# Pressure based finite volume method

Here are placed derivation, source codes, and examples of the numerical algorithm SIMPLE-TS (Semi Implicit Method for Pressure Linked Equations – Time Step). SIMPLE-TS is a finite volume method for the calculation of unsteady, viscous, compressible, and heat-conductive flows. It can be reduced straightforwardly to calculate unsteady, viscous, incompressible, and isothermal flows. SIMPLE-TS is part of SIMPLE-like algorithms. SIMPLE-TS calculates numerically equations of the Navier-Stokes.

In SIMPLE-TS for the first time is substituted density in the unsteady term in the pressure equation with pressure and temperature using the equation of state. In this way, we turned the numerical pressure equation into a stable that does not need a relaxation coefficient to ensure convergence. Furthermore, SIMPLE-TS is five times faster than SIMPLE and slightly faster than PISO.

A derivation of the SIMPLE-TS method and considerations are available here. The web page contains:

• a detailed derivation of the numerical equations and explanations;
• a detailed explanation of differences between SIMPLE-TS and other SIMPLE-like methods considering an example of one-dimensional unsteady, isothermal pressure-driven flow in a duct;
• a derivation of the numerical equations of the algorithm SIMPLE-TS using a first-order upwind scheme for the approximation of convective terms with Mathematica;
• a derivation of the numerical equations of the algorithm SIMPLE-TS using a second-order total variation diminishing (TVD) scheme for the approximation of convective terms with Mathematica;

SIMPLE-TS source codes and examples are placed here. The web page contains:

• parallel C++ source code with first-order approximation of convective terms;
• unsteady supersonic, compressible, viscous, heat-conductive fluid flow past a confined square in a micro-channel – Mach number 2.43 and Knudsen number 0.00283 (Reynolds number 1415);
• unsteady subsonic, compressible, viscous, heat-conductive fluid flow past a confined square in a micro-channel – Mach number 0.1 and Knudsen number 0.00194 (Reynolds number 85);
• increasing velocity at the channel inflow from Mach number 2.43 to Mach number 4.86, for Knudsen number 0.05;
• Rayleigh-Bènard flow of a rarefied gas;

The main work presented in this section is published in the following papers: