Academic literature on the topic 'Prandtl-Meyer angle'

A 2D aerodynamic study of the NASA’s X-43A hypersonic aircraft is developed using two different approaches. The first one is analytical and based on the resolution of the oblique shock wave and Prandtl–Meyer expansion wave theories supported by an in-house program and considering a simplified aircraft’s design. The second approach involves the use of a Computational Fluid Dynamics (CFD) package, OpenFOAM and the real shape of the aircraft. The aerodynamic characteristics defined as the lift and drag coefficients, the aerodynamic efficiency and the pitching moment coefficient are calculated for different angles of attack. Evaluations are made for an incident Mach number of 7 and an altitude of 30 km. For both methodologies, the required angles of attack to achieve a Vertical Force Balance (VFB) and a completely zero pitching moment conditions are considered. In addition, an analysis to optimise the nose configuration of the aircraft is performed. The mass flow rate throughout the scramjet as a function of the angle of attack is also presented in the CFD model in addition to the pressure, density, temperature and Mach fields. Before presenting the corresponding results, a comparison between the aerodynamic coefficients in terms of the angle of attack of both models is carried out in order to properly validate the CFD model. The paper clarifies the requirements needed to make sure that both oblique shock waves originating from the leading edge meet just at the scramjet inlet clarifying the advantages of fulfilling such condition.

Supersonic axisymmetric expansion flow is a typical and fundamental issue in gas dynamics. It plays a vital role in the high-speed external and internal flow fields regarding the contour design and performance evaluation of supersonic/hypersonic vehicles and their propulsion systems. The supersonic two-dimensional (2D) planar expansion flow is dominated by the well-known Prandtl–Meyer (P–M) theory. However, no similar explicit relation exists for the supersonic axisymmetric expansion flow, and only the computational fluid dynamics results could be employed at present. Therefore, this work focuses on developing the analytical solution of supersonic axisymmetric flow around a sharp convex corner on the basis of the generic gasdynamic functions in a newly established coordinate system for addressing the aforementioned issue. Theoretical derivations and numerical results prove that the flow deflection angle and Mach number in supersonic axisymmetric flow around a sharp convex corner obey the identical law to the 2D planar situation, that is, the P–M theory, while the local axisymmetric expansion fan is not the simple wave flow despite the conical flow. Meanwhile, the method of characteristics is employed to further explicate the intrinsic connection and difference between the 2D and axisymmetric sharp convex corner flow. The equivalence of sharp corner and curved surface flows with the identical deflection angle is discussed, and three limitations of the proposed analytical solution are clarified.

Stationary flows of an ideal plasma with translational symmetry along the (vertical) z axis are considered, and it is demonstrated how they can be described in the intrinsic (natural) coordinates (ξ, η, &), where ξ is a label of flux and stream surfaces, η is the total pressure and ϑ is the angle between the horizontal magnetic (and velocity) field and the x axis. Three scalar nonlinear equilibrium equations of mixed elliptic–hyperbolic type for ϑ(ξ, η), ξ(η, ϑ) and η(ϑ, ξ) respectively are derived. The equilibrium equation for ϑ(ξ, η) is especially useful, and has considerable advantages compared with the coupled system of algebraic–differential equations that are conventionally used for studying plasma flows. In particular, for this equation the location of the regions of ellipticity and hyperbolicity can be determined a priori. Relations between the equilibrium equation for ϑ(ξ, η) and the nonlinear hodograph equation for ξ(η, ϑ) are elucidated. Symmetry properties of the intrinsic equilibrium equations are discussed in detail and their self-similar solutions are described. In particular, magnetohydrodynamic counterparts of several classical flows of an ideal fluid (the Prandtl–Meyer flows around a corner, the spiral flows and the Ringleb flows around a plate, etc.) are found. Stationary flows described in this paper can be used for studying both astrophysical and thermonuclear plasmas.

The aim of this work is to develop a new numerical calculation program to determine the effect of the stagnation temperature on the calculation of the supersonic flow around a pointed airfoils using the equations for oblique shock wave and the Prandtl Meyer expansion, under the model at high temperature, calorically imperfect and thermally perfect gas, lower than the dissociation threshold of the molecules. The specific heat at constant pressure does not remain constant and varies with the temperature. The new model allows making corrections to the perfect gas model designed for low stagnation temperature, low Mach number, low incidence angle and low airfoil thickness. The stagnation temperature is an important parameter in our model. The airfoil should be pointed at the leading edge to allow an attached shock solution to be seen. The airfoil is discretized into several panels on the extrados and the intrados, placed one adjacent to the other. The distribution of the flow on the panel in question gives a compression or an expansion according to the deviation of the flow with respect to the old adjacent panel. The program determines all the aerodynamic characteristics of the flow and in particular the aerodynamic coefficients. The calculation accuracy depends on the number of panels considered on the airfoil. The application is made for high values of stagnation temperature, Mach number and airfoil thickness. A comparison between our high temperature model and the perfect gas model is presented, in order to determine an application limit of the latter. The application is for air.

Purpose The purpose of this paper is to design a double parabolic nozzle and to compare the performance with conventional nozzle designs. Design/methodology/approach The throat diameter and divergent length for Conical, Bell and Double Parabolic nozzles were kept same for the sake of comparison. The double parabolic nozzle has been designed in such a way that the maximum slope of the divergent curve is taken as one-third of the Prandtl Meyer (PM) angle. The studies were carried out at Nozzle Pressure Ratio (NPR) of 5 and also at design conditions (NPR = 3.7). Experimental measurements were carried out for all the three nozzle configurations and the performance parameters compared. Numerical simulations were also carried out in a two-dimensional computational domain incorporating density-based solver with RANS equations and SST k-ω turbulence model. Findings The numerical predictions were found to be in reasonable agreement with the measured experimental values. An enhancement in thrust was observed for double parabolic nozzle when compared with that of conical and bell nozzles. Research limitations/implications Even though the present numerical simulations were capable of predicting shock cell parameters reasonably well, shock oscillations were not captured. Practical implications The double parabolic nozzle design has enormous practical importance as a small increase in thrust can result in a significant gain in pay load. Social implications The thrust developed by the double parabolic nozzle is seen to be on the higher side than that of conventional nozzles with better fuel economy. Originality/value The overall performance of the double parabolic nozzle is better than conical and bell nozzles for the same throat diameter and length.

This thesis revisits the design of a planar supersonic nozzle for a supersonic wind tunnel using the method of characteristics (MOC) and Computational Fluid Dynamics (CFD). While design of a converging-diverging (CD) nozzle is in principle thought to be the outcome of a relatively straightforward procedure, the practical challenge is to design a nozzle that is free of waves in the test section of the wind tunnel. This thesis in particular studies how the waves generated due to the curvature discontinuity at the so-called point of inflection in the graphical method of MOC get modified and influenced by various aspects such as the growth of boundary layers in the nozzle and the test section, the curved nature of sonic line at the throat, the length of the initial expansion curve, the effect of various parameters of the subsonic contraction section upstream of the throat. The main objective of this effort was to design by MOC and validate using CFD a Mach 2 wind tunnel nozzle contour that results in a wave-free test section. First, the Subsonic and near sonic Mach number conditions are achieved by the use of an Error function profile for the contraction section using a procedure given by Ho and Emanuel [11] that can be used in conjunction with the divergent section of C-D Nozzle. The divergent section contour design consists firstly finding the Prandtl-Meyer angle ( ) corresponding to the Mach number at test section and then, starting at the throat and proceeding downstream, determining the flow field in terms of the local Prandtl-Meyer angle and wall angle ( and ), by MOC. MOC is based on the mathematical theory of characteristics of non-linear differential equations of the velocity potential for two-dimensional, steady, isentropic, irrotational compressible flow. The present graphical method determines the contour of the divergent walls that would transform a uniform sonic flow at the throat to a uniform shock-free flow at a desired supersonic speed. A MATLAB code was developed to calculate the nozzle wall coordinates based on MOC. The boundary layer correction was applied to the profiles to account for viscous effects which cause a Mach number reduction from the desired test section Mach number using a formulation given by Tucker [20]. The outcome of the present work is a realization of a CD nozzle with uniform and shock-free flow such that the Mach number variation on the center line of the test section was negligibly small that it can be deemed to be a constant. The design is assessed by numerical simulation using CFD software HiFUN. The final results are presented in terms of center line Mach number in the test section, which is the most significant parameters related to tunnel performance.

External supersonic gas flow in which changes in the fluid and flow properties are brought about by direction change is analysed in this chapter. In addition, it is shown that flow over a corner between two flat surfaces resulted in an oblique shockwave if the angle between the two surfaces is less than 180° (a concave corner). The analysis of flow through an oblique shockwave is based upon the superposition of the flowfield for a normal shock onto a uniform flow parallel to the shock. It is also shown that both weak and strong oblique shocks can occur. For an angle in excess of 180° (a convex corner), the flow is turned through an isentropic Prandtl-Meyer expansion fan. Analysis of a Prandtl-Meyer expansion fan starts from consideration of an infinitesimal flow deflection through a Mach wave.

Abstract Under-expanded jets have wide range of application from fuel injection to rocket propulsion. In the present work, a numerical model was generated to investigate the fluid mechanics behavior of under-expanded jet formation and wall interaction of a jet produced by exhausting a high pressure cylinder through a narrow tube into a low pressure cylinder. Axisymmectic, Reynolds Averaged Navier Stokes simulations were conducted employing the ANSYS FLUENT explicit, Coupled Pressure-Velocity solver to determine the stagnation pressure at the wall downstream of the orifice. Transient cases were conducted using timestep sizes of 1.0 × 10−8 s and 5.0 × 10−9 s. Various gases were investigated with Hydrogen being the primary working fluid with pressure ratios ranging from 10 to 100. This paper will focus primarily on the Hydrogen jets for pressure ratios of 10, 20, and 70. Numerical results were validated from both experimental results and higher fidelity Large Eddy Simulation results specifically analyzing the jet formation. Error between Mach disk height, Mach disk width, and Prandtl-Meyer expansion fan angles of the jet for pressure ratios of 10 and 70 were kept below 5%. The peak stagnation pressures at the center of the far wall for pressure ratios of 10, 20, and 70 were observed to be 86,843 Pa, 127,786 Pa, and 315,843 Pa, respectively. The predicted peak pressures show a linear relationship with respect to the initial pressure ratio existing between the high pressure and low pressure regions when the ratios are bounded between 10 and 70.