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Viscous Transonic Flow in Relaxing Gases

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Page 1: Viscous Transonic Flow in Relaxing Gases

M. SICHEL I Y. K. YIN: Viscous Transonic Flow 316

ZAMM 66,315 -329 (1976)

M. SICHEL / Y. K. YIN

Viscous Transonic Flow in Relaxing Gases

Es wurden die Gleichungen einer ebenen viskosen schallnahen Stromung eines entspannten Gases fiir Anstrom- geschwindigkeiten hergeleitet, die nahe eines Gleichgewichts und fester Schdlgeschwindigkeiten liegen. Eindimensionale Losungen wurden gefunden, aus wekhen der Einfluj der Druckabhikngigkeit der Viskoaitiit auf die Struktur einer Kompressionswelle eines entspannten Gases deutlich wird. Die visbse Struktur des Eckbereiches sowie der Beachleu- nigungswelle, die be; der Ubereinstimmung von Stromaufwartsgeschwindigkeit und fester Schallpchwindigkeit auf - tritt, wird im Einzelnen dargestellt. Die Struktur einer teilweise dispergierten Welle, welche den Ubergang des strom- aufwarts liegenden viskosen StoJes zur stromabwarts sich befindenden Entspannungszone einschliefit, wird ebenfalls bestimmt.

The equations for the plane viscous transonic flow of a relaxing gas are derived for free stream velocities near the equilibrium and frozen speeds of sound. One-dimensional solutions art? found which show the influence of the com- pressive viscosity upon the structure of a compression wave in a relaxing gas. The viscous structure of the corner region or acceleration wave which occurs when the upstream velocity just equals the frozen speed of sound is investigated in detail. The structure of a partially dispersed wave including the transition from the upstream viscous shock to the downstream relaxation zone is also determined.

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I. Introduction The present paper deals with the influence of compressive viscosity on the plane transonic flow of a relaxing gas, i.e. a gas with a lagging internal energy mode. Such gases are characterized by two sound speeds: a frozen speed of sound b and an equilibrium speed of sound a, and the character of the flow depends on the value of the gas velocity V relative to these two acoustic speeds.

The inviscid theory of the one-dimensional flow of such gases has been considered by LIGHTHILL [ll], BECKER [l], BECKER and BOERME [2], and CLARKE and MCCHESNEY [7] among others. This theory shows that when a < V < b steady compression waves can exist whose structure is determined by a balance between convection and relaxation. Such waves are called fully dispersed. When V reaches the frozen speed of sound b the theory predicts a corner or acceleration wave upstream of the wave front. In nature auch corners must be smeared out by viscosity, and this is one of the problems investigated here. When V > b the inviscid solution for relaxing flow becomes double valued, and physically meaningful results can only be obtained by introducing a frozen shock discontinuity upstream of the relaxation region. Such waves axe termed partially dispersed. Again, the initial growth and structure of this viscous shock can only be determined by including viscous terms in the equations describing the flow, and this problem is also studied here.

The influence of viscosity on the structure of shock waves in relaxing gases was also the subject of the early papers of BROER [4], and BROER and VAN DEN BERCEN [5]. Solutions for the shock structure were deter- mined numerically and analytical solutions were presented for the special cases of L,,/Le< 1 and L,/L,, > 1 where L, and L, are a viscous and relaxation length respectively. The present study is an extension of this early work in that the influence of viscosity is considered systematically as the velocity upstream of the wave increases from the equilibrium through the frozen speed of sound. SCALA and TALBOT [19] found numerical solutions for the structure of relatively strong shock wavw with relaxation, and their results reinforce the present work; however, they made no attempt to assess the influence of viscosity in various regimes of flow.

The flows described above are transonic in the sense that either (Mew - 1) or (Hfm - l), or both are amall compared to unity, where Me, and Mf, are the MAUH numbers upstream of the wave based on the equilibrium and frozen speeds of sound. For a relaxing gas it is necessary to distinguish between equilibrium and frozen transonic flow since different phenomena can be expected in each case. These distinctions have been discussed by NAPOLITANO [ 131 and PRUD’HOMME [ 161 who formulated the different equations valid in the many MACE number regimes which are possible in the flow of a relaxing gas. In the present paper the equations for viscous flow with relaxation are derived and the one-dimensional forms of this equation valid in different regimes of transonic flow are studied in detail.

Fully dispersed and partly dispersed waves have been observed in carbon dioxide by GRIFFITH and KENNY 1101, and by STREXILOW and MAXWEIL [MI. In each case the bending vibration of CO, provided the internal energy mode. The interferograms of STREHLOW and MAXWELL [18] show the transition from fully dispersed to partially dispersed wave structure, and since the observed waves were very thick the detailed structure

21*

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316 M. SXCHEL 1 Y. K. YIN: Viscous Transonic Flow

could be clearly distinguished. The pictures suggested an analogy between the structure of weak shocks and fully dispersed waves, which could be extremely useful since viscous shocks are a t least an order of magnitude thinner than fully dispersed waves and so more difficult to observe. This analogy was justified analytically, by NAPOLITANO and RYZHOV [14], and by RYZHOV [17], and is also verified below.

