Abstract
Full Text
UDC 533.6
AERODYNAMICS
E. A. KRASILSHCHIKOVA
VELOCITY FIELD EXCITED BY WING VIBRATIONS PROPAGATING OVER THE SURFACE AT SUPERSONIC SPEED
(Presented by Academician L. I. Sedov, 12 II 1968)
- Let us consider the rectilinear translational motion of a wing with constant velocity \(u\) within an unbounded volume of an ideal compressible medium. Beginning at some instant of time \(t_0\), small oscillations propagate over the elastic surface of the wing with supersonic speed \(v\). The normal component of the velocity due to the basic motion is prescribed on both sides of the wing in the form
\[ v_{0n}=-u\alpha, \tag{1} \]
where \(\alpha\) is the angle of attack of the elements of the surface being flowed around. The normal component of the velocity due to the vibrations is prescribed in the form
\[ v_{\Delta n}=A_{\Delta}, \tag{2} \]
where \(A_{\Delta}\) is a function of time and of points of the surface being flowed around. The functions \(\alpha\) and \(A_{\Delta}\) are small and may be arbitrary integrable functions of their arguments.
Fig. 1
Assuming that the medium is weakly disturbed, we consider the problem of determining the velocity field in a linearized formulation \((^{1,2})\). We shall suppose the motion of the medium to be irrotational and to occur in the absence of external forces. We take a moving coordinate system \(Oxyz\), rigidly attached to the moving wing (Fig. 1). The axis \(Oz\) is directed perpendicular to the plane of the figure.
The velocity potential \(\varphi\) satisfies the equation
\[ (u^2-a^2)\varphi_{xx}-a^2\varphi_{yy}-a^2\varphi_{zz}-2u\varphi_{xt}+\varphi_{tt}=0, \tag{3} \]
where \(a\) is the speed of sound in the undisturbed medium, and the boundary conditions are in the plane \(xOy\).
In the region \(\Sigma_0\)—the projection of the wing onto the plane \(xOy\) ahead of the front of vibration propagation (the line \(FF_1\) in Fig. 1)—the derivative is
\[ \varphi_z=-u\alpha(x,y)=A_0(x,y). \tag{4} \]
In the region \(\Sigma\)—the projection of the wing onto the plane \(xOy\) behind the front \(FF_1\)—the derivative is
\[ \varphi_z=A_0(x,y)+A_{\Delta}(x,y,t)=A(x,y,t). \tag{5} \]
In the region \(\Sigma_1\)—the projection of the vortex wake onto the plane \(xOy\),
\[ \varphi_t-u\varphi_x=0. \tag{6} \]
Everywhere in the plane \(xOy\), outside the region \(\Sigma_0+\Sigma+\Sigma_1\), the potential is
\[ \varphi=0. \tag{7} \]
In addition, at each instant of time the Chaplygin–Zhukovsky principle must be satisfied at the trailing edge of the wing.
Thus, the problem is as follows. Find, in the half-space \(z \geqslant 0\), a function \(\varphi(x,y,z,t)\) that satisfies equation (3), the boundary conditions (4)—(5), specified in regions with a moving boundary, and the boundary conditions (6)—(7), specified in regions with fixed boundaries. The solution of the problem in the half-space \(z<0\) is found from the condition
\[
\varphi(x,y,-z,t)=-\varphi(x,y,z,t).
\]
Fig. 2
2. To solve the problem we shall apply the method developed in the papers \((^3,^4)\). Let us turn to the space \(xyt\) and consider in it a region \(V\), inside which the derivatives \(\varphi_z\) are specified by the flow condition. The region \(V\) is bounded by the surface \(\Sigma^*\). The surface \(\Sigma^*\) is a cylindrical surface with generators parallel to the time axis \(Ot\), and with directrix given by the contour \(AOBD\), which is specified by the equation: \(\eta=\psi(\xi)\) (Fig. 2).
Let the line \(FF_1\) be the front of propagation of vibrations along the wing surface, moving according to the law: \(x=f(t)\), where \(f'(t)=v\). The surface \(F^*\), specified by the equation \(\xi=f(\tau)\), divides the region \(V\) into two parts \(V_1\) and \(V_2\) with different values of the derivative \(\varphi_z\), according to conditions (4) and (5). To the left of the surface \(F^*\), in the region \(V_1\), the derivative \(\varphi_z=A_0\); to the right—in the region \(V_2\), the derivative \(\varphi_z=A\) (Fig. 2).
