Exact solutions for flow of a perfect gas in a two-dimensional Laval nozzle

A family of exact solutions is found for the problem of steady irrotational isentropic shockfree transsonic flow of a perfect gas through a Laval nozzle in two dimensions. The hodograph method is used, whereby the position co-ordinates x , y are expressed in terms of the velocity variables; the expressions are infinite series in the subsonic part of the flow field, infinite integrals (analytic continuations of the series) in the supersonic part. An inversion is required to get the velocity as a function of position; in general, this requires detailed numerical calculations, but approximate formulae (62) are found for the neighbourhood of the sonic point on the axis.

1959 ◽  
Vol 1 (1) ◽  
pp. 80-94 ◽  
Author(s):  
T. M. Cherry

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.


1975 ◽  
Vol 97 (4) ◽  
pp. 482-491
Author(s):  
S. Popp

The wall effects for fully and partially cavitating flows are investigated for both compressible and incompressible two-dimensional jets. The exact solution for Roshko’s model in a channel with a wedge shaped body is obtained and some particular models are studied. The hodograph method as developed by S. V. Falkovitch [19] is used and the solutions are given as infinite series of Chaplygin’s functions. The exact expressions of the drag coefficients for the aforementioned configurations are also given. Numerical computations are carried out for wedges of all angles. Tables and diagrams are included.


1957 ◽  
Vol 24 (4) ◽  
pp. 497-500
Author(s):  
Toyoki Koga

Abstract A numerical procedure is proposed for solution of certain problems in steady gas flow where subsonic, sonic, and supersonic regions appear simultaneously. The difficulties that occur in analytical methods for taking into account the differences of the type of the fundamental equation (elliptic, parabolic, hyperbolic) are avoided. Given a streamline and the state of the gas along that streamline, the co-ordinates of the neighboring streamline and the state of the gas along it can be computed. The procedure can be applied successively to cover a flow field. The method is described in detail for two-dimensional, steady, irrotational flow (without shocks) of a perfect gas, and an example is given.


2019 ◽  
Vol 866 ◽  
Author(s):  
João A. Isler ◽  
Rafael S. Gioria ◽  
Bruno S. Carmo

Results are reported from linear and nonlinear stability analyses of pulsatile flow in a straight channel with a smooth constriction with 50 % occlusion. Physical mechanisms based on the energy variation of the two-dimensional modes are proposed to elucidate the equilibrium states and changes in the flow-field behaviour. When investigating how the energy of the two-dimensional modes varied with Reynolds number, an energy minimum was observed before the primary instability, so that it marked the Reynolds number value at which the flow field started to exhibit one pulse front per cycle downstream of the constriction. In addition, this flow is characterized by a bistable system in two dimensions, whose equilibrium states are related to the energy level of the two-dimensional mode. The energy difference between the equilibrium states surprisingly decreases from the lower to the higher Reynolds number, which indicates that the system should converge to a unique solution at high Reynolds number. Regarding the flow’s nature, the equilibrium state with higher energy presented a vortex pair soon after the constriction, with vorticity opposite to that of the base flow separating shear layers.


1992 ◽  
Vol 07 (33) ◽  
pp. 3071-3079 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI

An action for two-dimensional gravity conformally coupled to two dilaton-type fields is analyzed. Classically, the theory has some exact solutions. These include configurations representing black holes. A semiclassical theory is obtained by assuming that these singular solutions are caused by the collapse of some matter fields. The semiclassical equations of motion reveal then that any generic solution must have a flat geometry.


Author(s):  
Ioannis Templalexis ◽  
Pericles Pilidis ◽  
Vassilios Pachidis ◽  
Petros Kotsiopoulos

Two-dimensional compressor flow simulation software has always been a very valuable tool in compressor preliminary design studies, as well as in compressor performance assessment, operating under uniform and non-uniform inlet conditions. In this context, a new streamline curvature (SLC) software has been developed capable of analyzing the flow inside a compressor in two dimensions. The software was developed to provide great flexibility, in the sense that it can be used as: a) A performance prediction tool for compressors of a known design, b) A development tool to assess the changes in performance of a known compressor after implementing small geometry changes, c) A design tool to verify and refine the outcome of a preliminary compressor design analysis, d) A teaching tool to provide the student with an insight of the two-dimensional flow field inside a compressor and how this could be effectively predicted using the SLC method, combined with various algorithms and loss models, e) A 2-D compressor model that can be integrated into a conventional 0-D gas turbine engine cycle simulation code for the investigation of the influence of non-uniform radial pressure profiles on whole engine performance. Apart from describing in detail the design, structure and execution of the SLC software, this paper also stresses the importance of developing robust, well thought-out software and highlights the main areas a potential programmer should focus on in order to achieve this. This manuscript highlights briefly the programming features incorporated into the development of software before continuing to explain the internal workings of individual algorithms. The paper reviews in detail the set of equations used for the prediction of the meridional flow field. Numerical aspects of the application procedure of the full radial equilibrium equation are examined. The loss models incorporated for subsonic and supersonic flow are presented for design and off design operating conditions. Deviation angle rules are presented, together with the parameters for quantifying the diffusion process. Moreover, the methods used for the prediction of surge and choke are discussed in detail. Finally, the end wall boundary layer displacement thickness calculation is discussed briefly, in conjunction with the blockage factor computation. The code has been validated against experimental results which are presented in this paper together with the strong and weak points of this first version of the software and the potential for future development.


Author(s):  
Thomas K. Ogorzalek

This theoretical chapter develops the argument that the conditions of cities—large, densely populated, heterogeneous communities—generate distinctive governance demands supporting (1) market interventions and (2) group pluralism. Together, these positions constitute the two dimensions of progressive liberalism. Because of the nature of federalism, such policies are often best pursued at higher levels of government, which means that cities must present a united front in support of city-friendly politics. Such unity is far from assured on the national level, however, because of deep divisions between and within cities that undermine cohesive representation. Strategies for success are enhanced by local institutions of horizontal integration developed to address the governance demands of urbanicity, the effects of which are felt both locally and nationally in the development of cohesive city delegations and a unified urban political order capable of contending with other interests and geographical constituencies in national politics.


Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 27
Author(s):  
J. Barry Greenberg ◽  
David Katoshevski

A theoretical investigation of the influence of a standing wave flow-field on the dynamics of a laminar two-dimensional spray diffusion flame is presented for the first time. The mathematical analysis permits mild slip between the droplets and their host surroundings. For the liquid phase, the use of a small Stokes number as the perturbation parameater enables a solution of the governing equations to be developed. Influence of the standing wave flow-field on droplet grouping is described by a specially constructed modification of the vaporization Damkohler number. Instantaneous flame front shapes are found via a solution for the usual Schwab–Zeldovitch parameter. Numerical results obtained from the analytical solution uncover the strong bearing that droplet grouping, induced by the standing wave flow-field, can have on flame height, shape, and type (over- or under-ventilated) and on the existence of multiple flame fronts.


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