I n both inert and relaxing gas flows there is a close relationship between plane steady and one-dimensional unsteady flows. A key question is how fully and partially dispersed waves are formed when a, piston initially a t rest is accelerated to a finite velocity. The linearized problem was first studied by Chu [8] for a piston instantaneously accelerated to a velocity small compared to that of sound. The analysis showed that an atten- uating discontinuous wave front moves ahead of the piston at the frozen speed of sound followed by a con- tinuous wave moving at the equilibrium speed of sound. This latter wave continually grows thicker and ulti- mately is completely smeared out by the influence of relaxation. The linearized theory thus cannot explain the formation of fully and partially dispersed yaves. BLYTHE [3], and OCEENDEN and SPENCE [15] made a more comprehensive study of this problem using the nonlinear equations. Discontinuities form in the flow when the piston velocity exceeds a certain critical value. These discontinuities can be analyzed by using the RANKINE-HUGONIOT conditions or by introducing viscous terms in the equation and studying the limiting behavior of the solution as the viscosity approachev zero. The latter approach was employed by OCKENDEN and SPENCE [15] and their results bear a close relation to those reported below for the steady viscous flow of a relaxing gas. The propagation of one-dimensional nonlinear waves through a relaxing medium has also been investigated by RYZHOV [ 171 ; however, he did not consider the influence of viscosity.

The main objective of the present study is to assess the influence of compressive or longitudinal viscosity upon the one-dimensional transonic flow of a relaxing gas. Approximate equations are derived for the equili- brium and frozen transonic regimes of flow. One-dimensional or wave type solutions of these equations are then investigated in order to establish the main physical properties of the flow.

11. Derivation of the Basic Equations The basic equations for the viscous flow of & relaxing gas are derived below. The main purpose of the present investigation is to study the combined effect of viscosity and relaxation. Hence, the simplest possible model gas, i.e. an ideal gas with a single internal energy mode, will be used. The thermodynamics of such gases is considered in detail by BECKER [l], and BECKER and BOEHME [2].

The derivation starts with the equations for the conservation of mass, momentum and energy below:

Here e, p , and V are the density, pressure, and velocity vector, respectively. z’ is the viscous stress tensor with the components

The bulk viscosity is taken as zero since relaxation will be considered separately. I n planar flow the viscous dissipation y is given by

where u and v are the x and y components of the velocity vector V. The heat flux 8 has components due to thermal conduction and the diffusion of excited species, However, for a gas only slightly out of equilibrium diffusion can be taken into account by a suitable modification of the thermal conductivity [21] so that

8 = - % x , v l l ’ , (6) where x, is a modified coefficient of thermal conductivity and T is the translational temperature of the gas. dldt denotes the substantial derivative and reduces to the operator (V * 0) in steady flow.

BECKER and BOEHME [2] have shown that the properties of an ideal gas with a single internal state variable q can be described by the caloric equatidn of state

h = C,T*(p/p*)RICp (q/T*)-CRICp exp ((X - fJ*)IC,) + QRq

(WW,,, = T

h = C,T + CRq .

(7) for the enthalpy h as a function of the pressure p , entropy 8, and internal variable q. The quantities p*, T , and S* are reference values. From the relation

i t is readily shown that

(8) Thus CR is the specific heat and q the temperature corresponding to the internal energy mode. C, is the frozen specific heat a t constant pressure. From (7) it follows that the thermal equation of state is

1) = QRT (9)

Page 3: Viscous Transonic Flow in Relaxing Gases

M. SICEEL Y. K. Ym: Viacous Transonio Flow 317

where R is the specific gas constant. The frozen and equilibrium speeds of sound b and a are given by

= (appe)s,q = yfRT; as = (ap/ae)s,q-p = Y ~ R T (10) where yf and ye, the frozen and equilibrium ratios of specific heat, are

and ij is the equilibrium value of the internal variable q. Transport properties and specific heats are assumed constant.

Only small perturbations from equilibrium are considered here; hence, the simple relaxation equation

can be used [a] to relate p to its equilibrium value ij. The relaxation time z will be taken as a constant. Since q is a temperature corresponding to the internal energy mode, e.g., the vibrational temperature, a equals the translational temperature T.

Equations (1) to (3) together with the caIoric equation of state (7) and the rate equation (11) form a com- plete set; however, it is desirable to eliminate as many of the thermodynamic variables as possible. First, it is readily &own that p and e are related by

where the subscript denotes partial differentiation. The entropy X can be related to the other thermodynamic variables with GIBBS' Equation in the form

dh 1 dp dS dq -=-- dt e dt + T x + h * z

which when combined with the energy equation (3) yields

The first three terms on the right of (14a) represent entropy production due to viscous dissipation, heat con- duction and relaxation while the last term is due to entropy transport. Combining equations (12) and (14) with the continuity and momentum equations (1) and (2) now yields the relation

Using the momentum and energy equations (2) and (3) together with the relation (8) it foliows that dT dq dV 1 cp - + CR - + T' .- = - ( y - . e + v . v . q . dt dt dt e

The internal state variable p can be eliminated from (15), (16) and the relaxation equation (11) by first writing (16) in the form

(9 - v . e + v . ' i ~ . z) 'd dq dV 1 Cp&(T - 4) + (GR + C )-+ V*-=-

dt dt e and differentiating (11) so that

d/dt(T - 4) = z d/dt(dq/dt) . Combining (17) and (18) and eliminating dq/dt with (15). then results in the equation:

I n eqution (19) C, and C, are the frozen and equilibrium specific heats at constant volume.

to that for the inviscid flow of a relaxing gas if the last three transport terms are neglected, i.e. to Equation (19) is a dynamic equation involving mainly the velocities V, b, and a. The equation reduces

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318 M. SICHEL / Y. K. YIN: Viscous Transonic Elow

where z' = Cvr/Cve. When linearized, (20) reduces to the equation used by MOORE and GIBSON [12] and VIN- CENTI [20] to study the flow of relaxing gases. In the limit z' + 00, equation (20) reduces to that describing the inviscid flow of an inert gas, or to what is sometimes called the gasdynamic equation.

All thermodynamic variables have not been eliminated from (19) since b, a and 8 are functions of the temperature T while e is a function of p and T through the thermal equation of state (9). In transonic flow, however, the energy equation (16) can be used to express T in terms of the velocity and then (19) can be reduced to an equation involving only the components of the velocity 8.

