Page 156 - Computational Fluid Dynamics for Engineers

5.1 Introduction 143

t

y\ u + c

/

"I *~ x

Fig. 5.1. Characteristic lines for one-dimensional unsteady flow.

These eigenvalues give the shapes of the characteristic lines in the x-t plane, as
shown in Fig. 5.1. At a given point in the (x,£) plane, there are three charac-
teristic lines with shapes
dt 1 1 1 1 1 1 . ^
5 L9
;T = T- = - ' !- = —]—' ~r = ' ( -
dX X\ U A2 U + C A3 U — C
respectively. The eigenvalue Ai indicates that information is propagated by a
fluid element moving at velocity u; the eigenvalues A2 and A3 indicate that
information is propagated to the right and left, respectively, along the x-axis
at the local speed of sound relative to the moving fluid element. In Fig. 5.1,
the curve with shape equal to \/u is called a particle path and the curves with
shapes l/(u + c) and l/(u — c) are right- and left-running Mach waves.
The characteristic lines play a significant role in the development of numeri-
cal methods since information concerning a flowfield travels along the character-
istic curves (Section 2.6). Since the eigenvalues of the Jacobian matrix give the
shapes of the characteristics and their values represent the velocity and direc-
tion of propagation of information, the solution procedure should be consistent
with the velocity and direction which information propagates throughout the
flowfield.
In flows where the flow variables are smooth and continuous, it is gener-
ally satisfactory to use central-difference schemes to solve hyperbolic equations.
When the flow variables are not smooth, for example, when there are discon-
tinuities in the flowfield such as shocks, the central-difference schemes are not
satisfactory; the solutions exhibit oscillations which may be considerable in the
vicinity of the discontinuity and can lead to unacceptable results. Upwing differ-
ence schemes are devised to overcome the shortcomings of the central schemes
and, thus to obtain solutions of hyperbolic equations.