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REVIEW ARTICLES

# On Boundary Conditions for Incompressible Navier-Stokes Problems

[+] Author and Article Information
Dietmar Rempfer

Department of Materials, Mechanical and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL 60616

We note in passing that this problem is also discussed by Gresho in (12), who gives the asymptotic solution as $u=y$, which is obviously wrong since it does not satisfy the velocity boundary conditions.

Alternatively, we could assume that we have a numerical scheme that automatically enforces $∇∙ω=0$ by calculating one of the vorticity components from this constraint. For two-dimensional flow, the single component $ωz$ is always divergence-free.

Appl. Mech. Rev 59(3), 107-125 (May 01, 2006) (19 pages) doi:10.1115/1.2177683 History:

## Abstract

We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which often do not have immediately obvious “physical” boundary conditions. Most of the issues are discussed for the example of a primitive-variables formulation of the incompressible Navier-Stokes equations in the form of momentum equations plus the pressure Poisson equation. However, analogous problems also exist in other formulations, some of which are briefly reviewed as well. This review article cites 95 references.

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## Figures

Figure 1

Initial velocity field calculated from the stream function 50 with Aλ=2.64244, λ=0.349911

Figure 2

Pressure fields or Neumann and Dirichlet conditions, respectively: (a)pNeu(x,y) and (b)pDir(x,y)

Figure 3

Acceleration fields for Neumann and Dirichlet conditions, respectively: (a)∂uNeu∕∂t(x,y), (b)∂uDir∕∂t(x,y), (c) ∂vNeu∕∂t(x,y), and (d) ∂vDir∕∂t(x,y). Note that ∂uNeu∕∂t and ∂vDir∕∂t do not vanish at the wall.

Figure 4

Discrepancy between the pressures and velocity derivatives for the two choices of boundary conditions: (a)pNeu−pDir and (b)∂uNeu∕∂t−∂uDir∕∂t

Figure 5

Stokes flow in a driven cavity. Note that the pressure is singular in the upper corners for this flow: (a) velocity field and (b) pressure field

Figure 6

Distribution of the x component of velocity, without and with pressure: (a) u field for p≡0 and (b) Correct u field

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