Interdisciplinary Applied Mathematics

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= (uj ■ V)v +/iS72uj in Г2,    (2.11a)

V2v = -Vxw in Q,    (2.11b)

Vv = 0 in Q,    (2.11c)

ш = Vxv in Q,    (2.11d)

where the    elliptic    equation    for    the    velocity    v is    obtained    using a    vector

identity and the divergence-free constraint. We also assume here that the domain Q is simply connected. An equivalent system in terms of velocity and vorticity is studied in (Karniadakis and Sherwin, 1999). The problem with the    lack    of    direct    boundary    conditions    for    the    vorticity    also    exists

in the more often used vorticity-streamfunction formulation in two dimensions.

Finally, a note regarding nondimensionalization. Consider the free-stream flow U0 past a body of characteristic size D in a medium of dynamic viscosity p as shown in Figure 2.1. There are two characteristic time scales in the problem, the first one representing the convective time scale tc = D/U0, and the second one representing the diffusive time scale td = D2/v, where v = p/p is the kinematic viscosity. If we nondimensional-ize all lengths with D, the velocity field with U0, and the vorticity field with U0/D, we obtain two different nondimensional equations corresponding to the choice of the time nondimensionalization:

Incompressible High-Speed Flows:


Incompressible Low-Speed Flows:



where t*c and t*d are the nondimensionalized time variables with respect to tc and td, respectively, and Re = U0D/v is the Reynolds number. Both forms are useful in simulations, the first in high Reynolds number simulations (e.g., micronozzles, Section 6.6), and the second in low Reynolds number flows (e.g., microchannels).

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