HYDROMECHANICS

A. I. UTKIN

INVESTIGATION OF SUPERSONIC FLOWS BY MEANS OF THE VOLTERRA INTEGRAL

(Presented by Academician L. I. Sedov, 16 IV 1957)

The Volterra method \((^{1,2})\) is applied to the investigation of the linear problem of supersonic flow past axisymmetric surfaces. By a series of transformations the Volterra integral is reduced to a linear integral equation of the second order, the solution of which can be obtained by the methods of operational calculus or by series.

As a practical application, for a particular case, the problem proposed by L. I. Sedov concerning the profile of an annular wing of minimum wave drag is considered, and the shape of the generating curve of the bottom region is also determined if the pressure in it is prescribed.

General solution. Consider the flow past an axisymmetric surface \(S\) that differs little from a circular cylinder (Fig. 1). If we put

\[ \bar z=\frac{z}{\sqrt{M_\infty^2-1}}, \tag{1} \]

then the equation for the potential of the additional velocities \(\Phi(x,y,z)\) will be

\[ -\Phi_{\bar z\bar z}+\Phi_{xx}+\Phi_{yy}=0. \]

The boundary condition on the surface of the body will be, as usual,

\[ \left(\frac{\partial \Phi}{\partial n}\right)_S=V_0\delta, \]

where \(V_0\) is the velocity in the undisturbed flow, \(\delta\) is the angle between the tangent to the contour and the \(z\)-axis. We shall restrict the length of the surface by the condition

\[ L_0<D_0\sqrt{M_\infty^2-1}, \]

where \(D_0\) is the mean diameter of the surface \(S\).

The Volterra integral \((^1)\) makes it possible to find the value of the potential at any point of the flow \(\Phi(x_1,y_1,\bar z_1)\), if the values of \(\Phi(x,y,\bar z)\) and \(d\Phi/dn\) are known on some surface \(\Sigma\) carrying the initial Cauchy data, \(n\) being the normal to the surface \((^{1,2})\).

The Volterra integral has the form

\[ \Phi(x_1,y_1,\bar z_1)=\frac{1}{2\pi}\frac{\partial}{\partial \bar z_1} \iint\limits_{\Sigma} \left( V\frac{d\Phi}{dN}-\Phi\frac{dV}{dN} \right)d\Sigma. \tag{2} \]

The surface \(\Sigma=\Sigma_1+\Sigma_2\) is cut off by the characteristic cone on the initial surface of disturbance \(\Sigma_2\) and on the body surface \(\Sigma_1\) (Fig. 1).

Let us set \(\Phi=0\) on the surface \(\Sigma_2\); then \(d\Phi/dN\) on \(\Sigma_2\) also vanishes, since the conormal in this case coincides with the generator of the conical surface \(\Sigma_2\), where \(\Phi=0\). Thus, the integration need be carried out only over the surface of the body, i.e. \(\Sigma=\Sigma_1\).

In expression (2)

\[ V=\ln \frac{(\bar z_1-\bar z)-\sqrt{(\bar z_1-\bar z)^2-(x_1-x)^2-(y_1-y)^2}} {\sqrt{(x_1-x)^2+(y_1-y)^2}} \]

is Volterra’s fundamental function \((^{1,2})\).

Introduce cylindrical coordinates \(x=r\cos\varphi;\ y=r\sin\varphi;\ \bar z=z\). Differentiating expression (2) with respect to \(\bar z_1\), taking into account the dependence of the limits of integration on \(\bar z_1\), and carrying out a number of transformations, we obtain

\[ \Phi(\bar z_1,r_1)\left(1-\frac{1}{2}\sqrt{\frac{2r_1}{D_0}}\right) = -\frac{1}{\pi}\sqrt{\frac{D_0}{2r_1}} \int_0^{z_1-(r_1-D_0/2)} \frac{d\Phi}{dn}\,K(\lambda_1)\,d\bar z - \frac{1}{\pi} \int_0^{z_1-(r_0-D_0/2)} \Phi(\bar z_1,D_0) \left[ \frac{K(\lambda)}{D_0} + \lambda\frac{\partial K(\lambda)}{\partial z_1} \right]d\bar z, \tag{3} \]

where

\[ \lambda=\frac{\bar z_1-\bar z}{D_0}, \]

and \(K(\lambda)\) is the complete elliptic integral of the first kind \((^6)\).

