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7. Solution of the Navier-Stokes Equations
The fact that momentum and energy conservation are both governed by
the same equation makes construction of numerical approximations that conserve both properties difficult. As already noted, kinetic energy conservation
cannot be enforced independently. If the momentum equations are written
in strong conservation form and a finite volume method is used then global
momentum conservation is usually assured. The construction of energy conservative methods is a hit or miss affair. One selects a method and determines
whether it is conservative or not; if not, adjustments are made until conservation is achieved.
An alternative method of guaranteeing kinetic energy conservation is to
use a different form of the momentum equations. For example, one could use
the following equation for incompressible flows:
where ~ i j k is the Levi-Civita symbol (it is +1 if {ijlc) = (123) or an even
permutation of it, it is -1 if { i j k ) is an odd permutation of (123) such as
(321) and zero otherwise). w is the vorticity defined by Eq. (7.64). Energy
conservation follows from this form of the momentum equation by symmetry;
when the equation is multiplied by ui, the second term on the left hand side
is identically zero as a consequence of the antisymmetry property of ~ i j k .
However, because this is not a conservative form of the momentum equation,
construction of a momentum conserving method requires care.
Kinetic energy conservation is of particular importance in computing complex unsteady flows. Examples include the simulation of global weather patterns and simulations of turbulent flows. Lack of guaranteed energy conservation in these simulations often leads to growth of the kinetic energy and
instability. For steady flows, energy conservation is less important but it does
prevent certain types of misbehavior by the iterative solution method.
Kinetic energy is not the only quantity whose conservation is desirable
but cannot be independently enforced: angular momentum is another such
quantity. Flows in rotating machinery, internal combustion engines and many
other devices exhibit pronounced rotation or swirl. If the numerical scheme
does not conserve global angular momentum, the calculation is likely to get
into trouble. Central difference schemes are generally much better than upwind schemes with respect to angular momentum conservation.
7.2 Choice of Variable Arrangement on the Grid
Now let us turn to the discretizations. The first issue is to select the points
in the domain at which the values of the unknown dependent variables are
to be computed. There is more to this than one might think. Basic features
7. Solution of the Navier-Stokes Equations
The fact that momentum and energy conservation are both governed by
the same equation makes construction of numerical approximations that conserve both properties difficult. As already noted, kinetic energy conservation
cannot be enforced independently. If the momentum equations are written
in strong conservation form and a finite volume method is used then global
momentum conservation is usually assured. The construction of energy conservative methods is a hit or miss affair. One selects a method and determines
whether it is conservative or not; if not, adjustments are made until conservation is achieved.
An alternative method of guaranteeing kinetic energy conservation is to
use a different form of the momentum equations. For example, one could use
the following equation for incompressible flows:
where ~ i j k is the Levi-Civita symbol (it is +1 if {ijlc) = (123) or an even
permutation of it, it is -1 if { i j k ) is an odd permutation of (123) such as
(321) and zero otherwise). w is the vorticity defined by Eq. (7.64). Energy
conservation follows from this form of the momentum equation by symmetry;
when the equation is multiplied by ui, the second term on the left hand side
is identically zero as a consequence of the antisymmetry property of ~ i j k .
However, because this is not a conservative form of the momentum equation,
construction of a momentum conserving method requires care.
Kinetic energy conservation is of particular importance in computing complex unsteady flows. Examples include the simulation of global weather patterns and simulations of turbulent flows. Lack of guaranteed energy conservation in these simulations often leads to growth of the kinetic energy and
instability. For steady flows, energy conservation is less important but it does
prevent certain types of misbehavior by the iterative solution method.
Kinetic energy is not the only quantity whose conservation is desirable
but cannot be independently enforced: angular momentum is another such
quantity. Flows in rotating machinery, internal combustion engines and many
other devices exhibit pronounced rotation or swirl. If the numerical scheme
does not conserve global angular momentum, the calculation is likely to get
into trouble. Central difference schemes are generally much better than upwind schemes with respect to angular momentum conservation.
7.2 Choice of Variable Arrangement on the Grid
Now let us turn to the discretizations. The first issue is to select the points
in the domain at which the values of the unknown dependent variables are
to be computed. There is more to this than one might think. Basic features