System of Orthogonal Curvilinear Coordinates on the Isentropic Surface behind a Detached Bow Shock Wave
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em of Orthogonal Curvilinear Coordinates on the Isentropic Surface behind a Detached Bow Shock Wave G. B. Sizykh Moscow Aviation Institute, Moscow, 125080 Russia e-mail: [email protected] Received July 11, 2019; revised September 16, 2019; accepted September 24, 2019
Abstract—Three-dimensional ideal gas flows behind a detached bow shock wave near the nonsymmetrical nose part of a bluff body in a uniform supersonic flow are investigated. The vector lines of the vector product of the velocity and entropy gradient are considered on isentropic surfaces, which are stream surfaces starting on closed lines located on the shock wave and covering the leading point of the shock wave. It is shown that if these vector lines encircle the isentropic surface they are closed. This means that an orthogonal curvilinear coordinate system in which the coordinate lines coincide with the streamlines may be constructed on each isentropic surface. Keywords: curvilinear coordinates, isentropic surface, detached bow shock wave DOI: 10.1134/S0015462820070095
A detached shock wave with a curved surface forms before a blunt-nosed body flown around by a homogeneous supersonic flow, due to which the gas flow behind the body is vortex. The relation between the shape of the shock wave and the vorticity on its surface is known [1, 2]. In the general 3D case (without any symmetry assumed), the vortex flow regularities are true in the flow behind the shock wave. The most important ones are presented in [3–6]. All of them generalize the invariants known for barotropic flows. The generalized circulations and the fields of generalized velocity and vorticity are considered, which makes the mentioned conservation laws hard to oversee. However, under an additional assumption about the isoenergetic nature of the vortex flow (which is valid in flows behind a detached shock wave), the regularities found are more apparent, especially for plane and axisymmetric flows. First of all, we need to mention the Crocco result [7]: in the plane case, the vorticity to pressure ratio I1 = Ω/p is conserved along the streamlines, and in the axisymmetric case, the invariant I 2 = Ω/( pr ) , where r is the distance from the symmetry axis, is conserved. There is also a formula relating the vorticity on the surface of the axisymmetric body (at the zero angle of attack) to the minimum radius of the shock wave’s curvature [8]. In addition to the conservation laws, the regularities describing the shapes of the lines determined by the flow field are of interest. Thus, for example, for an incompressible fluid, the regularities of streamline shapes were studied [9, 10] and the regularities of vortex line shapes were used [11, 12] and investigated [13]. Recently, some findings have been published [14, 15] concerning the general (nonsymmetrical) 3D case of flows behind a detached shock wave. It was proposed [14] to call the point at which the vector of a normal to the shock wave surface is parallel to the velocity of the oncoming flow the leading point. For the general 3D case, it is sh
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