OPERATION OF LAVAL NOZZLES IN UNDESIGNED MODES*

0. S. RYZHOV

Oxford - ~~oscow

(Received 24 December 1966)

WHEN designing the gas flow conditions for Lava1 nozzles it is usual1.y stipulated that the velocity should change from subsonic at the input to supersonic at the output. The transonic flow and passage through the critical velocity occur close to the minimal channel cross-section. If the pressure-difference between the two ends does not reach the design figure, however, the nozzle operates in modes in which the velocity field is primarily subsonic, though supersonic regions may occur. The latter are either adjacent to the channel walls, while the flow remains purely subsonic close to the axis of symmetry; or else they occupy all the central part of the pipe, in which case their forward boundary is a sonic surface, and their rear boundary a density jump, from which a second sur- face can depart with the critical velocity. The first mode, with local supersonic zones at the channel walls, was first studied by Taylor, then later by many other authors; a detailed history of the topic will be found in [XI, where the relevant solutions of the differential equations of gas dynamics are given. The aim of the present paper is to examine the second type of undesigned operating mode, when the supersonic velo- city region in the central part includes a density jump cutting the axis of symmetry at right angles.

The gas particle velocity will be assumed nowhere to exceed sub- stantially the velocity of sound. The shock wave intensity is then negligible and the field of flow can be assumed irrotational to a first approximation. To investigate the flow close to the aperture of the nozzle, we use Karman’ s equation (see [21)

a2(T de2

1 - r

h-J - ar

0, (1)

* Zh. vychisl. Mat. mat. Fiz. 7. 4, 859 - 866. 1967.

187

188 03. Ryzhou

which holds in the transonic region; x, r, 8 denote the axes in a di- mensionless cylindrical coordinate system, and g is the disturbance potential. The main uniform flow is assumed to have the critical velo- city and to be directed along the pipe axis x, which is the line of intersection of its two mutually perpendicular planes of symmetry.

The solution

cp = ‘/z/W + As(*/& - I cos 28)sS + (2) + A3 (‘h - ili~l cos 20 + m cos 40) fl

of Karman’s equation with arbitrary constants A, 1, m, describes a flow with finite acoeleration at the point of intersection of the nozzle axis with the sonic surface (see [ll). A is numerically equal to the gas acceleration at the point just mentioned in the dimensionless system of units, so that A > 0. The solution (2) gives the velocity field in the input part of the nozzle, which is defined by 8cp /&r < 0 and stretches upwards relative to the flow from the sonic surface

- 2 = A (i/l - I co9 28) 9.

In addition, (2) describes the supersonic region with &p/&r > 0, stretching downwards relative to the flow from this surface and includ- ing a density jump at its rear.

To find the flow in the exhaust part of the nozzle behind the front of the shock wave, it is best to differentiate (1) as a preliminary with respect to n, and transform from the potential Q to the longitudinal component vr=&p /ax of the disturbed velocity vector. We have

The solution of (3) is

ux=4 < 2(l+a2+20cos28)P-4$f(E), ( > (4)

where the constants a, c, and d can take arbitrary values, while I’(<) satisfies the 1st order ordinary differential equation

f ;= @+f (b2= 1 +.uq. (5)

Lava1 nozzles in undesigned modes 189

It will be assumed in future, unless stipulated otherwise, that c > 0 and cl > 0. The integral (4) was used to obtain Taylor flows in plane nozzles by Tomotika and Tamada, and in nozzles of circular cross-section by Tomot ika and Khazimoto; its application to the study of general gas motions in local three-dimensional supersonic zones adjacent to the channel walls can be found in [Il.

It is only when they satisfy the Hugoniot conditions, simplified under the usual assumptions of transonic flow theory, that solutions (2) and (4) jointly describe the undesigned nozzle mode ‘with a density jur~p. Approximate relations for the shock front were derived by Busemann (see [21). They are best considered successively, starting with the relation- ship that must be satisfied by the longitudinal component uX of the dis- turbed velocity vector. Writing the equation of the front as n=xz(r,4), we have

(6)

where the subscript 1 of the function vZ refers to its values on the side of the discontinuity surface facing the incoming flow, while the subscript 2 refers to values on the other side of the surface.

The expression for uz in the region upwards into the flow from the density jump is found by differentiating (2)

vz = Az + D( i/l - I cos 28) fl (7)

It can be assumed in this approximation that the shock front is a second order surface < = c2 = const., i.e.

(8)

Substitution of (4), (7) and (8) in (6) gives two relations connecting the constants A and 1 with a, c, ci, and an initial condition for inte- grating (5).

