160
7. Solution of the Navier-Stokes Equations
In some cases the non-conservative terms, considered as body forces, dominate the transport equation ( e g when swirling flows are computed in polar
coordinates or when flows are treated in a rotating coordinate frame, for
example, in turbomachinery flows). The treatment of the non-linear source
terms and the variable coupling may then become very important.
7.1.3 Conservation Properties
The Navier-Stokes equations have the property that the momentum in any
control volume (microscopic or macroscopic) is changed only by flow through
the surface, forces acting on the surface, and volumetric body forces. This important property is inherited by the discretized equations if the FV approach
is used and the surface fluxes for adjacent control volumes are identical. If
this is done, then the integral over the entire domain, being the sum of the
integrals over the microscopic control volumes, reduces to a sum over the
surface of the domain. Overall mass conservation follows in the same way
from the continuity equation.
Energy conservation is a more complex issue. In incompressible isothermal
flows, the only energy of significance is kinetic energy. When heat transfer
is important, the kinetic energy is generally small compared to the thermal
energy so the equation introduced t o account for energy transport is a conservation equation for thermal energy. So long as the temperature dependence
of the fluid properties is not significant, the thermal energy equation can be
solved after solution of the momentum equations is complete. The coupling
is then entirely one-way and the energy equation becomes an equation for
the transport of a passive scalar, the case treated in Chaps. 3 t o 6.
An equation for the kinetic energy can be derived by taking the scalar
product of the momentum equation with the velocity, a procedure which
mimics the derivation of the energy equation in classical mechanics. Note
that, in contrast to compressible flow, for which there is a separate conservation equation for the total energy, in incompressible isothermal flows both
momentum and energy conservation are consequences of the same equation;
this poses the problems that are the subject of this section.
We shall be interested principally in the kinetic energy conservation equation for a macroscopic control volume, which may be either the entire considered domain or one of the small CVs used in a finite volume method. If
the local kinetic energy equation obtained in the manner just described is
integrated over a control volume, we obtain, after using Gauss' Theorem:
7. Solution of the Navier-Stokes Equations
In some cases the non-conservative terms, considered as body forces, dominate the transport equation ( e g when swirling flows are computed in polar
coordinates or when flows are treated in a rotating coordinate frame, for
example, in turbomachinery flows). The treatment of the non-linear source
terms and the variable coupling may then become very important.
7.1.3 Conservation Properties
The Navier-Stokes equations have the property that the momentum in any
control volume (microscopic or macroscopic) is changed only by flow through
the surface, forces acting on the surface, and volumetric body forces. This important property is inherited by the discretized equations if the FV approach
is used and the surface fluxes for adjacent control volumes are identical. If
this is done, then the integral over the entire domain, being the sum of the
integrals over the microscopic control volumes, reduces to a sum over the
surface of the domain. Overall mass conservation follows in the same way
from the continuity equation.
Energy conservation is a more complex issue. In incompressible isothermal
flows, the only energy of significance is kinetic energy. When heat transfer
is important, the kinetic energy is generally small compared to the thermal
energy so the equation introduced t o account for energy transport is a conservation equation for thermal energy. So long as the temperature dependence
of the fluid properties is not significant, the thermal energy equation can be
solved after solution of the momentum equations is complete. The coupling
is then entirely one-way and the energy equation becomes an equation for
the transport of a passive scalar, the case treated in Chaps. 3 t o 6.
An equation for the kinetic energy can be derived by taking the scalar
product of the momentum equation with the velocity, a procedure which
mimics the derivation of the energy equation in classical mechanics. Note
that, in contrast to compressible flow, for which there is a separate conservation equation for the total energy, in incompressible isothermal flows both
momentum and energy conservation are consequences of the same equation;
this poses the problems that are the subject of this section.
We shall be interested principally in the kinetic energy conservation equation for a macroscopic control volume, which may be either the entire considered domain or one of the small CVs used in a finite volume method. If
the local kinetic energy equation obtained in the manner just described is
integrated over a control volume, we obtain, after using Gauss' Theorem: