Discontinuous solutions for nonequilibrium supersonic flows

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DISCONTINUOUS SOLUTIONS FOR NONEQUILIBRIUM SUPERSONIC FLOWS

JUNE

1967

by

Duvvuri Tiruma1esa

UTIAS TECHNICAL

NOTE NO. 114

AFOSR

67-0687

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DISCONTINUOUS SOLUTIONS FOR NONEQUILIBRIUM SUPERSONIC FLOWS

Manuscript received June 1966

by

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ACKNOWLEDGEMENTS

The author wishes to thank Dr. G. N. Patterson for his encourage-ment and interest in the present work and Dr. I. I. Glass for many helpful dis-cussions and critical comments.

This work was supported by AFOSR under Grant No. AF-AFOSR

365-66,

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SUMMARY

Generalized solutions with weak or strong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for nonequilibrium supersonic flows. The curves with weak dis-continuities are shown to be the characteristics of the system of equations. The interdependance of the discontinuities for several variables are discussed.

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TABLE OF CONTENTS Page.." NOTATION v l . INTRODUCTION

1

2.

BASIC EQUATIONS

2 3·

ANALYSIS

3

3.1

Weak Discontinuities

3 3·1.1

Some General Results

4

3.1.2

Reacting Gas Flows

6

3.1.3

Summary of Results

7

3.2

Strong Discontinuities

10

4.

EXAMPLES

14

5.

DISCUSSION

16

6.

CONCLUSIONS

17

REFERENCES

18

.

,.,

APPENDIX A

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p p a: T h S -+-" q R Q w x,y 7jJ(x,y) L ~ cp(x,y)

ë

_n

S

-? n E u,v v ~2 f pressure density NOTATION

degree of dissociation (mass concentration of atoms) temperature

specific enthalpy specific entropy flow velocity vector

gas constant per unit mass referred to diatomic gas external heat added to or removed from the flow mass production rate

Cartesian coordinates stream function

characteristic distance for dissociation vorticity

equation of the curve C in Cartesian coordinates normal vector of curve C

unit vector along streamline unit vector normal to streamline

-? -? angle between C and n

n

velocity components in x and y directions

equal to 1 for plane flow and 2 for axially symmetrie flow equilibrium speed of sound

specific heat ratio for frozen flow

specific heat ratio for partially frozen flow lf

pip

,frozen speed of sound

partially frozen speed of sound,

(l-

_f

pip)

defined as any smooth test function which vanishes identically outside a subdomain R of G (Eq.

49)

M 2 - 1 f

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1. INTRODUCTION

The application of the method of characteristics for the computa-tion of a supersonic flow field in a reacting gas under nonequilibrium condicomputa-tions has been considered by several authors (Refs. 1 -

5).

In any discussion of the integration of systems of first order quasi-linear partial differential equations of the hyperbolic type, two important points arise:

1) Which of the . dependant variables can be prescribed arbitrarily along a given initial data line without over-determining the solutions?

2) Apart from the solution for which the dependent variables and their derivatives of various orders are continuous in the whole flow field,does this system admit any other solutions:

a) for which the first derivatives of the dependent variables are discontinuous in some part of the flow field while the variables themselves are continuous(these solutions are called generalized solutions with weak discon-tinuities),

b) for which some or all of the dependent variables themselves are discontinuous (these solutions are called generalized solutions with strong dis-continuities ) .

3) If such generalized solutions are available, where do these dis-continuities occur? Does the discontinuity for one of the dependent variables across a curve mean discöntinuities in all other dependent variables across the same curve? Are the discontinuities in the various dependent variables across any curve independent of each other or are they interrelated?

It is of ten thought that question 2a) can be answered directly from the characteristic equation along a given characteristic curve (e.g. Ref.

3).

This could lead to erroneous conclusions as will be shown in this paper since the only thing that the characteristic equation tells us is that the derivatives normal to the characteristic curve along which that particular equation is valid cannot be calculated from a knowledge of the variables on it; however, it

cannot be automatically concluded that these derivatives could be arbitrarily prescribed, that is they can be discontinuous. As will be seen later in this paper, this element of arbitrariness about the normal derivatives is eliminated by the basic set of differential equations themselves even without the intro-duction of initial and boundary conditions. (See the Appendix and Section

5

for further details.)

Thus, in this paper, it is proposed to answer questions (2) and (3) for supersonic flow of reacting gases in nonequilibrium by a systematic analysis starting from the basic set of differential equations defining the flow field. As question (1) was dealt with in considerable detail 'in Ref.

3,

it will not be considered in this note.

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2. BASIC EQUATIONS

Even though consideration of a number of reacting species or nonequilibrium in internal modes such as vibration does not introduce any special difficulties, for simplicity, the discussion will be restricted to the dissociative nonequilibrium of a pure dissociating diatomic gas. The vibra-tional degrees are considered to equilibrate instantaneously with the transla-tional and rotatransla-tional degrees of freedom (Ref.

5).

The basic steady flow equa-tions for the thermodynamic and flow variables p, P,

a,

T, h, S,

q,

which are the pressure, density, atomie mass fraction, temperature, specific enthalpy, specific entropy and velocity can be written in vector notation

as:-Mass: -_DP _Dt. + P div

ct

= 0 (1)

Momentum: grad q2j2 - -7 q x curl -7 q + -1 grad p

₌

0 (2) P

Energy: Dh 1 QJ2 = 0 (3)

Dt p Dt

T grad S

= -

-7 -7 (4)

Entropy: q x curl q + ~ grad a

Rate

ro

*(p,p,a)L(p,p,a) (5)

=

Equation: Dt Equation of p p RT (1 + a) (6) State:

Enthalpy: h h(p,p,a) or h(a,T) (7)

The expressions for specific enthalpy hand specific entropy S for the kind of model gas considered are given in Ref.

5.

In the above equations DjDt ~.grad and explicit expressions for Q, ~, L in terms of p, p,

a

can be written. ~

The basic derivation of the rate equation was considered in con-siderable detail in Ref.

5.

The momentum equation, Eq. (2) is written

explicitly in terms of the vorticity vector curl ~to facilitate the discussion.

". ~ .:..,:

We will further restriet our discussion to two dimensional or axially symmetrie flows. The latter differs from the former only in the addition of a term in the continuity equation while all others are unaffected. The con-tinuity equation will be of the form

~

(pu) +

dy

0

(pv) +

('\1-

1) pv 0 y (1' ) in Cartesian coordinates x, y or

0 oe

v -

1 _sin 0

ds

(pq) + pq

_dn

+ pq

e

= r

(1" )

in streamline coordinates, s, n (s along streamlines and n perpendicular to it). In Eqs. (1'), (1") u, v are the x,y components of the velocity, q the speed,

e

the streamline angle, ~

=

1 for plane flow and v

=

2 for axially symmetrie flows and r is the radial distance from the center line ofaxial symmetry.

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3 .

ANALYSIS

Leaving aside the solutions of this set of equations for which the dependent variables and their derivatives of various orders are continuous everywhere in a supersonic flow field, some interesting results about generalized solutions, that is, solutions with weak or strong discontinuities in certain pant's of the flow field can be derived (Refs. 6,

7).

3.1 Weak Discontinuities

As the system of equations Eqs. (1) to (6)is of first order, generalized solutions with weak discontinuities imply discontinuities in the

first derivatives of the dependent variables (Ref.

7).

y

Curve C

S reamline

L---~x

~

Let C be a curve in the flow field with unit normal Cn. Consi_{der two points Pl and P}2 in the immediate neighbourhood but on opposite

sides of this curve and along the normal. The relation between the normal derivatives of the pressure pand velocity q across the curve C can be

ob--7

tain_{ed by taking the dot product of the momentum, Eq. (2), with Cn •}

~ 2

~ ~ ~ l~

C

.

grad

3..._

q x curl q+-C

.

grad p 0

n 2 n p n

or d

q2/2.. +-1 dp

ë!

~ q x curl q = 0 ~ (8)

d Cn p dC n

n

where the differentiation d/dC n is along the normal to the cruve C.

Applying Eq. (8) at t_{he points Pl and P2' taking their difference and letting}

Pl' P2~ P on the cutve C,one

obtains:-[~n]

+.!

[Pen] -

[~

~ ~ (9) Cl _p • q x curl q]= 0 where _lim

_[Gd

j

(~)pJ

(10)

[~nJ

Pl,P2~

'

~

Pl 2

(10)

. -+ -+ and similarly for the other two terms~ The direct ion of the vector q x. curl q in plane or axially symmetrie flows is normal to the streamlines and in the plane of flow or the meridian plane.

If ~,

ti

denote unit vectors along and normal to streamlines, for ~n

=

~, the third term in Eq. (9) is zero, while for any other ~n

f

~, it is different from zero. In terms of the streamline coordinates s,n and the variables q, espeed and streamline angle,

-+ -+

q x curl q

=

q

(~

, - q ~

de )

-+ n

n os (11)

=

q

t

ti

where ~ is the vorticity. Thus

[

~C _{. q}-+ _K _{curl q}-+ ] ₌ [~C. q I" _~-+n ]

n n (12)

= q [~ C~.-+n] n

=

q cos E [~]

-+ -+

since the dependent variable q and the angle between Cn and n, E are continuous across the curve C, while

S

which contains derivatives of q and e need not be continous. Thus Eq. (9) is finally written as:

+

3.1.1 Some General Results

1

p - q cos E

[S]

o

Some results on the inadmissibility of discontinuous derivatives of certain variables across certain curves valid for reacting, non-reacting, rotational and irrotational flows will be first derived. It will be seen that these are a direct consequence of the basic set of partial diffexential equations defining the flow field and no initialor boundary conditions are introduced in their derivation.

(a)

[sJ

=

0 everywhere except across streamlines:

Let the equation of the curve C in Cartesian coordinates he cp(x,y) = O. cp must have continuous partial derivatives of first order and cp2n + cp2y

f

O. Then the derivatives of any quantity along and normal to Care given by the operators:

along C:

across C:

d/dY

(14)

Since the variables are continuous along the curve C and also their derivatives in the direct ion of C, the jumps in the derivatives of the x,y components of the velocity u, v and pressure pare related through

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CPy [u ] _x CPx [u ] _y 0 (15) CPy [v ] _x CPx [v ] _y 0

(16)

CPy

[px]

- cp _x [Py] 0

From the x,y components of the momentum equation,

(18) u [u ] + v [u ] + _[px]/p 0

x y

u [v ] + v [v ] + _[Py]/p

=

0

x y

Mu1tiplying Eq. (18) by cp and Eq. (19) by cp and subtracting, it can be shown with the aid of Eqs. (15)Yto (17), that x

(u cp + v cp ) [u - v ]

= (

q.

gr ad cp) [ s'] = 0

x y y x

or

q

ct .

~)

CS]

=

0 (20) -7 .

where s lS the unit vector along the stream1ines. Thus the jump in the

vortici~ [~ is zero except when the curve is the streamline itself in which case s . en 0 and thus

Cs]

need not be zero.

(b) [8 ,] = 0, [p] = 0 across streamlines:

n n

App1ying the continuity equation Eq. (1") in streamline coordinates on eitper side of a streamline and taking the limit, one has

+ P,q

[8]

_n

=

0

(21)

As thi last drops' out. streamlines

term in Eq. (1") contains only variables, but no derivatives, it Since P, q and so also their derivatives are continous along

[p ] , [q ] are zero leading to

s s

[8 ] _n

=

0 (22)

Applying the n component of the momentum equation in streamline coordinates s, n on either side of a streamline and taking the limit

(23) Sincè the stream1ine angle 8 and its derivative along the streamlines are con-tinuous

[8 ]

=

O. Thus

s

Equations (22) and (24) show that no discontinuities in p

. n across stream1ines. and 8 n (24) are admissible

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3.1.2 Reacting Gas Flows

(a)

.

