158
7. Solution of the Navier-Stokes Equations
where, for a Newtonian fluid and incompressible flow:
Because the momentum equations are vector equations, the viscous term is
more complicated than the generic diffusive term. The part of the viscous
term in the momentum equations which corresponds to the diffusive term in
the generic conservation equation is
This term can be discretized using any of the approaches described for the
corresponding terms of the generic equation in Chaps. 3 and 4 but it is
only one contribution of viscous effects to the ith component of momentum.
Equations (1.28), (1.26), (1.21) and (1.17) allow us to identify the others as
the contributions of the bulk viscosity (which is non-zero only in compressible
flows) and a further contribution due to the spatial variability of the viscosity.
For incompressible flow with constant fluid properties, these contributions
disappear (thanks to the continuity equation).
The extra terms which are non-zero when the viscosity is spatially variable
in an incompressible flow may be treated in the same manner as the terms
(7.4):
where n is the unit outward normal to the surface of the control volume and
summation on j applies. As noted above, for constant p, this term vanishes.
For this reason, this term is often treated explicitly even when implicit solution methods are used. It is argued that even when the viscosity varies this
term is small compared to the term (7.4) so its treatment has only a slight
impact on the rate of convergence. However, this argument applies strictly
only in an integral sense; the extra term may be quite large on any one CV
face.
7.1.2 Discretization of Pressure Terms and Body Forces
As noted in Chap. 1, we usually deal with the "pressure" in terms of the
combination p - pog . T + pidiv v. In incompressible flows, the last term is
zero. One form of the momentum equations (see Eq. (1.21)) contains the
gradient of this quantity which may be approximated by the FD methods
7. Solution of the Navier-Stokes Equations
where, for a Newtonian fluid and incompressible flow:
Because the momentum equations are vector equations, the viscous term is
more complicated than the generic diffusive term. The part of the viscous
term in the momentum equations which corresponds to the diffusive term in
the generic conservation equation is
This term can be discretized using any of the approaches described for the
corresponding terms of the generic equation in Chaps. 3 and 4 but it is
only one contribution of viscous effects to the ith component of momentum.
Equations (1.28), (1.26), (1.21) and (1.17) allow us to identify the others as
the contributions of the bulk viscosity (which is non-zero only in compressible
flows) and a further contribution due to the spatial variability of the viscosity.
For incompressible flow with constant fluid properties, these contributions
disappear (thanks to the continuity equation).
The extra terms which are non-zero when the viscosity is spatially variable
in an incompressible flow may be treated in the same manner as the terms
(7.4):
where n is the unit outward normal to the surface of the control volume and
summation on j applies. As noted above, for constant p, this term vanishes.
For this reason, this term is often treated explicitly even when implicit solution methods are used. It is argued that even when the viscosity varies this
term is small compared to the term (7.4) so its treatment has only a slight
impact on the rate of convergence. However, this argument applies strictly
only in an integral sense; the extra term may be quite large on any one CV
face.
7.1.2 Discretization of Pressure Terms and Body Forces
As noted in Chap. 1, we usually deal with the "pressure" in terms of the
combination p - pog . T + pidiv v. In incompressible flows, the last term is
zero. One form of the momentum equations (see Eq. (1.21)) contains the
gradient of this quantity which may be approximated by the FD methods