# Special Difficulties

The discussions in the previous subsection are for the solution of a general variable $\varphi$ with a known flow field. We now turn our attention to the solution of the flow field. For a two-dimensional incompressible flow problem without a body force, the continuity and momentum equations in the Cartesian coordinate system are

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
 $\rho u\frac{\partial u}{\partial x}+\rho v\frac{\partial u}{\partial y}=\frac{\partial }{\partial x}\left( \mu \frac{\partial u}{\partial x} \right)+\frac{\partial }{\partial y}\left( \mu \frac{\partial u}{\partial y} \right)-\frac{\partial p}{\partial x}$ (1)
 $\rho u\frac{\partial v}{\partial x}+\rho v\frac{\partial v}{\partial y}=\frac{\partial }{\partial x}\left( \mu \frac{\partial v}{\partial x} \right)+\frac{\partial }{\partial y}\left( \mu \frac{\partial v}{\partial y} \right)-\frac{\partial p}{\partial y}$ (1)

It is observed that the momentum equations in each direction could be expressed in the same format as eq. (4.200) by replacing the general variable $\varphi$ by the components of velocity in that particular (x-, or y-) direction. The source term for the momentum equations in the x- and y- directions can be expressed as $S_{u}=-\partial p/\partial x$ and $S_{v}=-\partial p/\partial y,$ respectively. It seems that we can directly apply the discretization schemes discussed in the preceding subsection in order to obtain the solution of the flow field. The only additional work necessary is to include the pressure gradient in each momentum equation. With the aid of Fig. 4.17, the pressure gradient in the x-direction can be expressed as

 $\left( \frac{\partial p}{\partial x} \right)_{P}=\frac{p_{e}-p_{w}}{\Delta x}$ (1)

If central difference is employed, the pressures at the faces of the control volume become

$p_{e}=\frac{p_{E}+p_{P}}{2},\text{ }p_{w}=\frac{p_{P}+p_{W}}{2}$

Substituting the above expressions into eq. (4.289), the pressure gradient in the x-direction becomes

 $\left( \frac{\partial p}{\partial x} \right)_{P}=\frac{p_{E}-p_{W}}{2\Delta x}$ (1)

Similarly, the pressure gradient in the y-direction can be expressed as

 $\left( \frac{\partial p}{\partial y} \right)_{P}=\frac{p_{N}-p_{S}}{2\Delta y}$ (1)
Figure 1: Checkerboard pressure field.

It can be seen that the pressure gradient at point P is related to the pressures of the neighbor grid points and is not related to its own pressure. Thus, the pressure gradient is obtained by using the pressures of two alternative points, so that the accuracy of the pressure gradient for grid size Δx (or Δy) is equivalent to that obtained for coarse grid size 2Δx (or 2Δy). Another consequence of eqs. (4.290) and (4.291) is that if we have a pressure distribution as shown in Fig. 1 – referred to as a checkerboard pressure field – the discretization scheme represented by eqs. (4.290) and (4.291) will obtain $\partial p/\partial x=0$ and $\partial p/\partial y=0$ throughout the computational domain, i.e., eqs. (4.290) and (4.291) cannot recognize the difference between a checkerboard pressure field and a uniform pressure field. In other words, if a checkerboard pressure field is supposed to be the real pressure field for any reason, the discretization scheme based on eqs. (4.290) and (4.291) can neither detect nor eliminate such an unrealistic effect.

While it is straightforward to use the momentum equations (4.287) and (4.288) to solve for the velocity components in the x- and y-directions, the pressure only appears as a source term in the momentum equations and there is not a separate equation for pressure itself. Since the continuity equation, (4.286), is not used, it would be logical to use the continuity equation to get the pressure field. Although a correct pressure field will yield a velocity field that satisfies the continuity equation, an algorithm must be developed to obtain the correct pressure field. The checkerboard pressure field problem and the lack of an equation for pressure are associated with the coupling between the pressure and velocity. An unrealistic (e.g., checkerboard) pressure field can be obtained if such a coupling relationship is not reflected in the algorithm (e.g., using central difference scheme to discretize the pressure gradient in the momentum equations).