In this paper a new particle-finite-volume hybrid algorithm for the joint velocityfrequency-composition PDF method for turbulent reactive flows is presented. This method is a combination of a finite-volume scheme and a particle method. The finitevolume scheme is used to solve the Reynolds averaged Navier-Stokes equations and the particle method to solve the joint PDF transport equation. The motivation is to reduce the bias and the statistical error and to have an algorithm which is more efficient than stand-alone particle-mesh methods. Therefore, in the particle method we use the smoother mean density ρ and Favre averaged velocityŨ fields computed by the finite-volume scheme: This scheme is an Euler solver for compressible flow with the turbulent fluxes and the reaction term, which are computed by the particle algorithm, as source terms. Since some of the quantities are computed twice (i.e., the mean density ρ and the Favre averaged sensible internal energyẽ s ), by the finite-volume scheme and by the particle method, the hybrid algorithm is redundant. Although the model differential equations are consistent, it was difficult to satisfy consistency numerically, and an accurate particle tracking algorithm is crucial. Therefore a new scheme to interpolate the Favre averaged velocity has been developed which is second-order accurate and quasi conservative; i.e., it is based on the fluxes at the volume interfaces. Another important issue is the coupling between the finite-volume scheme and the particle method. A new time-averaging technique adds stability to the hybrid algorithm, and it also reduces the bias and the statistical error enormously. The properties of the new algorithm are demonstrated by results for a nonpremixed piloted-jet flame test case. First it is shown that the solution becomes statistically stationary and that it is internally consistent. Studies of the asymptotic behavior show that, for a given error tolerance, the new hybrid algorithm requires much less computer time than the stand-alone particle-mesh method (for this pilotedjet flame test case a factor of 20 times less). Finally, grid convergence studies verify that the scheme is second-order accurate in space.

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The paper describes a new hybrid finite-volume (FV)/particle method developed for the solution of the PDF equations for statistically stationary turbulent reactive flows. In this approach, the conservation equations for mean mass, momentum, and energy conservation are solved by a FV method while a particle algorithm is employed to solve the fluctuating velocity-turbulence frequency-compositions joint PDF transport equation. The mean velocity and pressure are supplied to the particle code by the FV code which in turn obtains all the Reynolds stresses, the scalar fluxes, and the reaction terms needed in the FV code. An important feature of the method is the complete consistency between the set of equations solved by the FV and particle methods. The algorithmic and numerical issues arising in the development of the hybrid method are studied in the simple setting of the stochastic ideal flow equations. Numerical results are obtained for 1D reactive stochastic ideal flow to demonstrate numerical properties of the method. The total numerical error is identified as statistical error, bias, spatial truncation error, and temporal truncation error. In contrast to the selfcontained particle method, the bias is found to be negligibly small. It is shown that all the numerical errors converge at the expected rates. Finally, the global convergence of the hybrid method is demonstrated and the optimal strategy for time-averaging that gives the best global convergence rate is investigated.

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The chaotic mixing in a drop moving through a winding channel is studied computationally in a two-dimensional setting. The molecular mixing is ignored and only the mixing due to the chaotic advection is considered. Passive tracer particles are used to visualize the mixing patterns and mixing is quantified by two distinct methods. It is found that both the quantification methods are consistent with visual observations as well as with each other. The effects of various non-dimensional parameters on the quality of mixing are studied and it is found that the capillary number, the ratio of the drop phase fluid viscosity to that of the ambient fluid and the relative size of the drop compared to the average channel width are the most critical parameters influencing the mixing. The mixing is found to be weakly dependent on Reynolds number.

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The impact and spreading of a compound viscous droplet on a flat surface are studied computationally using a front-tracking method as a model for the single cell epitaxy. This is a technology developed to create two-dimensional and three-dimensional tissue constructs cell by cell by printing cell-encapsulating droplets precisely on a substrate using an existing ink-jet printing method. The success of cell printing mainly depends on the cell viability during the printing process, which requires a deeper understanding of the impact dynamics of encapsulated cells onto a solid surface. The present study is a first step in developing a model for deposition of cell-encapsulating droplets. The inner droplet representing the cell, the encapsulating droplet, and the ambient fluid are all assumed to be Newtonian. Simulations are performed for a range of dimensionless parameters to probe the deformation and rate of deformation of the encapsulated cell, which are both hypothesized to be related to cell damage. The deformation of the inner droplet consistently increases: as the Reynolds number increases; as the diameter ratio of the encapsulating droplet to the cell decreases; as the ratio of surface tensions of the air-solution interface to the solution-cell interface increases; as the viscosity ratio of the cell to encapsulating droplet decreases; or as the equilibrium contact angle decreases. It is observed that maximum deformation for a range of Weber numbers has (at least) one local minimum at We=2. Thereafter, the effects of cell deformation on viability are estimated by employing a correlation based on the experimental data of compression of cells between parallel plates. These results provide insight into achieving optimal parameter ranges for maximal cell viability during cell printing.

Three different PDF algorithms have been applied to investigate a constant-density bluff-body stabilized flow using the same turbulence models and the same boundary conditions. The objectives of this paper are to compare the three algorithms in terms of numerical accuracy and efficiency and to demonstrate the ability of PDF methods to calculate this type of flow accurately. While one of the three algorithms is a standalone particle-mesh method, the other two are consistent hybrid algorithms, i.e., both are particle methods coupled with finite-volume schemes. The motivation for hybrid algorithms is to reduce the statistical and bias errors. Since the coupling between the finite-volume scheme and the particle method is a major numerical issue, different approaches have been investigated. It is shown that the results obtained from the three numerical algorithms are in good agreement with each other and with the experimental data.