High-speed Compressible Flows about Axisymmetric Bodies

The approximate method for the account of compressibility, which was successfully applied earlier for 2-D problems, is developed for axisymmetric case in the present work. The offered method is analyzed with reference to two mediums: air and water. Comparison of the results of calculations on an offered method and a method of Dorodnitsyn integrate relations has shown acceptable accuracy of the proposed method for engineering calculations. The method allows determining parameters of compressible flow through the parameters of incompressible flow up to the velocity corresponding to critical Mach number. The method of the account of compressibility of a flow does not depend on mathematical model of calculation of incompressible flow. Calculations have shown that for the same body the value of critical Mach number in water is less, than in air. The effect of compressibility is shown in water more strongly and at smaller values of Mach numbers of the flow infinity, than in air. INTRODUCTION Modern development of hydrodynamics is characterized by achievement big subsonic and even transonic and supersonic speeds of movement of bodies in water [1, 2, 3]. Such big speeds until recently were inherent only for the bodies moving in air. The higher the Mach number of the free stream and the level of disturbances produced by the body at low Mach numbers, the stronger the effects of compressibility of the flow. The compressibility at movement of bodies becomes actual for such medium as water, which earlier traditionally was considered incompressible. As it is known, movement of bodies in water may occur both at advanced cavitation mode (supercavitation), and at a mode of continuous flow. The knowledge of critical value of Mach number at which somewhere on a surface of a body local speed of flow reaches local speed of a sound, will allow making a correct choice of the governing equations of hydrodynamics for movement of a body with the high speed. As it is known, at the transition of speed of sound the equations change one’s own type from elliptic at subsonic speeds to hyperbolic at supersonic. Value of critical Mach number may serve as the upper boundary of application of the equations of elliptic type. The approximate method for the account of compressibility, which was successfully applied earlier for 2-D problems [5] is developed for axisymmetric case in the present work. The offered method is analyzed with reference to two mediums: air and water. Methods of the account of compressibility in twodimensional flows enough well are developed [6-10]. The wide application was obtained by a method based on a model of Chaplygin gas [10]. The drawback of this method is the necessity to search for velocity potential and stream function of a flow, which should satisfy the linear equations of gas dynamics in a plane of a hodograph of velocity. These equations have received the name of Chaplygin equations. Usually the theory of function of complex variable is applied to the solution of Chaplygin equations. The solution of these equations in a case of adiabatic flows is connected to the big difficulties, as it is not always possible to formulate boundary conditions in a plane of hodograph of velocity. However, in case of barotropic gas model it is possible to consider the approximate model, which has received the name “Chaplygin gas model”. The solution of a problem for Chaplygin gas model is carried out by method of successive approximations. For zero approximation, the functions satisfying an incompressible liquid are accepted. Chaplygin method has obtained development in S. A. Khristianovich's works [11]. Analytical solution to the problem of a compressible gas flow around a single circle was obtained by A. I. Nekrasov [12] based on Legendre transformations method. For an elliptic contour, such problem with Legendre transformations method was considered by L. K. Kudriashov [13] and with a method of decomposition of velocity potential in a series in terms of powers of 2 ∞ M – by C. Kaplan [14]. S. A. Khristianovich, I. M. Yuriev [15] and L. I. Sedov [7] obtained solutions for ellipses by methods of hodograph of velocity. The drawback of these methods is that there is a deformation of a streamline contour, so the obtained solutions describe flows around some ovals instead of initial ellipses. The method of approximation of an adiabatic for the solution of a problem of a flow around a circle and an ellipse has been developed and applied by G. A. Dombrovski [9]. Numerical solutions based on a method of Dorodnitsyn integrate relations with the account of compressibility for twodimensional flows around single circular and elliptic contours, and airfoils were obtained in works of P. I. Chushkin [16, 17], R. Melnic and D. C. Ives [18], M. Holt and B. Meson [19]. The review of the methods which are taking into account compressibility in two-dimensional flows is contained in works of H. W. Liepmann and A. E. Puckett [6], Shih-I Pai [8], G. A. Dombrovski [9] and L. I. Sedov [7], but basically these methods were directed on solutions of problems of compressible gas flow around the thin bodies such as airfoils. If the body is bluff, the disturbances from it at the high Mach numbers will be significant, therefore all methods using the assumption about a little relative thickness can’t be applied. Chaplygin method is based on application of a theory of complex variable function, which remains the tool for the solution of two-dimensional problems only. The problem of axisymmetric compressible flows of liquid around spheroids was considered. These bodies were chosen because there are exact analytical solutions to a ve-