A turbulence bridging method purported for any filter-width or scale resolution—fully averaged to completely resolved—is developed. The method is given the name partially averaged Navier-Stokes (PANS) method. In PANS, the model filter width (extent of partial averaging) is controlled through two parameters: the unresolved-to-total ratios of kinetic energy (fk) and dissipation (fε). The PANS closure model is derived formally from the Reynolds-averaged Navier-Stokes (RANS) model equations by addressing the following question: if RANS represents the closure for fully averaged statistics, what is the corresponding closure for partially averaged statistics? The PANS equations vary smoothly from RANS equations to Navier-Stokes (direct numerical simulation) equations, depending on the values of the filter-width control parameters. Preliminary results are very encouraging.

The evolution of infinitesimal material line and surface elements in homogeneous isotropic turbulence is studied using velocity-gradient data generated by direct numerical simulations (DNS). The mean growth rates of length ratio (l) and area ratio (A) of material elements are much smaller than previously estimated by Batchelor (1952) owing to the effects of vorticity and of non-persistent straining. The probability density functions (p.d.f.'s) of l/〈l〉 and A/〈A〉 do not attain stationarity as hypothesized by Batchelor (1952). It is shown analytically that the random variable l/〈l〉 cannot be stationary if the variance and integral timescale of the strain rate along a material line are non-zero and DNS data confirm that this is indeed the case. The application of the central limit theorem to the material element evolution equations suggests that the standardized variables $\hat{l}(\equiv (\ln l - \langle \ln l\rangle)/({\rm var} l)^{\frac{1}{2}})$ and Â(≡(ln A − 〈ln A〉)/(var A)½) should attain stationary distributions that are Gaussian for all Reynolds numbers. The p.d.f.s of $\hat{l}$ and  calculated from DNS data appear to attain stationary shapes that are independent of Reynolds number. The stationary values of the flatness factor and super-skewness of both $\hat{l}$ and  are in close agreement with those of a Gaussian distribution. Moreover, the mean and variance of ln l (and ln A) grow linearly in time (normalized by the Kolmogorov timescale, τη), at rates that are nearly independent of Reynolds number. The statistics of material volume-element deformation are also studied and are found to be nearly independent of Reynolds number. An initially spherical infinitesimal volume of fluid deforms into an ellipsoid. It is found that the largest and the smallest of the principal axes grow and shrink respectively, exponentially in time at comparable rates. Consequently, to conserve volume, the intermediate principal axis remains approximately constant.The performance of the stochastic model of Girimaji & Pope (1990) for the velocity gradients is also studied. The model estimates of the growth rates of 〈ln l〉 and 〈ln A〉 are close to the DNS values. The growth rate of the variances are estimated by the model to within 17%. The stationary distributions of $\hat{l}$ and  obtained from the model agree very well with those calculated from DNS data. The model also performs well in calculating the statistics of material volume-element deformation.

In this paper a stochastic model for velocity gradients following fluid particles in incompressible, homogeneous, and isotropic turbulence is presented and demonstrated. The model is constructed so that the velocity gradients satisfy the incompressibility and isotropy requirements exactly. It is further constrained to yield the first few moments of the velocity gradient distribution similar to those computed from full turbulence simulations (FTS) data. The performance of the model is then compared with other computations from FTS data. The model gives good agreement of one-time statistics. While the two-time statistics of strain rate are well replicated, the two-time vorticity statistics are not as good, reflecting perhaps a certain lack of embodiment of physics in the model. The performance of the model when used to compute material element deformation is qualitatively good, with the material line-element growth rate being correct to within 5% and that of surface element correct to within 20% for the lowest Reynolds number considered. The performance of the model is uniformly good for all the Reynolds numbers considered. So it is conjectured that the model can be used even in inhomogeneous, high-Reynolds-number flows, for the study of evolution of surfaces, a problem that is of interest particularly to combustion researchers.

Many engineering calculations of turbulent reacting flows employ the assumed-pdf approach. On the form of the assumed pdf, however, there has been little consensus and the choice has often been ad hoc. The objective of this work is to investigate the validity of the assumed P-pdf model for the case of the inert mixing of two scalars and extend the applicability of the model to multiple scalar mixing.By comparing the p-pdf model favorably with the two-scalar mixing data obtained from direct numerical simulations (DNS), it is demonstrated that the use of this model is justified for all stages of the mixing process in statistically-stationary. isotropic turbulence. It is also shown analytically that during the final stages of mixing the model P-pdf reduces to a Gaussian-pdf, consistent with the observations from experiments and DNS.The suggested multivariate p-pdfmodel for multiple-scalar mixing relates algebraically, Ihe mean scalar concentrations and the turbulent scalar-energy-to the joint pdf of the scalar concentrations. The model requires that only one extra transport equation-ror the turbulent scalar-energy-e-be solved, apart from the mean equations. The model can be used in conjunction with any model for the turbulence velocity field and is expected to provide a simple method for calculating scalar mixing in turbulent flows.

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Partially-averaged Navier Stokes (PANS) is a suite of turbulence closure models of various modeled-to-resolved scale ratios ranging from Reynolds-averaged Navier Stokes (RANS) to Navier-Stokes (direct numerical simulations). The objective of PANS, like hybrid models, is to resolve large scale structures at reasonable computational expense. The modeled-to-resolved scale ratio or the level of physical resolution in PANS is quantified by two parameters: the unresolved-to-total ratios of kinetic energy (f k ) and dissipation (fε). The unresolved-scale stress is modeled with the Boussinesq approximation and modeled transport equations are solved for the unresolved kinetic energy and dissipation. In this paper, we first present a brief discussion of the PANS philosophy followed by a description of the implementation procedure and finally perform preliminary evaluation in benchmark problems.

A two-fluid lattice Boltzmann model for binary mixtures is developed. The model is derived formally from kinetic theory by discretizing two-fluid Boltzmann equations in which mutual collisions and self-collisions are treated independently. In the resulting lattice Boltzmann model, viscosity and diffusion coefficients can be varied independently by a suitable choice of mutual- and self-collision relaxation-time scales. Further, the proposed model can simulate miscible and immiscible fluids by changing the sign of the mutual-collision term. This is in contrast to most existing single-fluid lattice Boltzmann models that employ a single-relaxation-time scale and hence are restricted to unity Prandtl and Schmidt numbers. The extension of binary mixing model to multiscalar mixing is quite straightforward.

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