[oe ]

f

0 only across strearrilines: n

Since the dissociated mass fraction

a

as well as its derivative along a curve e are continuous, applying the operation in Eq.

(14)

(25)

Applying the rate equation, Eq. (5) on either side of curve e and taking limits

u [a ] + v [a ] =

0 ( 26 )

x y

Since ~ and L contain only the variables themselves but no derivatives. Equations

(25), (26)

show that

[a ] (uep +"Vq))

lep

=

[a ] q

t.

<f

lep

= 0

y x y y y n y

from which, along with Eq.

:

(25),

one obtains

[a]

=

0

=

[a ] for

t .

<f

f

0

y x n

Thus

[ac ]

_n == ep _x [a] _x + ep _y [a] _y

= 0

for

t .

<f

_n

f

0 (28)

It may be noted that since the rate equations for several reacting species and nonequilibrium in internal modes are all essentially of the type of Eq.

(5)

with the RHS being functions of tbe variables only, all of these can admit discontinuous normal derivatives across streamlines only.

(b) Other results:

For reacting gases, p can be written as a function of p, S,

a

by eliminating the temperature T between pand S, thus:

p = p(p, S,

a)

(29)

From this equation, one has

(30 )

By taking the dot product of the entropy equation Eq. (4) by

<f

_{n , one has} T Se _n == - q cos E

s+

Q

oe

_n

where Eq. (11) is made use of. Eliminating Se from Eqs. (30), (31), one has _n _.

[pen] - (ps)p,a q cos E[S]/T + { (Pa)p,s + (ps)p,a Q/T} [OCn] (32)

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---- - -

-which relates' the jumps in the normal derivatives across any cruve C. In the above equation a 2_f - (pp)S is the partially frozen speed of sound for a partially excited

dissoclà~ing

gas (Ref.

5).

Another relation between the jumps in the normal derivatives of p, P, Tand a can be derived from the equation of state Eq.

(6),

namely

+[TC ]/T + _n [~~ _-vn]/(1 + a)

For undissociated and frozen flows, the jump relations are obtained by putting

.a zero and constant respectively in Eqs. (32), (33). For equilibrium flows, a is a function of pand p i.e. a e =a(p,p) and thus the jump relations are

obtained by replacing derivatives of a in terms of p arid P. This will modify Eq. (32) such _{that a2r is replaced by a e2 , the equilibrium speed of sound.} 3.1.3 Summary of Results

The results derived in Sections 3.1.1 and 3.1.2(a) suggest the classification of the curves admitting discontinuous normal derivatives into

streamlines and others. For these two sets of curves, Eqs. (13, (31), (32) and (33) will be simplified and it will be shown later that the other curves are none else than the frozen Mach lines.

(a1 Curve C is a streamline (

-;

.

~

=

0 E

=

0):

n '

It is already shown that for streamlines

[e ]

n

o

(22)

(24) Replacing the differentiation with respect to _Cnby n, one has

From Eq. (31)

T [S ] -= '- q cos

E'

:

lO+

Q[a]

n n (31' )

Equation (13) reduces to

(34 ) Equation (32) reduces to

[Pn] = - (ps)p,a q[S]/T + { (Pa)p,s + (ps)p,a Q/T} [an]

(35)

E~ation (33) reduces to

+ [T ]/T + [a ]/(l+a)

(14)

One may note that for undissociated or frozen flows, the

a

term in Eqs.

(31')

(35)

and

(36)

disappears and the above equations reduce to T

Cs. ]

= - q cos E

[S]

n

(31")

(35')

+

[T ]/T

n = 0

(36' )

From Eqs.

(31"), (34), (35')

and

(36'),

it is seen that any jump in one variable determines all others uniquely, whereas for reacting flows Eqs.

(31'),(34),(35),

(36)

show that no one jump uniquely determines all others, but any two will determine all others uniquely. They may be taken as [qn] and

[an] .

Also note that streamlines are a set of characteri stics for reacting gas,., flows, and also that no initialor boundary conditions 'are utilized in deriving the above results.

One very interesting result 'for undissociated gas flows is that if one has an initial condition that

S

=

0,

S

remains zero everywhere and thus Eqs.

(31"), (34), (35'), (36')

show that none of the other variables admit any discontinuous normal derivatives across streamlines. Thus if one uses the de-fini ti on of characteristic curves as those along which discontinuities can pro-pagate, then the streamlines drop out as characteristic curves for undissociated flows with

S

=

0 initially i.e. irrotational flows. However from a mathematical point of view, streamlines are still characteristic curves for the set of Eqs.

(1)

to

(5).

(b) Curve C is other than a streamline

(s+ .

~n

f

0): In this case, as shown earlier,

Equation

(31)

reduces to Equation

(13)

reduces to Equation (32) reduces to Equation

(33)

reduces to

[s]

= 0

[aC ]

_n

=

0 [SC ] _n = 0 p q [~ _n] + [PC ] _n = 0 (Y_f - 1) [PCn]/p = Y_f [TCn]/T where Eq.

(39)

is made use of and

Y

f is defined by the relation

2

af

=

Y_f pip

(31'" )

(40)

(15)

to bring out the contrast with the undissociated gas flows. One may note that none of these results make use of any initialor boundary conditions and also that these equations themselves cannot determine whether [en] is different from zero or not across curves other than streamlines. Also Eqs.

