Tuesday, January 1, 2013

Near-Wall Boundary Conditions in FDS 6

Happy New Year!

This blog is a continuation in our series of articles on FDS 6 Release Notes.  Here we will focus on new wall functions for turbulent momentum and heat transfer.  In early versions of FDS, the near-wall boundary conditions were treated using a "slip condition" for the tangential component of velocity and, as is still the case, empirical correlations for heat and mass transfer.  In FDS 6, the default boundary condition for the tangential velocity is based on a log law wall function for both smooth and rough walls.  In addition, a new log law wall function for heat transfer, which accounts for variation in fluid Prandtl number, is in the testing phase and is available as an option by setting HEAT_TRANSFER_MODEL='LOGLAW' on the SURF line.

With FDS 5.4, the Werner-Wengle (1991) wall model became the default model for the tangential velocity component near smooth walls.  This model assumes a power law profile for the streamwise velocity component in the wall-normal direction and then further assumes that the tangential component of velocity near the wall is equivalent to the profile filtered over the height of the first grid cell.  Because of this improvement, FDS was able reproduce the Moody Chart (friction factor versus Reynolds number) for smooth walls over a very broad range of Reynolds numbers.  For rough walls, we implemented a wall function which took the maximum stress between the fully-rough log law given in Pope (Turbulent Flows, 2000) and the smooth wall stress from WW.  This formulation was practical, but the underlying mathematical inconsistency (filtered power law versus directly sampled log law) was not ideal. For compatibility with the wall functions used by the atmospheric community (outdoor flows for wildfires being of increasing interest), in v6 we decided to abandon the Werner-Wengle model and focus on the directly sampled log law wall functions.  Further, we no longer assume a fully rough wall---the transition from smooth to rough is accounted for by the new velocity wall function.

There are two other aspects of the near-wall treatment of velocity that deserve attention: one is how we handle the near-wall eddy viscosity and the other is how we handle the vorticity at the wall (in the FDS formulation, the vorticity resides in the advective term of the momentum equation).  For dynamic models of the eddy viscosity which require "test filtering," as we require for our Deardorff model, the near-wall treatment can be tricky.  When the dynamic Smagorinsky model was developed for channel flow, the standard practice was to average the model coefficient in the two homogeneous directions---the streamwise and spanwise directions---leaving the coefficient to vary only in the wall-normal direction.  But in a typical fire scenario we do not have the luxury of homogeneous directions.  Therefore, in the bulk flow, we use a test filter of size 2*dx in all three directions to smooth the velocity field.  This is problematic near the wall.  All attempts to retain the dynamic viscosity near the wall (special test filtering and so on) led to wildly erratic pyrolysis-based fire behavior unless very fine grid resolution was used.  This could not be tolerated, as it basically canceled out all the benefits of using the dynamic model to begin with.  To address this problem, we decided to simply use the constant coefficient Smagorinsky model in the first off-wall grid cell since this model does not require any test filtering.  To overcome the issue of convergence with the constant coefficient Smagorinsky model, we employ Van Driest damping of the mixing length near the wall.

As mentioned, a second important issue related to near-wall flow behavior is the treatment of the vorticity at the wall and at sharp edges.  Given that ventilation is a zeroth-order model parameter (translation: extremely important) and because common practice in fire protection engineering is to use relatively coarse grids, the near-wall value of vorticity has a surprisingly large impact on the overall model performance.  Think of doors and windows (which, let me add, should never be placed on the exterior of the computational domain) in the fire scenario, and imagine that the value of vorticity used at the edge may implicitly change the size of the opening by a cell width (this is basically the difference between a free slip and a no slip boundary condition).  The numerical approximations used to compute the vorticity on the wall or corner edge may effectively change the flow area of the opening.  To make matters worse, this effect is---by its very nature---grid dependent.  Our best attempt to deal with this dilemma has been to explore options for the vorticity "slip condition" while comparing FDS results for a simple test case which has many problematic features.  The case is a 3D flow over a square rib in a periodic channel (ribbed_channel test series in the Verification suite).  In this test we look at the location of the reattachment zone behind the rib, as well as the mean and RMS profiles for streamwise velocity in the wall-normal and streamwise directions.  With h being the height of the rib, we examine grid resolutions from h/dx=2 to h/dx=16.  The best result is found by applying a linear average between no slip and the slip value returned by the wall function.

Last but not least, the convective heat transfer model currently employed by FDS is based on taking the max of natural and forced convection Nusselt number correlations.  An often cited criticism of this approach is that the temperature and velocities used in these correlations are, of course, the "free stream" values in the correlation, but in FDS the values in the first off-wall cell are used.  Despite the obvious shortcomings, no one has yet systematically demonstrated a better alternative, and so these correlations remain the v6 default.  It is fair to say, in fact, that the subject of heat transfer wall functions is very much a research topic in the LES community.  Over the last year, some progress has been made in this area.  For practical fire simulations, the principal developer has been Ezgi Oztekin of the Fire Research Program at William J. Hughes Technical Center.  She implemented into FDS a basic log law heat transfer model and demonstrated the viability of the approach with compartment fire calculations.  On the verification front, Jung-il Choi's group from Yonsei University visited NIST last summer and modified Oztekin's model within FDS to include Prandtl number dependence and compared FDS results with the heated channel DNS results of Kim and Moin (1987).  This work represents the first serious verification study in development of the heat transfer model in FDS.  The results are in the FDS Verification Guide, and for this reason we are reasonably optimistic that the model will survive and become the default in the near future.