Pressure Velocity Coupling

Pressure Velocity couplings in steady flows

The convection of a scalar variable φ depends on the magnitude and direction of the local velocity field

In the previous chapter we assumed that the velocity field was some how known. But in general it is not known

The solution of equations presents us with two new problems:

The convective terms of the momentum equation contain non-linear quantities All three equations are intricately coupled because every velocity component

appears in each momentum equation and the continuity equation. The most complex issue to resolve is the role played by the pressure. It appears in both momentum equations, but there is evidently no (transport or

other) equation for pressure. If the flow is compressible the continuity equation may be used as a transport

equation for density and, in addition to momentum equations, the energy equation is a transport equation for temperature.

pressure may then be obtained from the density and temperature by using the equation of state p= p(ρ,T).

However, if the flow is incompressible the density is constant and hence by definition not linked to the pressure.

In this case coupling between pressure and velocity introduces a constraint on the solution of the flow field: if the correct pressure field is applied in the momentum equations the resulting velocity field should satisfy continuity.

• Both the problems associated with the non-linearities in the equation set and the pressure-velocity linkage can be resolved by adopting an iterative solution strategy such as

• SIMPLE algorithm

• SIMPLER algorithm

• SIMPLEC algorithm

• PISO algorithm, etc.,

The Staggered grid

• First we need to decide where to store the velocities. It seems logical to define these at the same locations as the scalar variables such as pressure, temperature etc.

• However, if the velocities and pressures are both defined at the nodes of an ordinary control volume a highly non-uniform pressure field can act like a uniform field in the discretised momentum equations.

• This can be demonstrated with the simple two-dimensional situation shown in Figure 6.1,

Fig.6.1 A checker-board' pressure field

The Staggered grid

It is observed that all the discretised gradients are zero at all the nodal points even though the pressure field exhibits spatial oscillations in both directions. As a result, this pressure field would give the same (zero) momentum source in the discretised equations as a uniform pressure field.

This behaviour is obviously non- physical.

• It is clear that, if the velocities are defined at the scalar grid nodes, the influence of pressure is not properly represented in the discretised momentum equations.

• A remedy for this problem is to use a staggered grid for the velocity components

• The idea is to evaluate scalar variables, such as pressure, density, temperature etc., at ordinary nodal points but to calculate velocity components on staggered grids centred around the cell faces.

• The arrangement for a two-dimensional flow calculation is shown in Figure 6.2.

Now the checker-board yields very significant non-zero pressure gradient terms.

The staggering of the velocity avoids the unrealistic behaviour of the discretised momentum

equation for spatially oscillating pressures like the 'checker-board' field.

A further advantage of the staggered grid arrangement is that it generates velocities at exactly the

locations where they are required for the scalar transport - convection-diffusion - computations.

Hence, no interpolation is needed to calculate velocities at the scalar cell faces.

u-control volume and its neighbouring velocity components

V-control volume and its neighbouring velocity components

The convective flux per unit mass F and the diffusive conductance D at v-control volume cell faces.

U momentum equation

where ΔVU is the volume of the w-cell, bi,j =S ΔVU is the momentum source term, Ai,j is the (east or west) cell face area of the w-control volume.

The pressure gradient source term in F.8) has been discretised by means of a linear interpolation between the pressure nodes located at the w-control volume boundaries.

Similarly v momentum equation

The SIMPLE algorithm

• The acronym SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations.

• The algorithm was originally put forward by Patankar and Spalding and is essentially a guess-

and-correct procedure for the calculation of pressure on the staggered grid arrangement

introduced above.

• To initiate the SIMPLE calculation process a pressure field p* is guessed.

• Discretised momentum equations are solved using the guessed pressure field to yield velocity

components u* and v* as follows:

The SIMPLE algorithm

The SIMPLE algorithm

The SIMPLE algorithm

Fig. 6.5 The scalar control volume used for the discretisation of the continuity equation

Equation represents the discretised continuity equation as an equation for pressure

correction p'.

The source term b' in the equation is the continuity imbalance arising from the incorrect

velocity field u*, v*

By solving equation 6.32, the pressure correction field p' can be obtained at all points.

Once the pressure correction field is known, the correct pressure field may be obtained

using formula 6.14 and velocity components through correction formulae 6.24-6.27).

The omission of terms such as Σanbu`nb, in the derivation does not affect the final solution

because the pressure correction and velocity corrections will all be zero in a converged

solution giving p*= p, u*= u and v*= v.

• The pressure correction equation is susceptible to divergence unless some under-

relaxation is used during the iterative process and new, improved, pressures p new are

obtained with

• where αp is the pressure under-relaxation factor.

• If we select αp equal to 1 the guessed pressure field p* is corrected by p'.

• However, the corrections p', in particular when the guessed field p* is far away from

the final solution, is often too large for stable computations.

• A value of αp equal to zero would apply no correction at all,which is also undesirable.

• Taking αp between 0 and 1 allows us to add to guessed field p* a fraction of the

correction field p' that is large enough to move the iterative improvement process

forward, but small enough to ensure stable computation.

The simple algorithm

SIMPLER algorithm

The SIMPLER (SIMPLE Revised) algorithm of Patankar A980) is an improved

version of SIMPLE.

In this algorithm the discretised continuity equation 6.29) is used to derive a discretised

equation for pressure, instead of a pressure correction equation as in SIMPLE.

Thus the intermediate pressure field is obtained directly without the use of a correction.

Velocities are, however, still obtained through thevelocity corrections 6.24-6.27) of

SIMPLE.

SIMPLER algorithm

The SIMPLEC algorithm

The PISO algorithm

• The PISO algorithm, which stands for Pressure Implicit with Splitting of Operators, is a pressure-velocity calculation procedure developed originally for the non-iterative computation of unsteady compressible flows.

• It has been adapted successfully for the iterative solution of steady state problems.

• PISO involves one predictor step and two corrector steps and may be seen as an extension of SIMPLE, with a further corrector step to enhance it.

Predictor step

Discretised momentum equations 6.12-6.13 are solved with a guessed or intermediate

pressure field p* to give velocity components u* and v* using the same method as the

SIMPLE algorithm.

Corrector step 1

The fields u* and v* will not satisfy continuity unless the pressure field p* is correct.

The first corrector step of SIMPLE is introduced to give a velocity field (u**, v**)

which satisfies the discretised continuity equation.

The resulting equations are the same as the velocity correction equations 6.21-6.22 of SIMPLE

but, since there is a further correction step in the PISO algorithm,

we use a slightly different notation:

As in the SIMPLE algorithm equations 6.50-6.51 are substituted into the discretised continuity equation F.29) to yield the pressure correction equation 6.32 with its coefficients and source term. In the context of the PISO method equation 6.32 is called the first pressure correction equation. It is solved to yield the first pressure correction field p'.Once the pressure corrections are known, the velocity components w** and v** can be obtained through equations 6.50-6.51).

PISO algorithm