Compressible Subsonic Flow in Two-dimensional Channels

1955 ◽  
Vol 6 (3) ◽  
pp. 205-220 ◽  
Author(s):  
L. C. Woods
Keyword(s):  
Conformal Mapping ◽  
Direct Problem ◽  
Subsonic Flow ◽  
Ideal Gas ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Step Functions ◽  
Curved Walls ◽  
Special Case ◽  
The Ideal

SummaryEquations for the calculation of the subsonic flow of an inviscid fluid through given two-dimensional channels (the “ direct” problem), and for the design (the “ indirect” problem) of such channels are derived. The method is based on conformal mapping, and in the special case of channels with walls made from a number of straight sections, or with wall pressure prescribed as step-functions, yields the same results as the well-known Schwarz-Christoffel mapping theorem technique. However, it is more general than this latter method, since it is capable of dealing with curved walls or continuously varying wall pressures. The compressibility of the fluid is allowed for only approximately, the ideal gas being replaced by a Kàrmàn-Tsien tangent gas.In Part II the theory is applied to various problems of aeronautical interest, perhaps the most important of which is to the setting of “ streamlined ” walls about a symmetrical aerofoil placed in the centre of the channel.

scholarly journals Collapsing Cavities and Converging Shocks in Non-Ideal Materials

2019 ◽  
Vol 72 (4) ◽  
pp. 501-520 ◽  
Author(s):  
Zachary M Boyd ◽  
Emma M Schmidt ◽  
Scott D Ramsey ◽  
Roy S Baty
Keyword(s):  
Gas Dynamics ◽  
Nonlinear Eigenvalue Problem ◽  
Ideal Gas ◽  
Test Problems ◽  
Infinite Dimensional ◽  
Cavity Collapse ◽  
Scaling Solution ◽  
Metal Deformation ◽  
Special Case ◽  
The Ideal

Summary As modern hydrodynamic codes increase in sophistication, the availability of realistic test problems becomes increasingly important. In gas dynamics, one common unrealistic aspect of most test problems is the ideal gas assumption, which is unsuited to many real applications, especially those involving high pressure and speed metal deformation. Our work considers the collapsing cavity and converging shock test problems, showing to what extent the ideal gas assumption can be removed from their specification. It is found that while most materials simply do not admit simple (that is scaling) solutions in this context, there are infinite-dimensional families of materials which do admit such solutions. We characterize such materials, derive the appropriate ordinary differential equations and analyze the associated nonlinear eigenvalue problem. It is shown that there is an inherent tension between boundedness of the solution, boundedness of its derivatives and the entropy condition. The special case of a constant-speed cavity collapse is considered and found to be heuristically possible, contrary to common intuition. Finally, we give an example of a concrete non-ideal collapsing cavity scaling solution based on a recently proposed pseudo-Mie–Gruneisen equation of state.

scholarly journals Statistical problem of ideal gas in general two-dimensional regions

2014 ◽  
Vol 29 (02) ◽  
pp. 1450243 ◽  
Author(s):  
Ci Song ◽  
Wen-Du Li ◽  
Pardon Mwansa ◽  
Ping Zhang
Keyword(s):  
Perturbation Theory ◽  
Conformal Mapping ◽  
Ideal Gas ◽  
Mapping Method ◽  
Numerical Results ◽  
Statistical Problem ◽  
Thermodynamic Quantities ◽  
Two Dimensional ◽  
Conformal Mapping Method

In this paper, based on the conformal mapping method and the perturbation theory, we develop a method to solve the statistical problem within general two-dimensional regions. We consider some examples and the numerical results and fitting results are given. We also give the thermodynamic quantities of the general two-dimensional regions, and compare the thermodynamic quantities of the different regions.

Subsonic plane flow in an annulus or a channel with spacewise periodic boundary conditions

1955 ◽  
Vol 229 (1176) ◽  
pp. 63-85 ◽  
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Direct Problem ◽  
Periodic Boundary ◽  
Subsonic Flow ◽  
Periodic Boundary Conditions ◽  
The Other ◽  
Closed Circuit ◽  
Inviscid Fluid ◽  
Plane Flow

Equations for the calculation of the subsonic flow of an inviscid fluid in channels with boundary conditions which are periodic in distance along the channel (for example flow in a closed circuit such as an annulus) are derived. Three types of boundary conditions are considered, namely, ( i ) shape of the walls given (‘direct’ problem), ( ii ) pressures or velocities on the walls given (‘indirect’ problem), and ( iii ) pressures on one wall and the shape of the other wall given (‘mixed’ problem). The theory, which is shown to have numerous aero-dynamic applications, is illustrated by several examples.

scholarly journals Solutions of the Noh problem for various equations of state using LIE groups

2000 ◽  
Vol 18 (1) ◽  
pp. 93-100 ◽  
Author(s):  
ROY A. AXFORD
Keyword(s):  
Equation Of State ◽  
Lie Groups ◽  
Equations Of State ◽  
Spherical Geometry ◽  
Ideal Gas ◽  
Explicit Solutions ◽  
Special Case ◽  
General Method ◽  
The Ideal

A method for developing invariant equations of state (EOS) for which solutions of the Noh problem will exist is developed. The ideal gas EOS is shown to be a special case of the general method. Explicit solutions of the Noh problem in planar, cylindrical, and spherical geometry are determined for a Mie–Gruneisen and the stiff gas equation of state.

scholarly journals Onset of Convection in Two-Dimensional Porous Cavities with Open and Conducting Boundaries

2021 ◽  
Vol 136 (3) ◽  
pp. 791-812
Author(s):  
Peder A. Tyvand ◽  
Jonas Kristiansen Nøland
Keyword(s):  
Critical Rayleigh Number ◽  
Numerical Results ◽  
Temperature Perturbation ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Order Problem ◽  
Fourth Order Problem ◽  
Thermally Conducting ◽  
Perturbation Pressure ◽  
Special Case

AbstractThe onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Bénard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.

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