HYDROMECHANICS

Yu. D. SHMYGLEVSKII

ON SOME PROPERTIES OF AXISYMMETRIC SUPERSONIC GAS FLOWS

(Presented by Academician A. A. Dorodnitsyn, 3 VI 1958)

Let us consider supersonic axisymmetric gas flows determined by a prescribed characteristic of the first family \(AC\) and by the generator \(AB\) of the body of revolution being streamlined (Fig. 1). Let the gas move from point \(A\) to point \(B\). Such gas flows obey the canonical system of equations (1)

\[ r_\eta=\tg(\vartheta+\alpha)x_\eta,\qquad r_\xi=\tg(\vartheta-\alpha)x_\xi, \]

\[ \vartheta_\eta-f_\eta+ \frac{\sin\vartheta\sin\alpha}{\sin(\vartheta+\alpha)} \frac{r_\eta}{r} -\sin2\alpha\cdot\varphi_\eta=0, \tag{1} \]

\[ \vartheta_\xi+f_\xi- \frac{\sin\vartheta\sin\alpha}{\sin(\vartheta-\alpha)} \frac{r_\xi}{r} +\sin2\alpha\cdot\varphi_\xi=0, \]

where \(\xi,\eta\) are characteristic variables, constant respectively along characteristics of the first and second families; \(x,r\) are Cartesian coordinates in the meridional plane of the flow; \(\alpha\) is the Mach angle; \(\vartheta\) is the angle of inclination of the velocity to the axis of the flow; \(f\) is a function of \(\alpha\), with

\[ df=-\frac{1+\cos2\alpha}{\varkappa-\cos2\alpha}\,d\alpha; \]

\(\varkappa\) is the adiabatic exponent;

\[ \varphi=\frac{\ln p-\varkappa\ln\rho}{2\varkappa(\varkappa-1)} \]

is the entropy function, depending only on the stream function \(\psi\); \(p\) is the gas pressure; \(\rho\) is the density. The stream function is defined by the relation

\[ d\psi=\frac{\omega}{\sin\alpha}\,(\cos\vartheta\,dr-\sin\vartheta\,dx), \]

where

\[ \omega= \sqrt{\frac{\varkappa+1}{2}}\,\rho_0 r \left( \frac{1-\cos2\alpha}{\varkappa-\cos2\alpha} \right)^{\frac12\frac{\varkappa+1}{\varkappa-1}}; \]

\(\rho_0\) is the density of the adiabatically decelerated gas.

Fig. 1

Let us compute the derivative \(\bar f=df/ds\) along an arbitrary characteristic of the second family at its point of intersection with \(AC\), where \(s\) is the distance along the line \(\eta=\mathrm{const}\). Let \(s\) increase when moving downstream. The symbol \(d/ds\) will have the same meaning in what follows. We have

\[ \bar f=\frac{f_\xi}{r_\xi}\sin(\vartheta-\alpha). \]

Find the partial derivative of \(\bar f\) with respect to \(\eta\):

\[ \bar f_\eta= \frac{f_{\xi\eta}}{r_\xi}\sin(\vartheta-\alpha) -\frac{f_\xi r_{\xi\eta}}{r_\xi^2}\sin(\vartheta-\alpha) +\frac{f_\xi}{r_\xi}\cos(\vartheta-\alpha)\cdot(\vartheta_\eta-\alpha_\eta). \tag{2} \]