111. Simplifications in Transonic Flow

Equation (19) above is quite general; the main restrictions on its applicability come from the simplified model used for the relaxation process. Considerable simplifications of this equation are possible in the case of transonic flow and these are developed below. The development is restricted to one-dimensional flows since these are the only flows which will be analyzed in detail. The one-dimensional solutions show the basic pro- perties of relaxing viscous flows; however, it is a relatively straightforward matter [21] to develop the equations of two-dimensional plane or axisymmetric flow from equation (19) to correspond to the cases treated below.

The variables are made dimensionless using U , To and eo as reference, velocity, temperature and densitv so that

where the bars denote dimensional quantibies. The X coordinate is stretched according to - 2 x=- AL'

L is a characteristic dimension of the problem under consideration, and 1 is a general stretching factor whose choice is determined by the nature of the flow being investigated. The viscous flow of a relaxing gas is charac- terized by two lengths; a relaxation length L,, and a viscous length L,, and these can be defined by

(4/3) F L, = -* eJJ L, = z 'U; (23) - ,u is the shear viscosity while (413) ,E corresponds to the compressive or longitudinal viscosity in a gas with zero bulk viscosity.

The appropriate expansions of u, a, b, T, and e in transonic flow are

@ = @(O) + Vl@(1) + V2@'2' + .*.;

c2 = YJRF = yJBT0(T(O) + v1T(l) + a * * )

q = q(0) + v l p + v 2 p + . . a . From equation (10) it then follows that the frozen and equilibrium speeds of sound are given by

eo and To are frequently chosen to make T(O) and $0) equal unity. Similarly, the reference velocity U can be chosen to make either a(O) or b(O) equal to unity. The small parameter vl can generally be related to upstream or boundary conditions. Substituting the expansions (24) above and keeping only the largest terms it is readily shown that the viscous term in (19) reduces to

ii2 = rsRT = yeRT0(T(O) + vlT(l) + b e * ) . (25)

The viscous dissipation q is of order v~(poU3Lv/A2L2) and so can be neglected.

mes With substitution of the dimensionless variables and the expansions (24) the energy equation (16) beco-

when only the largest terms are retained. PT" is the PRANDTL number based on the longitudinal viscosity ji". In the flows considered below (LJAL) < 1. Only those gases for which the energy of the internal mode is small compared to the total enthalpy will be considered. As indicated by BUGGISR [6], there are many gases which satisfy this condition so that (CRIG,) < 1. Then the energy equation (27) becomes

u2 au(l)

i l X C,T0 ax (28 1 _ _ _ ~ _ _ _ 3T(') -

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M. SIUHEL / Y. K. YIN: Viscous Transonio Flow 319

and if the flow originates in a region where u(l) = !W) = 0 integration of (28) yields US T(1) = - --u(1) .

CPTO Equations (28) and (28a) provide the relation between temperature and velocity mentioned in Section I1 above. Equation (28a) simply states that the frozen stagnation enthalpy is constant to first order.

With the relation (28a) it is possible to express the heat flux 8, the viscous terms, and the frozen and equilibrium speeds of sound in terms of u(l). Keeping only the dominant terms, equation (19) then reduces to:

(29)

whew

Equation (29) will now be used to study the effect of viscosity on one-dimensional fully dispersed and partially dispeirsed waves with the flow from left to right, i.e. from x = - 00 to x = + 00.

In analyzing the one-dimensional waves i t is necessary to use the RANKINE-HUGONIOT jump conditions across the wave. Using the conservation of mass, momentum, and energy across the wave i t is readily shown that [71

where the subscript- 00 refers to conditions upstream of the wave a t x = - 00 and the subscript s refers to equilibrium conditions downstream of the wave. Across a normal shock in which the internal energy.mode is frozeri, the upstream and downstream velocities are related by

where the subscript b refers to conditions downstream of a frozen shock.

IV. Fully Dispersed Waves

For fully dispersed waves with U, <b, it is most convenient to chose a*, the local talue of the equilibrium speed of sound when ii = a, as the reference velocity U. Then

d o ) * = 1.0; b(0)' = y f / y e . (32). It is adso convenient to introduce the parameter

Clearliy, the assumption (CRIC,) < 1 also implies that d < 1. For a limiting fully dispersed wave with uw=bw i t follows from the RANKINE-HUCONIOT conditions (30) that

urn = uw/a* = 1 + (1/2) d + O(d2); us = ;ii,/z* = 1 - (1/2) A + O(42).

1 - b(0)' = - (yf/ye) FJ .

(34)

(35) The parameter 1 - bQz which appears in (29) will be given by

Weak fully dispersed waves for which vl < A will be considered first. Then it is convenient to introduce a small parameter q to specify the wave strength such that y1 = qA so that the basic equation (29) becomes

The ratio of the relaxation length L, to the characteristic length% can be written as z'(v//L), and so is also the ratio of the relaxation time to a characteristic flow time. Hence, if Lc//L is sufficiently large the flow will be frozen while the flow will be in equilibrium if L&L is sufficientlpsmall. Neither of these extreme cases is of interest here. Thus, in inviscid flow, when only the first and third terms of (36) remain, the flows of interest are those for which these terms are of the same order of magnitude requiring a scale factor A such that

LJAL = 11. (37) It appears reasonable to retain this scaling for the viscoud equation (36).

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320 X. SICHEL / Y. K. YIN: Visooua Transoda Flow

The third term of (36) will now be of higher order so that equation (36) becomes

Only the larger of the two viscous terms in (36) has been retained in (38), (L,/AL,) < 1 in most cases of practical interest, and then choosing Y, = u, - 1 so that u s = + 1, the solution of (38) for a fully dispersed weak com- pression wave is readily shown to be

- with

- yg = r‘(& - a&) -” = z’Z*B.T,Ll .

fince (ye/y,) = 1 - ]led, and d < 1 it is possible to use (ye/y,) Srozen shock of strength vl is given by 1111 as

1.0 above. The structure of a weak viscous

Equations (40) and (39) are identical except for the fact that the diffusivity of sound Dj9’ in (40) is replaced by GB in (39). For this reason ye is often called the bulk viscosity. Comparisons of equations (39) and (40) thus indicates an analogy between fully dispersed waves and weak viscous shock waves, and this analogy can be shown to remain valid in plane flows [14,21].

The bulk viscosity and the ordinary viscous terms are additive in equation (38). The coefficient of the viscous term is readily shown to be

and so is, essentially, the ratio of the ordinary to the bulk viscosity. 9‘ can be associated with translational relaxation which occurs in only a few moleoular collisions while Ys ari& from the relaxation of the internal mode which requires many collisions. Hence, 9’ < j?B, justifying the neglect of the viscous term in (38). It now also follows from equations (39) and (40) that weak fully dispersed waves are much thicker than weak viscous shock waves, a result borne out by the experiments of STREEKLOW and MAXWELL [18].

The restriction to weak fully dispe~sed waves can be relaxed by allowing the parameter q to be of O(1) so that vl N O ( A ) . As before the first and third terms of (36) should be of the tame order which then requires that ;IL = L,, i.e. the relaxation length becomes tlie appropriate characteristic dimension in the x direction. Then equation (36) becomes

Now all terms of (36) must be retained. The solution of (42) satisfying the boundary conditions ua, = $1, u, = -1 is readily determined. The

viscous terms will be neglected since (L,/dL,,) = (5’ ’Tel j jB) < 1. Then, since (?,/ye) = 1.0, and re = I‘, with A < 1, the solution is given by

The constant C determines ,$he origin of the coordinate system, and is taken as zero for simplicity. The inviscid solution (43) describes a smooth transition from ug) = +1 to uil) = -1 as long as the

parameter q < (1/2), and is well known 171. It is clear from (43) that the use of a bulk viscosity is no longer appropriate. There appears to be no need to take viscous terms into account as long as q < (1/2).

This situation changes when 7 = (1/2), corresponding to urn = b,, for then the first term on the right of (43) vanishes, and the only solution satisfying the boundary conditions wg) = + 1, uil) = -1, is

(442

This solution has a corner a t x = -2 In 2 where the slope (du(1)ldx) changes discontinuously from 0 to -1. Near this corner, where the second derivative of becomes unbounded, the viscous terms in (42) must become important, and this “Corner problem” is cdnsidered in Section V below.

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M. SICBEL 1 Y. K. YIN: Viscous Transonio Flow 321

V. Partially Dispersed Waves

Here it is most convenient to use g*, the local value of the frozen speed of sound when ii = & as the reference velocity U. Then

Partially dispersed waves arise when G, 2 &, and, hence, it is convenient to introduce the parameter

which is a measure of the strength of the frozen viscous shock upstream of the relaxation region in the cllassical model of the partially dispersed wave. From the RANKINE-HUOONIOT conditions (30) and (31) it now follows that

ub= I-&; u,= 1 - E - A . (47) urn= 1 + & ;

The corner problem, with ii, = b,, which has already been discussed in Section N, corresponds to u, = 1.0 or E = 0. For a fully dispersed wave U, < b, so that (u, - 1) < 0 when 6* is used as the reference velocity. Then, for a one dimensional wave with E' = 1 - uw,

(48) urn = 1 - E l ; U, = I - ( A - E ' ) . The velocity Ub has no further significance since a frozen shock cannot exist with il, < 5,.

Substitution of (46) in (29) now yields

and this equation provides the basis for the treatment of partially dispersed waves.

A. The Corner Problem

The corner problem with E = 0 will be considered first. Since (u, - u8) = A across a plane wave in this oase, it is appropriate to choose v1 = A , and then having the first and third terms of (49) of the same order requires IL = L,. Equation (49) now becomes

As before (LJAL,) N (T'/FB) < 1. To study the influence of viscosity it is appropriate to introduce an expansion of the form

u(1) = uo +- 8,u, + ' a . (51) where 6, is a small viscous parameter. A natural choice for equation (50) would be 6, = (Lv/ALc). Since u(1)

is itself a coefficient of the expansion (24), useof (51) actuallymeans that adouble expansion in the small param- eters vl, v2, ... , a,, is being used. This can be seen if, for instance, the expansion for in (51) is substituted in (24) for then

A different notation has been used in (24a), i.e. u, has been replaced by %,, etc. Equations (49) and (50) are valid only to order vl. Thus the expansion (51) remains valid only if vl > v18. > va. A logical choice for vg might be dain the case of the corner problem. Then with 8, = (L,/A,L,), an expansion of u(l) to order 8, will be possible without considering effects of order vz only if A > L,/Lc > A2. In the corner problem treated below only the first term in (51) needs to be considered to obtain a meaningful solution SO that the first order equations are certainly sufficient. This situation changes for the partially dispersed wave, treated below, where it becomes necessary also to consider the terms of O(8,) in (61). The fully dispersed solutions discussed in Section N can also be considered as the zeroth order solutions of an expansion such as (61).

u = 1 -5 VI(%O + 8aUu + .-) + VZ(%O + 8&21 + *..) + *'. (244

Substituting (61) in (60) and keeping only the largest terms, u, satisfie8 d dx

with

The only solution of (62) satisfying these boundary conditions is the corner solution uo(-oo) = 0 ;

210 = 0.; lz - $0) < 0 u, = - - e-(UZ)(a-zo)]; (x - So) > 0

uo(+a3) = - 1 .

