Question

Describe:

Where the pressure and density relations are used in SIMPLE-based solutions.

The need for characterizing or defining the compressibility constant.

The most important computational rules in an iterative solution process.

Answer 1

1- The pressure-based approach was developed for low-speed incompressible flows, while the density-based approach was mainly used for high-speed compressible flows. However, recently both methods have been extended and reformulated to solve and operate for a wide range of flow conditions beyond their traditional or original intent."
"In both methods the velocity field is obtained from the momentum equations. In the density-based approach, the continuity equation is used to obtain the density field while the pressure field is determined from the equation of state."
"On the other hand, in the pressure-based approach, the pressure field is extracted by solving a pressure or pressure correction equation which is obtained by manipulating continuity and momentum equations."
The pressure-based solver traditionally has been used for incompressible and mildly compressible flows. The density-based approach, on the other hand, was originally designed for high-speed compressible flows. Both approaches are now applicable to a broad range of flows (from incompressible to highly compressible), but the origins of the density-based formulation may give it an accuracy (i.e. shock resolution) advantage over the pressure-based solver for high-speed compressible flows."

2- Compressibility factor, usually defined as Z = pV/RT, is unity for an ideal gas.The compressibility factor is used as a correction factor to ideal behavior. It is a measure of how much the thermodynamic properties of a real gas deviate from those expected of an ideal gas. It may be thought of as the ratio of the actual volume of a real gas to the volume predicted by the ideal gas at the same temperature and pressure as the actual volume.For an ideal gas, Z always has a value of 1.

Thus, v_real = Z v_id is used to calculate the actual volume, v_real, as the product of the compressibility factor and the ideal gas volume, all at the same pressure and temperature.

Z is most commonly found from a generalized compressibility factor chart as a function of the reduced pressure, p_r = p/p_c, and the reduced temperature, T_? = T/T_c where p_r and T_r are the reduced variables and the subscript 'c' refers to the critical point.

3- Iterative methods:

a) RF(Richardson's) Method

b) Jacobi Method

c) Gauss-seidel Method

d) Successive over-relaxation Method

e) Symmetric SOR method

In iterative process,the convergence property of an iterative solution method is the most important aspect.