Preconditioned Equations

To provide efficient solution of both compressible and incompressible flows at all speeds, a preconditioning matrix is incorporated into Eqn. 37 as follows [4]:

(41)

where:

(42)

and:

is the dependent vector of primary variables.

In Eqn. 42, is the derivative of density with respect to temperature at constant pressure and  = 0 or 1. For an ideal gas ,  = 1 and this matrix becomes a member of Turkel's family of preconditioners [2]. For an incompressible fluid, both and are zero. The parameter is defined as:

(43)

The reference velocity appearing in Eqn. 43 is chosen such that the eigenvalues of the system remain well conditioned with respect to the convective and diffusive time scales [4]. This is accomplished by limiting such that is does not go below either the local convection or diffusion velocities. An additional limitation on considers local pressure differences and is designed to increase numerical stability in stagnation regions by prohibiting amplification of pressure perturbations there. Thus, the restriction on becomes:

(44)

where is the inter-cell length scale over which the diffusion occurs and is the pressure difference between adjacent cells. For compressible flows is further limited such that it does not exceed the local acoustic speed c. The scaling parameter appearing in Eqn. 44 is set to 10^-3; which so far appears to be sufficient to ensure stability.

The non-preconditioned Navier-Stokes equations are recovered exactly from Eqn. 37 by setting to , the derivative of density with respect to pressure. In this case reduces to the Jacobian .