We take the solution of equation (3) in the form \((^3)\)
\[
\varphi(x,y,z,t)=\frac{u^2-a^2}{2\pi}
\iint_{S(x,y,z,t)}
\frac{\varphi_z\left(\xi,\eta,0,t-\dfrac{u(x-\xi)+ar}{u^2-a^2}\right)\,dS}
{\sqrt{(u^2-a^2)^2r^2+[a(x-\xi)-ur]^2+a^2k^4(y-\eta)^2}},
\tag{8}
\]
\[
r=\sqrt{(x-\xi)^2-k^2(y-\eta)^2-k^2z^2},\qquad
k=\sqrt{u^2/a^2-1},
\]
where the region of integration \(S\) is the surface of a hyperboloid determined by the equation
\[
(x-\xi)^2+(y-\eta)^2+z^2+2u(x-\xi)(t-\tau)+(u^2-a^2)(t-\tau)^2=0
\tag{9}
\]
and the inequality \(\tau<t\). In formula (8) we pass from the surface integral to double integrals with a plane region of integration in the plane \(xOy\). Let us consider the supersonic velocity of wing motion \(u>a\); then we obtain
\[
\varphi(x,y,z,t)
=-\frac{1}{2\pi}
\iint_{S^*(x,y,z)}
\frac{\varphi_z(\xi,\eta,0,\tau_1)}{r}\,d\xi d\eta
-\frac{1}{2\pi}
\iint_{S^*(x,y,z)}
\frac{\varphi_z(\xi,\eta,0,\tau_2)}{r}\,d\xi d\eta .
\tag{10}
\]
\[
\tau_1=t+\frac{u(x-\xi)+ar}{u^2-a^2},\qquad
\tau_2=t+\frac{u(x-\xi)-ar}{u^2-a^2}.
\]
The region \(S^*\) is bounded above by the Mach wave, and below by the Mach hyperbola or, for \(z=0\), by the Mach lines.
In constructing the solution, an essential role is played by the line of intersection of the surfaces \(S\) and \(F^*\). Let us denote the projection of this line onto the plane \(xOy\) by \(l\). The curve \(l\) divides the plane region \(S^*\) into parts with different values of the derivative \(\varphi_z\). If the front of propagation of the vibrations over the wing surface is a straight line \(FF_1\), moving opposite to the motion of the wing with constant velocity
Fig. 3
Fig. 4
\(v>u+a\), then the surface \(F^*\) is a plane defined by the equation \(\xi+v\tau=0\), and the curve \(l\) is an ellipse defined by the equation
\[ v^2(x-\xi)^2+v^2(y-\eta)^2+v^2z^2+ \]
\[ +2uv(x-\xi)(vt+\xi)+(u^2-a^2)(vt+\xi)^2=0. \tag{11} \]
We note that the ellipse \(l\) is always inscribed in the Mach hyperbola or, for \(z=0\), in the angle formed by the Mach lines.
Assuming that on the contour \(AOBD\) the condition
\[ \left|u\psi'(\xi)/\sqrt{\psi'^2(\xi)+1}\right|\ge a, \]
is satisfied, we represent the solution (10) in the form
\[ \varphi(x,y,z,t)=\varphi_0(x,y,z)- \tag{12} \]
\[ -\frac{1}{2\pi} \iint_{\sigma(x,y,z,t)+\sigma_1(x,y,z,t)} \frac{A_\Delta(\xi,\eta,\tau_1)}{r}\,d\xi d\eta -\frac{1}{2\pi} \iint_{\sigma_1(x,y,z,t)} \frac{A_\Delta(\xi,\eta,\tau_2)}{r}\,d\xi d\eta, \]
where the region of integration \(\sigma\) is the part of the region \(S^*\) that, at the time \(t\), lies inside the ellipse \(l\), while the region \(\sigma_1\) is the part of the region \(S^*\) lying outside the ellipse \(l\), below the arc of the ellipse \(K_1LK_2\). The points \(K_1\) and \(K_2\) are the points of tangency of the ellipse \(l\) with the Mach hyperbola (Fig. 3). The function \(\varphi_0\) in formula (12) is the solution of the known problem on the flow past the wing under consideration by a steady supersonic gas flow \((^5)\).
- The analytical form of the solution of the problem depends on the relative position of the ellipse \(l\) and the wing contour \(AOBD\), which determines the regions of integration \(\sigma\) and \(\sigma_1\) in the solution (12).