The expressions for \(f_{\xi\eta}\) and \(r_{\xi\eta}\) entering the last equality are found from the system of equations (1). Substituting these expressions into (2) and choosing \(\eta\) so that \(\eta=\psi\) on the characteristic \(AC\), we obtain

\[ \bar f_\eta=-\Phi_1\bar f^{\,2}-\Phi_2\bar f-\Phi_3, \tag{3} \]

where

\[ \Phi_1=\frac{\chi+1}{\omega\sin 2\alpha(1+\cos 2\alpha)}, \]

\[ \Phi_2=\frac{\chi+1}{1+\cos 2\alpha}f_\eta-\left(2+\chi-2\cos 2\alpha\right)\frac{d\varphi}{d\psi} -\frac{3\sin\vartheta\cos\alpha-\cos\vartheta\sin\alpha}{2\omega r}, \]

\[ \Phi_3=\frac{\sin^2\vartheta\sin 2\alpha}{\omega r^2} -\frac{\sin(\vartheta+\alpha)}{2r}f_\eta +\omega\sin 2\alpha\left[2\sin^2\alpha\left(\frac{d\varphi}{d\psi}\right)^2-\frac{d^2\varphi}{d\psi^2}\right] \]

\[ -\left[\frac{\sin(\vartheta+\alpha)\sin 2\alpha}{r} +\omega(\chi+1+\cos 2\alpha)f_\eta\right]\frac{d\varphi}{d\psi}. \]

Let us denote the derivative \(ds/df\) along the line \(\eta=\mathrm{const}\) by \(\gamma\). Taking into account that \(\bar f=1/\gamma\), from (3) we find the equation for \(\gamma\)

\[ \frac{d\gamma}{d\eta}=\Phi_1+\Phi_2\gamma+\Phi_3\gamma^2, \tag{4} \]

where \(d/d\eta\) denotes the total derivative along a characteristic of the first family.

We establish the boundary condition for the function \(\gamma\) at the point \(A\). Let the equation of the generator \(AB\) be \(r=F(x)\) (the angle of the tangent to \(AB\) coincides with the angle \(\vartheta\)).

From the equalities for the total derivatives along \(AB\) (defining \(\xi\) so that \(\xi=x\) on \(AB\)) we obtain at the point \(A\)

\[ x_\xi+x_\eta\frac{d\eta}{d\xi}=1,\qquad r_\xi+r_\eta\frac{d\eta}{d\xi}=F',\qquad \vartheta_\xi+\vartheta_\eta\frac{d\eta}{d\xi}=F''\cos^2\vartheta . \]

Adjoining equations (1) to the last equalities and carrying out the calculations, we obtain the boundary condition for \(\eta=\psi_A\)

\[ \gamma=S\mathfrak{R}, \tag{5} \]

where

\[ S=\frac{1}{\mathfrak{R}\omega\left(f_\eta+2\sin 2\alpha\,d\varphi/d\psi\right)-2\cos\alpha}, \]

\(\mathfrak{R}\) is the radius of curvature of the generator \(AB\) at the point \(A\) when approaching the point \(A\) from the side of the point \(B\).

We now determine the character of the dependence of \(\gamma\) on \(\mathfrak{R}\). To this end, differentiate equation (4) and the boundary condition (5) with respect to \(\mathfrak{R}\), taking into account that in the equalities (4) and (5) only the quantity \(\gamma\) depends on \(\mathfrak{R}\). We obtain that \(\delta\gamma\) is determined by the differential equation

\[ \frac{d\,\delta\gamma}{d\eta}=\left(\Phi_2+2\Phi_3\gamma\right)\delta\gamma \]

and by the boundary condition for \(\eta=\psi_A\)

\[ \delta\gamma=-2S^2\cos\alpha\cdot\delta\mathfrak{R}. \]

Integration gives

\[ \delta\gamma=-\left[2S_A^2\cos\alpha_A \exp\int_{\eta=\psi_A}^{\eta}\left(\Phi_2+2\Phi_3\gamma\right)d\eta\right]\delta\mathfrak{R}. \]

In what follows we shall assume that the functions \(f_\eta\), \(f_\eta d\varphi/d\psi\), \((d\varphi/d\psi)^2\), \(d^2\varphi/d\psi^2\) are integrable on \(AC\).

