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This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference ΔU across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L×±∞. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a "vortex gas" comprising N point vortices of the same strength (γ=LΔU/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32000 vortices, with ensemble averages evaluated over up to 103 realizations and integration over 104L/ΔU. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/ΔU. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via "violent" and "slow" relaxations [ P.-H. Chavanis Physica A 391 3657 (2012)], over flow time scales of order 102 and 104L/ΔU, respectively (for N=400). The final state involves a single structure that stochastically samples the domain, possibly constituting a "relative equilibrium." The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter λ close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166±0.0002 times ΔU. Regime II, extensively studied in the turbulent shear flow literature as a self-similar "equilibrium" state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term "explosive" as it lasts less than one L/ΔU. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers [ J. Sommeria, C. Staquet and R. Robert J. Fluid Mech. 233 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

The vessel inside and coil outside film heat transfer coefficients of water and dilute aqueous polymer solutions of sodium carboxymethyl cellulose (SCMC) and sodium alginate (SA) have been studied in a turbine-agitated vessel for standard and nonstandard vessel configurations with agitator diameter, depth of agitation, helix diameter, and coiled tube outside diameter as parameters. The jacket-and coil-side heat transfer results are correlated.

Although the free-shear or mixing layer has been a subject of extensive research over nearly a century, there are certain fundamental issues that remain controversial. These include the influence of initial and downstream conditions on the flow, the effect of velocity ratio across the layer, and the nature of any possible coupling between small scale dynamics and the large scale evolution of layer thickness. In the spirit of the temporal vortex-gas simulations of Suryanarayanan et al. [“Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics,” Phys. Rev. E 89, 013009 (2014)], we revisit the simple 2D inviscid vortex-gas model with extensive computations and detailed analysis, in order to gain insights into some of the above issues. Simulations of the spatially evolving vortex-gas shear layer are carried out at different velocity ratios using a computational model based on the work of Basu et al. [“Vortex sheet simulation of a plane canonical mixing layer,” Comput. Fluids 21, 1–30 (1992) and “Modelling plane mixing layers using vortex points and sheets,” Appl. Math. Modell. 19, 66–75 (1995)], but with a crucial improvement that ensures conservation of global circulation. The simulations show that the conditions imposed at the origin of the free shear layer and at the exit to the computational domain can affect flow evolution in their respective downstream and upstream neighbourhoods, the latter being particularly strong in the single stream limit. In between these neighbourhoods at the ends is a regime of universal self-preserving growth rate given by a universal function of velocity ratio. The computed growth rates are generally located within the scatter of experimental data on plane mixing layers and closely agree with recent high Reynolds number experiments and 3D large eddy simulation studies. These findings support the view that observed free-shear layer growth can be largely explained by the 2D vortex dynamics of the quasi-2D large-scale structures known to be a characteristic of plane mixing layers.

An alternate BGK type formulation of the Enskog equation has been recently proposed. It was shown that the new model has a valid H-theorem and correct thermal conductivity. We propose Lattice Boltzmann (LB) formulation of this new Enskog-BGK model. The molecular nature of the model is verified in case of shear flow by comparing the predicted normal stress behavior by the current model with the prediction of molecular dynamics simulations. We extend the model for multiphase flow by incorporating attractive part as Vlasov type force. To validate multiphase formulation, the results of 3D simulations of a condensing bubble in a periodic box are presented.