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11.4 Artificial Compressibility Method: INS2D 331

11.4 Artificial C o m p r e s s i b i l i t y M e t h o d : I N S 2 D

The artificial compressibility method INS2D*, widely made popular by Roger
and Kwak [8-10], is described in this section. For simplicity, the numerical
method is discussed with the equations expressed in Cartesian coordinates for
steady flows rather than in transformed form (subsection 2.2.4) for steady and
unsteady flows. In this case, the governing equations are given in Problem 2.17,

£ £
+
+
F F
f |( - "» |< - »'= O (P2.17.2)
where
\p~ " 0u ' " 0V '
D = u E = Ei = u 2 +p F = E2 = uv (P2.17.2)
V _ uv _v 2 +p_

— (P2.17.3)
E v F v
'xy
u
L xy
The viscous normal and shear stresses are given by Eq. (2.2.£
du ' du dv ~ 9v
&x >yx ~ ^ [ — + ~^Z = 2/i— (2.2.8)
^dy dxj dy

t
11.4.1 Discretization of he Artificial Time Derivatives
INS2D uses an implicit scheme to solve Eq. (P2.17.2) subject to the boundary
conditions to be described in subsection 11.5.1, and the equation is written as

dD_
+ i? = 0 (11.4.1)
dr
where
d_
R=^(E-E V) (F - F v) (11.4.2)
dy
Application of a first order backward Euler formulas to Eq. (11.4.1) in
pseudo-time r yields
r)n+l _ j-\n
-R n+1 (11.4.3)
r n+l _ q-n
where the superscript n denotes the pseudo-time iteration count. Application
of Taylor series expansion, and noting that R is a function of D, leads to
n
Rn+1 =Rn + f | ^ V (D n+1 - D ) (11.4.4)

INS2D and related software can be obtained from NASA Ames by signing a non-
disclosure agreement. For details, see http://people.nasa.goc/~rogers/home.html