An eddy-viscosity model is proposed and applied in large-eddy simulation of turbulent shear flows with quite satisfactory results. The model is essentially not more complicated than the Smagorinsky model, but is constructed in such a way that its dissipation is relatively small in transitional and near-wall regions. The model is expressed in first-order derivatives, does not involve explicit filtering, averaging, or clipping procedures, and is rotationally invariant for isotropic filter widths. Because of these highly desirable properties the model seems to be well suited for engineering applications. In order to provide a foundation of the model, an algebraic framework for general three-dimensional flows is introduced. Within this framework several types of flows are proven to have zero energy transfer to subgrid scales. The eddy viscosity is zero in the same cases; the theoretical subgrid dissipation and the eddy viscosity have the same algebraic structure. In addition, the model is based on a fundamental realizability inequality for the theoretical subgrid dissipation. Results are shown for a transitional and turbulent mixing layer at high Reynolds number and a turbulent channel flow. In both cases the present model is found to be more accurate than the Smagorinsky model and as good as the standard dynamic model. Unlike the Smagorinsky model, the present model is able to adequately handle not only turbulent but also transitional flow.

Direct numerical simulation databases have been used to study the effect of compressibility on mixing layers. The simulations cover convective Mach numbers from 0.2 to 1.2 and all contain a fully resolved turbulent energy cascade to small spatial scales. Statistical information is extracted from the databases to determine reasons for the reduced growth rate that is observed as the convective Mach number is increased. It is found that the dilatational contribution to dissipation is negligible even when eddy shocklets are observed in the flow. Also pressure-dilatation is not found to be significant. Using an accurate relation between the momentum thickness growth rate and the production of turbulence kinetic energy together with integrated equations for the Reynolds stress tensor it is shown that reduced pressure fluctuations are responsible for the changes in growth rate via the pressure-strain term. A deterministic model for the required pressure fluctuations is given based on the structure of variable-density vortices and the assumption that the limiting eddies are sonic. Simple anisotropy considerations are used to close the averaged equations. Good agreement with turbulence statistics obtained from the simulations is found.

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Direct numerical simulation (DNS) and large-eddy simulation (LES) of particle-laden turbulent channel flow, in which the particles experience a drag force, are performed. In this flow turbophoresis leads to an accumulation of particles near the walls. It is shown that the turbophoresis in LES is reduced, in case the subgrid effects in the particle equations of motion are ignored. To alleviate this problem an inverse filtering model is proposed and incorporated into the particle equations. The model is shown to enhance the turbophoresis in actual LES, such that a good agreement with the DNS prediction is obtained.

Turbulence characteristics of vertical air–solid pipe flow are investigated in this paper. Direct numerical simulations of the gas phase have been performed, while the solid particles have been simulated by a Lagrangian approach, including particle collisions. The modelling of wall roughness is shown to be important to obtain agreement with experimental data. Reynolds stresses and Reynolds stress budgets are given for both phases and for a wide range of solid–air mass load ratios (mass loads), varying from 0.11 to 30. Air turbulence intensities, Reynolds shear stress, and turbulence production reduce with increasing mass load. The mean air profile does not alter for low mass loads. In this regime, a simple theory predicts that the reduction of air turbulent production relative to unladen turbulent production is approximately equal to the mass load ratio. The insight that the solids Reynolds shear stress can be significant, even for low mass loads, is essential for this explanation. It is shown that at least two mechanisms cause the turbulence reduction. In addition to the classically recognized mechanism of dissipation of turbulent fluctuations by particles, there is another suppressing mechanism in inhomogeneous flows: the non-uniform relative velocity of the phases, created because particles slip at the wall, collide, and slowly react with the continuous phase. Investigation of the air turbulent kinetic energy equation demonstrates that the relative reduction of air pressure strain is larger than the reduction of turbulent production and dissipation, and pressure strain may therefore be a cause of the reduction of the other quantities. The fluctuational dissipation induced by the drag forces from particles is small compared to the other terms, but not negligible. For intermediate and high mass loads the air turbulence remains low. The relatively small turbulence intensities are not generated by the standard turbulent mechanisms any more, but directly caused by the particle motions. The particle–fluid interaction term in the turbulent kinetic energy equation is no longer dissipative, but productive instead. On increasing the mass load, the radial and azimuthal fluctuations of the particles grow. The corresponding reduction of solids anisotropy is an effect of the inter-particle collisions, which act as a solids pressure strain term. For intermediate and high mass loads, fluctuational drag force and particle collisions appear to be the relevant dissipation mechanisms in the solids fluctuational energy equation. In contrast to the air turbulent production, the solids ‘turbulent’ production term has the same level for low and high mass loads, while it attains a clear local minimum between. With increasing mass load, large-scale coherent turbulent fluid structures weaken, and eventually disappear. Simultaneously, the fluid fluctuations at relatively small length scales are enhanced by the motion of the particles. The highest particle concentration occurs near the wall for low mass loads, but on increasing the mass load, the concentration profile becomes uniform, while for the highest mass load particles accumulate in the centre of the pipe. Two-point correlation functions indicate that the addition of a small number of small solid particles to a clean pipe flow increases the streamwise length scale of the turbulence.

Statistical profiles of the first- and second-order spatial derivatives of velocity and pressure are reported for turbulent channel flow at Reτ = 590. The statistics were extracted from a high-resolution direct numerical simulation. To quantify the anisotropic behavior of fine-scale structures, the variances of the derivatives are compared with the theoretical values for isotropic turbulence. It is shown that appropriate combinations of first- and second-order velocity derivatives lead to (directional) viscous length scales without explicit occurrence of the viscosity in the definitions. To quantify the non-Gaussian and intermittent behavior of fine-scale structures, higher-order moments and probability density functions of spatial derivatives are reported. Absolute skewnesses and flatnesses of several spatial derivatives display high peaks in the near wall region. In the logarithmic and central regions of the channel flow, all first-order derivatives appear to be significantly more intermittent than in isotropic turbulence at the same Taylor Reynolds number. Since the nine variances of first-order velocity derivatives are the distinct elements of the turbulence dissipation, the budgets of these nine variances are shown, together with the budget of the turbulence dissipation. The comparison of the budgets in the near-wall region indicates that the normal derivative of the fluctuating streamwise velocity (∂u′/∂y) plays a more important role than other components of the fluctuating velocity gradient. The small-scale generation term formed by triple correlations of fluctuations of first-order velocity derivatives is analyzed. A typical mechanism of small-scale generation near the wall (around y+ = 1), the intensification of positive ∂u′/∂y by local strain fluctuation (compression in normal and stretching in spanwise direction), is illustrated and discussed.

Results of point-particle direct numerical simulations of downward gas-solid flow in smooth and rough vertical channels are presented. Two-way coupling and inter-particle collisions are included. The rough walls are shaped as fixed layers of tiny spherical particles with diameter much smaller than the viscous wall unit. The turbulence attenuation induced by the free solid particles in the gas flow is shown to be enhanced with increasing wall roughness. The so-called feedback force, the force exerted by the free particles on the gas, is decomposed into three contributions: the domain average of the mean feedback force, the non-uniform part of the mean feedback force and the fluctuating part of the feedback force. Since the non-uniformity of the mean feedback force increases with wall roughness, the effect of the non-uniform part of the mean feedback force is investigated in detail. For both smooth and rough walls, the non-uniform part of the mean feedback force is shown to contribute significantly to the particle-induced turbulence attenuation.