(63 )

22

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322 M. SICHEL / Y. K. YIN: Viscous Transonic Flow

with xo a constant of integration determining the location of the corner. It is readily shown that (53) and (44) are equivalent. Since the velocity gradient is discontinuous at the corner, there must be a small neighborhood near the corner where the viscous terms of (49) are of the same order as the other terms. Near the corner it is thus, appropriate to choose vl = q A where q < 1. The scaling factor il will also be different since equating the order of the first and third terms of (49) now requires that ilL = qLo. The third term of (49) will now be the largest viscous term, and the condition (L,/AL) = qA must be satisfied if this viscous term is to be of the same order as the inviscid terms. From these two conditions it follows that

which then determines the extent of the viscous region near the corner. Substitution of the inner variables

rp) = .ii o + & + . * , 2 = (ZIT) (55 ) in equation (49) now yields the following inner equations for Go:

.iio(-oo) = 0 . This equation can be integrated once, and when normalized with t’he transformation

becomes - - f - - f f O . d2i df dX2 dX

Equation (58) was appropriately called the “corner equation” by COLE 191. With p = df/dX the trajectories in the phase or p - f plane satisfy

Equation (59) has a saddle point a t f = 0, p = 0 and dpldf = 0 on the line p = -1. Typical phase plane trajectories are sketched in Fig. l(a) with arrows in the direction of increasing X.

The corner solution satisfies the boundary condition f = 0, p = 0, and so must be one of the singular solutions passing through the saddle point a t the origin. This solution, which is indicated by the heavy curve in Figure l(a), is asymptotic to the line p = -1 as X -+ + 00. As will become evident later, only this solution matches -- A

Figure 1 (a). Phase Plane - Corner Solution fiigure 1 (b). Typical Corner Solution (Qualitative)

the zero order inviscid solution (53). The singular solution curves have the slope (dpldf) = & 1 at f = 0, p = 0. Integration of (59) yields the equation

f”2 = p - ln ( p + 1) (60) for the trajectory of the corner solution. A typical solution trajectory in the f - X plane is shown qualitatively in Figure l(b).

As X -+ -my p + 0 so that expansion of (60) yields

(61) 1212 = p - (I, - p y 2 +- ..*) . Keeping only terms up to O(p2) in (61) the solution for f becomes

f = - - C e k x . (62)

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M. SICHEL / Y. K. YIN: Viscous Transonic Flow 323

Only the positive sign in (62) yields a bounded solution as X + -a. I n physical variables the asymptotic corner solution as x -+ --oo is

z - 2; uo = - exp Lc(DjF”/YB)1/2

The constant of integration, which has been absorbed in %;, must be determined by matching with the down- stream inviscid solution. From (63) it follows that the characteristic length of the corner region where uo decays to the upstream value is related to the physical parameters of the flow by

and so depends on both viscosity and relaxation. As noted above, p 3 -1 as X 3 $00 so that

go -+ - (17~217~) 2 + c = - (112) 1% - zo) , where Zo is a constant to be determined by matching. Since A < 1, re N rj as noted previously. To verify that the zeroth order inner and outer solutions match an intermediate variable x - (x/v(q)) = (q%/v(q)) will be intro- duced, such that v(q) + 0; v(q)/q + 00 as q 4 0. Then considering the limit 7 -+ 0 with x,, fixed q. -

Thus the inner and outer solutions match to zerot1’ order provided xna = Zn0, where xq0 is the value of xq cor- responding to xo in the outer solution while Zq, is x,, corresponding to 5. It is convenient to let xo = Z0 = 0 since this choice does npt affect the properties of the solution. In physical variables the inner solution as x -+ m, is

and depends only on parameters related to relaxation. Although asymptotic solutions of (58) have been found the complete solution of this equation can only

be found numerically. This solution, the outer solution (53), and a composite consisting of the inner and outer solution minus the common part, are shown in Figure 2 (a). In Figure 2 (b) the eolution obtained above by the method of matched asymptotic expansions is found to be almost identical to that obtained by numerical inte- gration [21] of equation (50) for the combined flow in both the corner and relaxation regions. Inclusion of a viscous term has thus made it possible to resolve the corner problem, and the corner equation (56) describes the details of the transition between the upstream region where viscosity and relaxation simultaneously affect the flow and the inviscid relaxation region downstream.

t -701

Figure 2 (a). Outer, Inner, and Composite Solutions for the Corner Problem &, = 6,, or am = 1.0

Figure 2 (b). Comparison of Aanlytical and Numerical Solu- tions for the Corner Problem, ie - or uoD = 1.0

B. Weak Viscous Shock

A partially dispersed wave such that (&/A) < 1 will be considered next. This flow should be similar to the cbrner flow above so that the choice vl = yA and the inner equation (56) should remain valid. The only change is that, since now u(- 00) = 1 + E , the upstream boundary condition becomes,

Go(- CO) = U == E / ~ A N 0(1) . (69)

The theory described below is thus restricted to (&/A) - O(q), while 7 remains of order ( f$’ /YB)l iZ as before. 22*

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324 M. SICHEL / Y. K. YIN: Viscous Tranaonio Mow

The zeroth order inner equation normalized according to (67) now becomes daf df h

-- f - - (f - U) = 0 , dXa dX

where

The trajectories in the p - f or phase plane then satisfy the equation

_- dP f P + ( f - 3) (71) h

df - (P

which reduces to equation (69) for the corner problem as U -+ 0. There is now a saddle point at p = 0, f = a corresponding to the upstream boundary X 4 -00, and the solution of interest must, as before, be one of the solutions passing through this singularity. The singular trajectories through this saddle point have the slope

- dp = (Q2) (1 rt df P=O,

f= U

which approaches & 1 as 6 -+ 0. Hence, the solution of interest corresponds to the positive sign in (72). Phaso plane trajectories are shown in Figure 3 for the particular case U = 1.0 with the arrows in the direction of increasing X and the weak shock solution indicated by the heavy line.

