<aside> 💡 The mathematical complexity of convection heat transfer is traced to the non-linearity of the Navier-Stokes equations of motion and the coupling of flow and thermal fields.

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Difficulties: Non-linearity in solving Eqns and pressure gradient.

The convective term in the momentum equation $(\nabla .(UU))$ is non-linear.

The pressure gradient behaves like a source term for momentum Eqn. For a specified pressure field, there is no difficulty in solving momentum Eqn. The challenging task is to determine the pressure field.

The pressure field is indirectly linked with continuity Eqn.

When right/correct pressure field is plugged into the momentum Eqn, the resulting velocity field should satisfy the continuity Eqn.

We need to get any algorithms to solve this.,

1. SIMPLE
2. PISO are the most commonly used ones.

SIMPLE:

1. Derive equation for pressure from the momentum and continuity equations.
2. Derive a corrector for the velocity field, so that it satisfies the continuity equation.

Deriving the equations

Matrix Decomposition

The first stage is to express the momentum equations in the general matrix form:

$MU = -\nabla p$

M is the matrix of coefficients for a given set of system, by discretizing the terms in the equation. The coefficients all known.

M is derived from decomposing the momentum equation, using FEM, FDM or FVM methods