Monday, November 5, 2012

Hydrodynamics and Turbulence


Following the announcement of the v6 beta release, we thought it might be helpful expand a little bit on some of the topics in the release notes.  Our aim here is to explain why certain changes were necessary or why certain choices were made in a way that might provide more insight than can be gleaned from just reading the formal documentation (though reading the guides is still highly encouraged!).

The focus in this article will be hydrodynamics and turbulence.  Hydrodynamics is the foundation of any CFD code, especially a reacting flow code like FDS.  Chemical species cannot react until they have mixed, and (in a diffusion flame) they cannot mix until they have been transported.  The velocity field, therefore, plays a critical role in fire dynamics.  In a model like FDS, other things being equal, the model for the turbulent viscosity determines the behavior of the velocity field and hence to a large extent the dynamics of the fire.

In v5, the turbulent viscosity was modeled by constant coefficient Smagorinsky (1963) (csmag), which has the drawback being overly dissipative (have you ever seen a 10-foot fire look like a candle flame?).  Thus, finer grid resolution was required to achieve accurate results.  But even highly resolved flames did not look qualitatively correct.  The reason: csmag is not convergent.  That is, the modeled term does not go away (as it should) when the grid is refined to the level of a DNS.

The original candidate to replace csmag was the dynamic Smagorinsky model (dsmag).  While this model added about 20% to the cost of computing the velocity field, in theory this cost could be recovered by achieving more accurate results (statistically) with coarser grid resolution.  So, dsmag was implemented and run through a battery of verification and validation tests with more or less satisfactory results.  On the plus side, we were getting very realistic-looking flames at moderate grid resolution.  But on the minus side, if the resolution was too coarse (often inevitable in engineering calculations), the flames were erratic---instead of a 10-foot flame looking like a candle, dsmag might produce an unstable ball of fire.

The solution to this dilemma came in the form of Deardorff’s model (1972), with a twist.  In the model Deardorff originally proposed, he solved a transport equation for the subgrid kinetic energy (sgs ke).  This strategy is expensive, but it allows for the inclusion of complex subgrid physics, like unresolved buoyancy production.  To avoid this cost, we employ a simple algebraic model for the sgs ke based on the scale similarity model of Bardina (1986).  The result is a model that is cheap and performs reasonably well at both coarse and fine resolution.  As an added benefit, the framework is now in place to utilize a transported sgs ke, a topic that might be worth exploring as FDS gets pushed to model large-scale outdoor flows in both wildfire and wind engineering applications.

While the new turbulence model is certainly significant, the change that really defines v6 is the new scalar transport scheme.  The ripple effects of this modification were not fully appreciated until a couple of years after the initial implementation.  What follows is the “saga of removing spurious temperature wiggles.”

Take FDS 5 and simulate a simple fire plume with open boundaries and 20 C ambient air temperature.  Now look closely at the temperature field near the corner of the burner.  You will see unphysical excursions of the local temperature to well below ambient, near 0 C.  Clearly this is undesirable.  What is not as obvious is that there are also unphysical excursions well above ambient, within the fire.  Of course, 1020 C versus 1000 C is not as dramatic a problem.

So why does this happen?  The root cause is central differencing in the solution of the density equation.  Purely centered schemes are notorious for generating dispersion errors---wiggles.  To combat this problem, v5 uses a boundedness correction which tries to prevent the scalar fields (density and species concentration) from going above or below defined limits.  With mass fractions the limits are easy to set (0 and 1).  With density there is only one hard limit (0).

In v6, we have decided to try another approach: total variation diminishing (TVD) scalar transport.  This is a fancy way of saying “we use just the right amount of upwinding.”  Pure upwind schemes are too dissipative (translation: inaccurate).  But TVD schemes are specially designed to track scalar discontinuities with minimal dispersion error and minimal dissipation.  This effectively solved the temperature wiggles problem.

Several TVD schemes are implemented in v6.  The default for LES is Superbee (Roe, 1986), so chosen because this scheme does the best job preserving the scalar variance in highly turbulent flows with coarse grid resolution.  The default scheme for DNS is Charm (Zhou, 1996) because the gradient steepening used in Superbee forces a stair step pattern at high resolution, while Charm is convergent.   A few other schemes (including Godunov and central differencing) are included for completeness; more details can be found in the Tech Guide.

These modifications were completed by the summer of 2009 (yes, you are reading that right).  It seemed at this point that we were very close to a v6 release.  Then… the wheels came off the bus.  In what we thought would be a routine pass through the verification suite, we hit a snag: our energy budget cases were off, by about 10%.  In addition, several of the validation cases now showed low upper layer temperatures---yep, by about 10%.

Hmmm.  A lot of head scratching ensued---two years, in fact, until Jason Floyd finally developed a test case that pinpointed the problem.  Imagine a T-mixer in which equal mass fluxes of a hot gas with a low cp (heat capcity) and a cold gas with a high cp mix and exit together.  The critical aspect of this case is that the flow paths are one cell thick and we artificially set the diffusivity and conductivity to zero.  Thus, any mixing is purely numerical.  What we found was that the outlet stream temperature was mass weighted instead of enthalpy weighted---there was no accounting for the variation in cp.  The pieces of the puzzle finally started to come together because another relatively recent development (actually in v5) was the inclusion of variable temperature specific heats.

The aftermath of this discovery was a torturous dissection of the energy equation (see appendix of the Tech Guide) illustrating that basic assumptions of the low-Mach number approximation can hide the effects of the numerical mixing from TVD transport schemes.  The bright side of the story is that through this analysis we were able to derive corrections for numerical mixing that force FDS to satisfy the discrete, conservative form of the sensible enthalpy equation, ultimately solving the temperature issues.

The benefit of the delayed release is that many other parts of the code were improved in the mean time.  So stay tuned for our next installment.