Consider a time \(t\) belonging to the interval \(0=t_0\le t\le t_1=d/v\), where \(d\) is the length of the segment \(OD\). In the space \(xy t\) we draw the plane \(\tau=t\). The projection of the line of intersection of the plane \(\tau=t\) with the plane-
by the surface \(F^*\) on the plane \(xOy\), and denote it by \(FF_1\) (Fig. 2). Let us consider in the space \(xyt\) a family of cones defined by the equation
\[ (X-\xi)^2+(Y-\eta)^2+2u(X-\xi)(T-\tau)+(u^2-a^2)(T-\tau)^2=0 \]
and by the inequality \(\tau>T\), with vertices on the line of intersection of the plane \(F^*\) and the surface \(\Sigma^*\). Denote the envelope surface of this family by \(\Omega\). Let the projections of the lines of intersection of the plane \(\tau=t\) with the envelope \(\Omega\) onto the plane \(xOy\) be denoted by \(\Omega_1\) and \(\Omega_2\). We find the equations of the curves \(\Omega_1\) and \(\Omega_2\) in parametric form
\[ v^2(x^*-\xi)^2+v^2[\psi(x^*)-\eta]^2-2uv(x^*-\xi)(x^*+vt)+ \]
\[ +(u^2-a^2)(x^*+vt)^2=0, \]
\[ v^2(x^*-\xi)+v^2[\psi(x^*)-\eta]\psi'(x^*)-uv(x^*-\xi)- \tag{13} \]
\[ uv(x^*+vt)+(u^2-a^2)(x^*+vt)=0, \]
where \(x^*\) is the parameter.
The lines \(FF_1\), \(\Omega_1\), \(\Omega_2\) separate regions with different analytic character of the solution of the problem. To the region enclosed between the straight line \(FF_1\) and the curve \(\Omega_1\) there corresponds the solution (12), in which the region of integration \(\sigma\) is the part \(S^*\) bounded by the ellipse \(l\), lying entirely inside \(S^*\) (Fig. 2). To the region enclosed between the curves \(\Omega_1\) and \(\Omega_2\) there corresponds the solution (12), in which \(\sigma\) is the part of \(S^*\) cut off by the ellipse \(l\), which intersects the wing contour. In this case the points of tangency \(K_1\) and \(K_2\) may lie both inside the wing (Fig. 3) and outside it (Fig. 4). To the region enclosed between the curve \(\Omega_2\) and the Mach wave there corresponds the solution (12), when in both integrals the integration extends over the whole region \(S^*\), i.e., the region \(\sigma\) is absent, and the region \(\sigma_1\) coincides with the region \(S^*\).
If the velocity of propagation of the vibrations along the surface in the flow satisfies the inequality \(v<u+a\), then the curve \(l\) is a hyperbola.
- In the particular case when harmonic oscillations with frequency \(\omega\) propagate along the elastic surface of the wing, the function
\[ A_\Delta(x,y,t)=A_1(x,y)\exp i\omega\bigl(t+\alpha_1(x,y)\bigr)=\operatorname{Re} A_2(x,y)\exp i\omega t. \]
Using the relation
\[ \exp \frac{i\omega a}{u^2-a^2}r+\exp \frac{-i\omega a}{u^2-a^2}r = 2\cos \frac{\omega a}{u^2-a^2}r, \]
we write solution (12) in the form
\[ \varphi=\varphi_0-\frac{1}{2\pi}\operatorname{Re}\exp(i\omega t)\exp\left(\frac{i\omega u}{u^2-a^2}x\right)\times \]
\[ \times \iint_{\sigma}\frac{A_2(\xi,\eta)}{r}\exp(i\omega t)\exp\left(\frac{-i\omega u}{u^2-a^2}\xi\right) \exp\left(\frac{i\omega a}{u^2-a^2}r\right)d\xi d\eta- \]
\[ -\frac{1}{2\pi}\operatorname{Re}\exp(i\omega t)\exp\left(\frac{i\omega u}{u^2-a^2}x\right)\times \]
\[ \times \iint_{\sigma_1}\frac{A_2(\xi,\eta)}{r}\exp\left(-\frac{i\omega u}{u^2-a^2}\xi\right) \cos\frac{\omega a}{u^2-a^2}r\,d\xi d\eta, \]
where the prescribed function \(A_2\) determines the amplitude and initial phase of the oscillations at each oscillating point on the wing surface.
Institute for Problems in Mechanics
Academy of Sciences of the USSR
Received
10 I 1968
CITED LITERATURE
- L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 1957.
- L. I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, 1966.
- E. A. Krasil’shchikova, DAN, 117, No. 5 (1957).
- E. A. Krasilshchikova, Arch. Mech. Stosowanej, 2, 16 (1964).
- E. A. Krasil’shchikova, A Wing of Finite Span in a Compressible Flow, 1952.