Page 340 - Computational Fluid Dynamics for Engineers

330 11. Incompressible Navier-Stokes Equations

results are consistent with those of Table 10.1 for the subsonic case since the
speed of sound is infinite for the incompressible flow equations.
When the artificial compressibility method is used, the hyperbolic system
in pseudo-time allows the use of the compressible boundary conditions dis-
cussed in Section 10.7 for inviscid and Section 12.2 for viscous flows. The one-
dimensional incompressible Navier-Stokes equations in vector-variable form,
with the pseudo-time derivative, are obtained from their two-dimensional coun-
terparts (Eqs. (P2.17.2)-(P2.17.4)),
D +
£
I J>- «> 0 (11.3.1)

where
~p~ 0u ' 0 "
D E = 2 , Ey — (11.3.2)
u i r + p .Trx _
Rewriting Eqs. (11.3.1) and applying the method of characteristics described
in Chapter 5, we obtain

d_ 6E_dD_ dE v
D (E - E v) =
dr dx dD dx dx
3D dE v xdD dE v
= A =
~ ~^—l~ ~^~~ -Al/llA 1 ——h —— (11.3.3)
ox ox ox ox
If the left-hand side is multiplied by X x l and the matrix moved inside the
spatial and pseudo-time derivative,
l
l
l
dX7 D A dX7 D dX7 E v
= —A\— 1 (11.3.4)
Or dx dx
a wave equation with an added viscous term is obtained. The sign of the eigen-
values in Ai determines the direction of the wave and it can be shown (Problem
11.1) that the eigenvalues are

A 1± = u± yju 2 + (3 (11.3.5)

These are real and have opposite signs so that the flow remains subsonic
with respect to the pseudo-sonic speed which depends on the local flow velocity
and the parameter (3
c = ^u ' 2 + (3 (11.3.6)
Chang and Kwak [7] have shown that a good choice for the parameter (3 is

2
/? 1 L 4 Y
> 1 (11.3.7)
^ref
where u and 6 are the velocity and length scales of the specific problem to be
solved.