Interdisciplinary Applied Mathematics

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2.1.1 Incompressible Flow

For an incompressible fluid, where Dp/Dt = 0, the mass conservation (or continuity) equation simplifies to

V-v = 0.

Typically, when we refer to an incompressible fluid we mean that p = constant, but this is not necessary for a divergence-free flow; for example, in thermal convection the density varies with temperature variations. The corresponding momentum equation has the form:

where the viscosity p(x, t) may vary in space and time. The pressure p(x, t) is not a thermodynamic quantity but can be thought of as a constraint that projects the solution v(x,t) onto a divergence-free space. In other words, an equation of state is no longer valid, since it will make the incompressible Navier-Stokes system overdetermined.

The acceleration terms can be written in various equivalent ways, so that in their discrete form, they conserve total linear momentum fn pv dQ and total kinetic energy fQ pv ■ v dQ in the absence of viscosity and external forces. In particular, the following forms are often used:

   Convective form: Dv/Dt = dv/dt + (v ■ V)v,

Conservative (flux) form: Dv/Dt = dv/dt + V-(vv),

Rotational form: Dv/Dt = dv/dt — v x (Vxv) + 1/2V(v ■ v),

Skew-symmetric form: Dv/Dt = dv/dt + 1/2[(v ■ V)v + V-(vv)].

The incompressible Navier-Stokes equations (2.9), (2.10) are written in terms of the primitive variables (v,p). An alternative form is to rewrite these equations in terms of the velocity v and vorticity ш = Vxv. This is a more general formulation than the standard vorticity-streamfunction, which is limited to two dimensions. The following system is equivalent to equations (2.10) and (2.9), assuming that p, p are constant:

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