Total Pressure - Boundary Conditions

For incompressible flows, the density of the fluid is assumed to be constant. Therefore, Total Pressure is typically applied at the inlet and outlet boundaries of the domain. This condition allows the solver to determine the velocity field consistent with the prescribed total pressure and boundary conditions.

Mathematically, Total Pressure can be expressed as:

\(p_{0} = p_{s} + 0.5 |\vec u|^2\)

where, \(p_{s}\) is static pressure and \(p_{0}\) is total pressure.

Based on the above formula, the velocity \(U\) can be easily calculated as:

\(U = \sqrt{2(p_0-p)}\)

Please remember that for incompressible flows, the flux is expressed in [\(m^3/s\)] and pressure is expressed in [\(m^2/s^2\)] - the pressure is called the kinematic pressure.

The pressure definition includes density and is expressed as below:

\(p_{0} = p_{s} + 0.5 \rho |\vec u|^2\)

For transonic compressible flow, the relation between static and total pressure looks as follows:

\(p_{s} = \frac{p_{0}}{1 + 0.5 \psi |\vec u|^2}\)

where \(\psi\) is compressibility (for fixed composition of gas) expressed as \(\psi = (RT)^{-1}\).
\(R\) represents gas constant and \(T\) represents temperature.

Supersonic compressible flows are using 1D isentropic flow equation:

\(p_{s} = \frac{p_0}{(1 + \frac{\gamma-1}{2 \gamma} \psi |\vec u|^2)^{\frac{\gamma-1}{\gamma}}}\)

where \(\gamma = C_p/C_v\) is the ration of specific heats for a given fluid.

Example applications: pipe flow

These types of simulations can be solved using the pimpleFoam (solver) This solver has two basic independent variables: pressure and velocity. Additionally, turbulence-related variables can be defined. Total Pressure can be applied to the domain inlet if the velocity is unknown.

Example applications: free surface flows (free/entrainment boundaries), for example ship hull motion

The interFoam (solver) can be used for free surface modeling problems. When the free boundary exists in the domain, both inflow and outflow may occur on such a patch. zeroGradient for velocity and Fixed Value for pressure are not recommended as they may lead to instability. Instead, Total Pressure supports inflow and outflow through the boundary as well as improves the stability of the solution.

Example applications: fire modelling in compartments, flame propagation

The reactingFoam (solver) can be used for flame propagation. To model stable flame behavior without risk of backflows, it is recommended to apply Total Pressure at the outlet.

For incompressible flows, the only required parameter to be defined is total pressure \(p_{0}\). However, in SimFlow, all incompressible solvers are using pressure divided by density. Such pressure is called the kinematic pressure and is expressed in \(m^2/s^2\) - Figure 1

For compressible flows, the required parameter is total pressure \(p_{0}\) expressed in \(Pa\). The density \(\rho\) and heat capacity \(c_{p}\) are taken from material properties defined in Thermo section - Figure 2.