For \(r>0,\ 0<\alpha<\pi/2\) we can conclude that \(\delta\gamma<0\) for \(\delta\mathfrak{R}>0\), or that \(\delta\alpha/ds<0\) for \(\delta\mathfrak{R}>0\). The last equation of system (1) shows that,

that is, \(\delta\, d\vartheta/ds < 0\) for \(\delta \mathfrak{R} > 0\). Thus, the following property holds.

Property 1. In axisymmetric supersonic flow past a body of revolution, an increase in the radius of curvature \(\mathfrak{R}\) of the generatrix \(AB\) of this body at the point \(A\) leads to a decrease in the derivatives \(d\alpha/ds\) and \(d\vartheta/ds\) on the specified characteristic \(AC\) (with signs taken into account).

It follows from Property 1 that, when the curvature of \(AB\) at the point \(A\) is equal to \(-\infty\), the derivatives \(d\alpha/ds\) and \(d\vartheta/ds\) have minimum values.

Let, further, the characteristic \(AC\) intersect the shock wave \(DE\) at the point \(C\), while ahead of the shock wave there is a uniform gas flow with velocity equal to \(V\) (the velocity is referred to the critical flow speed). The quantities \(\alpha, \vartheta, p, \rho\) behind the shock wave, for a fixed value of \(V\), are determined from the known relations
\[ \alpha = A(\sigma), \quad \vartheta = \Theta(\sigma), \quad p = P(\sigma), \quad \rho = R(\sigma), \]
where \(\sigma\) is the angle of inclination of the shock wave to the flow axis.

The derivative \(d\vartheta/ds\), taken along the characteristic \(CG\) at the point \(C\), is equal to
\[ \frac{d\vartheta}{ds} = \frac{\sin \vartheta \sin^2 \alpha \cos(\sigma-\vartheta)} {r \sin(\vartheta+\alpha-\sigma)} + \frac{1}{\mathfrak{R}_b} \frac{\sin 2\alpha}{\sin(\vartheta+\alpha-\sigma)} \left[ \frac{1}{2}\frac{d\Theta}{d\sigma} + \frac{2(\varkappa-\cos 2\alpha)}{(\varkappa+1)^2} \left( 1-\frac{\varkappa-1}{\varkappa+1}V^2\cos^2\sigma \right) \operatorname{ctg}\sigma\,\operatorname{ctg}\alpha \right], \tag{6} \]
where \(\mathfrak{R}_b\) is the radius of curvature of the shock-wave line \(CE\) at the point \(C\).

If the velocity of the gas that has passed through a shock wave of finite intensity remains supersonic, then on the shock wave, in particular at the point \(C\), the known inequalities are satisfied
\[ 0<\alpha<\frac{\pi}{2}, \quad \sigma<\vartheta+\alpha, \quad \sigma<\frac{\pi}{2}, \quad \frac{d\Theta}{d\sigma}>0. \]

From these inequalities it follows that the expression multiplying \(1/\mathfrak{R}_b\) in formula (6) is positive, and a decrease in \(d\vartheta/ds\) leads to an increase in \(\mathfrak{R}_b\). Hence, from Property 1 there in turn follows:

Property 2. In axisymmetric supersonic flow past a body of revolution, an increase in the radius of curvature \(\mathfrak{R}\) of the generatrix \(AB\) of the body of revolution at the point \(A\) leads to an increase in the radius of curvature \(\mathfrak{R}_b\) of the shock-wave line \(CE\) at the point \(C\), if the points \(A\) and \(C\) are connected by a characteristic of the first family.

Property 2 permits one to conclude that, when the curvature of \(AB\) at the point \(A\) is equal to \(-\infty\), the curvature of the shock-wave line \(CE\) at the point \(C\) is minimal.

Computing Center
Academy of Sciences of the USSR

Received
24 V 1958

CITED LITERATURE

1. A. A. Dorodnitsyn, Collection of Theoretical Works on Aerodynamics, Moscow, 1957, p. 77.