shapes, finenesB ratios and Maoh numbers for whioh th~se theories give aooeptable results, is limited. It should ba borntin mind that this applies only to the pressure distzibution along

6

the body. Researches on the validity of the linearized theory for the determination of the whole flow field, show that ths deviations between linearized and exaot results beoome largsr in the outer flow field. This makes it rather doubtful if linearized theory oan be used to solve pro-blèms of interferenoe in areliabIe sense. One important representative of these problems is the searoh for optimum shapes with respeot to wave drag. In this case a oertain part of the fuselage has to interfere with all the other parts in auoh a way that the wave drag is as low as possible •

_.

The point of view suggested by the results of the mentioned researoh-es oan be summarized now as followsl

Evidently the linesrized potential theory is the simplest tooI available for analysing s~personio flow around a oertain oonfiguration. However, in praotioe no measure of the quantitative error made is pOBsible, if no oomparison can be made with exaot resulte. Up to now, there are only very few problems whioh can be solved by using exact methods.

On the other hand the aPl'lioa tion of exact methods for the num'3rioal determination

ot

a flow field, which required a large amount of time beoauss no eleotronio computers were'avàilable when they were proposed for the first time, has become much ~implèr :due to the rapid development

ot

these devices. Therefore it seeme ~dvisable to usa these exact methods wherever possible, in order fo avoid the unoertainty of the values

obtain-ed by using the linearizobtain-ed theory. This implies the development of ap-propriate methode tor a variety of problems.

The task set forward by these oonaiderationa oan therefore be described as follows:

1. Ä method should be found to meaeure the quantitative error in the results of the linearized theory which would not require making a direct comparison between these results and the results obtained by using other more exact theories.

2. For oases where it has been ehown that linearized theory oannot ba applied, methods should be developed which would be both numarioally applicaqle to, as weIl as based on the exaot diff3rential equations of supersonio flow.

It is the purpose of this thesis to investigate along these lines a rather small domain of the theory of supersonic flow.

Two classes of problems will be considered: In the first place the flow around axially-symmetric configurations where the axis is aligned with the direction of the undisturbed !ree stream. in the second p~ace the flow around a quasi axially-symmetric body will be considered (suoh a body is obtained by deforming the axis of an ini tially axially-symmetric _body.

A configuration at an angle of attack is one of the most simple examples). To achieve.a systematio representation the paper has been divided into three main parts.

The first part ·gives a genera I account of the equations' governing eupersonio flow and shock waves, deriving thereby the frequently used equations needed in ths other two parts.

The secpnd part oontaine the resulte of investigations on the super-sonic flow around axially-symmetric configurations. First a quantitative measure for the error in using the method of linearized theory is given by considering mass flow and momentum flow through conveniently ohosen control surfaoes. Espeoially for the flow around a cone simple results are obtained, but the method is equally applioable to more general axially-symmetrie bodies. For greater understanding of the usefulness of this oon-cept, a direot oomparison is systématioally given between the results of flow phenomana obtained by calculating with an exact method of charaeter-istios and those results obtained by using the linearized theory. The re-sults obtained sh~N that linearized ,theary is of only limited value, especially when it is used on those problems where interferenoe occurs.

When using the non-linear equations of supersonic flow this insight leads to the inveetigation of·optimum shapes ofaxially-symmetrio bodies with a given base area. The dieoussion will be restrioted here to the case where the flow in a certain part of the flow is isentropio.

In the third part the supersonic flow around quasi axially-symmetric configurations will be investi?B-ted. Here also will be. given a quant i tative measure of the error made by using linearized theory. Here, however, the

situation is less favourable for a direot comparison, sinee only the flow around acone at an angle of attack has been solved by using more advanced methods. Therefore aftar hav~ng shown with momentum transport consider-ations, that this analysis of the flow around a cone is fully consistent, an attempt is made to forward a theory which enables the numeri cal cal-culation of the flow field around a quasi axially-symmetrio body. The

8

method proposed is, in fact, analoeous to that of Stane, a perturbation theory superposed on the purely axially-symmetric flow field. The in-vestigation"is limited to the determination of the perturbatione up to the first order of a small deformationparameter.

2 The basic equations.

Here a rather detailed derivation will be given of the basic

equations. Subsequently the equations for a supersonic domain and for a three dimensional shock wave will be given. The equations valid tor the 1inearized potential flow will be summariz"ed.

2.1 The field equationa.

In this seotion the basio equations will be given which are valid for a domain of supersonic flow not containing shook waves.

It will be assumed that the effects of viacoaity, thermal conduction and diffusion oan be neglected, with the medium cansidered an ideal gas.

A Cartesian coordinate system xl' _{x2' x}

3

will be uaed (see fig.I.). The velocities in the directions of the respective axes are given as

uI' u_{2 ,} u_3•

Using the summation convention of minstein the equationà of motion

and the equation"of aontinuity aan be written as dUi I

a

(2.1)

_-+-_2.E...=

₀ dt P 3x i ~ a~ _{0 (2.2)} dt

-+-

fa~

:: d where p is the densi ty, P the pressure and t denotes time. The symbo I

TI

ia the substantial derivative. Tbe assumptions about the physical pro-perties of the medium give rise to the equations

dQ dS 0 (2.3) dt :: T dt ~ and p = TIT

(2.4)

P

where Q is the heat added, S the entropy and T the absolute temperature.

Use wiJl be made of the fact that tor a reversib1e procese

Tbe internal energy dE tor an ideal gas is given by

dE

=

c dT _v (2.6)

where c is a constant, viz. the specific heat with constant volume.

v

Introducing eq.

(2.ó)

into

(2.5)

the -entropy aan be written as:

where a is the speaific heat ;yi th c ons tant preas-ure.

p

It is preterable to define a specific entropy by

s

e = -c v

(2.7)

(2.8)

If the values in the undisturbed stream, which is aBsumed to be uniform, are given by p ==

Poo

t P

=

Poo and s

=

0,

equation

(2.7)

aan be

~Titten by using eq.

(2.4)

as -y C +s PP

=

e

(2.9)

c -y p where C '= p p. and Y "" - -00 co a • v de

It 'should be remarked that whereas dt

=

0, the value of s is not in general equal te zero, beaause sh~kwavesmay have occurred outside the domain considered.

Differentiating sq.

(2.9)

and using ths relation for the veloaity of sound and eq. (2.3), there is ebtained

dp 2 dp

dt - a dt

=

0

(2.10)

where _•

(2.11)

rrhe analysis will be rsatric.ted to the case of steady flow hencs (2.12 )

10

If account is taken of

eq"

(2.12) equation (2.10) together with eq. (2.1) and eq. (2.2) multiplied by u., gives rise to the fundamental

). re1ation

=

0 (2.13)

Tc obtain another set of equations, use wil1 be·made of the entha1py H. This is defined by

R=E+R P

Using eqs. (2.3) and (2.5) and substituting eq. (2.1) times u. into _). the differentia1 equation which can be obtained from eq. (2.14), there is obtained aftar integration

R +

~

uiu_i

=

constant along a stream line.

Since the flow is assumèd to ba uniform far upstream, there holds:

H +

~ ~iUi ~

constant in the who1e flow field. (2.15) Differentiating this equation with respect to ~ and subtracting eq. (2.1) there follOWSt

u.

J

~ui

_-

_-

~l

_.

_-

.

ént

1 à~ _{0 (2.16)}

). l

d~ dX_i

J

.

_{~~ p} a~ •

F'rom eqs. (2.14) and (2.5) the following gen~ra1 relation can be obtained

dH -

1:.

dp

=

T d S

P • (2.17)

This means that the va1ue of a contour integral has to be zero i.e.:

f

{<IR -

%

dp - T dS}" 0 • (2.18) Frcm this result, with the aid of eqs. (2.3), (2.5) and (2.14)

together with the condition that the flow is unifo~m far upstream, it can be derived that

(2.19)

Introduèing this relation into eq. (2.16) there is fina11y obtained (2.20)

or in vector notation

2

- -+ a

u x rot u

= -

-yr(Y-_~l~)- grad s

(2.21)

which equation is known.as Crocco's theorem.

By using eq.

(2.15)

and observing thai for an ideal gas the enthalpy is equal to c T, it follows that

P

(2.22)

where Moo is the Mach number and _Uoo is the velocity of the uniform undisturbed flow.

Substituting eq.

(2.22)

into eq.

(2.13)

and into the system of eqs.

(2.20)

there is obtained a set of four non-linear differential equa ti.ons for the four unknown quant i ties ui and s.

This system of equations will be investigated further in the remainder of this section.

It is. of advantage to use a oylindrical coordinate system x,r,~

and associated velocity components u, v, w, because here our main interest is the study of ~xially-symmetrio bodies (see fig.I).

and

The transformation formulae are given by

Xl

=

x

x₂ '" r sinq>

x

3 = r cos <IJ

u₂ = v sin q, + W oos

<i'

u

3 = v cosq, - w sinq>

(2.23)a

On using these equations, eq.

(2.13)

and eqs.

(2.20)

can be transform-ed into the following system of four equations

u2 au y2

av

1

",2

alF:' y

(1--

₂ )-+

_ax

(1--)-

_{2 ar} +-_r

(1--)

₂ ~

_at"

+-+

_.

_r

a a a "

vu (au av) _ .u'W(~

1

~)_ ~

(1

av ~) '"" 0

- 2'

ar'"

äX

2 é}x + r aq> 2 r aq, + é}r

12

v(2Y _

~)

.w

(i!. _12.!!.) . a

2

l!

=

0 ax ar ax r a~ Y(Y-l ax _ w(l av _ Uw ... !!)-u

{~

_ au} + r a~ ar r ax or 2 a a~

y(y-l)

ar :z 0

(1

av

~

w)' {

aw

1

au}

a,2

1

OB v

r

aq, -

är -

r

-u

äX -

r

ä<P

+ Y (Y -1)

r

ä<P

=

0

Thia system will now be brought into the farm of a set of relations valid along oharaoteristio surfaoes.

These surfaoes are thus defined that the re1ations that are valid along them contain on1y' derivatives along the surface.

Henoe, it is not possible to oonstruct a solution tor the flow field starting from quantities given along auch a oharaoteristic surface. ,

To find the oharaoteristic surfaces it will be assumed that auoh a surface oan be written as

r

=

f (x,lj»

The derivatives along this surfac!3 or the so called "inner" deri-vatives are given then by

af where

crI

=

ax 1 af where Cï 2

=

r

aep

(2.26)a

(2.26)b

$ubstituting these equations into the system

(2.24),

the result can be written aSI

(2.27)

This set ot equations has been given in tul 1 on the following page.

ac.oi

This is a system ot ~quations trom whioh the quantities

ar-

oan be solved, provided that the surfaoe and the :flow quant i ties w{ along the surface are given. In that oase t~e right hand side is known, to-gether ;,i,th the ooeffioients ot the unknown derivatives.

NcW as has already been remarked_f the requirement for r ~ t(x,~) to be a oharaoteristic surfaoef is that it is not possible to continue

- W(J"2 + v -u uCJ 2 t V(j'" 1

, -ua:

-wü. 1 2 , vU₂ , WÛ l

,

t -u(J. 1 (1- U 2 )-ÖU _

!!!

