1. Solution Algorithms for Pressure-Velocity Coupling in Steady Flows
Lecture 07

2. Issues with pressure-velocity calculations
The convection of a scalar variable  depends on the magnitude and direction of the local velocity field.
It was assumed that the velocity field was somehow known.
In general the velocity field is, however, not known and emerges as part of the overall solution process along with all other flow variables.
Transport equations for each velocity component - momentum equations – can be derived from the general transport equation by replacing the variable  by u, v and w respectively.
The velocity field must also satisfy the continuity equation.
Let us consider the equations governing a two-dimensional laminar steady flow:
(1)
(2)
(3)

3. Issues with pressure-velocity calculations…
The pressure gradient term, which forms the main momentum source term in most flows of engineering importance, has been written separately to facilitate its importance.
The solution of equation set 1-3 presents us with two new problems:
The convective terms of the momentum equation contain non-linear quantities, for example the first term of equation (1) is the x-derivative of u2.
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.

4. Issues with pressure-velocity calculations…
If the pressure gradient is known, the process of obtaining discretised equations for velocities from the momentum equations is similar to that for any other scalar, and schemes (e.g. hybrid, upwind etc.) are applicable.
In general purpose flow computations we also wish to calculate the pressure field as part of the solution so its gradient is not normally known beforehand.
If the flow is compressible the continuity equation may be used as a transport equation for density and, in addition to eq. 1-3, the energy equation is a transport equation for temperature.
The pressure may then be obtained from the density and temperature by using the equation of p=p(, 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.

5. Issues with pressure-velocity calculations…
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 the SIMPLE algorithm of Patankar and Spalding (1972).
SIMPLE: Semi Implicit Pressure Linked Equation
In this algorithm the convective fluxes per unit mass F through cell faces are evaluated from so-called guessed velocity components.
Furthermore, a guessed pressure field is used to solve the momentum equations and a pressure correction equation, deduced from the continuity equation, is solved to obtain a pressure correction field which is in turn used to update the velocity and pressure fields.
To start the iteration process we use initial guesses for the velocity and pressure fields.
As the algorithm proceeds our aim must be progressively to improve these guessed fields.
The process is iterated until convergence of the velocity and pressure fields.

6. Issues with pressure-velocity calculations…
The solution procedure (central difference, upwind etc). for the transport of a general property  be enlisted to solve the momentum equations.
Matters are, however, not completely straightforward since there are problems associated with the pressure source terms of the momentum equations that need special treatment.
The finite volume method starts, as always, with the discretisation of the flow domain and of the relevant transport equations 1-3.
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 , where a uniform grid is used for simplicity.

7. A 'checker-board' pressure field
Issues with pressure-velocity calculations…
Let us assume that we have somehow obtained a highly irregular 'checker-board' pressure field with values as shown in Figure.
If the pressures at ‘e’ and 'w' are obtained by linear interpolation the pressure gradient term dp/dx in the w-momentum equation is given by
A 'checker-board' pressure field
(4)
Similarly, the pressure gradient dp/dy for the v-momentum equation is evaluated as
(5)

8. Remedy: the staggered grid
The pressure at the central node (P) does not appear in 4 and 5. Substituting the appropriate values from the 'checker-board' pressure field in Figure into formulae 4-5, we find 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 (Harlow and Welch, 1965). 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.

10. Two-dimensional flow calculation in staggered grid arrangement

11. Remedy: the staggered grid
The scalar variables, including pressure, are stored at the nodes marked (•).
The velocities are defined at the (scalar) cell faces in between the nodes and are indicated by arrows.
Horizontal () arrows indicate the locations for w-velocities and vertical () ones denote those for v-velocities.
In addition to the E,W,N,S notation Figure also introduces a new system of notation based on a numbering of grid lines and cell faces.
For the moment we continue to use the original E, W, N, S notation; the u- velocities are stored at scalar cell faces 'e' and 'w' and the v-velocities at faces n and s.
In a three-dimensional flow the w-component is evaluated at cell faces 't' and ‘b’.
The control volumes for u and v are different from the scalar control volumes and different from each other.