(37)

to

(40)

are equally valid for undissociated or frozen flows. While in the former case the frozen speed of sound and l f are to be replaced in Eqs.

(39)

and

(40)

by the unambiguous speed of sound and the isentrqpic exponent, for frozen flows the appropriately frozen values of l f and af are to be used. Also for all flows, reacting or nonreacting, the above equations indicate that any one jump uniquely determines all others except [en] as already noted. In other words, for both reacting and nonreacting flows, two jumps of which one should invariably be[enJ are required to uniquely determine all other jumps, in contrast with the

situation for streamlines where any two could be chosen.

One interesting result when one makes use of the boundary condi-tion [en]

=

0 (for example flow around a smooth convex bend) , is that these curves can still propagate discontinuities even for non-reacting flows. This seems to be the reason for indicating Mach lines (it will be shown below that these other curves are none el se than Mach lines) as the only characteristic directions for irrotational flows in all Gasdynamics textbooks dealin~with the application of the method of characteristics to gasdynamics problems . .

To show that these curves are none other than Mach lines, one proceeds as follows:

Equations

(38), (39)

_{with the aid of the definition of d/dc n given} in Eq. (14) lead to

~

u [u ] +

~

u [u J +

~

v [v ] +

~

v [v ]+

a~ {~

[p ] +

~

[p ]}!p = 0 x x y y , x x y y x x y y

(42)

Also applying the continuity equation Eq. (1') in cartesian coordinates on both

sides of the curve C and taking the limit

(43)

The last term in Eq. (1') drops out since it contains only variables but no derivatives. Multiplying Eq. (15) by ~ and Eq. (16) by ~ and adding

y x

o

(44)

since

[sJ

=

0 for all curves C other than streamlines. Since p is continuous along

e,

so also its derivative along C and thus

With the aid of Eqs.

(43)

to

(45),

Eq.

(42)

can be reduced to (u

2 _

a

f

2)

~

x

2

+

2

u v

~

x (46)

which is the equation giving the frozen Mach lines as the other curves where discontinuous normal derivatives are admissible. As is already pointed out, for nonreacting flows, the frozen speed of sound is to be replaced by the ordinary speed of sound.

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3.2 Strong Discontinuities

For a system of equations or nth order, generalized solutions with strong discontinuities are defined as solutions with discontinuities in the variables themselves or in their derivatives up to the (n-l)th order (Ref.7). Let u be a function of x,y and have continuous derivatives of all orders

every-where in a domain G except on a curve u(x,y)

=

O. Let the function u satisfy

the linear partial differential equation

L{u) = A u + B u + Cu

x y

o

where A, B, and Care functions of x and y only.

Define an adjoint operation L* such that (see Ref.

8

for a dis-cussion of generalized adjoint operator)

?TJ L{u) - u L* (TJ)

,

is a divergence expression. u is said to be a generalized solution

of the differential equation L{u)

=

0, if C

R2 cp(x,y) =

I IR

[TJL(u) -u L*(11) ] dx dy = 0

(48 )

where TJ is any smooth test function which vani.shes identically outside a subdomain

R of G. This definition can be extended to the case of quasi-linear equation

with one dependent variable and also to systems of quasi-linear equations for

several dependent variables provided they have the form of divergence equations

or conservation laws, for example,

L{u)

=

px(x,y,u) + ~(x,y,u) + n(x,y,u)

=

0 (50)

where p,q,n are twice continuously differentiable function vectors of x,yin G

and of u in some domain: Let the curve C divide the domain R into two sub-domains Rl and R2 and then applying Eq. (49) in Rl and R2 and using Gauss's theorem the relation between the jump discontinuities p and q are connected with

cp ,cp as

(17)

o

.

J J

[11 L(u) R - uL* (11}J d.x dy

=

rr

(:P11), + (411) d.xdy

JJ

"x y

=

0

R

.

J

11

(~

-

x[p]

+

~y[q])dS

C

where ['11] again denotes jump eonditions . As this should be trule for any 11

cp/,

[p]

+ ~

[q]

= 0

x y

(Further details may be obtained from Ref.

7).

Equations (1) to (3) may be written in Cartesian eoordinates as

pu u + pv u + p = 0 x y x pu v + pv v + p

=

0 x y y u v u h + v h - -p - - p 0 x y p x p y

Equations (54) and (55) with the aid of Eq. (53) ean be rewritten as

(p + pu2) + (puv)

=

0 x y (puv) + (p + pv2)

=

0 x y (54' ) (55') Eliminating p from Eq. (56) with the aid of Eqs. (54) and (55) and making use of Eq. (53),

2 2

uh + vh + u u + uv u + uv v + v v

=

0

x y x y x y

puh + pvh + pu(u2j2) + Pv(u2j2) + Pu(v2j2) + pv(v

2

j2) = 0

x y x y x Y

u2+ v2 u2 + v2

epu (h + ~ ) _Jx+ [pv (h + 2 )Jy

=

0 (57)

or

ComparingEqs. (53), (54), (55), (57) with Eq. (50) and app1ying

the eriterion of strong solution, the fo1lowing re1ations between the jumps ean

be obtained in a simi1ar way to th at used to obtain (52). For examp1e

pU'l -h pV11 ]) d.x dy

x

Y

(18)

Hence

cp [ pu]

+

cp [ pv] = 0

x y

Similary from Eqs.

(54'), (55'), (57)

cp

[p

+ pu2] + cp [puv] = 0 x y (60) CPx [puv] + CPy

[p

+ pv2_]=₀ ₍₆₁₎ [ u2 + v 2 ] [ u2 + v2 ] mx pu (h + 't' 2 ) + m 't'y pv (h + 2 )

=

0 (62)

To understand the meaning of Eqs. (59) to (62), let the curve be cp(x,y) = x so that cp =

0,

cp = constant

"0.