1.

bu_

~ ~

-

~

1.

~

_

~ ~

+ (1-

i)

.!

~

2 öx 2 r b~ 2 bx ~ r b~ 2 bx ~ r ó~ a a a a a - a 1 bu bv ....,,--+v-r

b'tJ

öx 1 éu u

r

bq> CV 1 bv u - -Jfol-~

ex

r 0.,. 1 óv +v

r

ó<p

bw + w bx u bw öx a2

~8

+ r{r-l)

bx

~

The matrix equation a ij

-

or

J

bi (2.27) written in full.

2 a cr: , Y(r-i) 1 2 - a -IJl , r(r-l) 2 a +v , r(r-l)(f2

ov

ar Ow

or

AS

--

or V + -_r 2 w + -_r 2 _ a " 1 bs vw

-

+

Y{r~11

r

bcp -

r

(2.27)a

...

V4

14

aw.

means that there cannot be found unique solutions for the quantities ~.

The system shou1d be either incompatib1e or dependent. Then there ho1ds

Det.

a.

_1.J

.

=

I

a··1

_1.J = 0 (2.28) This requirement leads to an equation for the unknown quantities

crI

and' cr 2 •

Now the on1y physically possi bIe cas'e is tha t the equa tions are de-pendant. This maans that for each associated, pair of va1ues for Û l' and

Û 2 obtained from aq. (2.28) an "annuling vector" can be found such that

y.

a .. = 0

1. 1.J

This is on1y possib1e if at the same time the following relation is satisfied.

Y.b.

= 0

1. 1. (2.3 0 )

This equation is the compatibility equation, for it is valid if equation (2.29) is valid. Now b. is an expreasion containing on1y the

1.

functions and the inner derivatives along the surface r = f(x,<P), and thus eq. (2.30) is arelation which satisfies the requirements for f to be a characteriatic surface.

The characteristic directions can be found by applying eq. (2.28). If the operation of determining the determinant of a

ij is performed, the result obtained is:

(V-UÛ

1

....

Û

2)2{Ü/

+ ü/+

1 -

a~

(V-

UG

i-<i2)2} - 0

Now tbe vector

(U

l ,-I,CJ2) is proportional to tbe unit normal vector (nI' n2, n3 ) of the surface r

=

f(x,~).

Equation (2.31) gives as characteristic directions therefore

v-ua: -wG:

1 2 = ~ + vn2 + wn₃ = 0 (2.~2)a and _v-uGi-wG2 = _±a

V

_([12 _{+ Û2}2 ₊ I

or

unI + _vn2 + wn

3 :% - a (2.3 2 )b

It shou1d be observed that the re1ation (2.32)a has tb be counted twice,according to eq. (2.31). The interpretation of the eqs. (2.32)a and (2.32)b is in fact .quite simpIe. Equation (2.32)a states that the normal vector in a certain point P of the surface r

=

f(x,~) shou1d be perpendicular to the vector (u, v, w). The set of characteristic sur-faces obtained in this case is therefore the set of stream sursur-faces. The etream1ine can be considered as a characteristic 1ine in this case.

Equation (2~32)b states that the velocity normal to the character-istic surface is equa1 to the 10cal velocity of sound. This means that this surface is 10cally a cone with a half top angle u with respect to the vector u, v,

w,

where ~ is defined by

1

tan IJ.

= i3

= .. 1 (2.33)

where M is the 10cal Mach number.

To find the relations (2.30) which are valid along the character-istic surfaces, first the annu1ing vectors

Y

.

have to be determined.

1

If sq. (2.32)a is va1id, ths matrix a

ij of eq. (2.27) reduces to

<Tl

-

~ _{û2_} 0

uGï

VÛ l w()1 ccrl -u -v -w -c v(J2 wU_{2 cG2} 2 a where c ".

r(Y-l)

It aan be seen immediately that the annuling vector,has to satisfy ths the reléi.tions

If the aompon~ts ~2' V3 and

y

4 are considered to be the compo-nents of a vector

y ,

equation (2.35) can be written ana1ogous to eq.

(2.32)a as

-.

...

16

This equation has the, two independent solutions

...

_'\>

₌

-

_u

-

--

-and

y

= u x n

as follows by nsing eq. (2.32)a.

The two annuling vectors for this case are therefore given by

'and

Tbe given by

Vi

={o, u, v,

wf

~

1 -

{o,

_vU2 + V', WV₁ -

uu

_{2 ,} -U-v<J₁} compatibility equation for the annuling vector

u~+w.!.~= 0 bx r

b<l'

(2.38)a is

Prom tbis equation, by using the eqs. (2.26)a and t2.26)b it follows, tbat ds or dt

=

0 (2.37)a (2.37)b (2.38)a (2.39)

Thus the result is found that the entropy has to be'constant along a streamline. This cannot be too surprising, since in fact this is a direct consequence of the assumption made about thephysical behaviour of the medium. Equation (2.40) is the same as eq. (2~3) as it ought to be

Tbe compntibility equation for the annuling veotor (2.38)b is given by 2 u bu _ u<r ~ + .!. by _ u bw _ ~

cr

--r~a-=-T

r

ó<p 2 óx

ua

l r b~ bx r I + Y(Y-I) 1 bs

r

b~ = 0 where uae has been made of eq. (2.39).

It oan be shawn that,this equation expresses the taot that the oomponent of the ::rotation veotor, normal to a stream ~urfaoe for whioh the entropy is oonstant, vanishes.

Now the annu11ng vectors and the oompatibility equations will be determined in case that eq. (2.32)b ,holds.

