Some Types Of Flow Patterns Of Euler Equations For Compressible Flow In Two Dimensions

In this paper, we are mainly concerned with some types of flow patterns of Euler equations for compressible flow in two-dimensions. These flow patterns are simple waves, von Neumann reflection configuration, pressure delta waves and critical transonic shock.The second chapter consider two-dimensional isentropic ir-rotational steady simple waves. The two-dimensional isentropic ir-rotational steady simple wave is a type of flow whose flow region is covered by a one parametric family of straight characteristics, along each of which u, v and consequently p, p, c remain constant. A significant property is A non-constant state of flow adjacent to a constant state is always a simple wave. On the basis of the direction of wave propagation, we obtain the prior first-order derivative estimates of the solutions to some types of Goursat problems. By using these estimates the global solutions to the problems of two rarefaction simple waves interaction, rarefaction simple wave reflection on sonic curve, and the inverse problem of two compression simple waves interaction are constructed.The third chapter study two-dimensional isentropic ir-rotational pseudo-steady sim-ple waves. Similar to the two-dimensional isentropic ir-rotational steady simple wave, a two-dimensional isentropic ir-rotational pseudo-steady simple wave flow region is also covered by a one-parametric family of straight characteristics, along each of which u, v, c and consequently p, p remain constant;and a non-constant state of flow adjacent to a constant state is always a simple wave. Geometrically, if a simple wave and its images are represented in the same coordinate, the images of the simple wave can be represented by a hodograph curve equipped with sonic circles centered at (u(s),v(s)) with radius c(s), c(s) satisfy Each straight characteristic is tangent to the corresponding sonic circle and its direction is perpendicular to the direc-tion of the hodograph curve at the corresponding point. We also construct simple wave flow construction along a pseudo-stream line with a bend part and the global solutions to two rarefaction simple waves interaction.In order to resolve von Neumann triple point paradox, a new reflection, called von Neumann reflection (vNR), was proposed by Colella and Henderson (J. Fluid Mech.,213, 1990,71-94) in investigating numerically the weak shock reflection. In this shock reflection flow pattern, there is on discontinuity in the slop between the incident shock and the Mach stem, and the triple point is not exist but degenerate to a curved band of region, the flow in this region is compressed. A theoretical problem about, the von Neumann reflection is if the reflection configuration is a mathematically possible flow pattern. In the fourth chapter we prove that this flow pattern is impossible for Euler equations.In the fifth chapter we present in investigating two-dimensional Riemann problems for Euler equations for a Chaplygin gas a new type of wave:pressure delta waves, which is absent in one space dimension, but appear in the solutions to two-dimensional Riemann problems. This type of Delta wave is a Dirac type concentration in the pressure variable. This type of discontinuities are different from delta shocks for the pressureless gas flow model, for which the delta shocks are associated with convection and concentration of mass. By writing the the Chaplygin gas system into a new form we are able to define dis-tributional solutions that include the pressure delta waves. Generalized Rankine-Hugoniot conditions for the pressure delta wave are derived.Sheng, Wang and Zhang " Critical transonic shock and supersonic bubble " introduce a concept of critical transonic shock (behind the shock wave the flow is pseudo-sonic) in investigating numerically climbing ramp problem of a rarefaction simple wave. In the last chapter we prove that this type of shock wave ia mathematically possible.