Then Eqs. (59) - (62)

simplify to y x or

Epu]

=

0 2

[p

+

pu ]

=

°

[puv] = 0

Epu

(h

+

.

l/2) ]

= 0 y x = 0 discontinuity curve ---~--- x Pl u_{l = P2 u2} P ₁ +p _{1 1}u 2 =p +p u 2 ₂ _{2 2} Pl ul v 1 = P2 u₂': v 2 q12 qd2 (63) (64) (65) (66) (67) (68)

(19)

By virtue of Eq. (67), Eqs. (69) and (70) reduce to

Taking the dot product of pq and Eq.

4

for the entropy

pu S + pv S

x y ua x

which may be rewritten with the aid of Eq. (53) as

[p~

(S -

~

) Jx +[pv (S -

~

) J y from which as before

For cp 0, [pu S- pQua

_J

₌

°

Y T or ul Ql al u2 Q2

a

₂ Pl u l Sl - P2 u2 S2 Pl _Tl P2 T 2 Using Eq. (67), Eq. {76) gives

From the equation of state, Eq. (6) and Eq. (68)

or (71) (72) (73' ) (74) (75)

Equations - (67), (68), (71) and (72) are the well-known oblique shock-wave relations; while Eqs. (59) to (62) give the relations for any curved shock. This section may be closed by noting that in the case of strong dis-continuities, the curves where they can occur are not characteristics. It is also not possible simply from these equations to determine the shape of such curves.

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4.

EXAMPLES

Examples will be given for the simpler case of weak discontinuities. As an illustration of the results of section 3.1, consider the flow around a

smooth convex corner AB with constant flow properties upstream of the corner.

Uniform flow

•

y~

I

I p,p,q etc. constant

At points A and B while the streamline angle 9 is continuous, its derivative mayor may not be continuous. The curvature is definitely discontinuous. Thus there can be discontin.uities in the normal derivatives of the flow variables across the frozen Mach lines emanating from the points A and B across which Eqs.

(68)

to

(40)

apply. This is so since it was shown in Section 3.1.3(b) that whether [gen] is different from zero or not, Eqs.

(38)

to

(40)

would still give

jumps if one of the other variables has a jump discontinuity in its normal derivative •

. f ' ,,: Equation (38) may be written in terms of the x,y components of

the velocity u, v as

u

[UC ]

n + v [vC] n +

Further u,v,9 derivatives are related as

u [ven] - v

[UC

_{n ]}

2 2

u + v

=

0

Since v

=

0 at point A, Eqs. (79) and (80) give

(80)

(81)

(82) In this case the geometry of the problem, that is, a knowledge of the variation of 9 is enough to calculate [vC]. But there does not seem to be any way by which [PC] or

[UC]

can be ca~culated purely from the equations. One may have to aBpeal to sgme physical reasoning.

(21)

Consider as a further example the flow around a sharp convex corner with the flow variables again cQnstant upstream of the corner. In this

,.

-_-x

case 8 is discontinuous at the corner since 8 changes abruptly from zero to 8 at O. Also the derivative of 8 at 0 is not defined therefore Eq.

(81)

has no meaning. However, by inserting an expansion fan centered at 0 whose head and tail are inclined to the flow at the Mach angles appropriate to the upstream and downstream uniform flows, one may say that the flow is smoothed Qut far

from the corner. For the flow above some streamline ~, one may apply the criterion as before. However, the corner itself is a forbidden zone.

Consider further the flow in a di~ergant nozzle ABA'B' where the upstrea~ and downstream portions/are parallel, that is, the

B C

B' C'

flow is parallel upstream of AOA~ where A 0 and A'~ are the frozen Mach lines emanating from A and A' on the walls and inclined to the flow at the upstream frozen Mach angle • . Also downstream of CO'C', CO' and C'O' are Mach lines emanating from 0' and inclined at the downstream frozen Mach angle. At points A, A', C, C' one may have a discontinuity in the derivatives of the streamline angle 8 and Eqs.

(81)

and

(82)

again hold.

Finally, consider the free-jet expansion of a reacting supersonic flow through a nozzle. For the nozzle exit pressure Pe slightly greater than the plenum chamber pressure p , expansion fans will be set up at the corners at the

e'ki

1è

·

12

· b:tnè

~c

-s.b~

r

:dis:t&:ce_ downStrearii.::of' the corner at·. tlie.. .. free-j et boundary, which is a streamline, all the variables are

(22)

Plenum chamber

Free jet boundary Flow

direction

continuous. The vorticity

S

on either side of this boundary streamline will be different making

CS]

f

O. Thus from Eqs. (31')-and (34) to (36), the1

dis-continuities in the derivatives 'of the other variables can be determined pro-vided one other quantity like [an] or [T n ] are known.

5.

DISCUSSION

It was pointed out in the introduction that one can arrive at erroneous results by inferring from the characteristic equations as to which variables can have discontinuobs normal derivatives across a given

characteristic curve. As an example, consider the s component of the momentum e quat ion

p q qs + Ps

=

0 (83)

which is in characteristic form along streamlines. From this one may conclude that pand q can admit discontinuous normal derivatives across streamlines as do ne in Ref. 3. But the results of Section 3.1.1(a) show that p cannot

admit any such discontinuities across streamlines.

As a further example by choosing S, q, p, a as basic variables, one gets the following charact~ristic equation valid along frozen Mach lines (see the Appendix for details):

~f(T

dS - Q da + q dq)

+

q2 de

(84) where Al' A2,A are known functions of the variables. From this equation one may

conclu~e

th at S, a, q, e can have discontinuous normal derivatives across frozen Mach lines. The results of Sections 3.1.2(a) and 3.1.3(b) show that a and S cannot admit any such discontinuities.

(23)

Thus the analysis given in this paper is the correct approach to find out as to which variables can have weak discontinuities. This analysis further gives the relation between the discontinuities in the several variables.

