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328 11. Incompressible Navier-Stokes Equations

is called SIMPLE (Semi-Implicit Method for Pressure Linked Equations). A
discussion of the method and its derivatives (SIMPLER, SIMPLEC and PISO)
can be found in [4]. The pseudo or artificial compressibility method developed
by Chorin [5] introduces an additional term in the continuity equation which
transforms it from elliptic in space only to elliptic in space and hyperbolic in
pseudo-time, hence its name. The artificial waves originating from the additional
derivative term provide a mechanism for propagating information throughout
the domain and drive the divergence of velocity towards zero at steady state.
Since the methods of solving a system of travelling waves have been discussed
in Chapters 5 and 10 (upwind schemes or central schemes with added artificial
dissipation), the artificial compressibility method is discussed in Section 11.4.
The discretization procedure of the time derivatives, convective and diffusive
fluxes is detailed, and the use of the ADI procedure explained. In Section 11.5,
its application to a sudden expansion laminar duct flow is discussed and the
convergence of the solutions as a function of time steps and pseudo compress-
ibility parameter /3 is studied. The application of the method to a laminar and
turbulent flow over a flat plate is discussed in Section 11.6 and to multi-element
airfoils in Section 11.7.

t
11.2 Analysis of he Incompressible Navier—Stokes Equations
The incompressible Navier-Stokes equations in tensor notation are given by
Eqs. (P2.2.1) and (2.2.2), that is,

^ = 0 (P2.2.1)
dxi

d l
-£ +u^ = - - ^ + pL +fi (P2.2.2)
Ot OXj Q OXi OXj
where the viscous stresses GIJ are given by Eq. (2.2.7). The continuity equation
which remains the same for unsteady and steady incompressible flows has an
elliptic character. The conservation of mass appears as a kinematic constraint
to the solution of the momentum equation. Hence, there is no obvious means to
couple the velocity and the pressure, which is done in the case of compressible
flows through the density and an equation of state relating the density and
the pressure. One way to obtain a relation is to take the divergence of the
momentum equation while making use of the continuity constraint; this results
in
2
I y P = -V(i7- V)v + V • / (11.2.1)
Q
the Poisson equation for the pressure once the velocity field is known. This
equation forms the basis of the pressure correction methods, which first solve the