12. Remedy: the staggered grid
In the staggered grid arrangement, the pressure nodes coincide with the cell faces of the w-control volume.
The pressure gradient term dp/dx for the w-control volume is given by
(6)
where xu is the width of the w-control volume. Similarly dp/dy for the v-control volume shown is given by
(7)
where yv is width of the v-control volume.

13. Remedy: the staggered grid
If we consider the 'checker-board' pressure field again, substitution of the appropriate nodal pressure values into equations 6 and 7 now 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.

14. Staggered grid: Momentum Eq.
Advantages:
No need to Interpolate pressure
Pressure Checker boarding avoided

15. Discretization of Momentum Eq.
Discretization of other terms are similar to the scalar transport equation
Discretization

16. Discretization of Continuity Eq.
Discretize continuity on main control volume
No need to interpolate velocities
No checkerboarding

17. Calculation of velocity and pressure fields
Velocity field
Pressure field

18. In a two dimensional analysis, the x and y momentum equations are:
Convective Acceleration Term
This term causes the Navier-Stokes equations to be nonlinear. To circumvent this problem, we will provide an initial guess ( 𝑢 , given) for the fluid velocity, such that the steady state momentum equation takes the following form:
This arrangement is the basis of the SIMPLE method in which 𝑢 is provided via a guess or an earlier iteration. The velocity field resulting from the x and y momentum equations are then used to correct the pressure field

19. Starred Velocity Field
Guess the pressure field p*
Approximate velocity field is solved based on a guessed pressure field p* can be called as ‘Starred’ velocity field

20. The Pressure and Velocity Corrections

21. Velocity Correction Formula

22. Pressure Correction Equation

23. Pressure Correction Equation…

24. Simple Algorithm

25. Sequence of Operations

26. Sequence of Operations

27. Discussions
The pressure correction equation is a vehicle by which the velocity and pressure fields are nudged towards a solution that satisfies both the discrete continuity and momentum equations.
Iterating with an algorithm which satisfies continuity repeatedly – b term in the continuity equation is a measure of resulting imbalance

28. Discussions…

29. Discussions…

30. Boundary Conditions
 P  b Ab

31. Pressure BC

32. Relative Nature of Pressure
Assume a constant density steady situation, in which normal velocities are given at the all boundary locations.
All the boundary coefficients such as ab will be zero at the boundary.
This is something like a heat conduction with unknown boundary temp which says that if T is a solution then T + C will also be a solution.
This can only be avoided if boundary temperature is prescribed in some form.
Similarly, we got to know absolute pressure at some boundary, otherwise both p’ and p’ + C will satisfy p’ equation
Computationally, N control volumes and N - 1 equations. Don’t have enough N equations for pressure corrections.

33. Relative Nature of Pressure…
Hence if density is unaffected by pressure, relative nature does not cause any problem in the momentum equation as we need the pressure difference, not the absolute pressure
That is why we can say pressure is a relative variable, not an absolute one
In an iterative method, absolute value of pressure is decided by an initial guess to solve for p’ equation (for velocity BC) (for pressure BC initial guess could be the known boundary pressure)
Value of absolute pr. is much higher than the pressure difference could cause round-off errors
Good to assume p = 0 as reference value at some grid point.

34. SIMPLER Algorithm Its rate of convergence is faster than SIMPLE
In most cases, it is reasonable to suppose that the pressure-correction equation does a fairly good job of correcting the velocities, but a rather poor job of correcting the pressure.
If we employ pressure correction equation only for velocity corrections and provide some other means of obtaining a improved pressure field, we get an improved algorithm - that is the essence of SIMPLER.
SIMPLER: Semi Implicit Pressure Linked Equation Revised

35. The Pressure Equation

36. The Pressure Equation…
Substituting these values of velocities in the continuity equation as before, we get the pressure equation as
Term b is only difference if we compare this pressure equation from the previously derived pressure correction equation
But there is a major difference – what is that ?

37. The SIMPLER Algorithm

38. Discussions For known velocity field, SIMPLER is better
Unlike SIMPLE, SIMPLER does not require a guess pressure field. It generates a good pressure field from a good guess of velocity field, which is easier to guess.
More computational effort as one extra equation for pressure has to be solved apart from the calculation of pseudo velocities.
But less number of iteration for convergence compensates that.
The pressure correction don’t have to be under-relaxed as in SIMPLE – as the pressure is not corrected with that.

39. End