In that oase the matrix a

,

o

u 2 2

v_/~t

2-

'

w/2-

2

a;.

.±.

'ä

UI +ü2 +1 , -I.±.

'ä'fJi

+Ü2 +1

,cr

2

±

äVGi

+(f2 + 1

uGi

.±.

_{a lü1}2 +

c:r/-

+11, v(j"l

,

-u

-w

,

-0

,

where the + sign refers to v-Uül-wü2

" 2 2 1

the - sign to v-uCil -w<T_{2 :: -aVÛl + <1'2 +1}

Again the components )) 2' Y3 and ~ 4

satisfy the relation

of the annuling vector have to

or

---

n.

V ::

0

Now according tó eq. (2.32)b there holds

--

n.u = + a

These two equations together with eq. (2.42) give

-

--

v=

_~+-u n

a - a

v

1 ::-1 . •

The complete expressions for the annuling veotors are then given bYI

(2.44)a and

V'

{-1'

2'-

u

crI

,

""2+ v 1

,

2"-

w af12+Û₂2₊ i

o

2 2 i · a _{1 -} a a a 0"1 +02 +1

CJ"2

!(2.

44

)b

,J.

2 2 i a V

<1i

+02

+1

The compatibility equation for the annuling veotor (2.44)a is given by

18 1 2 ' bs w bs a bs 1 ba v-ua:

-wa: {

}

2 { } +

r(r-I)

u bx +

r

blP"

rCr-1) Û1

bi

+

Û2

r

b<f

+

-(v-ucr:-wcr: )1(1- u2

)''lU _

~

1.

~

_

~ ~v

_!:!

1.

~

_

~ ~

+ 1 ,2 _{. ' a} 2 bx _a 2 r by _a 2 bx _a 2 r b~ _a 2 bx 2 1 bw +(1- w₂₎

r

blP a

+ -

.

v-}

r

=

0

As oan be seen, the oharaoteristio equation (2.45) in this form is equally valid for the annuling veotor (2.44)b. ~e equation thue ie valid along the two different surfaoee gi~en by eq. (2.32)b. The differenoe lies in the faot that t~e quantitiee

G

1 and

cr

2 are re1ated by a different formu1a in the two ~ases.

Thus the original set of four partial differentia1 equations has been transformed into a system of four oharaoteristio equations given by eqs. (2.40), (2.41) and (2.45) together with the oharaoteristio direotions given by the eqs. (2.32)a and (2.32)b. It is this set of relations whioh wi1l play an important role in the following investigat1on8.

2.2 The equat10ns for shook transition.

Sinoe in the following paragraphe the notion of a shook wave wil) be used frequent1y, here an aooount will be given of the equations valid for the transition. In fact a shook wave 'is a surfaoe where the flow quantities oan be oonsidered to ohange disoontinuous1y. In reality it ie in genera I a domain of the flow with a thiokness of a few mean free moleou1ar pathes, where due to visoosity and thermal oonduotion rapid ohanges oeour.

In the treatment given here, it wi1l be assumed as befere that the gas is ideal, and that outside the shook the effeets of visooeity and therma1 eonduetion are neg1igible. The general eonditions for shock transition are given by

10 eonserv~tion of mass 20 conservation of momentum

30 conservation of'energy.

in accordance with the seoond fundamental law of thermodynamica.

With the assumptions made here, the resulting equations get a rather simple form. To derive these equations it will be assumed that in a cer-

...

tain arbitrarypoint of the shock surface the normal vector n and two

--+ ~ ...

tangent vectors tI and t? are given. The component -of the ve100ity q 1n the dlrection of these vectors will be denoted by u_n' Ut and Ut

1 2

respeotively.

The shock wave itself is assumed to have zero velooity.

If the index f refers to the state in front of the shock and the in-dex a to the state aft of the shook the relations oan be written aSI

PfUf "" P U a a (2.46)a n n 2 2 (2.4 6 )b Pf+Pfuf =: Pa+Pa U a n n Pfuf uf :c Pa u _a u (2.4 6 )c at n tI n ₁ Pfuf uf = _{Pa u} u (2.4 6 )d n t 2 a n at 2 1 2 2 2 ) 1 2 2 2 ) (2.46)e Hf+

2'

(u f +uf +uf =: H + 2'(u +u +u a a _{at at} n tI _{t 2} n 1 2

These equations together with the equation of state (2.4) and the equation for the ohange in entropy (2.9) suffioe to determine all the quantities aft of the shock wave, if those in front of it are given. It must be noticed that equation (2.46)e has already been derived (see eq. (2.15

».

The system (2.46) oan be greatly simplified by observing that from eqs. (2.46)0 and (2.46)d follows by using eq. (2.46)a

U f

=

u tI at ₁ and u f _t

=

u 2 at 2

20

If now the Maoh number M is introduoed by

n u f n M -n af

the system gives, by elimtnating pand P , rise to the following

a a

equation

Y;1 U;nMn2 - UanU

fn

i"1+

YMn 2 }+

U~n-

{1+ Y;1 M_n2.} • 0 or

.

'

l+YM 2.(1-M 2) n - n M 2 n

As oan be ~hown the oond1tion (2.46)f allows only the + sign in

eq.