The expression obtained does not make it possible to find the solution directly, since the value of \(\Phi\) on the surface \(\Sigma_1\) is unknown. To determine \(\Phi(\bar z,D_0)\) we use equality (3), putting in it \(D_1\to D_0\), i.e. we shall consider the point \(M\) (Fig. 1) on the surface \(S\); then equality (3) becomes the integral equation

\[ \Phi(\bar z_1,D_0) = -\frac{2}{\pi}\int_0^{\bar z_1}\frac{d\Phi}{dn}K(\lambda)\,d\bar z - \frac{2}{\pi}\int_0^{\bar z_1} \Phi(\bar z,D_0) \left[ \frac{K(\lambda)}{D_0} + \lambda\frac{\partial K(\lambda)}{\partial z_1} \right]d\bar z. \tag{4} \]

The integral equation (4) is a linear integral equation of the second kind—a Volterra equation.

If we put \(D_0\to\infty\) and pass to Cartesian coordinates, then the last integral in equation (4) vanishes, and the first term, on the basis of expression (2), can be represented in the form of the well-known formula \((^{2,3})\) for a flat wing:

\[ \Phi(x,z) = -\frac{1}{\pi} \iint_{\Sigma} \frac{d\Phi}{dn} \frac{d\eta\,d\xi} {\sqrt{(x-\xi)^2-(M_\infty^2-1)(z-\eta)^2}}. \]

The solution of equation (4) is easily obtained by operational methods. Denote

\[ F(p)\div \Phi(\bar z_1); \qquad R(p)\div -\frac{2}{\pi}\int_0^{\bar z_1}\frac{d\Phi}{dn}[K(\lambda)]\,d\bar z; \]

\[ Q(p)\div \left[ \frac{1}{D_0}K(\lambda)\frac{\partial K(\lambda)}{\partial \lambda} \right], \]

where \(F(p)\), \(R(p)\), and \(Q(p)\) are the “images” of the indicated “originals.” Taking into account that the kernel of the integral equation will be symmetric, we obtain \((^4)\)

\[ F(p)=\frac{pR(p)}{p-Q(p)}; \]

passing again to the “originals,” we find the solution. We note that the solution cannot be written in the form of a finite combination of elementary functions, since the kernel of the integral equation is expressed in terms of elliptic integrals.

The solution of the equations may also be sought in the form of a series

\[ \Phi(\bar z_1)=\Phi_0(\bar z_1)+\eta\Phi_1(z_1)+\eta^2\Phi_2(z_2)+\ldots, \]

where

\[ \eta=-2/\pi, \]

\[ \Phi_0(\bar z_1)=-\frac{2}{\pi}\int_0^{\bar z_1}\frac{d\Phi}{dn}K(\lambda)\,d\bar z, \]

\[ \Phi_{n+1}(\bar z_1)=\int_0^{\bar z_1} \left[ \frac{K(\lambda)}{D_0}+\lambda\frac{\partial K(\lambda)}{\partial \bar z_1} \right]\Phi_n(\bar z)\,d\bar z. \]

The indicated series converges uniformly in \(z_1\) and in \(\eta\) for finite \(\eta\) and for \(\bar z_1<D_0\) \((^5)\). Both methods rapidly lead to the goal in the cases considered by us.

As a particular case of the problem of the profile of an annular wing of minimum wave drag, let us consider an internal triangular profile with a constant value \(c/L_0=\bar c\), and let us find the optimal position of its vertex \(\bar a=a/\sqrt{M_\infty^2-1}\) (Fig. 2). On the surface \(AB\) the potential, if terms of higher order of smallness are neglected, beginning with \(\frac{1}{16}\bar z^4/D_0^4\), as follows from the solution of equation (4), will be

\[ \Phi=\delta V_0\bar z\sum_{n=0}^{3}\frac{1}{2^n}\left(\frac{\bar z}{D_0}\right)^n, \]

where \(\delta=c/a\).

