166
7. Solution of the Navier-Stokes Equations
when contravariant (or other grid-oriented) components of vectors and tensors are the working variables. This complicates the equations by introducing
curvature terms that are difficult to treat numerically, and may create nonconservative errors when the grid is not smooth as will be shown in Chap.
8. When improved pressure-velocity coupling algorithms were developed in
the 1980's, the popularity of the colocated arrangement began to rise. The
advantages will be demonstrated below.
7.2.2 Staggered Arrangements
There is no need for all variables to share the same grid; a different arrangement may turn out to be advantageous. In Cartesian coordinates, the
staggered arrangement introduced by Harlow and Welsh (1965) offers several
advantages over the colocated arrangement. This arrangement is shown in
Fig. 7.2. Several terms that require interpolation with the colocated arrangement, can be calculated (to a second-order approximation) without interpolation. This can be seen from the x-momentum CV shown in Fig. 7.4. Both
the pressure and diffusion terms are very naturally approximated by central
difference approximations without interpolation, since the pressure nodes lie
at CV face centers and the velocity derivatives needed for the diffusive terms
are readily computed at the CV faces. Also, evaluation of mass fluxes in the
continuity equation on the faces of a pressure CV is straightforward. Details
are presented below.
Fig. 7.2. Fully staggered arrangement of velocity components and pressure (right)
and a partially staggered arrangement (left) on a FV grid
Perhaps the biggest advantage of the staggered arrangement is the strong
coupling between the velocities and the pressure. This helps to avoid some
types of convergence problems and oscillations in pressure and velocity fields,
an issue that will be further discussed later.
The numerical approximation on a staggered grid is also conservative of
kinetic energy which has advantages that were discussed earlier. The proof
of this statement is straight-forward but lengthy and will not be given here.
7. Solution of the Navier-Stokes Equations
when contravariant (or other grid-oriented) components of vectors and tensors are the working variables. This complicates the equations by introducing
curvature terms that are difficult to treat numerically, and may create nonconservative errors when the grid is not smooth as will be shown in Chap.
8. When improved pressure-velocity coupling algorithms were developed in
the 1980's, the popularity of the colocated arrangement began to rise. The
advantages will be demonstrated below.
7.2.2 Staggered Arrangements
There is no need for all variables to share the same grid; a different arrangement may turn out to be advantageous. In Cartesian coordinates, the
staggered arrangement introduced by Harlow and Welsh (1965) offers several
advantages over the colocated arrangement. This arrangement is shown in
Fig. 7.2. Several terms that require interpolation with the colocated arrangement, can be calculated (to a second-order approximation) without interpolation. This can be seen from the x-momentum CV shown in Fig. 7.4. Both
the pressure and diffusion terms are very naturally approximated by central
difference approximations without interpolation, since the pressure nodes lie
at CV face centers and the velocity derivatives needed for the diffusive terms
are readily computed at the CV faces. Also, evaluation of mass fluxes in the
continuity equation on the faces of a pressure CV is straightforward. Details
are presented below.
Fig. 7.2. Fully staggered arrangement of velocity components and pressure (right)
and a partially staggered arrangement (left) on a FV grid
Perhaps the biggest advantage of the staggered arrangement is the strong
coupling between the velocities and the pressure. This helps to avoid some
types of convergence problems and oscillations in pressure and velocity fields,
an issue that will be further discussed later.
The numerical approximation on a staggered grid is also conservative of
kinetic energy which has advantages that were discussed earlier. The proof
of this statement is straight-forward but lengthy and will not be given here.