Know Thy Solver – Part III: Pressure Equation Methodologies for Incompressible Flow

In compressible flows the continuity equation can be used to determine
the density and the pressure can be calculated from an equation of state. This approach is not appropriate for incompressible or low Mach number flows.
When considering the incompressibility assumption as valid, Navier-Stokes equations supplemented by continuity take the following recognizable form (body forces neglected):

Finding a solution for the above is complicated by the lack of an independent equation for the pressure, whose gradient contributes to each of the three momentum equations. Furthermore, the continuity equation is a kinematic constraint on the velocity field rather than a dynamic equation, stating that the velocity vector field is solenoidal (divergence free).

As we saw in “Know Thy Solver- PART II: Projection Methods” one way out of this difficulty is to construct the pressure field so as to guarantee satisfaction of the continuity equation.
The underpinning for such methodologies is recognizing that the momentum equations determines the respective velocity components so their roles are clearly defined. Furthermore, as far as the incompressibility assumption is valid, absolute pressure is of no significance but only the pressure difference (gradient of the pressure) affects the flow.
This leaves the continuity equation which in its incompressible form does not contain the pressure explicitly – to determine the pressure.

A hint for what comes next we saw in “Know Thy Solver- PART II: Projection Methods”.
The basic philosophy behind segregated algorithms stems from the fact that although we can solve the momentum equations for the velocity field at time step n+1 using the velocities and pressure gradient from the previous time step, there is no reason why this should lead to mass conservation at time level n + 1.
Our route then to obtain a solenoidal (divergence-free) velocity field at the advanced time step is to try and construct a pressure equation such that
if satisfied enforces continuity at the advanced time step.

Before considering commonly used methods represented by (implicit) segregated algorithms for solving the steady state Navier-Stokes equations, let us look at a method for the unsteady equations that illustrates how the numerical Poisson equation for the pressure is constructed and the role it plays in enforcing continuity.

The choice of type of approximation for the spatial derivatives will not be important for illustrative purposes, so for what follows I shall use δ to represent discretized spatial derivatives (it could even be different kind of approximation for different terms, i.e. without loss of generality).
Furthermore, B will be a shorthand notation for the advective + viscous terms of which computation is generally essential but shall not be important to expand on for the illustration.
Now let us write the momentum equations, discretized in space but not in time:

(We are interested in incompressible flows, but these include flows with variable density)

We shall assume an explicit (i.e. the only n+1 term in the equation is what we are calculating) Euler method:

Now, say we have the velocity and pressure from the previous time step n, we can calculate B and the pressure gradient for that time step. This gives us an estimation for the (ρu) at time step n+1 which in general does not satisfy the continuity equation:

To illustrate how continuity may be enforced I will take the divergence (the numerical spatial approximation δ) of the equation marked in red:

I —> The divergence of the new velocity field which we want to be zero if continuity is to be satisfied.

II —> The divergence of the velocity field from previous time step which we is zero since we assume continuity was enforced at previous time step.
(Retaining this term is necessary when an iterative method is used to solve the Poisson equation for the pressure and the iterative process has not fully converged).

Following I and II the result is the (discrete) Poisson equation for the pressure at time step n which if satisfied would mean that the velocity field for time step n+1 will be divergence free (in other words if III is zero than I is zero):

The above illustrates just how solving a pressure poisson equation can enforce a divergence free velocity field (i.e. satisfy continuity).
The fact of matter is that similar methods as described above could be used when accurate time history solution for NSE is required, the difference being the use of higher order time advancement methods rather than the simplest explicit Runge-Kutte method as first order Euler.

Steady-state problems may be regarded as solving an unsteady problem until steady state is reached, but while in unsteady problems there is a need to choose a time step small enough such that the time history is obtained accurately, for a steady state solution large time steps may be used to reach a steady state solution if possible.
Explicit methods are faster to converge than implicit ones for fast-transient flows for which the time step needed to accurately capture physical phenomena is comparable to that of the time step the Courant–Friedrichs–Lewy (CFL) condition restriction allows. However, for steady-state or slow-transient flow, a much less stringent restriction on time step of implicit methods make them the preferred choice.
Implicit algorithms such as SIMPLE, SIMPLEC and PISO are of the common types used in the calculation of slow-transient and steady incompressible flows, and may be regarded as variations on the methodology presented above, i.e. they use a pressure equation to enforce a (numerically) divergence-free velocity field (continuity) at each outer iteration (which is somewhat synonymous to the time step illustrated above).

The following post “Know Thy Solver – Part IV: Implicit Pressure-Correction Algorithms” the above illustrated concept will be adapted for implicit algorithms (along with some additional difficulties that rise from the implicitness of the discretized equations), and three pressure equations (pressure-correction) variations, those of SIMPLE, SIMPLEC and PISO shall be described along with their implications for steady and slow-transient incompressible NSE calculations.

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