Plane and Axisymmetric Bodies in a Flow with the Greatest “Critical” Mach Number

We construct two-dimensional (plane and axisymmetric) bodies in a flow of an ideal (inviscid and non-heat-conducting) gas with the greatest “critical” Mach number М* under certain additional restrictions. If the freestream Mach number М₀ < М*, then in the entire flow, including the surfaces, М < 1, shock waves are absent, and, as a consequence, the wave drag is zero. At М₀ = М* the equality М = 1 is fulfilled at least at one point in the flow, while at М₀ > М* there appear supersonic zones, generally, with the formation of shock waves and the wave drag increasing with increase in М₀. It is known that maximum critical Mach numbers М* are realized by two-dimensional configurations such that, when being in a flow with М₀ = М*, their contours are partly segments of sonic lines. The trivial examples of such configurations are furnished by a flat plate at zero incidence and a segment of a straight line (“axisymmetric needle”) in a uniform flow with М ≡ М₀ ≡ М* ≡ 1; these configurations do not disturb the flow. The area of their longitudinal section divided by the square of a fixed chord S = 0. If, in addition to the chord length, we preassign an area S > 0, then the critical contours of these bodies consist of the forward and rear faces and the upper and symmetric lower sonic streamlines connecting the faces without bends. As S → 0, the face heights tend to zero, while М₀ and М* tend to unity, so that we arrive at the trivial solutions. In order to avoid at S > 0 the almost inevitable separations behind the bodies constructed in the assumption of separationless flow, the restrictions on the angles of inclination of the contours of their rear parts are introduced. As a result, the rear faces are replaced by inclined rectilinear segments and a plane critical configuration becomes a symmetric wing airfoil. Although the structure of the two-dimensional critical configurations is in principle simple, the available methods of their construction are rather complicated. The numerical “tools” applied in this study turned out simpler. They are based on the genetic algorithm of “direct” optimization with the representation of the unknown segments of sonic streamlines by the Bernstein—Bézier curves, together with the integration of the equations of ideal gas flow by means of the modified, higher-order (on smooth solutions) Godunov scheme and the procedure of steady solution attainment. Earlier, these tools were developed and applied by the authors in constructing a wide range of optimal aerodynamic shapes.