If electronic computers are employed, there is no point in using these methods, since accurate numerical methods for solving these problems belong to the simplest problems of computational mathematics. In the spatial case the formal solution of the Neumann problem for the velocity potential will correspond to the physical picture of the flow only in very exceptional cases. In a viscous fluid a solid is followed by a turbulent trail.
As the Reynolds number increases, this trail becomes thinner for a continuous flow and in the limit it becomes an infinitely thin turbulent surface, the intensity of the turbulent layer of which vanishes only in exceptional cases e. Accordingly, the problems involving discontinuous velocity potentials in the region of the flow zone represent the real spatial problems of an incompressible fluid.
The location of this discontinuity surface is unknown, and for this reason the exact problem of flow around solid bodies in the presence of a discontinuous potential surface is a very complicated non-linear problem. It is only in the linear approximation, i. In such a case the surface of turbulence can be considered as horizontal, and it may be assumed that the potential along the stream has a constant discontinuity.
The normal derivatives of the velocity potential are given on the projection plane of the solid. Analytic solutions have been obtained on these assumptions for a circular and an elliptic wing.
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Numerical methods of solving this problem for other shapes of wings are available and simplify the task to a considerable extent cf. Wing theory. The principal difficulty in problems involving a heavy fluid is that the boundary conditions for the potential on the free surface are non-linear.
Exact solutions have been found for planar problems of the theory of waves only. Little progress has been made on the spatial problem of waves with a finite amplitude.
On the other hand, the linearized problem small wave amplitudes has been largely solved, and the calculation of the wave resistance to the motion of a ship is actually based on the linear theory. The possibility of aircraft travelling at near-sonic and supersonic speeds stimulated the search for solutions of aerodynamic problems of compressible fluids. The problems involving subsonic, transsonic and supersonic velocities are all treated in substantially different manners.
In the subsonic range the aerodynamic equations remain elliptic and the solutions qualitatively resemble those obtained for incompressible fluids. In the first period of research on the aerodynamics of compressible fluids much research was done on the effect of compressibility on the aerodynamic characteristics of the solids around which the flow takes place. All these methods involve some sort of linearization of the velocity of the unperturbed flow. Khristianovich's method is an exception, permitting as it does to obtain accurate solutions of the equations of subsonic aerodynamics, though for a solid whose shape is unknown in advance.
In practice, the difficulties involved in the problems of subsonic aerodynamics are the same as those to be dealt with when investigating incompressible fluids: problems involving plane-parallel and axi-symmetric flows may be fairly easily solved by numerical methods cf. Integral-relation method ; Adjustment method. In the case of spatial flows with discontinuous velocity potential, solutions can only be obtained for linearized equations.
These equations can be converted, by an elementary transformation, into equations for incompressible fluids in which the boundary conditions and the conditions on the surface of discontinuity are the same as those for incompressible fluids.
The zone of transsonic aerodynamics is, from the point of view of mathematics, including numerical mathematics, the most difficult to study. The presence of local supersonic parabolic zones which almost always terminate in shock waves precludes any analytic approximation.
Moreover, these problems display the principal drawback of equations of elliptic type — the propagation of the effect of any particular disturbance throughout the space. The most suitable method for solving the problems of transsonic aerodynamics is the adjustment method, in which a non-stationary aerodynamic problem is solved starting from an arbitrary initial state , and the solution of the stationary problem of transsonic flow around a solid is obtained as a limit solution of the non-stationary problem as the time tends to infinity. This method is very laborious, but is quite feasible if electronic computers are employed.
As regards mathematical methods of study, supersonic aerodynamics may be subdivided into three fields. At higher velocities and, consequently, at higher temperatures ionization takes place and radiation processes become significant. This part of hypersonic aerodynamics may be said to form a separate field, since the mathematical problems involved are quite different from those arising in the case of "transparent" gases.
Problems in pure supersonic flow have been most thoroughly investigated. This included the development of numerical methods of characteristics, finite difference methods and methods of semi-characteristics, by which relatively simple solutions may be obtained not only for plane-parallel or axi-symmetric but also for spatial flows. The linear theory of supersonic flow has also been exhaustively studied. Analytic solutions can be obtained of numerous practical problems.
The latter may be somewhat complicated by the occurrence of weak shock waves, but even these difficulties are of a computational and not of a principle character. Practically speaking, these problems are limited to flows around sharply pointed solids and to the problems of internal aerodynamics calculations of jet nozzles.
