Hydrostatic Total Pressure - Boundary Condition Description
Hydrostatic Total Pressure is a fixed-value boundary condition for the pseudo-hydrostatic pressure \(p \_ \rho gh\) - the “reduced-gravity” pressure that many buoyant and multiphase solvers solve instead of the raw static pressure p. The static pressure \(p \_ \rho gh\) is assigned based on the User-provided total pressure \(p_0\), and it takes into account local dynamic head \(0.5 \rho U^2\) minus the hydrostatic head \(\rho g (h-h_{ref})\). The boundary condition is designed for cases where there is a hydrostatic pressure variation in the flow due to gravitational effects, particularly for buoyant and multiphase flows.
Hydrostatic Total Pressure works similarly to Total Pressure, but instead for solving for the static pressure \(p\), the pseudo-hydrostatic pressure field is solved \(p \_ \rho gh\).
Typical applications of Hydrostatic Total Pressure are:
- Open boundaries in free-surface (VOF) flows,
- Compressible buoyant flows,
- Hydraulic-head / open-channel problems,
- HVAC.
Hydrostatic Total Pressure - Boundary Condition Understanding Hydrostatic Total Pressure
Mathematically, Hydrostatic Total Pressure can be expressed as:
\(p \_ \rho g h = p - \rho g (h - h_{ref})\)
where:
- \(p \_ \rho g h\) – the pseudo hydrostatic pressure \(Pa\),
- \(p\) - static pressure \(Pa\),
- \(\rho\) – density,
- \(g\) – gravity vector,
- \(h\) – height in the opposite direction to gravity,
- \(h_{ref}\) – reference height in the opposite direction to gravity.
To calculate the static pressure \(p\), the User-defined total pressure \(p_0\) is used:
\(p = p_0 - 0.5 \rho |U|^2\)
The reason why \(p \_ \rho gh\) term is used is driven by several factors:
- numerical conditioning - the hydrostatic term \(\rho g h\) can be orders of magnitude larger than the dynamic pressure variations that really matter to the flow. Removing it shrinks the range of numbers being solved for, so the Poisson matrix is far better conditioned and round-off error is reduced. Convergence is noticeably faster and more robust, especially for tall domains or high-density fluids,
- cleaner momentum equation - in the Navier–Stokes momentum equation the gravity term and the hydrostatic part of \(\nabla p\) cancel exactly. Writing the equation in terms of p_rgh makes that cancellation explicit, leaving only the buoyancy contribution due to density differences \(\nabla (\rho) g\). This isolates the true driving force of natural-convection flows.
- straight-forward boundary conditions - many real boundaries are specified by a known static pressure, (e.g. 0 Pa gauge at an opening). With \(p \_ \rho gh\), users can still impose that familiar static value through the Hydrostatic Total Pressure, which converts it automatically at every face, accounting for local height and changing density. You don’t have to hand-build a tricky, height-dependent profile yourself
Hydrostatic Total Pressure - Boundary Condition Application & Physical Interpretation
Hydrostatic Total Pressure is particularly important in simulations where gravity plays a significant role, especially in multiphase or buoyancy-driven flows.
Hydrostatic Total Pressure in HVAC applications
Example applications: room cooling and heating
The buoyantPimpleFoam (solver) can be used for room cooling and heating simulations.
| Physics | Modified Pressure | Velocity | T |
|---|---|---|---|
Inlet | Fixed-Flux Pressure | Fixed Value | Fixed Value |
Outlet | Hydrostatic Total Pressure | Pressure Inlet-Outlet Velocity | Zero Gradient |
Hydrostatic Total Pressure - Boundary Condition Hydrostatic Total Pressure in SimFlow
The definition of boundary conditions in SimFlow is both simple and intuitive. To specify the Hydrostatic Total Pressure boundary condition, the user must navigate to the Boundary Conditions panel, select the appropriate boundary for the pressure , and choose the correct option from the drop-down menu Figure 1.
Hydrostatic Total Pressure - Boundary Condition Hydrostatic Total Pressure - Alternatives
In this section, we propose boundary conditions that are alternative to Hydrostatic Total Pressure. While they may fulfill similar purposes, they might be better suited for a specific application and provide a better approximation of physical world conditions.
| Boundary Condition | Description |
|---|---|
fixed static pressure with hydrostatic correction but without the stagnation term | |
defines total pressure condition at the boundary | |
defines static pressure condition at the boundary |