The solution trajectory is once again asymptotic to the line p = -1 as X + 00, indicating that the zeroth order weak shock solution matches with the zeroth order corner solution as before. Now, however, the trajectory has a minimum in p so that the physical solution will have an inflection point in the f - X or to - 53 plane, which is absent in the corner solution. For comparison, the corner solution is also shown in Figure 3. The slope at the inflection point increases with increasing U, as is evident from Figure 4 where qualitative phase plane trajectories are sketched for 6 = 0, 0.5, 1.Oj 1.5.

A

*

i P i

1 inf/ectian 1

1 A- 4

paint

Figure 3. Phase Plane Trajectories with Weak Viscous Shock

P

Figy 4. Phase Plane Trajectories for Different Values of u

-The relaxation region is preceded by a frozen normal shock in the classical or inviscid theory of partially dispersed waves. Hence, it is of interest to compare the structure of a weak normalshock of strength (u, - 1) = E

to the structure of the viscous region described here. In normalized variables the phase plane trajectory of the weak normal shock structure described by equation (40) is the parabola

passing through the points p = 0, f = f U corresponding to upstream and downstream infinity, and

This trajectory is also shown in Figure 3 for U = 1, and the drastic difference between the corner like and frozen shock solutions is evident. The phase plane trajectory of the corner like solution approaches the frozen

(73)

(dpldf) = f ;IT at p = 0 , f = * U . (74)

A p = ( fa/2) - (m)

A

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M. SIUHEL / Y. K. YIN: Viscous Transonic Flow 326

A

shock trajectory near upstream infinity as 6 increases since then the slope of the corner trajectory (dpkdf) -+ 27 at the upstream singularity according to (72). From (72) it follows that the asymptotic behavior of the inner solution as X * --co is

Go = I? - C exp [( 6/2) (1 + vm)) Z(Te/Dl)1/2] . (75) For large 6 the argument of the exponential function in (76) becomes &(Te/D,)112 = 2I''/Dl(Lv/e) so that the viscous decay occurs over a distance of the order of L*/E, i.e. the thickness of a weak shock. On the other hand, for c< 1 the argument in (76) becomes (re/Dl)lI2/qLc so that the decay occurs in a distance of the order of qLc, as in the corner problem. The characteristic length of the viscous region thusis a function of the param- eter 6 and changes from that for the corner problem to the thickness of a weak frozen shock with increasing 6.

The zeroth order outer solution, i.e. the solution of equation (62) bears further discussion. As was noted above, the zeroth order inner solution for U > 0, i.e. the weak shock, can be matched to the corner solution (63) as Z -+ co. Hence the corner solution still appears to be the appropriate outer solution, at least to zeroth order. From the RANKINE-HUOONIOT condition (47) on the other hand uo(+co) = - (1 + x ) , where x = € / A , while the corner solution satisfies uJ+co) = -1. With uo(+co) = - (1 + x ) one integration of equation (62) yields

(76) duo - - (1/2) (4 + uo - x - $1 -- dx UO

and it would appear natural to match the inner and outer solutions at u, = Uob = - x . However, it follows from (76) regardless of how small x is that duo/dx = -1 for uo = -x which does not match the slope of the inner solution (66) as 5 -+ 00. For the corner flow, with x = 0, (76) again yields the appropriate result du,/ds = - - - (1/2) when uo == 0.

These difficulties can be explained by studying the phase plane trajectories of the outer solution, which from (52) eatisfy

where p = duo/dx. The trajectories have a saddle point at u,, = 0, p = - (1/2) and are shown qualitatively in Figure 5(a). The corner solution, U = 0, passes through the saddle point where the slope is discontinuous. As soon as U or x > 0, the solution trajectory no longer passes through the saddle point and this behavior accounts for the sudden change in (duo/dx) noted above when x > 0. Trajectories for fully dispersed and weak fully dispersed waves are also shown in Figure 6 (a). As is well known [7] with U > 0, the inviscid solution reaches an infinite slope and doubles back on itself as is evident from the phase plane.

P I P

! weak futkdspersed

+I fullv dispersed i:, wave

I

I lnvisc/d corner so/uflon

soluflon, Y =o viscous inner solutlon, PO

Figure 5 (a). The Phase Plane for Corner Flow Figure 6 (b). The Viscous Solution in the Phase Plane

In Figure 6(b) the inner solutions and the inviscid corner solutions are sketched qualitatively in the p , uo plane. It is evident that the inner solution replaces the discontinuity in slope of the inviscid solution with a, viscous boundary layer. It can also be seen from Figure 6(b) that only the value of p is matched to zeroth order. More accurate matching requires evaluation of higher order solutions, but this h t ~ not been done here. It also appears appropriate to match the inner solution for U > 0 to the downstream corner solution, for since x = qZJ N O(q), the corner solution does satisfy the downstream RANKINE-HUOONIOT condition to zeroth order.

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326 M, SICHEL / Y. K. YIN: Viscous Transonic Flow

C. Viscous Shock of Arbi t rary S t rength

The theory above remains valid only for very weak upstream shock waves with &/A N 0 ( I ' $ / Y B ) 1 / 2 . When the upstream shock, while still weak, is of arbitrary strength so that E < 1 but x = (&/A) N 0(1) or > 1 it becomes appropriate to choose vl = E. For the outer inviscid region the scaling is again determined by AL = L, so that equation (49) becomes

The small parameter which multiplies the viscous terms is now the ratio of LV/&, the thickness of a weak viscous shock, to L,, the relaxation length. Equation (78) is almost identical to the outer equation (50) of the corner problem.

