# Interdisciplinary Applied Mathematics

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ALE Formulation

In this section, we consider domains that are arbitrarily moving in time. This is not a trivial generalization, because we have to discretize the time-dependent operators as well as the time-dependent fields. In the ALE formulation, the local elemental operators are formed at every time step. This is necessary in order to handle the mesh shape variations in time. The high-order ALE formulation implemented with spectral element discretizations exhibits the usual advantages of low dispersion and robustness to large deformations, as we will demonstrate below. This, in turn, implies that no remeshing is required during the simulation, which frequently dominates the computational cost.

We consider the nondimensionalized incompressible Navier-Stokes equations with a passively advected scalar field 0(x,t). The domain is time-dependent (Q(t)),    and    it    is moving    with    velocity    w.    The    governing    equa

tions are

dv

~dt

дв

dt

+ (v + (v

w) • Vv

w) • Vd V • v

1

-Vp+ ——V2v + f Re

—V2<9 in Q(t), Pe

0 in Q(t).

in Q(t),

The Peclet number Pe is the Reynolds number Re multiplied by either the Prandtl number Pr or the Schmidt number Sc for heat transfer or species transport applications, respectively. For heat transfer problems, the nondimensional temperature is given as

в

T — To

AT where T0 is a reference temperature and AT is a predefined or desired temperature difference. For the species transport applications, в can be identified as the concentration density normalized by a reference value.

To discretize the equations in time we use a high-order stiffly stable scheme (see (Karniadakis and Sherwin, 1999) in two passes:

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