(2.49)~The

final result, is thereforel

(2.50)

From this ,equation it is readily derived that

( Y"'1)M 2 , Pa n - , . ----~-Pf (Y-1)M 2+2 n (2.51)a and Pa 2Y 2 Pf' ,. 1+r+ï (Mn -1) (2.51)b

Equation (2.9) gives then: AS,

fn

(:;)(:;f

or

The genera1 shook oond1t1ons f'or an idea1 gas and a shook ve100ity zero are thus derived. The equations that are important for the

follow-ing investigations are the four re1at1ons (2.47)a, (2.47)b, (2.50) and

2.3 The equations for lineari·zed potential flow.

Here a short derivation will be given of the equations valid for a linearized potential flow. To that end i~ will be assumed that the pertur-bation velocities are smal~ as compared with the velocity Doo of the free stream. Rence u

=

0 (:Voo) v

«U

_oo w

«

U_oo (2.53)a (2.53)b (2.53)C Furthermore it will be assumed that the effects of entropy production

J

can be neglected. According to the interpretation given of eq. (2.41) this means that the rotation vector is identically zero in the whole flow field. Thus there holde:

1

av

1

awr

r

a<jI-

r

ar-

=

0

From eq. (2.24)a together with eqs. (2.53) there follows by neglecting pröducts of small quantities

2

au

av

law v 0 -~co

äX

+

ar

+

r

a~ +

r

=

where _{~oo =}

\!

_V

_{Moo -1} 2 '

(2.54 )a

The eqs. (2.54) allow the introduction of a velocity potential

<fJ by u

=

v

=

i î

_ar

1 i}ql w

=

r

a<p (2.56)a

(2.5

6

)c

The flow is then goyerned by one linear partial differential

equation of the second order. ·This equation follows by inserting eqs. (2.56) into eq. (2.55). The result is

22

=0 0

This is the well-known linearized potential equation for supersonio flow. It should be observed that, eqs. (2.56) are valid in every flow doinain where the entrop,. is a oonstant throughout this dQmain.

I

Studies on supersonio flow aroUJld a::dall,. symmetrio bodies •

To stud,. the charaoterietics of supersonio flow past a certain oon-figuration, in most of the oases use has been made of the linearized potential theor,.. Rowever, as has been al ready indioated in the

intro-, ,

duotion, this theor,. bas the disadvantage of being onl,. approximate, the approximation bei~ poarer if the oonfiguration is less slender and the Maoh, number is highsr •

Bo direot estimates, however, are known about the 11mits of appli-oability ofthis theory, other than b,. oomparison with the results of exaot theory. This is onl,. possiblein ver,.few oases, for instanoe in the oase of flow around oones.

In this ohapter, the validity of linearized theory as applied to the study' of supersonio flow around axial1! symmetrio oonfigurations will first be investigated. It will be s'hown that the 1inearized thear,. is inadequate in predictiDg the flow field around bodies of praotioal importanoe for nearly every Maoh number. Espeoiall,. in the oase of inter-ferenoe no other result oan be expeoted than the oorreot order of magni-tude, sinoe on the basis of the 'present investigations, i t appears to be that the flow quantities at a oertain distanoe trom the oonfiguration ~~~

more in error than those ,nearer to the body.

Aooording to these arguments, the determination of optimum bod,.

'shapes by using l1nearized methods should be suspeoted. Therefore it seemed wanted to dèv1ae a method using the non-linear equations for deriving optimum conditions. In the seoond part of tbis, ohapter suoh a method wi11 be disoussed for a body with a presoribed value of the base area and for a g1ven Maoh number. The method of ana1ysis 1s 010se1,. re-lated to the stud7 of 11nearized flow sinos 1n both oases use wi11 be made of the-notion of a oontro1 surfaoe.

3.1 On the validity of linearized theory for axially symmetrie flow. The present investigation, whose aim it is to give a quantitative value of the error made by using linearized theory, was undertaken af ter eertain ineonsistencies were discovered by applying the theory of ref.17.

There a method is given, based on linearized theory, which aims at construetirig axially symmetrie eonfigurations of optimum shape with a given base area,by preseribing the value of the disturbanee velooities on the forward eharacteristie emanating from the base (see fig.2).

The method of characteristies was used to construct these bodies and due to the properties of the configuration investigated, the first part of the body contour could be ehosen freely. It then·proved, however, to be impossible to reaeh the proper value of the radius of the base area, and moreover the drag as found by integrating along the body contour was not equal to the preseribed value. The differences were rather large and this seemed very surprising since the preseribed disturbance velocities were sueh as to give the correct mass- and momentum transport.

It was found along the lines outlined below, that this differenc~

was due to the use of the linearized theory, in partieular beeause of the rather th~ck nose of the configuration a~d the interference of flow

between various parta of the configuration.

A method to study the validity of linearized theory, can befound by -observing that ths body area at a eerta-in distance from the nose of

t~e body and the drag on that part of the body can be expressed as inte-grals of functions of the disturbance veloeities over a control surfaee.

This surface emanates from the section considered and intersects the shock wave from the nose of the body (see fig.3). In most of the cases it is convenient to take as a control surface the forward directed

characteristic surface.

The method of eomparison between these integrals and body area, and integrated drag offered itself as a natural tooI to study the applioabi-lity of. linearized theory.

The order of magnitude of the average error in the flow quantities ean be predicted correetly on the basis of this comparisón. It should be remarked that this method of estimation of error is independent of ths use of more exact theory.

Here, the case of an axially-symmetrie body will ~e eonsidered, where the free stream is in alignment with the axis of the bOdy. The simplest case of sueh a body is a eone, and mueh attention will be foeuased upon this configuration.

To give more insight, a detailed oomparison of flow fields and pressure distributions is preaented for oertain eonfigurations.

The investigation will start with a derivation of'integral expressions for body area and drag.

3.1.1 Integral expressions for body area arid drag.

In this seotion a derivation will be given for oertain integral ex-pressions whioh are suitable to disouss,the validity of linearized theory. The derivation will be given first without making any a3sumptions about the order of magnitude of the quantities oeeurring.

