# Interdisciplinary Applied Mathematics

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FIGURE 6.11. Simulated step (top) and frequency (bottom) responses of an accelerometer.    The    number    of    holes in    the    mass    greatly    affects    the    damping

properties and the settling time. (Courtesy of T. Veijola.)

shear stress affecting the velocity slip directly (Us ~ dU/dn). In contrast to the straight channel flow, cross-flow variations are significant here, so that the rarefaction effect is truly two-dimensional. In addition, the change of characteristic length scale from the inlet channel height to the larger length downstream presents extra difficulties in choosing the proper scaling.

We first investigate compressibility effects in backward-facing step flow and compare with corresponding incompressible flow simulations. The geometry used in    this    study    is    given    in    Figure 6.12    along    with    a    typical

spectral element discretization (see Section 14.1). It corresponds to S/h = 0.467, where S is the step height and h is the height of the channel.

0    1    2    3    4    5

FIGURE 6.12. Backward-facing step geometry. Spectral element discretization with 52 elements and tenth-order polynomial expansions in each direction (see Chapter 14).

FIGURE 6.13. Variation of nondimensionalized separation distance XR/S (S is the    step    height)    as    a    function    of    Re1y    for    incompressible, compressible, and

rarefied flows. For the rarefied flow simulations Kn0 = 0.04 based on the channel exit conditions.

The high-order program p,Flow that solves the compressible Navier-Stokes equations with    and    without    slip    at    the    wall    is employed    (see chapter    2).

The computational domain is discretized with 52 spectral elements, where each element is further discretized with Nth-order polynomial expansions in each direction; here we use N = 10.

In Figure 6.13 we present the variation of nodimensionalized separation

distance (Xr/S) as a function of Reynolds number Re^ for no-slip flows, where Re^ is based on the mass flowrate (M) per unit width (W) of the channel

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