Interdisciplinary Applied Mathematics

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—- = -on • (V x uj) dn



dv wn

dt



If the slip condition is applied and the nonlinear effects are nonnegligible, then this term should also be added, appropriately, on the right-hand side of the above equation. In addition, the boundary condition in equation (14.3b) should be corrected appropriately.

14.1.2 Compressible Flows


We present here the two-dimensional formulation in order to discuss some important issues associated with the interface and boundary conditions. Both Galerkin and discontinuous Galerkin projections can be employed, but here for simplicity we present the standard Galerkin approach. For an introduction to discontinuous Galerkin methods with emphasis on compressible flow simulations the interested reader can consult (Cockburn et al., 2000).


The compressible Navier-Stokes equations in nondimensional flux form are (see Section 2.2, equation (2.16))


( p


f pu


( pv


d


pu


d


2


pu + p


d


pvu


dt


pv


dx


puv


+dy


pv + p


E


(E + p) • u)


(E + p) • v)


/


0



1


d


2 /0 du dv Зг’У^дх dy)


Re


dx


vi


P (2


du


dx


dv dy >


• v +

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