How to set the open boundary condition in simplified 2D version of FDS?

[prk...@gmail.com]

Hello, I am Haejun, a grad student at WPI Fire Protection
Engineering.
I am writing a matlab code of the hydrodynamic model of FDS. It is
only two dimensional(two side walls, floor and ceiling) and covers
less than 200 nodes. Instead of using fire, I set one cell temperature
higher than the other cells, such that buoyancy can be simulated. I
followed predictor-corrector scheme in general, and used line
iterative method to solve the poisson equation of H. I refered to the
FDS tech guide version 4 and 5 together. By the way, THANKS for the
thorough explanations from governing equations to the discretization
in the FDS tech guide.
Anyway, the matlab code seems to work well for a closed room
envrionment(all boundaries are closed) with some limitations in terms
of the maximum temperature of the heated cell, and time step. Beyond a
certain values, it was not stable. I set 'half-slip' condition for
velocity boundary condition, and H as the tech guide says with LES
model.
Now, I am working on the room configuration with an opening. I set the
entire right side wall as open boundary and applied boundary condition
of H using the SAME velocity boundary conditions(half slip) that I
used for closed boundary. For ghost cells on the open boundary, I set
H(i+1, k) = u(i,k)^2+w(i,k)^2 - H(i,k) for the outgoing flow and H(i
+1,k)= - H(i,k) for the incoming flow as technical guide says. The
code seems to work( flow pattern makes sense to me).

However, I am not sure if any change of velocity boundary condition is
requried for open boundary as compared to closed boundary. And if not,
please let me know the reason. And any explanation about the boundary
setting of H or reference material related to it would be
appreciated.

Thank you for your consideration in advance.

-Haejun-

[Kevin]

The open boundary requires a Dirichlet boundary condition, that is, H
is specified, and its value depends on whether the flow is going out
or coming in, as you have described. A closed boundary requires a
Neumann BC, where the normal derivative of H is specified so that du/
dt (u is the normal component of velocity) remains zero. In other
words, you do not specify u=0, but rather du/dt=0 by setting dH/dn=-F
at the closed boundary. This assumes you have written the normal
component of the momentum equation at the wall

du/dt + F + dH/dn = 0

[prk...@gmail.com]

Kevin,

Thank you very much.

-Haejun-

[prk...@gmail.com]

Hello,
Following the discussion above, I have another question about the
value of H at the ghost node for open boundary condition.
FDS tech guide v.5 says,
H(i+1/2,k) = ( u(i,k)^2+w(i,k)^2 )/2 for u(i,k)>0 ---------(1)
H(i+1/2,k) = 0 for u(i,k)<0 ---------(2)

Therefore for the H values at the ghost nodes are,
H(i+1, k) = u(i,k)^2+w(i,k)^2 -H(i,k) for u(i,k)>0 -------(3)
H(i+1,k) = -H(i,k) for u(i,k)<0 -------(4)

In eq.(1) and(3), for outgoing flow, pressure perturbation is set zero
at the boundary whereas kinetic energy is conserved. Could you let me
know why it is set like this?

And For eq.(2) and (4), for incomming flow, asssuming the same
streamlines, the total pressure(kinetic and static pressure) at the
infinite distance is maintained to be zero from Bernoulli equation at
the boundary. If flow at (i,k) and (i+1) are in the same stream line,
H(i+1,k)=0 for u(i,k)<0, instead H(i+1,k) = -H(i,k). I tried both
cases, and both flow patterns look different,but make sense to me.

Thank you.

-Haejun-

[Kevin]

We assume that H=constant along streamlines, and pressure_pert=0 at
the OPEN boundary.

This is the basic idea. The real reason that we do what we do is that
it works. Try something different and you'll probably run into
trouble.