Theoretical background

In the preceeding example and excercises, we have used the simpleFoam solver to compute steady-state incompressible flow. This solver computes flow according to a variant of our simplified equations.

(9)\[\nabla \cdot \vec{U} = 0\]

(10)\[\nabla \cdot (\vec{U} \vec{U}) = \nabla \cdot (\nu \nabla \vec{U}) - \nabla p\]

Let’s have a closer look at how this solver actually works.

Solution strategies - the SIMPLE algorithm

The 3-D Navier-Stokes equation has four unknowns (three velocity components and one pressure) but the standard formulation does not include an explicit “pressure equation”, which complicated solving the system of equations.

Tip

There is a very nice youtube video by FluidDynamics101 (Aidan Winshurst) that explains the SIMPLE algorithm. We are here following his derivation and notation.

The SIMPLE algorithm solves the equations stated above by deriving a pressure equation by combining the momentum and continuity equations and by formulating a corrector equation to make the velocity field satisfy the continuity equation.

Step 1 is to formulate the momentum balance as general matrix equation:

(11)\[\nabla \cdot (\vec{U} \vec{U}) = \nabla \cdot (\nu \nabla \vec{U}) - \nabla p\]

(12)\[M \vec{U} = - \nabla p\]

Next we split the matrix \(M\) into a diagonal part and remaining part.

(13)\[M \vec{U} = - \nabla p\]

(14)\[A \vec{U} -H = - \nabla p\]

The nice thing about diagonal matrices is that they are easy to invert.

Step 2 is to solve for \(\vec U\) using an initial guess for the pressure field.

(15)\[\vec{U} = A^{-1} H - A^{-1}A \nabla p\]

and plug it into the continuity equation:

(16)\[ \nabla \cdot \vec{U} = 0\]

(17)\[ \nabla \cdot (A^{-1} H - A^{-1}A \nabla p) = 0\]

to get

(18)\[ \nabla \cdot A^{-1} \nabla p = \nabla \cdot(A^{-1}H)\]

Equations equation (12) and equation (17) form a set of equations that are solved in sequence.

The actual solution sequence in the SIMPLE algorithm is this. First, we solve the momentum balance using an initial guess for pressure:

(19)\[M \vec{U} = - \nabla p\]

The we update pressure by solving the pressure equation equation (17)

(20)\[ \nabla \cdot A^{-1} \nabla p = \nabla \cdot(A^{-1}H)\]

Finally, the pressure is used to “correct” the velocity solution, so that it fulfills the continuity equation equation (9) .

(21)\[ \vec{U} = A^{-1} H - A^{-1}A \nabla p\]

This is the SIMPLE algorithm for the steady-state incompressible Navier-Stokes equation. Later we will also learn about other algorithms for more general forms of the N-S stokes like the PISO and PIMPLE algorithms.