In the region behind the density jump the function uZ is more com- plicated; the relationships for the constants are therefore solved for 4 and 1:

d5-Iu+4w A=-+-1(1+~~~2))~ z=-4(3+2a2_-)/(1+4b2)) . (9)

The construction of the flow thus starts from the exhaust part, and the

190 0-S. Ryzhov

acceleration A at the point of intersection of the axis of this part, with the sonic surface is determined by the constant u and the ratio c/a. A similar remark holds for the coefficient 1. The initial data for the function f (5) also include the coordinate Ej2 of the shock front

f = fi = ‘/*A&/d. (10)

Further boundary conditions on the shock front include continuity of the tangential components of the velocity vector on passing through this surface. Following Busemann (see [21), they can be simplified for the range of transonic velocities:

iiere, ur and v8 denote the r- and e-components of the disturbed velocity vector, connected with the potential 9 by v,. = drp/dr and va = rid~~/de. From (2), we have for the incoming flow

vT = A2(l/2 - 21 cos 28) xr + A3 (l/i6 - i/s cos 20 + 4m cos 48) 13, (12) ve = 2A211t.r sin 28 + A3(i/~l sin 28 - 4n sin 40)?.

The functions vr and ve corresponding to the solution (4) are found by using the conditions for the flow to be irrotational

dv, au, 1 au, ave au, dr ve -c- dr dx '

----_- rat3 ax' Tc=TiY-

and the fundamental equation of mot ion (1). The result can be (see [II):

easily

written

vT= 8 0 f Ztr(b2+2ac0s2~)+8~f(f)r(1+acos2~)-

- 4fl [(>

b2 d 3 c ( t)3cos

+ b/3 ab2 20 + i/t, h cos 40 I

,

ve = -16 0 t 2axrsin26-8~3af(~)rsin26+ (13)

+ + [ 8J3 ab2 (t)’ sin 20 + h sin 401.

These relationships include an extra arbitrary constant h.

It now remains to substitute (4), (7), (8)) (12) and (13) in (ll),

Lava1 now les in undes irned nodes 191

which leads to seven new conditions connecting the constants A, 1, m and a, c, ti, n. The interesting point is that five of them can be trans- formed to the form (9), in other words, they impose no extra restrictions on the equations derived earlier. The remaining two conditions are identical and define the coefficient m in terms of the constants U, C, ;I, h:

These assertions can easily be verified directly.

The derivation of (14) completes the construction of the flow both in front of and behind the density jump. Consider further (IO), snecify- ing the initial data for.f(<). From the first of (9), it can be re- written as

f2 = 42(1 -I’(1 +4b2))~2 > 0,

since (see [zI), if 4, c, and Ld are positive, <z must also be positive. Compare this last expression with

f=w1+71(1+4mE (E>O), (16)

which gives (see [II) the velocity field behind the critical section of the nozzle in a Taylor flow with supersonic zones merging on the channel axis. If c2 = 0, then fz = 0, and we get (16) as the solution of (5). The flow expressed by (16) thus possesses dual limiting properties: on the one hand, it can be obtained from Taylor flows with local supersonic zones originating and developing at the channel walls, and on the other, it is the limit of the undesigned type of flows with density jumps. In the limiting flow itself, the velocity field is continuous, but contains the characteristic surface (see [ll)

which carries discontinuities of the first derivatives of the velocity components with respect to the coordinates. From (ES), (16), the dis- turbances of the gas parameters behind the density jump are all less

than in the limiting continuous flow, though at a great distance behind

the throat, the flow through the shock front is almost shockless. The same conclusions can be drawn from the elementary one-dimensional t!leory.

The flows in plane nozzles are obtained by putting a = k 1 in (4) and later formulae; flows in nozzles of circular cross-section correspond

192 O.S. Ryzhov

to a = 0. From (3)) the surface of the shock wave front is concave in either case towards the exhaust side, and not towards the incoming flow, as is usually assumed. This is likewise true if 1~~1 < 1; the variations of the coefficient 1 are only restricted by the common inequality I II <5/M. Th e sonic surface is an elliptic paraboloid, extending up- wards into the flow, with la I < l/3, Ia ( > 1 and / 1 I < l/4. In the range

l/3 < Id I < I. and l/4 < I 11 < 5/16, the sonic surface is a hyperbolic paraboloid.

It can actually be assumed in the present approximation that we are given the flow at the nozzle input and the pressure (or velocity) at a point x = x3 on the axis in the exhaust part; when specifying these, the corresponding values of 1‘3 = f(<3) should not be too different from (16). ‘Using (9), (1.4) and (15) and integrating (5), the velocity field can be continued into the region behind the density jump. (Integral curves in the <f plane will be found in [ll). An important point is that all the geometric flow characteristics are determined by the coefficients a, c, U, and A, the expressions for which in terms of the constants A, 1. and m do not contain the value c2 of the shock wave front coordinate. In view of (8) and (17), this means that the shape of the density jump re- mains unchanged when the pressure difference between the input and out- put increases; the surface of the shock front merely moves gradually in the direction of the exhaust part. It is only necessary to specify f3 = f(53) if we want to find the exact position of the shock front and the velocity vector of the particles behind it. When the second of expressions (9) is transformed, 1 is not uniquely expressed in terms of a.

