HYDROMECHANICS

L. E. STERNIN

ON THE BOUNDARY OF THE DOMAIN OF EXISTENCE OF SHOCK-FREE OPTIMAL NOZZLES

(Presented by Academician V. P. Glushko, November 9, 1960)

The solution of the variational problem on the best shape of the supersonic part of an axisymmetric jet nozzle is given in papers (¹–⁴). In constructing the solution, a prescribed bundle of characteristics arising in the flow about a blunt corner \(A\) is assumed (Fig. 1). The best contour \(AB\) is obtained as the result of solving Goursat’s problem between the characteristic \(AC\) and the extremal characteristic \(CB\).

Fig. 1. Layout of characteristics in an extremal nozzle

Let us consider the equations on the extremal \(CB\), given in paper (⁴):

\[ m_1 \cos \alpha + 2\pi w \cos(\alpha-\theta)=0, \tag{1} \]

\[ m_2 + 2\pi y \rho w^2 \tan \alpha \sin^2 \theta =0, \tag{2} \]

where \(\alpha\) is the angle between the velocity and the characteristic; \(\theta\) is the angle of inclination of the velocity to the \(x\)-axis; \(w\) is the velocity; \(\rho\) is the density; \(m_1\) and \(m_2\) are constant Lagrange multipliers determined from the results of calculating the bundle of characteristics and from formulas (1) and (2).

From equations (1) and (2) we obtain that on the extremal \(CB\)

\[ y=-\frac{4\pi m_2}{\rho \sin 2\alpha} \left( \frac{m_1 \sin \alpha \pm \sqrt{4\pi^2 w^2 - m_1^2 \cos^2 \alpha}} {m_1^2 - 4\pi^2 w^2} \right)^2, \tag{3} \]

where the plus sign is taken for \(\alpha(C)\leqslant \theta(C)\), and the minus sign for \(\alpha(C)\geqslant \theta(C)\), owing to the fact that at the point \(C\) equation (3) must be satisfied automatically.

As is easy to see, in the case \(\alpha(C) > \theta(C)\), along the extremal, as \(y\) increases, \(\alpha\) and \(\theta\) decrease; in the case \(\alpha(C) < \theta(C)\), along the extremal \(\alpha\) increases and \(\theta\) decreases. At some point of the extremal \(\alpha = \theta\). For the further calculation of the extremal it is necessary to change the sign before the radical in formula (3). We note in passing that, as a result of this, the velocity distribution along the extremal characteristic becomes nonmonotone.

It follows from equation (3) that when the point \(C\) is moved along the characteristic \(ACP\) toward smaller \(x\) (in the region \(\alpha < \theta\)), the derivative in the direction of the extremal \(\left.\dfrac{dy}{d\alpha}\right|_{C}\) decreases. The value of this derivative passes, at some \(C_0\), through zero and then becomes negative.

Geometrically, this phenomenon is interpreted as a loop at the base of the extremal. Physically it means the impossibility of constructing shock-free solutions of the variational problem for all points of the characteristic \(ACP\) (lying to the left of the point \(C_0\)) which could serve as initial points for constructing extremals.

If, however, along the extremal, as \(y\) increases, \(\alpha\) is decreased, then, as calculations show, we arrive at nozzles that do not possess the greatest thrust.

Equating to zero the derivative of the right-hand side of equation (3), using conditions (1) and (2), as well as the known formulas expressing \(\rho\) and \(w\) in terms of \(\alpha\), we obtain, for a gas flow with constant adiabatic exponent \(\varkappa\), the following extremely simple expression relating \(\alpha\) and \(\theta\) on the line \(AC_0\), to the left of which no shock-free solutions of the variational problem exist:

\[ \varkappa \sin\theta \sin(\alpha+\theta) - \cos\alpha \sin^{2}2\alpha + \sin\theta \sin(\theta-\alpha) - \sin\alpha \sin^{2}\theta \cos2\alpha = 0. \tag{4} \]

The author takes this opportunity to express deep gratitude to Yu. D. Shmyglevskii, who rendered great assistance in carrying out the work.

REFERENCES

1. G. Guderley, E. Hantzsch, Mechanics, 53, No. 4 (38) (1956).
2. Yu. D. Shmyglevskii, Applied Mathematics and Mechanics, 21, 195 (1957).
3. G. V. R. Rao, Jet Propulsion, No. 6, 377 (1958).
4. L. E. Sternin, Izv. AN SSSR, Mechanics and Machine Building, 41, No. 1 (1959).