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7. Solution of the Navier-Stokes Equations
motion; Eq. (7.63) is a special case of the more general definition that applies
in 3D as well:
In two dimensional flows, the vorticity vector is orthogonal to the plane of
flow and Eq. (7.64) reduces to Eq. (7.63). The principal reason for introducing the streamfunction is that, for flows in which p, p and g are constant, the
continuity equation is identically satisfied and need not be dealt with explicitly. Substitution of Eqs. (7.62) into the definition of the vorticity (7.63) leads
to a kinematic equation connecting the streamfunction and the vorticity:
Finally, by differentiating the x and y momentum equations with respect to
y and x, respectively, and subtracting the results from each other we obtain
the dynamic equation for the vorticity:
The pressure does not appear in either of these equations i.e. it has been
eliminated as a dependent variable. Thus the Navier-Stokes equations have
been replaced by a set of just two partial differential equations, in place of
the three for the velocity components and pressure. This reduction in the
number of dependent variables and equations is what makes this approach
attractive.
The two equations are coupled through the appearance of u, and u,
(which are derivatives of I )) in the vorticity equation and by the vorticity
w acting as the source term in the Poisson equation for q. The velocity
components are obtained by differentiating the streamfunction. If it is needed,
the pressure can be obtained by solving the Poisson equation as described in
Sect. 7.3.1.
A solution method for these equations is the following. Given an initial
velocity field, the vorticity is computed by differentiation. The dynamic vorticity equation is then used t o compute the vorticity at the new time step;
any standard time advance method may be used for this purpose. Having the
vorticity, it is possible to compute the streamfunction at the new time step
by solving the Poisson equation; any iterative scheme for elliptic equations
may be used. Finally, having the streamfunction, the velocity components are
easily obtained by differentiation and we are ready to begin the calculation
for the next time step.
A problem with this approach lies in the boundary conditions, especially
in complex geometries. Since the flow is parallel to them, solid boundaries
and symmetry planes are surfaces of constant streamfunction. However, the
values of the streamfunction at these boundaries can be calculated only if
7. Solution of the Navier-Stokes Equations
motion; Eq. (7.63) is a special case of the more general definition that applies
in 3D as well:
In two dimensional flows, the vorticity vector is orthogonal to the plane of
flow and Eq. (7.64) reduces to Eq. (7.63). The principal reason for introducing the streamfunction is that, for flows in which p, p and g are constant, the
continuity equation is identically satisfied and need not be dealt with explicitly. Substitution of Eqs. (7.62) into the definition of the vorticity (7.63) leads
to a kinematic equation connecting the streamfunction and the vorticity:
Finally, by differentiating the x and y momentum equations with respect to
y and x, respectively, and subtracting the results from each other we obtain
the dynamic equation for the vorticity:
The pressure does not appear in either of these equations i.e. it has been
eliminated as a dependent variable. Thus the Navier-Stokes equations have
been replaced by a set of just two partial differential equations, in place of
the three for the velocity components and pressure. This reduction in the
number of dependent variables and equations is what makes this approach
attractive.
The two equations are coupled through the appearance of u, and u,
(which are derivatives of I )) in the vorticity equation and by the vorticity
w acting as the source term in the Poisson equation for q. The velocity
components are obtained by differentiating the streamfunction. If it is needed,
the pressure can be obtained by solving the Poisson equation as described in
Sect. 7.3.1.
A solution method for these equations is the following. Given an initial
velocity field, the vorticity is computed by differentiation. The dynamic vorticity equation is then used t o compute the vorticity at the new time step;
any standard time advance method may be used for this purpose. Having the
vorticity, it is possible to compute the streamfunction at the new time step
by solving the Poisson equation; any iterative scheme for elliptic equations
may be used. Finally, having the streamfunction, the velocity components are
easily obtained by differentiation and we are ready to begin the calculation
for the next time step.
A problem with this approach lies in the boundary conditions, especially
in complex geometries. Since the flow is parallel to them, solid boundaries
and symmetry planes are surfaces of constant streamfunction. However, the
values of the streamfunction at these boundaries can be calculated only if