For incompressible flows, density is constant, and the continuity equation operates as a kinematic constraint on velocity rather than a thermodynamic equation of state:

\[\nabla \cdot \vec{u} = 0\]

The **SIMPLE** algorithm couples pressure and velocity by formulating a pressure correction equation from the continuity constraint.

Mathematical Derivation

The discretized momentum equation for a velocity component \(u^*\) computed with a guessed pressure field \(p^*\) is:

\[a_P u_P^* = \sum a_{nb} u_{nb}^* + b + (p_W^* - p_E^*) A_e\]

We define the correct velocity as \(u = u^* + u'\) and correct pressure as \(p = p^* + p'\). Subtracting the guessed equations from the true momentum equations yields the velocity correction formula:

\[u_P' = d_e (p_W' - p_E') \quad \text{where} \quad d_e = \frac{A_e}{a_P}\]

Substituting this correction velocity into the continuity equation results in the Pressure Correction equation:

\[a_P p_P' = \sum a_{nb} p_{nb}' + b'\]

Where \(b'\) is the mass divergence of the guessed velocity field: \(b' = (\rho u^* A)_w - (\rho u^* A)_e\).

References

  • Patankar, S. V., & Spalding, D. B. (1972). A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer, 15(10), 1787-1806.