Subsonic to Supersonic Nozzle Flows

Abstract

Abstract. Steady, irrotational flow of a compressible fluid through a two-dimensional planar and axisymmetric nozzle is formulated and solved numerically in the hodograph or velocity plane. The Legendre potential is used to express the governing equation which for planar flow is linear and for axisymmetric flow nonlinear. The hodograph transformation method is extended to solve the nozzle problem for supersonic flow. A rectangular numerical domain for the supersonic case results from the way the information travels in the region that is the image of the flow that occurs around the lip of the nozzle surface. Knowledge of the characteristic's exact location in the supercritical region of the domain is not required, but only the general direction in which information propagates. 1. Introduction. The study of air flow through a convergent nozzle is well established and has important practical applications such as in rocketry, the converging propulsive nozzle of a subsonic turbojet engine, and jet engines. This paper discusses two forms of compressible flow: two-dimensional planar and axisymmetric flow. The geometry of the nozzle for planar flow consists of two convergent nozzle walls which extend to upstream infinity, while the axisymmetric nozzle surface is conical. For both cases, the angle δ which the nozzle surface makes with the nozzle center-line is assumed to be within the range −π/2 ≤ δ < 0. Only the upper half of the physical plane is considered due to symmetry about the center-line.The hodograph transformation was first applied by Chaplygin [1] to obtain analytical solutions for the nozzle problem in the form of a series of hypergeometric functions. Chaplygin's method is limited to planar subcritical and critical flow, in general. Frankl [2] obtained the value of the discharge coefficient analytically for supersonic choked flow. Norwood [3] solved the two-dimensional supercritical problem numerically in the hodograph plane. He considered the subsonic and supersonic regions separately by modifying iteratively the solution along the sonic line to achieve a continuous solution from one region to the other. The method relies on the linearity of the planar equations in that characteristics are fixed and easily determined in the hodograph plane. Alder [4] developed a numerical method for planar and axisymmetric flow in which the subsonic region is treated in the hodograph plane and the supersonic region in the physical plane by applying the method of characteristics. A matching technique is used to obtain smooth transition of streamlines from one region to the other.The approach here follows Cook et al. [5], who express the governing equation in terms of the Legendre potential and solve the two-dimensional and axisymmetric nozzle problems for subsonic and sonic flows. Introducing the Legendre potential has

Generation of high concentrations of respirable solid-phase aerosols from viscous fluids

“…As the aerosol exit orifice is sharp-edged, at high mass gas flows the sonic gas velocity constraints of choke flow are not met and supersonic gas flow, i.e. Mach number >1, was evident (Kryeziu 2013) in all the nozzles used herein at both 275 kPa (40 psi) and 414 kPa (60 Psi). Using exit orifices smaller than 400 μm in diameter an axial fluid jet centered within a high velocity gas jet as depicted in Figure 3 was similar to that reported by Ganan-Calvo (2005) and Gañán-Calvo and Montanero (2009).…”