Pressurevelocity

Pressure-velocity refers to the relationship and coupling between the pressure field and the velocity field in fluid dynamics, especially in the numerical solution of the Navier–Stokes equations. In incompressible flows, the velocity field must satisfy mass conservation, and the pressure acts as a Lagrange multiplier that enforces this constraint. The result is a coupled system where pressure and velocity are determined together through iterative procedures.

In incompressible flow, the Navier–Stokes equations are written as rho (du/dt + u·∇u) = -∇p + μ∇^2u + f with ∇·u = 0. Because the velocity field does not uniquely determine pressure, a Poisson equation for pressure is derived by taking the divergence of the momentum equation and applying the continuity condition. Solving this pressure Poisson equation provides a pressure correction that updates the velocity field so that mass conservation is satisfied. This pressure-velocity coupling is central to many computational fluid dynamics (CFD) schemes.

Common pressure-velocity coupling algorithms include SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), SIMPLER, and PISO (Pressure-Implicit with Splitting of Operators). These segregated approaches alternate solving momentum and pressure equations and apply corrections to adjust velocity and pressure. For transient problems, PISO adds additional correction steps for improved accuracy. Fully coupled solvers or hybrid methods such as PIMPLE combine aspects of these strategies to handle complex flows.

Applications span engineering and science, including internal pipe and duct flows, external aerodynamics, hydraulics, and multiphase problems. The choice of coupling strategy influences stability, accuracy, and computational cost and is guided by flow regime, mesh quality, and time-stepping requirements.

See also: Navier–Stokes equations, Computational fluid dynamics, Poisson equation, Pressure correction methods, Lagrange multiplier.