6.

CONCLUSIONS

In conclusion, one finds that

1. Across frozen Mach lines, the atomic mass fract10n a, and entropy Scannot admit discontinuous derivatives while all other flow variables can. Also the vorticity ~ cannot have any discontinuity. These discontinuities are interrelated and any two, of which one should invariably be [en]' determine all others uniquely.

2. For streamlines, the analysis shows that pand e cannot admit discontinuous derivatives across them while all others can and again only two of them are independent while all others are uniquely determined by these. ~hese two quantities may be chosen as [qn] and [an~

3.

For undissociated flows, the initial condition of zero vorticity, i.e. irrotational flows, eliminates the streamline completely as a possible curve for discontinuous derivatives. The boundary condition [eCn]

=

0, however, leaves the Mach lines as possible curves of discontinuities.

(24)

1. Broer, L.J.F., 2. Wood, W.W. Kirkwood, J.G. 3. Der, J.J. • 4. Brainerd, J.J. Levinsky, E.S. 5· Glass, lol. Takano, A. 6. Courant, R. Friedrichs, K.O.

7.

Courant, R. Hilbert, D. 8. Morse, P.M. Feshbach, H. REFERENCES

Characteristics of the Equations of Motion of a

Reacting Gas,

J.

Fluid Mech., Vol. 4, Part 3, pp.

276,

July 1958.

Hydrodynamics of a Reacting and Relaxing Fluid,

J. Chem. Phy. Vol. 28, No. 4, April 1957.

Theoretical Studies of Supersonic Two-Dimensional and Axi-Symmetric Nonequilibrium Flow, NASA TRR-164, 1963 .

Viscous and Nonviscous Nonequilibrium Nozzle Flows, AIAA Journal, Vol. 1, No. 11, November, 1963.

Nonequilibrium Expansion Flows of Dissociated Oxygen Around a Corner, UTIAS Report No. 91, June 1963. Supersonic Flow and Shock Waves, Interscience

Publishers (1948).

Methods of Mathematical Physics, Vol. 11, Interscience

~blishers (1962), pp. 486 - 490, Chap. V.

Methods of Theoretical Physics, Part I, McGraw Hill Book Co., Inc., (1953).

(25)

APPENDIX A

It was pointed out in the introduction in the main text that one can arrive at erroneous results by trying to conclude directly from the

characteristic equations along given characteristic directions as to which variables could admit discontinuous derivatives across these curves. This point was already discussed to a certain extent in the Section

5

of the main text. It is proposed to give some more details in this Appendix.

First of all let us note that Eqs. (1) to (7) given in the text are the general equations defining the flow of a pUre dissociated diatomic gas in dissociational nonequilibrium. They can be specialized to undissociated gas flow by p~tting all terms containing the dissociated mass fraction

a

equal to zero. This is apart from any initialor boundary conditions such as the vorticity ~-

= 0

initially or that all the streamlines emanate from the same reservoir such that the stagnation enthalpy h is the same everywhere. The prescription of these initialor boundary

con~itions

further specialises to a particular kind of problem.

For the general problem (i.e. without the introduction of initial or boundary conditions) either with or without dissociation, one may ask: what is the number of characteristic directions? Which are the characteristic directions? What are the characteristic equations along these directions? Can these

characteristic equations be written such that they contain differentials of different combinations of the fundamental variables along a given characteristic direction (for example combination of pressure p, streamline angle 8 or speed q and streamline angle 8)? In this last question, one is not interested in the utility of these alternate equations in the numerical computation of the flow by the method of characteristics even though this point will be discussed.

For the purposes of this discussion the following definition of characteristic equations is adopted, namely,

"Characteristic equations are those containing all differentiations in a single directiQn; this direction is called the characteristic direction".

Let us restriet our attention to two-dimensional .or axially symmetrie steady flows. Equations

(1)

to (6) written in thestreamline coordinates are:

0 08 + v~l

-Mass continuity: - (pq) + pq pq:sin

OS

Ön

r

s momentp.m:

pq~

+

~

0 n momentum:

pq~

pq

.c

+ op

Ön

0

S

=

~

-

q'ds 08

Equation (A3) may also be written without introducing 2 08 +

SE -

0 p q

as

on-8

=

0 (Al) (A2) (A3) (A4)

S

the vortici ty as (A3')

(26)

Energy: s-Entropy: n-Entropy: or Dissociation rate: or

o

(A5)

(A6) T

~

=

-qS

+ Q

~

(A7) _q (

Qg, _

q

ce)

+ Q

~

cn cs on (A7' ) ca . q

ds

=

7/JL (AB) ca _

'!l!J:.

=

ds -

q w (AB' )

One has further the following algebraic equations

Equation of state: Enthalpy: Entropy: p = pRT (l+a) h h(p,p,a) or h(a,T) S S(p,T,a) (A9) (AlO) (All) All the variables occurring in the above equations are already defined in the main text. Explicit expressions for

Q,

7/J, L, h, S are given in Ref.

5

for a pure dissociated diatomic gas. In the above, Eqs. (Al) to (A9) can be treated as eight equations for the eight unknowns p,p, T, a, h, S, q

e.

If one wishes

to make use of the definition of enthalpy h given in Eq. (AlO) and of Entropy S given in Eq. (All) one can show that Eqs.

(A5)

and

(A6)

are essentially the same and thus one of them may be dropped and also remove 'h' from the set of unknowns. Also one may note that the entropy Sappears only in Eqs.

(A6)

and (A7) and thus S or CS/cs or cS/cn can be computed from Eqs. (All),

(A6), (A7)

af ter solving the remaining set of equations for the remaining six variables

p, P, T, a, q,

e.

However, we shall eliminate only the enthalpy h from Eq.

(A5)

through the use of Eq. (AlO) and fruther eliminate cp/cs, cp/cs in the resulting equation with the aid of Eqs. (Al) and (A2). These manipulations lead to

-pq [P(hp)p,a- l ]

~

+ P(hP)p,a

~

+

p(~)p,p

~

0

v-l

sin r

or by the definition of the partially frozen speed of sound one has

)

Qg

ce

-1 cs - q

dn

+ qh w

a

php v-l q sin

e

r

sJ

+

'(ha)

t

= 0

p,P (A12) a;2=_ (hp

~pl/P)

o

(A13)

(27)

Also one may note that Eq. (A5) can be integrated with the aid of Eq.

(A2)

to yield the energy integral

(A14) where ho(n) is to stress that it is a function of n only. However, this equation is no better than (A5) as long as one does not wish to use any

initial conditions (e.g. that initially the flow emanates from a region where ho is the same on all streamlines and thus ho is a pure constant).

Thus for the general problem, one has seven equations (A2),

(A3'),

(A13) ,

(A6), (A7'), (A8'), (A9)

for the seven unknowns p, p,T,a,S,q,e. One of these equations is an algebraic equation; the remaining six are first order partial differential equations • . According to the theory of partial differential equations of the hyperbolic type, this system can be written into six characteristic equations along six characteristic directions (Ref.

7).

Some of these characteristic directions may have multiplicity.

It is readily seen from the definition of characteristic directions adopted above that Eqs.

(A2),

(A6),

(A8')

are already in characteristic form

along the streamlines. Thus streamlines are characteristic curves for this system. Also it may be noted that it has a multiplicity of 3 so faro To determine the remaining characteristic directions and the characteristic equations along them, let us define the coordinate ~ along this direction so that a total differential of any quantity along this direction may be written in terms of the partial differentials along s, n as

d ds è dn è

~ =~

ds

+ ~

dri

(A15)

Thus one has

dp ds èp + dn èp

d]"=d]"

_·ds

_dJdri

(A16) dq ds èq + dn èq ~ ~

· ds

~

dn

(A17) de ds èe + dn èe ~= _~

· _ds

_~

_dn

(A18) dp ds èp + dn èp ~ =~

· ds

~

dri

(A19) dS ds ès + dn ès ~ =~

· d'S

~dri (A20) da ds (Xx + dn (Xx

d]"=d]".

_ds

_d]"

_dri

(A21)

Equations

(A3'),

(A13),

(A7')

together with Eqs. (A16) - (21) give the characteristic directions and the characteristic equations valid along them as

(28)

follows:-. E (3

t)

ds dn d . E (A 8) d

Multlplying q. A by

a..e

• dJ

an uSlng qs. 1 an

(A19) , one has

or

dp ds 2 de dn [ 2 (dn)2 oe +

(~~)2 ~Ps

] -_ 0

~ • ~ + pq ~. de - pq ~

Ön

U.{I 0

or eliminating

~

with the aid of Eq.

(A2)

~. ~

+ pq2 :

.~ -[pq2(~)2 ~

_

pq(~)2 ~]

= 0

(A22)

Further eliminating

~

with the aid of Eq. (A13), one obtains,

~.~

+

pq2

~ .~

_

p~(:)2q ~

_

(~f[l~ ~

+

;~:p- V~l

i?

sine

J}

=

0

Simplifying Eq.

(A23)

gives

dp

dI

(A23)

From Eq.

(A24),

the 'condition for it to be of characteristic form is that the

coefficient of the partial differential oe/on be zero, i.e.

Le. where

(":;)2 _

~2 (~)2

=

0

f

~~7~

= +

~f

~2

=

M 2 _ 1 f f

Similarly multiplying Eq. (AT) by

~

Eqs. (A17) , (A18) , (A20) and (A21), one has

T ds [dS ds èS] + ds [dq ds è q ]

dl dJ- dJ

ds

q

dJ

d]-

dJ

ds

o

(A25)

(A26)

dn

(29)

or (dS)2L_[T

de _

_Q

Ca ]

\d1

ds

(A27) The third term on the LHS in Eq. (A27) is zero due to Eq.

(A6).

in Eq. (A27) with the aid of Eq. (A13), one has

Replaeing

~

ds [T dB Q dCt +

~]

2 dn

~

+ (dn

\2

2 oe

dI

d$ - d$ q d$ - q

dJ·

dJ

\.dl)

q

Ön

q~tW

v ..

i

'

.}

---:-- + - q Sl.ne = 0 Php

r

:

: Rearranging the terms in Eq. (A28), one has.

+

:

q2

(~~'f

ruw

_

~

sine]

=

0

. ~2

\"u,,;)

[Ph

.

r

f P

The condition for Eq. (A29)'to be of the charaeteristic form is that the coefficient of oe/on in it be zero i.e.

(~)2

_

1

(~~S ~

0 ~2 f i.e. dS

7

d$ dn d$

=

:t

~f

For the direction ~ satisfying Eq. (A25), one ean show that ds +

dJ

=

~f/Mr

and dn + 1

dJ=-

Mr

(A28) (A29) (A30) (A31) Substituting Eqs. (A25), (A30) and (A31) in Eqs. (A24) and (A29) one obtains two sets of eharacteristic equations valid along the frozen Mach lines, namely

~-l.

]

- -r-

Sl.n e

=

0 (A32)

·- ~.:

In the literature, this is the equation normally given. Since these equations contain differentials only· in two variables pand e, they ean be readily solved by numerical methods for pand e at a third point knowing these values at two other points.

