From: Bob Roach <rroachJUNK@cvd-proJUNK.com>
Subject: Re: boundary layers in slightly viscous flow
Date: Sun, 13 Jun 1999 07:41:16 +0200
Newsgroups: sci.math.num-analysis

You don't say whether you actually wish to resolve the boundary or not.  If you
do then you will be forced to place at least a few points within it and this will
most effectively be done with a grid stretched normally to the layer.  Then use
the curvilinear coordinate form of the NS equations.  This has been done lots and
lots of times in the CFD community where we routinely have small viscosity.
An alternative is to use something other than Taylor series for discretization in
the normal direction.  Taylor series is equivalent to fitting polynomial basis
functions.  One that works better for steep gradients is:  u = c1*exp(lambda*y) +
c2*y + c3.  This gives a combined convection-diffusion difference stencil for
equal spacing as:

       rho*u*du/dy - mu*d^2u/dx^2 =
           rho*u*[u(i,j+1) - (1+exp(Redy))*u(i,j) +
exp(Redy)*u(i,j-1)]/(dy*(1-exp(Redy))

where Redy = rho*u(i,j)*dy/mu is the cell Reynolds number in the vertical
direction.  Note that the argument will get large and the exponentials will choke
as you leave the boundary layer.  So what I have used is to simply freeze the
argument if it is larger than, say 50.  The FD coeffs are largely frozen above
Redy>10 anyway so you don't hurt anything by this device.  The other limit, Redy
->0 is also a bit of a problem since it goes to zero over zero.  However, this
too has an expansion which can be frozen for values of Redy<.01.

If you don't wish to resolve the boundary layer, then don't bother with a fine
grid.  But you'll likely need some artificial diffusion somehow to allow the
no-slip condition to be captured.

good lucks,
Bob


"Volker W. Elling" wrote:

> Hello,
>
> I have trouble discretizing the compressible Navier-Stokes equations with
> small viscosity. Most of the flow is Euler-like, only a thin layer (like:
> 1 mm compared to 1 m pipe width) at the boundary is slowed down due to
> viscosity. I see three options:
>
> ++ refine uniformly at the boundary: this produces a lot of cells
> ++ refine non-uniformly - only in the normal direction: this produces less
>    cells but a lot of stiffness because density/cell volume is very sensitive
>    with respect to coordinate movement normal to the boundary but not
>    with respect to tangential movement.
> ++ use something non-grid-like, for example a Galerkin approach.
>
> Does anybody have experiences with thin boundary layers?
>
> -- Volker Elling        Computer science/Mathematics, RWTH Aachen, Germany
> -- http://lem.stud.fh-heilbronn.de/~elling