Fig. 2

The potential on the surface \(BD\), as can be determined using equation (4), in the same approximation has the form

\[ \Phi_2=-\gamma V_0\bar \xi\sum_{n=0}^{3}\frac{1}{2^n}\left(\frac{\bar \xi}{D_0}\right)^n+ \]

\[ +\delta V_0\left[ Q_0\left(1+\frac{\bar \xi}{D_0} +\frac{\bar \xi^2}{2D_0^2} +\frac{5}{12}\frac{\bar \xi^3}{D_0^3}\right) +\frac{1}{12}\frac{\bar z^3-\bar \xi^3}{D_0^3} \right], \]

where

\[ \gamma=\frac{c}{L_0-\bar a},\qquad \bar \xi=\bar z-\bar a,\qquad Q_0=\sum_{n=1}\frac{1}{n!}\frac{\bar a^n}{D_0^{\,n-1}}. \]

The polynomials written above are the initial terms of series that converge rapidly absolutely and uniformly (for \(\bar z/D_0\leqslant 1\)), since they have been obtained by integrating series expressing elliptic integrals.

Knowing the potential, we find the pressure and then, by integration, the wave-drag coefficient (referred to the area \(\pi D_0L_0\)):

\[ C_x=\frac{8\bar c^2}{\sqrt{M_\infty^2-1}} \left( 1-\frac{\bar L_0-2\bar a}{4D_0} -\frac{\bar L_0^{\,2}-\bar a^2}{4D_0^2} +\frac{\bar a\bar L_0}{8} -\ldots \right). \]

Obviously, \(C_x\) will be smallest if \(\bar a/D_0 \to 0\) and \(\bar L_0/D_0 \to 1\), and will be approximately half the wave-drag coefficient of a triangular symmetric wing profile of infinite span.

Fig. 3

From physical considerations it is clear that \(\bar a\) cannot be very small, but decreasing \(\bar a\) gives a noticeable gain in drag and is advantageous up to certain limits (possibly up to the formation of a detached shock wave).

Let us consider, as the next application, the shape of the generatrix of the base region \(r=f(z)\) (Fig. 3). The dependence \(r=f(z)\) can be found by means of equation (4), if the pressure coefficient in the base region \(\bar P_{\text{д}}=\mathrm{const}\) is specified. From the condition \(\bar P=\mathrm{const}\) and equation (4) it follows that along the boundary of the region:

\[ \frac{\Phi}{V_0} = \bar P_{\text{д}} \frac{\sqrt{M^2-1}}{2}\,\bar z_1 = -\frac{2}{\pi}\int_0^{\bar z_1} \frac{\partial r}{\partial z} K(\lambda)\,d\bar z - \]

\[ -\frac{2}{\pi}\int_0^{z_1} \bar P_{\text{д}}\frac{\sqrt{M^2-1}}{2} \left[ \frac{K(\lambda)}{D_0} + \lambda \frac{\partial K(\lambda)}{\partial z_1} \right] d\bar z, \tag{5} \]

where, evidently, it is required to find \(dr/dz\). Consequently, the equation written down is a linear integral equation of the first kind. Differentiating (5) with respect to \(\bar z_1\) term by term, we pass to a linear integral equation of the second kind; solving it by the methods presented above, we find

\[ \frac{dr}{dz}=f(z) \]

and, integrating, finally obtain the equation of the generatrix of the base region (in the linear formulation):

\[ r=\frac{D_0}{2} - |\bar P_{\text{д}}|\,\frac{M^2-1}{2} \left( \bar z + \frac{\bar z^2}{2D_1} - \frac{1}{12}\frac{\bar z^3}{D_0^2} + \frac{1}{24}\frac{\bar z^4}{D_0^3} -\cdots \right). \]

The series obtained converges absolutely and uniformly as the integral of a series expressing the sum of elliptic integrals. By substitution in (5) one verifies the correctness of the solution.

Received
10 IV 1957

CITED LITERATURE

1. É. Goursat, Course of Mathematical Analysis, 3, part 1, 1933.
2. S. V. Falkovich, Applied Mathematics and Mechanics, 11, no. 3 (1947).
3. E. A. Krasilshchikova, A Wing of Finite Span in a Compressible Flow, 1952.
4. A. I. Lur’e, Operational Calculus, 1951.
5. I. G. Petrovsky, Lectures on the Theory of Integral Equations, 1951.
6. E. Jahnke, F. Emde, Tables of Functions with Formulae and Curves, 1949.