Abstract
Full Text
HYDROMECHANICS
Yu. D. SHMYGLEVSKII
A VARIATIONAL PROBLEM OF GAS DYNAMICS FOR AXISYMMETRIC SUPERSONIC FLOWS
(Presented by Academician A. A. Dorodnitsyn, 23 X 1956)
The purpose of the work is to determine the shape of a part of a body of revolution having the least wave drag in a supersonic flow, and of a nozzle with minimal losses.
Fig. 1
The line \(AB\) (Fig. 1) is the sought generator of the body of revolution; \(AC\) is the given characteristic of the first family; \(BC\) is the sought characteristic of the second family.
The method of A. A. Nikolskii \((^1)\) makes it possible to formulate the problem under consideration for functions on the characteristic \(AC\).
We introduce the notation: \(x, r\) are Cartesian coordinates in the meridional plane of the flow; \(\alpha\) is the Mach angle; \(\vartheta\) is the angle of inclination of the velocity; \(\kappa\) is the adiabatic exponent. The Mach angle and the angle of inclination of the velocity on \(AC\), being known functions, will be denoted, respectively, by \(A\) and \(\theta\), and the coordinate \(r\) by \(R\). Finally, the magnitude of the wave drag divided by \(2\pi \rho_0 \sqrt{\dfrac{\kappa+1}{2}}\) (where \(\rho_0\) is the stagnation density) will be denoted by \(\chi\), the difference \(x_B-x_A\) by \(X\), and the total gas flow through the contour \(ACB\) by \(\Psi=0\).
In the inviscid case the following variational problem arises:
Given constants \(r_A, r_B, X\) and functions \(A(R)\), \(\theta(R)\), find functions \(\alpha(r)\), \(\vartheta(r)\) that realize the extremum of \(\chi\):
\[ \chi = \int_{R=r_A}^{r_C} \sqrt{\frac{\kappa+1}{\kappa-\cos 2A}} \left(\frac{1-\cos 2A}{\kappa-\cos 2A}\right)^{\frac{1}{2}\frac{\kappa+1}{\kappa-1}} \left[ \frac{\sin A}{\kappa} + \frac{\cos\theta}{\sin(\theta+A)} \right] R\,dR - \]
\[ - \int_{r=r_B}^{r_C} \sqrt{\frac{\kappa+1}{\kappa-\cos 2\alpha}} \left(\frac{1-\cos 2\alpha}{\kappa-\cos 2\alpha}\right)^{\frac{1}{2}\frac{\kappa+1}{\kappa-1}} \left[ \frac{\sin\alpha}{\kappa} - \frac{\cos\vartheta}{\sin(\vartheta-\alpha)} \right] r\,dr \]
under the isoperimetric conditions
\[ X = \int_{R=r_A}^{r_C} \operatorname{ctg}(\theta+A)\,dR - \int_{r=r_B}^{r_C} \operatorname{ctg}(\vartheta-\alpha)\,dr, \]
\[ \Psi=0 = \int_{R=r_A}^{r_C} \left(\frac{1-\cos 2A}{\kappa-\cos 2A}\right)^{\frac{1}{2}\frac{\kappa+1}{\kappa-1}} \frac{R\,dR}{\sin(\theta+A)} + \int_{r=r_B}^{r_C} \left(\frac{1-\cos 2\alpha}{\kappa-\cos 2\alpha}\right)^{\frac{1}{2}\frac{\kappa+1}{\kappa-1}} \frac{r\,dr}{\sin(\vartheta-\alpha)}, \]
differential condition
\[ \frac{d\vartheta}{dr} - \frac{1+\cos 2\alpha}{\chi-\cos 2\alpha}\, \frac{d\alpha}{dr} - \frac{\sin\vartheta \sin\alpha}{r\sin(\vartheta-\alpha)} =0 \]
and boundary conditions
\[ \alpha(r_C)=A(R_C), \qquad \vartheta(r_C)=\vartheta(R_C). \]
Admissible functions are continuous functions \(\alpha(r)\), \(\vartheta(r)\) that give physically attainable flows.
The problem as formulated is a degenerate one, since the derivatives of the functions enter linearly, and it can be solved by the method of D. E. Okhotsimskii \((^2)\).
The solution leads to the following results. The generator of the body \(AB\) has, generally speaking, a corner at the point \(A\), from which characteristics of the first family issue as a pencil up to \(AD\). The required characteristic \(BC\) consists of the non-extremal segment \(CD\) and the extremal segment \(DB\). On the latter, the functions \(\alpha\) and \(\vartheta\) are determined from the explicit expressions
Fig. 2
\[ r\left(\frac{1-\cos 2\alpha}{\chi-\cos 2\alpha}\right)^{\frac12\frac{\chi+1}{\chi-1}} \sqrt{\frac{\chi-\cos 2\alpha}{\chi+1}} \left[ \frac{(\chi+1)\cos\alpha}{\chi-\cos 2\alpha} \right. \]
\[ \left. -\mu^2\cos 2\alpha\cos\alpha -\mu\sin 2\alpha \sqrt{\frac{\chi+1}{\chi-\cos 2\alpha}-\mu^2\cos^2\alpha} \right] =|\lambda|, \]
\[ \vartheta = \operatorname{sign}\vartheta_D \left| \arcsin \left[ \sqrt{\frac{|\lambda|\cos\alpha}{r}} \left( \frac{1-\cos 2\alpha}{\chi-\cos 2\alpha} \right)^{-\frac14\frac{\chi+1}{\chi-1}} \left( \frac{\chi-\cos 2\alpha}{\chi+1} \right)^{\frac14} \right] \right|, \]
where \(\lambda,\mu\) are Lagrange multipliers, whose values can be determined from the last two equalities from the known \(\alpha,\vartheta,r\) at the point \(D\).
Fig. 3
The angle of inclination of the velocity does not change sign on the extremal segment of the characteristic. All streamlines in \(ABD\) are extremals.
The generator \(AB\) is found as the solution of the Goursat problem between the characteristics \(AC\) and \(CB\).
The solution of the problem of the optimal nozzle (Fig. 2) is given by the same formulas for \(\alpha\) and \(\vartheta\).
The search for the optimal nozzle is the subject of the work \((^3)\), in which the authors reduce the problem to numerical integration of a system of ordinary differential equations. An explicit solution was not found in that work.
The same problems are solved analogously for the case of vortical flows. In this case the functions on the characteristic \(BD\) obey the following equations:
\[ |\lambda|\left(\sin 2\vartheta+\frac{1}{\varkappa}\sin 2\alpha\right) +\mu(1-\cos 2\vartheta)=0, \]
\[ \frac{|\lambda|\,\omega(\alpha)\varphi(\psi)\cos\alpha}{\sqrt{\varkappa}\,r} -\sqrt{\frac{\varkappa+1}{\varkappa-\cos 2\alpha}}\, \sin^2\vartheta=0, \]
\[ \frac{d\mu}{d\psi} = -\frac{\omega(\alpha)\varphi(\psi)}{\sqrt{\varkappa}\,r^2} \left[|\lambda|\cos(\vartheta-\alpha)+\mu\sin(\vartheta-\alpha)\right], \]
\[ \frac{dr}{d\psi} = -\frac{\omega(\alpha)\varphi(\psi)}{\sqrt{\varkappa}}\sin(\vartheta-\alpha), \]
where
\[ \omega(\alpha)= \left( \frac{\varkappa+1}{2\varkappa}\, \frac{1-\cos 2\alpha}{\varkappa-\cos 2\alpha} \right)^{ -\frac{1}{2}\frac{\varkappa+1}{\varkappa-1} }, \]
Fig. 4
\(\psi\) is the stream function, while the function \(\varphi\) is prescribed and is expressed in terms of the pressure \(p\) and density \(\rho\) by the formula
\[ \varphi=p^{\frac{1}{\varkappa-1}}\rho^{-\frac{\varkappa}{\varkappa-1}}. \]
Figures 3 and 4 give the results of some computations. For bodies having a nose cone with a half-opening angle of \(35^\circ\), Fig. 3 shows four different generators \(AB\) that ensure minimum drag at the Mach number of the oncoming flow \(M=3\). Different afterbodies possessing minimum drag for \(M=2\) are shown in Fig. 4.
Computing Center
Academy of Sciences of the USSR
Received
20 X 1956
REFERENCES
- A. A. Nikol’skii, “On bodies of revolution with throughflow possessing the least wave drag in a supersonic flow,” Tr. TsAGI (1950).
- D. E. Okhotsimskii, Prikl. matem. i mekh., 10, 251 (1946).
- H. Guderley, E. Hantsch, Sborn. Mekhanika, Foreign Literature Publishing House, 4, 1956, p. 53.