There af ter a simplified version will be given, by making the same assumptions used' for deriving ~ha linearized potential equation. Inparti-eular 1t is this latter version which will be used to disouss the validity

of the results obtained by using linearized theory.

To derive the integral expressions use will be made of the e~ept of a oontrol surface. This is a surface whichenvelops a eertain volume, in which a part of ths body is imbedded. The control surfaoe which will ba used here oonsists of two opposing parts, one of the two being part of the shockwave, the other emanating trom the body section which is considered to intersect the shookwave in a cirole withacertain radius.

The integral expression for 'the body area is found by observing that ths ingoing maas has to be aqual tothe outgoing maas. This oan'be written as

J9r

P

n _Vn

do -

0

(3.1)

where V

n is the outward direoted oomponent of the velooity along a narmal to the surtaoe and dO is an element of the control surfaoe.

In faot, eq.

(3.1)

is the maorosoopie form of the continuity equation. If the part of the shook wave belonging to the control surfaoe is danoted by

01'

and the rest of the oontrol surface by

02'

eq.

(3.1)

oan be written as

• (3.2)

If now~l is the angle batween the tangent to the first part ·of the oontrol surface and the axis of the body at a point at the radial distanoe r and

~2

the supplement of the corresponding angle for the aft part of the oontrol eurfaoe (see fig.3), eq. (3.2) can be written as

J

Ra

JRa

2npoo TI"" r

dr~

2" TI

ooRB P2{U2

where RB is the radius of tha body .seation oonsidered and Re is the radius of interseotion of the two parts of the control surface (see fig.3). It should be observed that the velocities u and vare made dimensionless with respect to the free stroo.m veloai ty UOC) •

Equation (3.3) aan be brought in a more elegant form by choosing for the aft part of the control surface a oharacteristic surfaCe. It will be shovnl later that in this case there holds (according to eq. (2.32)b)

•

Moreover from equations (2.9) and (2.22) there follows 1

2 2 y-l P 2 = P co (a₂ M~ P

Here "a" is the velooi ty of sound, made dimensionless wi th respeot to lJ 00.

The funotion P, whioh is in fact the ratio of the stagnation pressures of the disturbed and the undisturbed flow, is given by

e - Y-l

P

=

e _• (3.6)

Substituting eqs.

(3.4)

and

(3.5)

into eq. (3.3) there is obtained

R

_B

2

=

2l

c

{(a2~/1 U~P

P -

I} r

dr (3.7)

where q2

=

u2 + v2, and where the subsoripts 2 have been dropped. This is the required equation, expressing the body area as an integral of a

26

funotion of the velocities and the entropy only.

Now this equation will be simplified by using the assumptions that the disturbances are small and that the effeots of entropy are negligible. In this oase eqs. (2.53)a-b are valid while P = 1.

As can be seen immediately eq. (3.4) can be written then as

(3.8)

indicating that in this case the characteristics are straight parallel lines, which is in accordanae vrith eq~ (2.57).

Moreover, as can be found by expanding eq.

(3.5),

ths density can be written as

where Ui is the perturbation velocity defined by

UI

=

u - I _•

Inserting these equations into equation (3.3) gives

R 2

=

2

~C{ _~2

Ui

+

~

VI}

rdr

B R 00 00

B

where Vi has been written instead of v.

(3.10 )

(3.11)

Equation (3.11) can be considered as a first order expression for i;he body area, and thus shou1d be consistent wi th the use of linearized theory.

The second integral expression aan be obtained by applying the momenturn equation in an axial direction to the air within the control volume. If D is the drag force exerted by ths air on the body and if it is assurned that the pressure p of the undisturbed stream is acting on the base of the body the momentum equation aan be written as

D+1t Ptt;) RB 2 +

J/

P2 sin

J

2d02 -

J/

Plsin

J

_{l dOl}

=-02 01

Sinae UI is equal to the free stream velocity Uoo ' eq. (3.12) can be simplified by using the mass flow relation (3.2). The result obtained is

Now this equation will be brought into a form where the integrand is dependent on the velocities only. To this end it is observed that by using eq. (3.5) together with eq. (2.9) there follows

y

2-2 Y-1 P2 = poo( ~ 14~ P

Introduc.ing this equation into eq. (3.13) and using eqs. (3.5) and (3.10) there is found

•

Taking again for the aft part of the control surface a characteristia surface, the final result aan be obtained by using aq.(3.4) together with the following evident relation

1

YM2 _co

(3.16 ) If the subscripts 2 are omitted, the integral expression can be written then as

Y R 1

1 . [ 2 2 Y-l

J

jC

2 2 y-l · 2 .

---;~ l-(a M~ P rdr-2n (u-l)(a lÇ) P U~V(3 rdr

Y~

RE

(3.17) Ifthe flow field is calculated correctly this equation has to give the same value of ths drag as found by integrating the axial oomponents of the pressure forces working on the body.

28

A1so here, .the first order form of this equation wi11 be given. To do this, the Tay1or-expansion of eq. (3.14) wi11 be given. It turns out that up to the squares of the disturbance veloeities

P2-P"" •

poou!

{-u'-

~

v·

2 +

~ ~!

u·

2 } • (3.18) Substituting this equation together with eqs. (3.8), (3.9) and (3.10) into eq. (3.13) there is found

2 (~ U I - V i ) rdr

00 .

One of the interesting features of this simple expression is the faet that the integrand is quadratio, thus 1eading to the resu1t that the drag is at least zero. A far more important remark must, however, be made. 'rha usual approximation for tha .pressure eoaffieient is given by the first term of eq. (3.18), while eq. (3.19) has been derived by using also the quadratio terms. In fact the drag wou1d be identieal1y zero if only the first term of eq. (3.18) had been used. This result indieates the necessi-ty of using the form for the pressure eoeffièient given by eq. (3.18). This. statement wi1l be further~eDmmented' on.

