Page 319 - Computational Fluid Dynamics for Engineers

10.7 Boundary Conditions for the Euler Equations 309

2 2
where q = Vu + v , which can also be written as
2
2
(a 2 - q )cp^ + a (f vr] = 0 (10.6.4)
with
2 2
= u <p xx + 2uv(p xy + v <Py Xy CIO 6 5)

and
2
2
_ V <Pxx ~ 2uV(f xy + U (p yxy . ,
(fr]r] — 2 ^lU.O.DJ
When a 2 > q 2 (subsonic flow) is present, central differences are used just as
2
in the elliptic TSD case. However, if a 2 < q , then upwinding is more complex
than in the hyperbolic TSD case. Introducing the forward, backward and central
differencing operators A, V, <5, respectively, the discretized form of Eq. (10.6.4)
can be written as

2
2
, 2 _ 2\ (U V XV X + 2uvV xVy + V \7yVy\
K a Q ) \ 2 I Wj
/ \ (10.6.7)
2
/ ^ V x 4 + + 2 \ _
2 2uv8 x8 y u V yA y
where the first term represent upwinding, and the second term central differenc-
ing. When u < 0, then upwinding is achieved with forward-differencing (A XA X).
The same applies to the ^-derivatives if v < 0.
The solution of the conservative full potential equation requires the introduc-
tion of explicit artificial dissipation, as discussed by Jameson [4]. Alternatively,
the lagged density approach developed by Hoist [5] provides the necessary up-
winding in the supersonic region while simplifying the implementation. The
reader is referred to the original papers for details on the subject.

10.7 Boundary Conditions for the Euler Equations

The mathematical formulation of the boundary conditions for the Euler equa-
tions is more complex than the scalar wave equation or even the TSD or po-
tential equation, since there are now more than one wave travelling along char-
acteristics. The eigenvalues Ai, A2, A3 of the one-dimensional Euler equations
are respectively u, u + c and u — c, as discussed in Section 5.1. For super-
sonic flows (u > c), all waves travel in the same direction (u > 0, u + c > 0,
u — c > 0) whereas for subsonic flows, two waves are travelling in the flow di-
rection (u > 0, u + c > 0) while the third one travels against the flow direction
(u-c<0) (Fig. 10.11)
Consistent boundary conditions must be imposed for both types of con-
ditions. If the supersonic waves of Fig. 10.11a travel into the computational