# Interdisciplinary Applied Mathematics

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dQ

dn

Yov At.

w

This relation shows that the time-differencing error of the velocity field is an    order smaller    in    At    than    the    corresponding error    in    the    boundary-

divergence. For the inviscid-type pressure boundary condition, we are therefore limited to first-order accuracy, since the boundary-divergence flux is 0(1). However, in general, we obtain

v ж (At)Jp+1

if a high-order time integration scheme is used to advance the velocity field. In numerical experiments, it was found in (Karniadakis et al., 1991) that with Jp = 2 we obtain a third-order-accurate velocity field. Note that the boundary-divergence scales as

Qw oc yQ/(At)Jp.

These heuristic arguments have been more rigorously documented in (Karniadakis et al., 1991) and confirmed by numerical results. To demonstrate the effect of the incorrect inviscid pressure boundary condition versus the correct rotational form in the boundary condition, we consider a

FIGURE 14.2. Divergence of velocity field across the channel for a Stokes flow. At = 10-2. The spatial discretization is based on 20 spectral elements of order 10, which eliminates any spatial errors (Tomboulides et al., 1989). The inviscid-type boundary condition leads to large divergence errors at the boundaries.

decaying Stokes channel flow subject to compatible initial conditions (Kar-niadakis and Sherwin, 1999). In Figure 14.2 we plot the divergence of the velocity field across the channel. It is seen that incorporation of the rotational form of the pressure boundary condition almost eliminates the artificial boundary layer. High-order treatment produces smaller boundary divergence errors consistent with the aforementioned arguments.

Remark: The above pressure boundary condition assumes that the nope netration condition at the wall holds. However, if the surface moves perpendicular to its plane with velocity vw • n, e.g., squeezed film applications in Section 6.1, then a time-dependent term is present, i.e.,

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