The integra1 expressions (3.7), '(3.11), (3.13) and (3.19) are the

basic tools which wi11 be used in the fo1lówing investigations on the super-sonie axially symmetrie flow.

3.1.'2 fi~ . linearized flow around a eone.

In this seetion a study will. b.e made of the supersonie flow around a cone with the aid of 1inearized theory. By using the already derived integral expressions, the validity of this theory for a eone wi11 be investigated.

Aeeording to eq. (2.57) the linearized potential equation for super-sonie flow ~the ease of a ey1indrieal eoordinate system, is given by

o

Here, q> is defined in such a way, that ,the disturbance veloei ties, made dimensionless with respect to the free stream veloeity Uoo are given by

Ui

=2!.

ax (3.21)a Vi

=~

ar and

w'

=

1 2l'..

r

a'iJ

(3.21)c

Since the flow is conina1, ths disturbance velocities are constant along rays through tha origine Introducing

and

t

=

~

r

q> '" rtG(t)

aquation (3.20) can ba written, aftar obsarving that q>~~ vanishas dua to tha axial1y symmatrica1 character of the flow, as

t (t2

-~~)

::g

+ { 2( t2

_p~)_t2} ~~

=

0 Solving this equation there is obtained

Vt2_~~

i -1 to

G

=

-K --t"'--- + K cosh ~oo + _{Q •}

Tha disturbance valocities are given by

Ui :: G + t dG dt

Vi ==

Sinoe along the firs~ charaoteristio t

=

~co the quantity

~OOUI+V' has to be aqual tó zero,' .it follows that Q ...

o.

Equations (3.25)a and (3.25)b oan then be written as

-1 t

u'

=

K cosh

i3;,

v'

=

-K

Vt2_fJ~

j

(3.26)a

Tha integration constant oan ba determined f~om ths condition that the body has to ·be a stream surface, or if

~

is the tangent to the body contourl

30

•

. It should be observed that here the exact farm of the bounAary oon~ dition will be used.

If the semi-top angle of the oonical body is denoted by

J

s" i t can be shown that

K ::I _ _ _ _ -l~ _ _ _ _ _

,/ 2 2 I -1 to tovto -~co + cosh ~oo where to == 'óot'

r\)s~

, ..

The equations (3.26)a-b and (3.28) ~hus give the flow field, around a cone acoording to linearized theory.

The question that will be raised now, is t

"What ·is the range of Mach-numbers Moo and semi-top angles

J

s for which this result is approximately Yalid?"

Thls qU8stion will be answered by using the integral expressions

'derived in the foregoing seotion.

First use will be made of the expression for the square of the radius of the body oross seotio~ eq. (3.11). The integral at t~e r1ght hand side of this equation oan be caloula.ted by using the exPress'ions for the perturbation velooities u' and v'. The result is

I

Ro }

~

oosh

~1

; :

2

I

~~

u '+PQC)v , rdr ""

~2

1

-RB

l

' t ,lt2 2 ' -1 to

oV o~co +oosh

p;

It is evident from eq. (3.29) that eq. (3.11) is not satisf'ied. This oould be expeoted since only an approxim~te theory is ueed. The important point oonóerning these two equations, however, is tru;t sq. (3.29) gives the possibility to obtain a judgement on the ~lidity of the linearized theory. Due to the form of the integrand

ot

eq. ·(3.ll), the differenoe between the left-hand side and the right-hand side of this equation gives a measure ot,the average error in the flow quantities.

If a differenoe of ~percent between the left-hand side and the right-hand side is ooneidered as permissible, there can be oaloulated

the following equation

-1 0

X

2 2 -1 0

lfc:o

cosh

t

~oo

=

Wo

{~

t o t 0

-~oo

+oosh

~cio

t}

In fig.4 the limits for

X

=

5

and

X

=

10 have been given. AB oan be Been from these oUrves the region of applioability is very smalle If the flow field has to be known accurately, the lower bound has to be applied. This indioates that only the flow around very slender cones oan be

calou1ated by using linearized theory. For a praotical semi-angle, say 100, linearized the,ory is unable to give the flow field aocurate1y.

One important aspeot of the curves given in fig.4 is, that for Moo

very ne ar to unity, the va1ue of ~ s which is allowed decreases rapidly, showing that linearized theory is invalid around Moo= 1. This faot about the linearized theory, long sinoe known, oan' thus be shown tb be true in a very simple way.

If ){ is oaloulated as a funotion of Mach number Moo and of semi-top angle

,J

i t appears that wi th inoreasing Mach number and inoreasing

s '

semi-angle the average error in the flow field charaoterized by

X

in-oreases rapidly, as is shown in tab1e la.

To substantiate these results an analogous investigation will be per-formed by using eq.

(3.19).

Tbe right-hand side of this equation proves to be

R

n

[0

R:s

2 2 2{ 1 2 2 ' -1 to [

,[27

1 2 -1 toJ}

(~OOUi_V') rdr

=

nK

R:s -

2(to-~)+oosh

1Ç,

tovto-~oo-

2'

~ctposh

iÇ

(3.31) Ths right-hand sidet'of eq. (3.19) oan be obtained, as has a1ready been remarked, by integrating the axial oomponent

aoting on the body. Thus it is found that

of the pressure foroe

D

tJ2

PO) ClQ

~

.. n

J

0 rdr o p

where op is the pressure ooeftioient, whioh iB given by

o

p

P-Pco

-32

Ncw, different expressions can be obtained for D, according to the different approximations, used for the pressure coefficient. Here use will be made of the formula given in eq. (3.18), which was used also when de-riving eq. (3.19). Performing the integration indicated in eq.

