(This is rather a FEM question)

I have listened several times that, for the use of outflow BC there must be imposed the normal component of the stress tensor AND their tangential component.

In the FEM variational formulation normal component appears in a natural way (divergence theorem). In consequence, this is really a "do nothing" BC. But how about the tangential BC? I mean, how deal with it in the Finite Element framework?

I am interested in this question because I do not have Boundary conditions for the pressure (but now I begin to wonder if this ir really validid or not...)

Any help is grateful.

Carlos.

I will share my experience with you. I do not use any boundary condition on outflow (and sometimes none on inflow). In the step diffuser problem (say Re ~ 50 for faster non-linear convergence), I can cut off the mesh through the middle of the first recirculation bubble and get the same result as with a longer mesh which includes the full bubble. (Can anyone tell me what happens with other codes when you do this?)

First, I argue that the pressure gradient term doesn't belong in the equation for the motion of an incompressible fluid. In the derivation of the equation from first principles, one considers the momentum of the fluid in a small volume, examining the transport of momentum by convection and diffusion and the forces on the fluid, and applies Newton's second law. One force assumed to be present is a pressure difference across opposite faces of the fluid, leading to the pressure gradient term. BUT, since pressure disturbances propagate at infinite speed in an incompressible medium, no dynamic pressure difference can exist for a finite time, and so no pressure gradient can appear. This is consistent with the Helmholtz decomposition of the NS equation into a pressureless solenoidal part for the fluid motion, and an irrotational part for the consistent pressure gradient which is just the usual "pressure-poisson" equation. This decomposition has been used by mathematitions for over 65 years, but has been regarded by them as a "mathematical trick." In this view, the pressureless equation of motion is a KINEMATIC equation with incompressibility playing the role of a conservation law.

Next I use divergence-free Hermite vector finite elements (defined on a rectangle in 2D), derived by taking the curl of a Hermite stream function which is sufficiently continuous that the normal component of the flow is continuous across element boundaries (the degrees of freedom are the stream function and the components of the solenoidal field). The resulting vector field is necessarily pointwise divergence-free.

So how does one control the flow in "pressure-driven flow" problems? One specifies the difference in stream function across the inlet. No boundary conditions are used other than the Dirichlet BCs on no-flow surfaces, and on inflow if that is desired.

Other computational algorithms may require more BCs to keep them stable.

I reported on this method at the 14th US National Congress on Theoretical and Applied Mechanics held in Blacksburg, VA last summer, and showed results for many classical problems. The code is implemented in Matlab.

(I am basing most of this reply on the text "Incompressible Flow" by R. L. Panton)

In a moving fluid, you should distinguish between a mechanical pressure defined as the average normal surface force,

p_mech = -1/3(T11 + T22 + T33)

where Tii are the stress tensor components (which include viscous plus pressure effects)

and

the thermodynamic pressure, which for a simple compressible substance is a function of two thermodynamic variables, such as

p_thermo = function(e,s) = -T*rho*ds/d(rho) at fixed e

where

e = specific internal energy

s = specific entropy

rho = density

For an incompressible fluid (constant density), the density rho is no longer a thermodynamic variable and the fundalmental equation for an incompressible fluid becomes

s - so = Cv ln(e/eo) or T*ds = de

Since s is no longer a function of rho, we cannot define a thermodynamic pressure for an incompressible fluid. However, we can define a mechanical pressure (as shown above) for any fluid (compressible or incompressible). Thus the pressure gradient term in the incompressible momentum equations is the gradient in mechanical pressure.

For a compressible fluid, the typical assumption is that

p_mech - p_thermo = - k/rho * D(rho)/Dt

where

k = bulk viscosity (or 2nd coefficient of viscosity)

D/Dt = subtantial derivative

Typically k is assumed to be zero and

p_mech = p_thermo

is used for compressible flow calculations (Stoke's assumption).

A correction to my previous post:

For a compressible fluid, the typical assumption is that

p_mech - p_thermo = - k/rho * D(rho)/Dt

where

k = bulk viscosity = (lambda + 2/3*mu) where lambda is 2nd viscosity and mu = viscosity)