However, if the supersonic velocities are very high, sharp-edged solids are not employed charring of the edges , while a local subsonic flow always occurs at the blunt end. Having computed this region, the remaining pure supersonic flows are calculated by the methods of supersonic aerodynamics. As compared with the aerodynamics of transsonic flow, the advantage of the problems of aerodynamics of supersonic flow with local subsonic zones even though such zones are mixed consists in the fact that such zones usually represent bounded narrow domains in the vicinity of the blunt ends.
This is why it was possible to develop effective numerical methods for the computation of local subsonic zones the method of integral relations, the method of inverse problems and the adjustment method. It should be noted, however, that up to the time of writing no rigorous mathematical investigations of these problems have ever been performed. There is also no proof that their solutions do exists and are unique.
Accordingly, numerical methods are developed on the assumption that the conditions of continuous velocities and accelerations during the transition from the subsonic to the supersonic zone ensure the uniqueness of a solutions, which is physically realistic.
This hypothesis is confirmed by all the results of numerical calculations. Up till here, it has been said that the computation of the local subsonic zone including the supersonic part of the region of influence is followed by calculations of the flow by methods of purely supersonic aerodynamics.
There is another type of problems, which are connected with hypersonic flow around thin solids prior to the initiation of chemical reactions. Hypersonic flows around thin solids excepting the blunt-shaped regions are distinguished by small changes in the velocity component along the main stream.
For this reason the equations may be simplified so that the problem of flow around a thin body of a given shape in the planar and axi-symmetric cases becomes identical to the one-dimensional stationary problem. Many important qualitative characteristics of hypersonic flows have been obtained in this way and approximate similitude laws have been established; these are also extensively employed in the analysis of the results of numerical calculations, which are then reduced to very compact relations in a very wide range of Mach numbers and geometric parameters of the solid.
Aerodynamic problems concerned with gases undergoing chemical reactions involve the simultaneous solution of the equations of motion and the equations of chemical kinetics. While the resulting systems of equations are more complicated, the numerical methods employed in their solution are essentially the same as those employed in the aerodynamics of an ideal gas, except that they are much more laborious. Such calculations are extensively carried out in practical work, and since modelling natural conditions in aerodynamic tubes cannot be realized in this temperature range, the aerodynamic characteristics of hypersonic instruments are mainly obtained by numerical methods.
There are two principal trends in the theory of viscous fluids — the theory of the complete equations of a viscous fluid Navier—Stokes equations , and boundary-layer theory. The boundary-layer equations represent the principal term of the asymptotic expansion of the Navier—Stokes equations in the vicinity of the boundary of the solid to which the fluid particles adhere, completely or in part. It is this fact which radically disturbs the solutions obtained for ideal fluids near the surface of the solid around which the flow takes place. The order of the error involved in the boundary-layer equations is , which means that the theory is valid for large Reynolds numbers only, and even then only in the regions of smooth continuous flow.
Despite the fact that the boundary-layer equations contain all the main terms of viscous stresses, their mathematical structure is much simpler. While the complete equations are of elliptic type, the boundary-layer equations are parabolic with characteristics directed along the normal to the surface of the solid. It is therefore possible to conduct the computations "layer by layer" , i. One consequence of the practical importance of the theory of boundary layers computations of the resistance, of the surface temperatures, of the rate of destroyment of the surface under hypersonic flight conditions was the development of numerous approximate methods for performing such calculations the Pohlhausen method, the one-parameter Kochin—Loitsyanskii method, etc.
However, if electronic computers are employed, all these approximate methods become superfluous, since an accurate numerical solution of the boundary-layer equations presents no difficulties, even in the difficult case of a hot gas in which chemical reactions are taking place. The numerical routines may be so constructed that at each step the system of equations is split, with respect to the -variable along the tangent to the surface of the solid , into a number of separate second-order differential equations which, from the computational point of view, is highly advantageous.
What was said above applies to plane-parallel and to axi-symmetric flows. The structure of three-dimensional boundary layers is more complicated. The equations of the three-dimensional layers themselves vary significantly with the geometry of the solid around which the flow takes place.
The development of methods of calculation of three-dimensional boundary layers is much less advanced, partly because calculations on plane sections are often used in practice; this, despite the lack of a theoretical basis, often yields sufficiently accurate results. There are no principal difficulties in solving the exact equations of a three-dimensional layer. Except for a few special cases, in which analytic solutions could be obtained, the development of methods for solving the complete system of equations for a viscous liquid is due to the introduction of electronic computers.