174
7. Solution of the Navier-Stokes Equations
the beginning of each outer iteration, the terms on the right hand side of Eq.
(7.30) are evaluated using the variables at the preceding outer iteration.
The momentum equations are usually solved sequentially i.e. the set of
algebraic equations for each component of the momentum is solved in turn,
treating the grid point values of its dominant velocity component as the sole
set of unknowns. Since the pressure used in these iterations was obtained from
the previous outer iteration or time step, the velocities computed from Eqs.
(7.30) do not normally satisfy the discretized continuity equation. To enforce
the continuity condition, the velocities need to be corrected; this requires
modification of the pressure field; the manner of doing this is described next.
The velocity at node P, obtained by solving the linearized momentum
equations (7.30), can be formally expressed as:
As already stated, these velocities do not satisfy the continuity equation,
so u? is not the final value of the velocity for iteration m; it is a predicted
value, which is why it carries an asterisk (*). The corrected final values should
satisfy the continuity equation. For convenience, the first term on the right
hand side of the above equations is called fir;:
The velocity field iiy* can be thought of as one from which the contribution of
the pressure gradient has been removed. Because the method is implicit, this
is not the velocity that would be obtained by dropping the pressure gradient
entirely from Eq. (7.30).
The next task is t o correct the velocities so that they satisfy the continuity
equation:
which can be achieved by correcting the pressure field. The corrected velocities and pressure are linked by (see Eq. (7.32)):
Continuity is now enforced by inserting this expression for u y into the continuity equation (7.33), to yield a discrete Poisson equation for the pressure:
7. Solution of the Navier-Stokes Equations
the beginning of each outer iteration, the terms on the right hand side of Eq.
(7.30) are evaluated using the variables at the preceding outer iteration.
The momentum equations are usually solved sequentially i.e. the set of
algebraic equations for each component of the momentum is solved in turn,
treating the grid point values of its dominant velocity component as the sole
set of unknowns. Since the pressure used in these iterations was obtained from
the previous outer iteration or time step, the velocities computed from Eqs.
(7.30) do not normally satisfy the discretized continuity equation. To enforce
the continuity condition, the velocities need to be corrected; this requires
modification of the pressure field; the manner of doing this is described next.
The velocity at node P, obtained by solving the linearized momentum
equations (7.30), can be formally expressed as:
As already stated, these velocities do not satisfy the continuity equation,
so u? is not the final value of the velocity for iteration m; it is a predicted
value, which is why it carries an asterisk (*). The corrected final values should
satisfy the continuity equation. For convenience, the first term on the right
hand side of the above equations is called fir;:
The velocity field iiy* can be thought of as one from which the contribution of
the pressure gradient has been removed. Because the method is implicit, this
is not the velocity that would be obtained by dropping the pressure gradient
entirely from Eq. (7.30).
The next task is t o correct the velocities so that they satisfy the continuity
equation:
which can be achieved by correcting the pressure field. The corrected velocities and pressure are linked by (see Eq. (7.32)):
Continuity is now enforced by inserting this expression for u y into the continuity equation (7.33), to yield a discrete Poisson equation for the pressure: