An Overview of the Probability Density Function Method

PDF methods offer the significant advantage over conventional moment closure methods primarily because the advection and chemical source terms in the PDF equations occur in closed form. Whilst exact treatment of advection is always advantageous, for a hybrid particle-finite-volume scheme, such as that used by the PDF method, the primary source of advection error is due to the systematic error in the RANS turbulence model rather than the linearization and subsequent discretisation of the advection terms in the thermo-chemical transport equations. The exact treatment of the thermo-chemical source term by the PDF method is an enormous benefit, as accurate treatment of the chemical source term in

turbulent combustion has been and still is one of the fundamental challenges for accurate and predictive combustion models. Few other modelling methodologies are capable of treating the chemical source term in closed form as the PDF method does. Essentially the only other modelling methodologies that treat the chemical source term in closed form are direct numerical simulation (DNS) and the LES implementation of the PDF method, namely the filtered density function (FDF) method.

Despite its advantages there are a number of challenges that need to be addressed when using the TPDF methodology. By definition, the joint PDF methodology is a high dimensional method. The joint PDF of composition as used in this study has Ns+1 dimensions, where Ns represents the number species and the +1 is due to the enthalpy equation. Finite-volume based methods are not efficient in handling problems with high dimensionality as the computational cost scales to the exponent of the problem dimensionality. Clearly for 22 chemical species and enthalpy transport equations a 23 dimensional problem becomes intractable for finite volume methods.

A solution to the high dimensionality of PDF methods is to recast the PDF transport equations as stochastic differential equations and solve them using a Monte Carlo particle-based methodology. The advantage of Monte Carlo based solution methods is that the computational cost is linearly proportional to the dimensionality of the problem;

hence the high dimensionality of PDF methods becomes manageable for Monte Carlo based solution methodologies. There is however a drawback to the recasting of the problem as stochastic differential equations and subsequent solution using a Monte Carlo solution method, two new error sources are created: statistical error and bias. These error sources are not present in finite volume methods and are unique to the Monte Carlo solution technique. The new error sources can be controlled and partially quantified in a simulation. Because of these additional error sources additional work is required for Monte Carlo simulations to gain confidence in the accuracy of the calculations.

Although advection and reaction are treated exactly in the TPDF method a new unclosed problem that does not occur in traditional moment closure based methods arises: this is

the need for a micro mixing model. As the stochastic particles advect and react throughout the domain they must interact or mix in some manner. It is the role of the micro mixing model to govern how and which of the stochastic particles mix. In certain geometries such as the auto-igniting Cabra burner [63], the solution will display very little sensitivity to the mixing model but high sensitivity to the kinetics. In other geometries such as the prediction of extinction and re-ignition in the piloted non-premixed flames (such as Sandia flame F [64]) the sensitivity to the mixing model and the time scale ratio constant C_φ is very large. The sensitivity of a particular simulation to the mixing model and chemistry can be evaluated by examining the local stochastic particle sensitivities as shown by Ren and Pope [65].

Another challenge for TPDF methods is the computational cost. For a 2D axisymmetric configuration with detailed chemistry the computational cost in terms of time and memory requirements can be comparable to a non-reacting moderate resolution 3D LES of the same flow. Almost all of the PDF computational cost stems from the exact treatment of the chemical source term. In order for chemical reactions to be treated exactly, they must be explicitly calculated for each particle. This creates an enormous computational burden as the stochastic Monte Carlo methods require many particles per finite volume cell (typically 100) to minimise statistical noise and bias. Chemistry acceleration techniques such as in situ adaptive tabulation (ISAT) maybe used to speed up the chemistry computation by up to two to three orders of magnitude in some cases over direct integration. Compounding the overhead due to chemistry computation is the fact that stochastic Monte Carlo particle based methods typically take far longer to converge than standard finite-volume methods and once converged require a time averaging procedure to minimise statistical noise in the final solution. These computational overheads do not make PDF computations insurmountable, but rather in comparison to other RANS based combustion models for the same geometry, PDF methods require at least an order of magnitude greater computational time. For comparative purposes, a reacting 2D symmetric TPDF solution will typically require far less computational time than the corresponding reacting 3D LES of the same geometry, even if a tabulated chemistry method is used in the LES calculation. This is an interesting

comparison to make, as the success of the both of these methods can be roughly gauged from the results of recent TNF workshops. The results from these recent workshops indicate that the current state of the art TPDF simulations predict finite-rate chemistry effects well in non-premixed flames but struggle with the predictions of the turbulence fields, in contrast for non-premixed flames exhibiting strong finite-rate chemistry effects state of the art LES using a tabulated reaction rate approach typically predict the turbulence field well but struggle to predict the finite-rate chemistry effects of the flame sufficiently.

The thermochemical composition PDF simulation methodology utilised in this study requires the velocity and turbulence fields to be accurately solved. It is well known that RANS based turbulence models such as the k-ε model do not accurately simulate many flow geometries due to the many model deficiencies. Errors in the mean velocities translate into errors in the transport terms for the TPDF method, however so long as the mean velocity field is not grossly in error its impact is typically minimal. The fluid mechanical time scale k/ε is used by the mixing model in the calculation of the scalar time scale. This means that any flow that is sensitive to the mixing model will be equally sensitive to the turbulence model which governs the calculation of the mean turbulent kinetic energy and the mean turbulent kinetic energy dissipation fields. Indeed it is the error in the predicted mean turbulent kinetic energy and the mean turbulent kinetic energy dissipation fields that can often be the weak link in TPDF-RANS simulations.