It may appear at first sight that there is a good deal of arbitrari- ness in constructing the flow in the exhaust part, since it is only the ratio c/d, and c and d individually, that is expressed in terms of A by the first equation of (9). But the value of c is not in fact important, since it merely affects the scale along the n axis in the region behind the density jump. By a suitable choice of c, we can arrange, for instance, that t3 = 1 whatever the value of x3. The inequality that the function

vx3 = vx (x3, 0, 6) has to satisfy and which ensures that the flow passes through the critical velocity in the neighbourhood of the orifice in the channel, should likewise be independent of c. For,

whence, from (161,

vr9 ~ 3 + 2s + -01 +*4bz) 2b2

As 3.

Lava1 nozzles in undesigned modes 193

From the formal point of view, a solution other than the above can be considered, namely, the integral given by (4) when the constant c remains positive, while d is negative. The following relationships are now obtained for the constants A and 1 from the boundary conditions of the shock front, in place of (9):

A= a(5+‘)/(1 +4b2n I=- 4(3+2u2+,‘)‘(1+4b2))

( (18)

while (10) and (14) for fz = f((z) and XI remain unchanged. It can be seen from (g) and (17) that, in these flows, the density jump is concave towards the nozzle input, provided \CX I< 1 and / 11 < l/4. In particular, the density jump is concave in this direction in plane-parallel and rotationally symmetric flows. The shock waves can thus extend in the same flow both in the direction of motion of the gas particles, and in the opposite direction. As regards the shock waves travelling upwards into the flow, it seems impossible to use them to construct the velocity field in undesigned Lava1 nozzle modes. We obtain in this case, from (10) and (18):

f2 = -‘h(f +w +4wE2 < 0,

while the equation for the throat, with local becomes

the structure of the limiting Taylor flow behind supersonic zones merging on the axis of symmetry,

Comparison of these last two expressions shows that, behind the density jump, the gas parameters suffer more disturbance than their values in the limiting continuous flow. ‘Ihis is in contradiction to the one- dimensional theory. If it were true, there would be two distinct operat- ing modes corresponding to the same flow at the input and pressure (or velocity) in the terminal section x = x3: one with local supersonic zones at the channel walls, the other with a density jump passing ortho- gonally through the axis of symmetry. The gas motions behind the criti- cal section would be depicted in either mode by the same integral curves in the <f plane. The solution (4) with ii < 0 might possibly be used to analyse the velocity field when a stream subsonic at infinity passes through a profile and there are local supersonic zones.

If a density jump originates in the orifice of the nozzle, in which csse its intensity is negligible, the particle velocity behind it will only be subsonic close to the axis of symmetry. For, behixfd the shock front,

194 0.S. Ryzhov

If r = 0 we must have ux2 < 0, but after a certain distance, vx2 > 0 whatever the value of a. Figure 1 shows schematically the velocity field

FIG. 1. FIG. 2.

in plane-parallel flows, corresponding to positive values of the con- stant ci; Fig. 2 shows the velocity field for negative U. The shock wave is denoted by 7, while CT_O and C+O denote the characteristics touching the sonic surface. For d < 0 and la] < 1, the particle velocity behind the shock front increases at first on moving away from its surface along a streamline, and only gradually starts to fall. This assertion can be derived from (51, according to which u~/u< = 0, if

For /Q/ < 1 the straight line (20) travels above the line (19) in the half-plane 5 > 9, whence ~f/u< > 0 at points 4 = gz. For U < 0 and a = f 1, these lines coincide. i.e. in such plane-parallel flows the gas acceleration vanishes behind the shock front. As the oressure difference between the ends of the nozzle increases, the density jump moves Sac!<- wards, and the sonic surfaces, shown in Figs. I and 2 between the shock front and the walls, disappear.

Finally, it is worth comparing the formation of shock waves when the nozzle operates in undesigned modes and the gas acceleration remains finite at any point of the axis, with their formation in special nozzles with fairly shallow transition sections and zero acceleration at the centre (such nozzles were examined in detail in [ll). A typical feature in both processes is that, in the limiting case, a flow is obtained which separates shockless gas motions from motions with density jumps in which a characteristic surface is formed, carrying discontinuities of

Lava1 nozz les in undesigned modes 195

the first derivatives of the velocity components with respect to the coordinates. Though the method of passage to the limit is quite differ- ent in the two types of process, the shock front generation is always preceded by the occurrence of weak discontinuities along the character- istic surface. This feature supports the Frankl’ hypothesis (see [31) of the existence of stable (in a sense) stationary transonic flows with

j umps, close to unstable continuous flows.

Translated by D.E. &own

REFERENCES

1. RYZHOV, O.S. Study of transonic flows in Lava1 nozzles (Issledovanie

transzvukovykh techenii v soplakh Lavalya), Moscow, VTs AN SSSR,

1965.

2. GUDERLEY , K. G. Theory of transonic flow, Pergamon Press, 1962.

3. FRANKL’ , F.I. On the existence of “weakly” stable stationary tran-

sonic flows with jumps, close to unstable continuous flows, Usp.

mat. Nauk 15, 6, 163 - 168, 1960.