(3.32)

there follows

Again the~e is an apparent difference between the two expressions for the drag, eqs.

(3.31)

and

(3.34).

I t should be observed that both are cal-culated by using approximate values for the perturbation velocities. If both the drag according to the integral expression and that found by direct integration of the axial f'orces along the fuselage are equal, then no other conclusion can be reached than that both contain an error ofequal order. But they are not aqual, and therefóre, this difference must be a measure for the consistency qf linearized theory. Thus again limits of applicability ean be caleulated by solving the equation

Xl -1 to { I 2 -1 to

\~}

1 2 2 2 -1 to 2

Wo

eosh ~to (2" ~cO +l)cosh ~co +toyto-~co - 2(to-~~)= Moo(cosh ~c)

,

(3.35)

where

><

1 is the difference i.n pereents between the expressions

(3 .

:

U)

and

(3.34).

In table 1 b the quantity Xl has been given as ,a function of M 00 and ,.Js. The results are in oomplete agreement with those of table

1 a, leading thus to the same oonolusions about the validity of linearized theory.

A detailed oomparison between the flow fields as determined by linearized theory and exact methods respeetively shows how large the actual error is at each point. This will be done in seetion 3.1.4.

How-ever, first the flow field around pointed axially symmetrie bodies will be studied along the same lines ;B.S given here, to see if the eonelusions

reached for a eone have to be altered with more general oonfigurations.

3.1.3

The linear,ized flow around a body.

To obtain the flow field around a genera 1 axially-symmetric confi-guration in the linearized approximation, a solution of eq.

(2.57)

must be obtained which fulfills the boundary condition along thè bOdy contour as given by eq.

(3.27).

This problem can be solved, by means of an analytical method, such as a distribution of sources along the axis, or by a numerical methode

An excellent numeri cal method for the solution of hyperbolic

di·fferential equations is the method of characteristics, where the flow field is calculated step by step by using the characteristic equations along characteristic surfaces. A detailed description of the derivation of such equations has been given in chapter 2. It can be shown by using the results of this chapterthat in the linearized approximation these equations take the following farm

dut 1 dv' 1 VI

+ +

-dr ~co dr ~oo r

o

along the

~acteristic

wi th

~~

= -

~:

and dut I dvi 1 _Vi

- - - 0

dr ~oo ~r ~oo r

-along the characteristic wi th

~~

:a:

~~

•

The flow field can be dete~mined by using these relations if besidé the boundary, the flow around the nose of the body is also given. The shape of a pointed nose can always be considered as eonical. This gives the possibility to use the results of the preeeding section. In that case the flow quantities are known along the last eharacteristio of the oonieal region (see fig.5). The method of charaeteristics to be used here is straigbtforward and the most advisable forquantitative results if the flow field has to be known.

For the study of the applicability of linearized theory the flow field around two bodies is calculated for different Mach numbers. In fact each body represents a whole family, since the base ean be seleeted at arbitrary values of the axial coordinate x betweèn the nose and the base, because the flow is supersonie. This means that the flow aft of a given value of x eannot influence the flow field before the·backward directed eharaeteristic emanating from the eross section at x.

The two bodies ohosen have a conioal nose over 0.025 of the length and are followed by a parabolio shape whioh is .symmetrie with respect to the line x

=

0.5125. The base lies at x :z l.O. The oonical nose of

34

the first body has a semi-top ang1e

~s

=

7.5°'and that of the aeoond body has

J

=

12.50 • If the slenderness ia given by

s

~

a == ~--_{2 r}

max

the bodies have s 13.7 and a = 8.4 respeotive1y. The flow field around

the body with

Jr

_s

=

7.50 and s

=

13.7 has been determined for the Maoh numbere Yoo = 2 and Yoo= 5. The flow field around the body withJ_s

=

12.50

and s =- 8.4 bas been determined for the l,'taoh numbers Moo= 2 and Moo=

4.

Along several oharaoteristios for different values of x, the right-hand side of the inte~a1 expression for the mass flow eq.(3.11) has been oaloulated. The resulta faund thus' have to be oompared with the funotion

2 '

RB

as presoribed by,the bd~,oontours. Tbe results for the various oases, have therefore been given in figs

6

(a-d) together with the presoribed distribution of tbe oross seotional area of the bodies.

The dsviation between ths two ourves gives, just as for the oone, a measure of the average error in the flow field. It is found that the differ-enoes are re1ative1y the 1argest at the nose and at the end of the body. Tbe ourves indioáte that for bodies whioh have a positive slope over most of ths length, the error deoreases with inoreasing slenderness. However, if a negative slops is present over an appreoiable 1ength of the body the dsviation grows rapidly with inoreasing negative slope.

From the figures

6

(a-d) it can bs Been that the general trend of in-creasing deviations with deoreasing slenderness is very striking. The great importanoe of these figures, however,'is that they give a quantita-tive answer to the question of the validity of lin~arized theory for the ca1oulation of the flow field around these partioular configurations. Seen in this light, ~lthough in general the deviationa for bodies are leas pronounoed than for cones, the only oaae that may be given areliabie numerioal value, is the oase where

~s

- 7.50 and Moo= 2. Tbe differenoes

for the othe~ oases are so large that linearized theory determines apparent-ly onapparent-ly ths order of magnitude of the flow quantities.

Aa wa·s done for the oone, a oompari,son will also be made of the drag

as found by integrating the axia1 preaaure forces working on the body, and the drag as found by oalculating the loss of momentum through a Bui-tably ohosen oontrol aurfaoe. The r~speotive expressions aregiven by