(30)

Equation (A29) reduces with the aid of Eqs. (A25), (A30) , (A3l) to

±

[Tdi-QcLe+qdi dS da

3J

2 +~ M 2 f

[

~~

_

v~l

Sine] .P~p r

=

0 (A33)

This is the other set of equations valid along frpzen Mach lines. But this contains differentials in four variables S,a,q,e while there are only two

equations. Thus if one wishes to use this combination of variables it will be

difficult to solve the problem numerically. This is the equation given in the discussion in the main text where it was pointed out that one may conclude that S,a,q,e can have discontinuous derivatives across Mach lines. This is erroneous since it was shown in the text that S and a cannot admit any such

discon-tinuities across frozen Mach lines.

Let us._summarise all the characteristic

equàtions:-Along streamlines:

Along frozen Mach lines: + +

t3

dp +

fdl

pq~

+

~

o

T

ès

d'S

èa

ds

Q

_d'S

ècx

0

=

w

-o

sin eJ

=

0 ~-l r sine J

=

0 (A34) (A35) (A36) (A37) (A38) (A39)

If one chooses p,q, ,a as the basic set of variables, Eqs. (A34) (A36) and (A38) are to be used whereas if one chooses S,q,e,a as the basic variables, Eqs. (A35), (A36) and (A39) are to be used in the numerical com-putation of the flow field by the method of characteristics. Also one may note that one does not have a characteristic equation involving differentials of q and e alone along frozen Mach lines, but one has in addition the

differentials of S and a also. Some statements in Ref. 3 seem to imply that one can have a characteristic equation involving differentials of q and e alone along frozen Mach lines.

For non-reacting flows, the above equations reduce to (by putting a terms equal to zero),

(31)

•

Along streamlines:

Along frozen Mach lines:

±

~ ~

₊ pq 2

_dl-

de f d

±

~f

[T

~

+ q

* ]

-oS = 0

ds

v-l

~

-

_r _M f 2 de q

dl-(A4o) sine = 0 (A41) (A42) v-l

:l

_{sin e} ₌₀ r Mf

Note that Eqs. (A34) and (A4o) show that for even irrotational flows, the streamlines are still characteristics. Only that as pointed out in the text, the initial condition of zero vorticity removes streamlines from the directions which can admit discontinuous derivatives •

(32)

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Discontinuous S oluti ons for Nonequilibrium Supersonic Flows

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Tirumalesa, D.

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13 ABSTRACT

Generalized solutions with weak or strong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for nonequilibrium supersonic flows. The curves with weak discontinuities are shown· to be the characteristics of the system of equations. The interdependance of the discontinuities for several variables are discussed.

(33)

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lJrIAS TECIINICAL NarE NO. 114

Institute for Aerospace Studies, University of Toronto

Discontinuous Solutions for Nonequilibrium Supersonic Flows D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces :3 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. 11. lJrIAS Technical Note No. 114

Generalized solutions with weak or strong discontinuities (that is, discontinuities

in the derivatives or the variables themselves, respectively) are considered for

non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the oharaeteristios ot the aystem ot equations. The inter4.pen4anoe ot th.

dl.oon-tinuities for several variables are discussed.

Available oopies of this report are limited. Return th is oard to UTIAS, if you require a oopy.

Discontinuous Solutions for Nonequilibrium Supersonic Flows D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces :3 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. _{11. lJrIAS Technical Note No. 114}

Generalized solutions with weak or st rong discontinuities (that is, discontinuities

non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the characterist1cs ot the system ot equatio"s. The interdependanoe ot the

disoon-t1nuities for several variables are discussed.

Discontinuous Solutions for Nonequilibrium Supersonic Flows D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces 31 pages 2. Shock Waves 4. Superson1c FlowB

~

I. Tirumalesa, D. 11. lJrIAS Technical Note No. 114 Generalized solutions with weak or st rong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for

non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the character1st1cs ot the system ot equations. The interdependance ot the

discon-t1nuities for several variables are discussed.

Available copies of th is report are limited. Return this card to UTIAS, if you--<.equire a copy. lJrIAS TECIINICAL NOTE NO. 114

Discontinuous Solutions for Nonequilibrium Supersonic Flows

D. Tirumalesa 1. Method of Character1stics 3. Contact Surfaces :3 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. 11. lJrIAS Technical Note No. 114

Generalized solutions with weak or strong discontinuities (that is, discontinuities

non-equilibrium supersonic flovs. The curves with weak discontinuit1es are shown to be the character1st1cs ot the system ot equat10ns. The interdependance ot the

(35)

UTIAS TECHNICAL NOTE NO. 114

Discontinuous Solutions for Nonequilibrium Supersonic Flows D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces ] 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. 11. UTIAS Technical Note No. 114

Generalized solutions with weak or st rong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the charaeteriatics ot the syatem ot equations. The inter4epen4ance ot the di.con-tinuities for several variables are discussed.

Available copies of th is report are limited. RetlKfl this eard to UTIAS, if you require a eopy.

Institute for Aerospace Studies, Universlty af Toronto

Discontinuous Solutions for Nonequilibrium Supersonic Flows

D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces ] 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. 11. UTIAS Technical Note No. 114

Generalized solutions with weak or strong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the characteristics of the system of equationa. The interdepen4ance of the 4iacon-tinuities for several variables are discussed.

Available eopies of this report are limited. RetlKfl this card to UTIAS, if you require a eopy.

Inrtitute for Aerospace Studies, University of Toronto

Discontinuous Solutions for Nonequilibrium Supersonic Flows D. Tirumalesa 1. Method of Characteristics 3. Contact Surfaces ] 1 pages 2. Shock Waves 4. Supersonic Flows

~

I. Tirumalesa, D. 11. UTIAS Technical Note No. 114

Generalized solutions with weak or strong discontinuities (that is, discontinuities in the derivatives or the variables themselves, respectively) are considered for non-equilibrium supersonic flows. The curves with weak discontinuities are shown to be the characteristics of the system of equat10ns. The 1nterdependance ot the 41scon-tinuities for several var1ables are discussed.