The method of characteristics is a classical technique for solving problems involving the dynamics of a supersonic two-dimensional steady Inviscid Flow of a compressible fluid, which, in the general case of a plane irrotational flow of an ideal gas, can be described by a nonlinear system of two differential equations with partial derivatives of the first order
where u, v are the components of velocity vector in the xy plane, and u_{sound} is the sound velocity.
In the case of a supersonic flow, when (u^{2} + v^{2})/u^{2}_{sound} = v^{2}/u^{2}_{sound} = Ma^{2} > 1(Ma is the Mach number) the system of Equations [1] is of a hyperbolic type; in the case of a subsonic flow when Ma < 1, it is elliptic.
A characteristic feature of hyperbolic equations is the presence of characteristic surfaces (lines). For a system of equations (1) in the case of supersonic flow (Ma > 1), the characteristic lines (characteristics) used in the xy plane are the Mach lines, i.e., the lines which at a given point form with the velocity vector angles equal to the Mach angles for perturbation propagation
and the velocity vector itself is directed along the bisector of the angle between the characteristics. Thus, the projections of the velocity vector on a normal to the characteristics at a given point are equal in absolute value to a local sound velocity.
The dependent variable satisfies key reciprocity relationships for the calculation method
where
is the Prandtl-Mayer function, γ = C_{p}/C_{v}, θ is the angle between the velocity vector and the axes x, R and Q are the values (Rieman's invariant) constant along the characteristics of the first and second families, respectively,
For axisymmetric flows and other cases, the reciprocity relationships can be written in differential form which cannot be solved by quadratures, as in the plane case—but can easily be represented in finite-difference form in a numerical solution. A characteristics method of solution of the Cauchy problem, in which the components of the velocity vector at some point P must be found from the known boundary values of a certain noncharacteristic line, is shown schematically in Figure 1a.
The solution of the problem in variables (σ, θ), which can be transformed into physical variables (u, v), is done with the help of two characteristics of different families—AP and BP, according to which the value of the invariants Q_{A} and R_{B} are translated into point P. Hence,
However, because of the absence of a solution to the problem inside the region APB—as a consequence of which the characteristics AP and BP cannot be given beforehand in the numerical solution—this region is divided into small triangular elements (actually, this region is constructed from such triangles) and the solution at point P' (Fig. 1b) is found when the problem for points (1-5) is solved.
The accuracy of the solution obtained depends on the accuracy of construction of a net from the elements of characteristics, and can be increased by various means, for instance, by decreasing the size of the cells.
The region bounded by the characteristics of different families, drawn from the end points of the curve on the boundary (in Figure 1a, the region APB and the curve AB, respectively), is called the effect region
Through the use of various modifications, the method of characteristics allows the solution of problems with fixed and free boundaries, with shock waves, with nonisentropic flows, etc.
The advantage of the method is that it makes it possible to follow the breaks (shock waves) and carefully calculate them. However, the procedure of break separation differs from the procedure of solving the problem in the remaining region and if numerous breaks occur, the efficiency of the method drops.
At present, a number of finite-difference methods successfully compete with the characteristics methods. These include the methods of Lax-Vendroff and Godunov; TVD-methods, flow charts with flux splitting, etc., which are applied to more difficult problems, three-dimensional flows included.
Characteristics methods are also used for solving problems of one-dimensional, nonstationary isentropic motion of a compressed gas.
REFERENCES
Liepmann, H. W. and Roshko, A. (1957) Elements of Gas Dynamics, New York, London.
References
- Liepmann, H. W. and Roshko, A. (1957) Elements of Gas Dynamics, New York, London.