Incompressible Flow

#+CATEGORY: cfd

Incompressible Flow :: The Divergence of u Equals Zero Constraint

Fluid density is constant in incompressible flow because variation is negligibly small. The Navier-Stokes equations simplify structurally. The continuity equation decouples from energy. Pressure ceases to be an equation-of-state variable and becomes a Lagrange multiplier enforcing divergence-free velocity. This note covers the implications, approximations, and numerical strategies.

The Mathematical Change

Starting from the general governing equations, the incompressible limit is:

1. Setting rho = rho_0 (constant) in the continuity equation 2. Result: divergence of u equals zero 3. Pressure is no longer determined by an EOS. Pressure is determined by the momentum equation through the pressure Poisson equation (PPE)

The PPE is obtained by taking the divergence of the momentum equation. Apply divergence of u equals zero:

This elliptic pressure equation has global influence. A perturbation anywhere propagates instantaneously, the mathematical manifestation of infinite sound speed in the incompressible limit.

When Is Incompressible Flow a Valid Approximation?

The Mach number M = U/a (flow speed over sound speed) is the standard criterion:

This rule of thumb assumes no strong temperature gradients. Thermal effects change this picture. Natural convection and buoyancy-driven flows are examples. The Boussinesq approximation provides a midpoint between full compressibility and constant density.

Boussinesq Approximation

Natural convection and buoyancy-driven flows use small temperature differences. The differences are small relative to absolute temperature:

• Density is constant everywhere except in the buoyancy term. rho = rho_0[1 - beta(T - T_0)]
• Beta is the thermal expansion coefficient
• The momentum equation is: rho_0(partial u/partial t + u dot nabla u) = -nabla p + mu nabla^2 u + rho_0 g[1 - beta(T-T_0)]
• The energy equation solves independently for T. T feeds back only through the buoyancy term

This approximation powers many natural convection solvers. It is the default in most multiphase configurations with temperature.

The Pressure-Velocity Coupling Problem

No equation of state links pressure and density in incompressible flow. No explicit pressure equation exists. The velocity-pressure coupling must be handled algorithmically by one of two approaches:

1. Staggered-grid methods: Velocity and pressure are stored at different locations in the cell. The pressure correction (SIMPLE, PISO) iterates until continuity is satisfied. See Pressure-velocity coupling. 2. Projection methods: Compute an intermediate velocity from momentum without enforcing continuity. Project it onto a divergence-free space via a Poisson solve.

Modern OpenFOAM solvers (see Solver selection) fall into category 1 or derivatives thereof.

Numerical Implications

The pressure equation is the dominant cost. It requires global MPI communication across all processors. It needs careful boundary condition specification.

Neumann conditions on p come from normal momentum balance at boundaries.

The absence of acoustic waves means no CFL restriction from sound speed. CFL depends solely on convective speed, enabling larger time steps than compressible flow. Explicit compressible solvers are expensive by comparison.

Common Incompressible Solvers (OpenFOAM)

LES or DNS with incompressible flow uses a PISO or PIMPLE solver. The solver has a subcycle structure. The subcycles enforce continuity at each inner loop iteration.

Turbulence in Incompressible Flow

The RANS equations use the same form as the laminar equations. The variables are fluctuating. Reynolds averaging leads to the Reynolds-Averaged Navier-Stokes equations. The Reynolds stress tensor requires modeling.

See Solver selection for RANS versus LES guidance.

Summary

1. Incompressible flow eliminates the energy-momentum coupling (unless Boussinesq) 2. Pressure becomes a global constraint enforced via the PPE 3. No acoustic CFL limit allows larger time steps 4. Pressure-velocity coupling is the central algorithmic challenge 5. The incompressible regime handles the vast majority of engineering CFD

See Also

• Governing Equations
• Compressible Flow
• Pressure-Velocity Coupling
• Turbulence Models
• OpenFOAM Solver Selection