7.1 Special Features of the Navier-Stokes Equations
161
Here S stands for the viscous part of the stress tensor whose components are
rij defined in Eq. (1.13), i.e. S = T+pl. The first term in the volume integral
on the right hand side disappears if the flow is inviscid; the second is zero if
the flow is incompressible; the third is zero in the absence of body forces.
Several points relating to this equation are worth mentioning.
0 The first three terms on its right side are integrals over the surface of the
control volume. This means that the kinetic energy in the control volume is
not changed by the action of convection and/or pressure within the control
volume. In the absence of viscosity, only flow of energy through the surface
or work done by the forces acting a t the surface of the control volume can
affect the kinetic energy within it; kinetic energy is then globally conserved
in this sense. This is a property that we would like to preserve in a numerical
method.
0 Guaranteeing global energy conservation in a numerical method is a worthwhile goal, but not an easily attained one. Because the kinetic energy
equation is a consequence of the momentum equation and not a distinct
conservation law, it cannot be enforced separately.
0 If a numerical method is energy conservative and the net energy flux
through the surface is zero then the total kinetic energy in the domain
does not grow with time. If such a method is used, the velocity a t every
point in the domain must remain bounded, providing an important kind
of numerical stability. Indeed, energy methods (which sometimes have no
connection t o physics) are often used t o prove stability of numerical methods. Energy conservation does not say anything about the convergence or
accuracy of a method. Accurate solutions may be obtained with methods
that are not conservative of kinetic energy. Kinetic energy conservation is
especially important in computing unsteady flows.
0 Since the kinetic energy equation is a consequence of the momentum equations and not independently enforceable in a numerical method, global
kinetic energy conservation must be a consequence of the discretixed momentum equations. It is thus a property of the discretization method, but
not an obvious one. To see how it might arise, we form the kinetic energy
equation corresponding to the discretized momentum equations by taking
the scalar product of the latter with the velocity and summing over all
control volumes. We shall consider the result term-by-term.
0 The pressure gradient terms are especially important, so let us look into
them further. To get the pressure gradient term into the form displayed in
Eq. (7.8), we used the following equality:
v . gradp = div (pv) - p div v .
(7.9)
For incompressible flows, pdiv v = 0 so only the first term on the right hand
side remains. As it is a divergence, its volume integral can be converted to a
surface integral. As already noted, this means that the pressure influences
the overall kinetic energy budget only by its action at the surface. We
161
Here S stands for the viscous part of the stress tensor whose components are
rij defined in Eq. (1.13), i.e. S = T+pl. The first term in the volume integral
on the right hand side disappears if the flow is inviscid; the second is zero if
the flow is incompressible; the third is zero in the absence of body forces.
Several points relating to this equation are worth mentioning.
0 The first three terms on its right side are integrals over the surface of the
control volume. This means that the kinetic energy in the control volume is
not changed by the action of convection and/or pressure within the control
volume. In the absence of viscosity, only flow of energy through the surface
or work done by the forces acting a t the surface of the control volume can
affect the kinetic energy within it; kinetic energy is then globally conserved
in this sense. This is a property that we would like to preserve in a numerical
method.
0 Guaranteeing global energy conservation in a numerical method is a worthwhile goal, but not an easily attained one. Because the kinetic energy
equation is a consequence of the momentum equation and not a distinct
conservation law, it cannot be enforced separately.
0 If a numerical method is energy conservative and the net energy flux
through the surface is zero then the total kinetic energy in the domain
does not grow with time. If such a method is used, the velocity a t every
point in the domain must remain bounded, providing an important kind
of numerical stability. Indeed, energy methods (which sometimes have no
connection t o physics) are often used t o prove stability of numerical methods. Energy conservation does not say anything about the convergence or
accuracy of a method. Accurate solutions may be obtained with methods
that are not conservative of kinetic energy. Kinetic energy conservation is
especially important in computing unsteady flows.
0 Since the kinetic energy equation is a consequence of the momentum equations and not independently enforceable in a numerical method, global
kinetic energy conservation must be a consequence of the discretixed momentum equations. It is thus a property of the discretization method, but
not an obvious one. To see how it might arise, we form the kinetic energy
equation corresponding to the discretized momentum equations by taking
the scalar product of the latter with the velocity and summing over all
control volumes. We shall consider the result term-by-term.
0 The pressure gradient terms are especially important, so let us look into
them further. To get the pressure gradient term into the form displayed in
Eq. (7.8), we used the following equality:
v . gradp = div (pv) - p div v .
(7.9)
For incompressible flows, pdiv v = 0 so only the first term on the right hand
side remains. As it is a divergence, its volume integral can be converted to a
surface integral. As already noted, this means that the pressure influences
the overall kinetic energy budget only by its action at the surface. We