Within the viscous shock upstream of the relaxation region the viscous terms must be of the same order as the other terms of the equation so that 1L = L,/E, i.e. the characteristic length is the thickness of a weak shock. From (49), the inner equation is then

Equation (79) is quite different from (56), the inner equation of the corner problem. Here v, is the same for both the inner and outer expansions, and all the terms associated with what is sometimes called the "equilibrium operator" [13] are of higher order. This suggests that, unlike the corner solution, the zeroth order inner solution will be independent of relaxation effects, a result which is borne out below. With vl = E the solution must satisfy the boundary conditions

u(1) --f u(1) = + 1; 5 3 -m W

and "y = -1 . In the inner region it is convenient to introduce the variable

and the expansion

It is assumed that 6 < 1, i.e. that the viscous shock thickness is always much smaller than the relaxation length, and this is a physically reasonable assumption as long as x - O(1). The first and second order inner equations are then

with zo= +I, Gl = 0; g - t - 0 0 .

It is necessary here to find both the zeroth and first order solutions, for unlike the corner problem, matching of the zeroth order inner and outer solutions is no longer possible. The solution of (83) satisfying the boundary condit,ions (85) is

ii, = - tan11 f (86) and this solution, which describes the structure of a weak frozen shock wave is identical to that given by equation (40). To zeroth order relaxation, thus, has no effect on the upstream viscous region as was suggested above. Using (83) and the boundary conditions, equation (84) can be integrated twice to give

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M. SICHEL / Y. K. YIN: Viscous Tntnsonio Flow 327

Using the solut,ion (86), this equation can be integrated to yield the first order solution

i& = Ci sech2 f - tanh $ (In cosh $ + F + In 2) - 2rfx

1 _ _ _ ~ t a n h ~ . d ~ + ~ ~ + - 2 4 + Z l n 2 + 1

(1 + tanh &-+ fsech2 g) The constant of integration Ct multiplies the homogeneous solution of (87). Since relaxation effects must vanish as d 3 0 or x 3 co, i.e., as the inhomogeneous part of (87) vanishes, Ct must be O(l/x) or less.

From matching it follows that the appropriate expansion for the outer equation is u(1) = u 0 + 6% + *.. (89)

and it is convenient to use

The zeroth and first order outer equations then become

(92)

with

21 0 - - - 1 ; 5 = 5 0 .

Here 5, is the location of the inner viscous shock, and its value must be found from matching. The solution of the zeroth order equation satisfying the above boundary conditions is:

Near 5' = to, i.e. for (5 - 6,) < 1, it is readily shown that

2- On the other hand, as the inner variable 3 approaches infinity Go -+ - 1 + 2e- from which it is clear that U, and Go will not match. The zeroth order solution (93) for u, can be simplified when E > A ao that x > 1, and then

The first order equation (92) can be integrated once using (91) and the boundary conditions with the result :

The solution of this linear first order equation is

where C, is a constant integration and /I = F$,/I'?. This solution can be simplified further by using (95) when x > 1, with the result

wherea = 1 + (l/x).

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328 M. SIUEIEL / Y. K. Ym: Viscous !Ikansonic Flow

For purposes of matching an intermediate variable 5, = 5/v(8) = 8[/v(6) will be introduced with v such -+ co as 8 --f 0 so that the inner that v ( S ) -+ 0 ; v(S)/8 --t co as S 3 0, with tv fixed. In the intermediate region

solution can be written .ii, = - 1 + 2 e-2r+ 6. . tw

In the intermediate region 5 --f 0 as B 3 0 and the zeroth order outer solution in the case x >> 1 becomes 1 e@o

u, = - 1 - -+- (1 - pvtv + . . a ) .

x x

Comparing (100) and (101) it can be seen that for matching, 6, = 0. The first order outer solution is

The exponential terms in the inner solution can be dropped as can be matched term by term. Matching requires that

+ ca and then the inner and outer solutions

'co="rf ( 1-- ye) -- rf $+ ___ 1) 4 x + 1 '

As discussed above, the inner and outer expansions (82) and (89) mean that, overall,'a double expansion is being used. Unlike the previous cases it was, here, necessary to use the first order viscous solutions in order to obtain matching so that the condition vI > y1dv > yZ must be satisfied. Thus, with vl = E, yZ = es, and & = = Lv/rJL&, the first order viscous theory developed here will remain valid provided that E > (Lv/I"'Lo) >> ea,

TI. Discussion Equations have been derived for the viscous transonic flow of a relaxing gas valid as the upstream velocity ranges from the equilibrium sonic speed to some value above the frozen speed of sound. The simplest possible model gas has been chosen, i.e. a perfect gas with one internal energy mode. However, the one-dimensional analysis indicates that this model represents the main features of the flow.

Three main regimes of flow, each requiring a different formulation can be distinguished. I n the near equilibrium flow region viscous effects are of higher order and so can essentially be neglected. When the velocity lies very close to the equilibrium speed of sound so that (u - a) /@ - a)< 1, the flow is analogous to the viscous transonic flow of an inert gas; however, the compressive or longitudinal viscosity is replaced by the bulk viscosity. This analogy, which has also been mentioned by others [14], [17], suggests the use of a relaxing gas flow for the study of the structure of weak curved shock waves. The simple replacement of the compressive viscosity by the bulk viscosity is no longer valid when (u - a)/@ - a) becomes O(1).

In the near frozen flow regime where &/A - O(1) the one-dimensional calculations indicate that there will be a viscous shock layer upstream of the region where most of the relaxation takes place. To zero order this shock layer is completely independent of relaxation, a result which reinforces the use of a frozen shock dis- continuity followed by a relasation zone to approximate a partially dispersed wave [ll]. Higher order viscous effects must be taken into account to represent the details of the transition from the viscous shock layer to the relaxation region.

The regime of flow with upstream velocities very near the frozen speed of sound is best denoted as the corner regime. Even to zeroth order, both relaxation and viscosity are then important upstream of the main relaxation zone. In this paper the flow which arises when the upstream velocity is equal to or slightly above the frozen speed of sound has been studied in detail in the one dimensional case. However, the transition from the corner flow to the equilibrium flow as the upstream velocity drops below the frozen speed of sound remains to be investigated. It is of special interest to note how the corner flow gradually tends toward a frozen shock flow as the upstream velocity increases beyond the frozen speed of sound.

The detailed structure of plane waves in the three regimes of flow above could also have been computed by integrating the appropriate ordinary differential equation numerically, rather than using matched asympto- tic expansion. The detailed picture of the physical processes which take place in different parts of the flow is then, however, lost. In a few cases numerical calculations have been made [21] for purposes of comparison.

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M. S I U ~ / Y. K. Ym: Viscous Transonia Flow 329

The equations for viscous transonic relaxing flow were derived by reducing the conservation and relaxa- tion equations to a single equation for the velocity, and the anlysis described above was concerned with the solutioq of these equations. Other parameters of the flow can, of course, also be determined. The temperature is readily found from the relation (28a) between temperature and velocity. The internal variable q can be found from (16) once the velocity distribution is known.

Acknowledgment Part of the work reported here was supported by the Army Research Office under Contract DAKC0468C0008. The senior author is grateful to Professor E. BECKER and his co-workers for their many valuable comments and suggestions while the author was on sabbatical at the Technical University of. Darmstadt, and to the helpful suggestions of Professor T. C. ADAMSON, Jr., of The University of Michigan.

References 1 BECKER, E., “Chemically Reacting Flows,” Annual Reviev of Fluid Mechanics, 4, pp. 155-194 (1972). 2 BECKER, E. and BOHME, G., Gasdynamics, a Series of Monographs, ed. P. P. WEOENER, Vol. I, Non-Equilibrium Flows,

3 BLYTHE, P. A., “Non-Linear Wave Propagation in a Relaxing Gas’’ J. Fluid Mech. 37, Pt. 1, pp. 31-50 (1969). 4 BROER, L. J. F., “On the Influence of Acoustic‘plaxation on Compressible Flow,” Appl. Sci. Res, A2, pp. 447-468 (1950). 5 BROER, L. J. F. and VAN DEN BERGEN, A. C., On the Theory of Shock Structure 11,” Appl. Sci. Res., A4, pp. 157-170

6 BuaQIsCH, H., “The Steady Two-Dimensional Reflexion of an Oblique Partly Dispersed Shock Wave from a Plane Wall,”

7 CLARKE, J. F. and MCCHESNEY, M., The Dynamics of Real Gases, Butterworths; London (1964). 8 CHU, B. T., “Wave Propagation in a Reacting Mixture,” Heat Transfer and Fluid Mechanics Institute, Stanford, Cali-

9 COLE, J. D., Perturbation Methods in AppIied Mathematics, Blaisdell, Toronto and London (1968). 10 GRIBEITH, W. C., and KENNY, A., “On Fully Dispersed Shock Waves in Carbon Dioxide,” J. Fluid Mech. 3, pp. 286-288

(1957). 11 L I Q E ~ , M. J., “Viscosity Effects in Sound Waves of Finite Amplitude,” Surveys in Mechanics (eds. BATCHELOR and

DAVIS), Cambridge University Press (1965). 12 MOORE, F. K. and GIBSON, W. E., “Propagation of Weak Disturbances in a Gas Subject to Relaxation Effects,” J. Aero.

Sci., 87, 117-127 (1960). 13 NAPOLITANO, L. G., “Transonic Approximation for Reacting Mixtures,” Israel J. of Tech., 4, pp. 159-171 (1966). 14 NAPOLITANO, L. G. and RYZHOV, 0. S., “The Analogy between Non-Equilibrium and Viscous Inert Flows a t Near Sonic

Velocities,” USSR Comput. Math. and Math. Phys., 11, No. 5, Pergamon Press (1971). 15 OCKENDON, H. and SPENCE, D. A., “Non-linear Wave Propagation in a Relaxing Gas,” J. Fluid Mech., 39, pp. 329-345

(1969). 16 PXUD’HOWE, R., “Study of a Transonic Flow with Chemical Reactions by the Small Perturbation Method - One and

Two Dimensional Problem,” ONERA-TP-749 (1969). 17 RHYZOV, 0. S., “Non-Linear Acoustics of Chemically Active Media,” Prikl. Mat. and Mech., 36, pp. 1023-1037 (1971). 18 STREEU~W, R. A. and MAXWELL, K. R., “Two-Dimensional Diffuse Shock Waves in Carbon Dioxide,” Amer. Inst. of

Aeronaut. and Astronaut. J., 6, p. 2431 (1968). 19 SCALA, S. M. and TALBOT, L., “Shock Wave Structure with Rotational and Vibrational Relaxation,” Rarefied Gas Dyna-

mics, Proc. of the Third Intl. Sympoisum on Rarefied Gas Dynamics (J. A. LAURMAN, ed.), Academic Press, New York (1963).

20 VINCENTI, W. G., “Non-Equilibrium Flow over a Wavy Wall,” J. Fhid Mech., 6, pp. 481 -496 (1959). 21 YIN, Y. K., “Viscous Transonic Flow in Relaxing Gases,” Ph. D. Thesis, The University of Michigan, Department of

Part I, Marcel Dekker, New York, London (1969).

(1954).

J. Fluid. Mech., 61, Pt. 1, pp. 159-172 (1973).

fornia (1958).

Aerospace Engineering, Ann Arbor, Michigan (1971).

Eingereicht am 8. 11. 1974, revidierte Fassung 23.5.1975

Anschrift: Prof. Dr. MARTIN SICHEL, University of Michigan, College of Engineering, Ann Arbor 48 109, Michigan, USA