7.1 Special Features of the Navier-Stokes Equations
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meshes. If a method is not energy conservative on uniform regular grids,
it will certainly not be so on more complicated ones. On the other hand,
a method which is conservative on uniform grids might be nearly so on
complex grids.
A Poisson equation is often used to compute the pressure. As we shall see, it
is derived by taking the divergence of the momentum equation. The Laplacian operator in the Poisson equation is thus the product of the divergence
operator in the continuity equation and the gradient operator in the momentum equation. The approximation of the Poisson equation cannot be
selected independently; it must be consistent with the divergence and gradient operators if mass conservation is to obtain. Energy conservation adds
the further requirement that the divergence and gradient approximations
be consistent in the sense defined above.
For an incompressible flow without body forces, the only remaining volume
integral is the viscous term. For a Newtonian fluid, this term becomes:
Inspection reveals that the integrand is a sum of squares so this term is
always negative (or zero). It represents the irreversible (in the thermodynamic sense) conversion of kinetic energy of the flow into internal energy
of the fluid and is called viscous dissipation. As incompressible flows are
usually low speed flows, the addition to the internal energy is rarely significant but the loss of kinetic energy is often quite important to the flow. In
compressible flows, the energy transfer is often important to both sides.
The time differencing method can destroy the energy conservation property. In addition to the requirements on the spatial discretization mentioned
above, the approximation of the time derivatives should be properly chosen.
The Crank-Nicolson scheme is a particularly good choice. In it, the time
derivatives in the momentum equations are approximated by:
If we take the scalar product of this term with u ~ + " ~ ,
which in the CrankNicolson scheme is approximated by (u:+'+uY)/2, the result is the change
in the kinetic energy:
where v2 = uiui (summation implied). With proper choices of the approximations to the other terms, the Crank-Nicolson scheme is energy conservative.
163
meshes. If a method is not energy conservative on uniform regular grids,
it will certainly not be so on more complicated ones. On the other hand,
a method which is conservative on uniform grids might be nearly so on
complex grids.
A Poisson equation is often used to compute the pressure. As we shall see, it
is derived by taking the divergence of the momentum equation. The Laplacian operator in the Poisson equation is thus the product of the divergence
operator in the continuity equation and the gradient operator in the momentum equation. The approximation of the Poisson equation cannot be
selected independently; it must be consistent with the divergence and gradient operators if mass conservation is to obtain. Energy conservation adds
the further requirement that the divergence and gradient approximations
be consistent in the sense defined above.
For an incompressible flow without body forces, the only remaining volume
integral is the viscous term. For a Newtonian fluid, this term becomes:
Inspection reveals that the integrand is a sum of squares so this term is
always negative (or zero). It represents the irreversible (in the thermodynamic sense) conversion of kinetic energy of the flow into internal energy
of the fluid and is called viscous dissipation. As incompressible flows are
usually low speed flows, the addition to the internal energy is rarely significant but the loss of kinetic energy is often quite important to the flow. In
compressible flows, the energy transfer is often important to both sides.
The time differencing method can destroy the energy conservation property. In addition to the requirements on the spatial discretization mentioned
above, the approximation of the time derivatives should be properly chosen.
The Crank-Nicolson scheme is a particularly good choice. In it, the time
derivatives in the momentum equations are approximated by:
If we take the scalar product of this term with u ~ + " ~ ,
which in the CrankNicolson scheme is approximated by (u:+'+uY)/2, the result is the change
in the kinetic energy:
where v2 = uiui (summation implied). With proper choices of the approximations to the other terms, the Crank-Nicolson scheme is energy conservative.