Direct numerical simulations of turbulent shear flows

The present study sheds light on the subgrid modeling problem encountered in the large eddy simulation (LES) of practical flows, where the turbulence is both inhomogeneous and anisotropic due to mean flow gradients. The subgrid scale stress (SGS) tensor, the quantity that is key to the success of LES, is studied here in such flows using both analysis and direct numerical simulation (DNS). It is shown that the SGS tensor, for the case of inhomogeneous flow, where the filtering operation is necessarily performed in physical space, contains two components: a rapid part that depends explicitly on the mean velocity gradient and a slow part that does not. The characterization, rapid and slow, is adopted by analogy to that used in the modeling of the pressure–strain in the Reynolds-averaged Navier–Stokes equations. In the absence of mean flow gradients, the slow part is the only nonzero component and has been the subject of much theoretical study. However, the rapid part can be important in the inhomogeneous flows that are often encountered in practice. An analytical estimate of the relative magnitude of the rapid and slow components is derived and the distinct role of each component in the energy transfer between the resolved grid scales and the unresolved subgrid scales is identified. Results that quantify this new decomposition are obtained from DNS data of a turbulent mixing layer. The rapid part is shown to play an important role when the turbulence is in a nonequilibrium state with turbulence production much larger than dissipation or when the filter size is not very small compared to the characteristic integral scale of the turbulence, as in the case of practical LES applications. More importantly, the SGS is observed to be highly anisotropic due to the close connection of the rapid part with the mean shear. The Smagorinsky eddy viscosity and the scale-similarity models are tested by performing a priori tests with data from DNS of the mixing layer. It is found that the scale-similarity model correctly represents the anisotropic energy transfer between grid and subgrid scales that is associated with the rapid part, while the eddy viscosity model captures the dissipation associated with the slow part. This may be a physical reason for the recent successes of the mixed model (Smagorinsky plus scale similarity) reported in the literature.

Analysis of data from direct numerical simulation of turbulent shear flows has shown the importance of large-scale, coherent turbulent structure on optical propagation. A basic approach is taken that incorporates both an experimental and a computational effort to develop a fundamental understanding of fluid-optical interactions. A flow generator and diagnostic instrumentation have been designed and are being built to measure the effect that a `simple' canonical turbulent flow has on a transmitted beam. This `simple' flow retains the essence of actual cases of interest, determining the effect of turbulent flow on optical transmission, and yet is tractable. The diagnostics include standard sensors to characterize the flow and optical diagnostics, including tomography, to characterize both the flow and the effect of the flow on beam propagation. A new dynamical system model for passive-scalar transport in a turbulent flow is also being developed to guide the experimental effort and to provide insight on the fluid-optical interactions.

The numerical simulation of a flow through a duct requires an externally specified forcing that makes the fluid flow against viscous friction. To this end, it is customary to enforce a constant value for either the flow rate (CFR) or the pressure gradient (CPG). When comparing a laminar duct flow before and after a geometrical modification that induces a change of the viscous drag, both approaches lead to a change of the power input across the comparison. Similarly, when carrying out direct numerical simulation or large-eddy simulation of unsteady turbulent flows, the power input is not constant over time. Carrying out a simulation at constant power input (CPI) is thus a further physically sound option, that becomes particularly appealing in the context of flow control, where a comparison between control-on and control-off conditions has to be made. We describe how to carry out a CPI simulation, and start with defining a new power-related Reynolds number, whose velocity scale is the bulk flow that can be attained with a given pumping power in the laminar regime. Under the CPI condition, we derive a relation that is equivalent to the Fukagata–Iwamoto–Kasagi relation valid for CFR (and to its extension valid for CPG), that presents the additional advantage of naturally including the required control power. The implementation of the CPI approach is then exemplified in the standard case of a plane turbulent channel flow, and then further applied to a flow control case, where a spanwise-oscillating wall is used for skin-friction drag reduction. For this low-Reynolds-number flow, using 90 % of the available power for the pumping system and the remaining 10 % for the control system is found to be the optimum share that yields the largest increase of the flow rate above the reference case where 100 % of the power goes to the pump.

We compare the space-time correlations calculated from direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent channel flows. It is found from the comparisons that the LES with an eddy-viscosity sub-grid scale (SGS) model over-predicts the space-time correlations than the DNS. The overpredictions are further quantified by the integral scales of directional correlations and convection velocities. A physical argument for the overprediction is provided that the eddy-viscosity SGS model alone does not includes the backscatter effects although it correctly represents the energy dissipations of SGS motions. This argument is confirmed by the recently developed elliptic model for space-time correlations in turbulent shear flows. It suggests that enstrophy is crucial to the LES prediction of space-time correlations. The random forcing models and stochastic SGS models are proposed to overcome the overpredictions on space-time correlations.

Simulation and Reynolds stress modeling of particle-laden turbulent shear flows

Zonal Embedded Grids for Numerical Simulations of Wall-Bounded Turbulent Flows

Direct numerical simulation of turbulent free-surface high Prandtl number fluid flows in fusion reactors

Flow around a wall-mounted cube is an example of a turbulent flow around a three-dimensional bluff body attached to a surface. The main experimentally observed feature of this type of flow is the appearance of complex vortical structures, e.g. a horseshoe vortex originating in front of the body and enveloping it. The current paper is a follow-up to Yakhot, Liu & Nikitin (2006) in which we presented results of direct numerical simulation (DNS) of turbulent flow around a cube. Here, it is shown that unsteadiness of the considered flow is caused by inviscid–viscous interaction between the horseshoe vortex and the narrow band of positive vorticity attached to the surface in front of the cube. Details of the spatio-temporal evolution of large-scale vortical structures, including samples of long-term visualization and turbulence statistics, are presented. For the normal-to-the-wall velocity, in the vicinity of the cube's front face, the results reveal an anomalous probability distribution, namely, a bimodal distribution and one with high kurtosis.

A numerical simulation of a complex turbulent shear flow using large-eddy simulation techniques is carried out. The filtered Navier-Stokes equations are solved with a finite-volume method. The subgrid model is a local adaptation to the physical space of isotropic spectral eddy-viscosity models. The statistics of the mean field are in good agreement with the experimental data available, corresponding to low step. Calculations in a high-step case show that the eddy structure of the flow presents striking analogies with the plane shear layers, with large billows shed behind the step, and longitudinal hairpin vortices strained between these billows.

Chapter Two - Direct numerical and large–eddy simulation of complex turbulent flows

An analysis of the near-wall behavior of the proper orthogonal decomposition (POD) eigenfunctions (see Lumley 1967) derived from direct numerical simulation (DNS) of channel flow of Dinavahi and Zang (1993) is performed. This work is driven in part by a desire to include structure in turbulence models. Although this particular application is in the near wall region (because of availability of the data base), the ideas discussed here are applicable in free shear flows as well. Consistent with previous studies, a low order multi-mode reconstruction of the kinetic energy and Reynolds shear stress suffices. A similar reconstruction of the isotropic dissipation rate is shown to be insufficient, however. An analysis is performed of the multi-mode composition of the dissipation rate in the near-wall region, and it is shown that a significant number of higher-order modes are required to achieve the correct asymptotic consistency in the near-wall region. In an attempt to avoid this problem, a length scale definition is proposed in terms of an integration of the two-point correlation tensor which factors in the presence of the wall. The wall is accounted for by only integrating out to 2y+and not over the entire domain. Viscous and inviscid estimates for the dissipation rate are used in the near-wall and core regions, respectively, in conjunction with this length scale representation, to obtain an estimate of the dissipation rate throughout the domain. The resulting dissipation rate exhibits the proper behavior near the wall and in the inertial layer. Higher-order POD mode estimates of the length scale are computed and compared to that computed using the full correlation tensor from the DNS results of Dinavahi and Zang (1993).

The acceleration of inertial particles in a homogeneous shear flow may vigorously respond to the intense flow structures induced by the mean shear. In this study, by direct numerical simulation (DNS) of particle-laden shear flow, we observe the statistical properties of those accelerations, and then we assessed the recently proposed simulation approach in which the effect of intermittency on residual scales is linked directly with coarsely resolved flow turbulence. First, we focused on the acceleration statistics of fluid particles in a homogeneous shear flow. Consistent with previous findings in homogenous isotropic turbulence, the norm and the direction of fluid acceleration in shear flow are shown to be conditioned by the dynamics of intermittent long-lived vortical structures. The averaged acceleration norm of the fluid particle exhibits a pseudo-cyclic behaviour, which is a signature of the periodic action of the largest confined vortices against the mean shear. The long correlation time of the acceleration norm, of the order of the integral time, the high level of fluctuations of the acceleration norm (very close to the magnitude of the mean norm of acceleration) and the log-normality in its statistical distribution reflect the impact of intense flow structures in shear flow. The presence of these zones results also in highly non-Gaussian statistics of the acceleration of a fluid particle and its velocity increments at small time lags. Contrary to the acceleration norm, the acceleration direction of a fluid particle is observed to be short, of the order of the Kolmogorov time, and to be statistically independent of the acceleration norm. The short-time correlation of the acceleration direction is attributed usually to effects of centripetal forces in intense vorticity filaments. We suggest that, in homogeneous shear turbulence, there may be a supplementary effect on the acceleration direction: the vortex, stretched by the mean shear, may exert the preferential direction of fluid particle acceleration. As evidence, our DNS shows that fluid particles are accelerated preferentially in the direction of longitudinal vortical tubes, effectively stretched by the imposed mean shear. Concerning simulations with heavy point-wise particles, it is shown that, when the inertia of a particle is not high, its acceleration closely follows all the aforementioned properties of the fluid particle acceleration. Particularly, an inertial particle is also entrained by accelerating motion in the direction of the effectively stretched vortical tubes. Although with increasing Stokes number of the inertial particle, the effects of strong intermittency in the flow are filtered, it is shown that the alignment of the particle acceleration direction with vortical tubes is amplified – particles with higher inertia respond to solicitations of stronger vortical structures. An alternative to the Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441–465) preferential sweeping mechanism is discussed in this paper. When the shear turbulence is under-resolved, we employed the large-eddy simulation (LES) equations with a forcing term on the smallest resolved scales in order to simulate stochastically the effects of the dynamics on the residual scales. The forcing term is expressed with two independent stochastic processes, one for its norm and another for its direction. While the norm of acceleration is modelled using Pope’s log-normal process with the integral time for correlation, its direction is modelled in the framework of an Ornstein–Uhlenbeck process on the unit sphere. Consistently with our DNS, the latter process contains two presumed times: the homogeneous strain rate is specified as typical time of relaxation towards the direction of resolved vorticity, and the Kolmogorov time is presumed as a typical time of the diffusion process on the unit sphere. The high efficiency of this approach is demonstrated in prediction of the small-scale dynamics observed in DNS, even in the case when the shear length scale is not resolved by LES.

Direct numerical simulations (DNS) of turbulent flows provide a complete data base to develop and to test turbulence models. In this article, the progress made in developing models for the dissipation rate equation is reviewed. New scaling arguments for the various terms in the dissipation rate equation were tested using data from DNS of homogeneous shear flows. Modifications to the epsilon-equation model that take into account near-wall effects were developed using DNS of turbulent channel flows. Testing of new models for flows under mean compression was carried out using data from DNS of isotropically compressed turbulence. In all of these studies the data from the simulations was essential in guiding the model development. The next generation of DNS will be at higher Reynolds numbers, and will undoubtedly lead to improved models for computations of flows of practical interest.

Error propagation and conditioning analysis of DNS data of turbulent viscoelastic channel flows

A Direct Numerical Simulation (DNS) of turbulent channel flow of dilute suspensions of small, Brownian fibres in a Newtonian solvent is presented. The DNS investigates the potential of drag reduction under situations, where no internal elasticity of the additives is present. The DNS is solving the microscopic equations for the suspended fibres and couples the resulting stresses into a (macroscopic) DNS of the solvent. The microscopic equations for the conformation of the fibres as well as the resulting stresses are derived by the rheological theory of dilute suspensions of Brownian particles in Newtonian solvents. These equations are solved by a Monte-Carlo method. First results show a dramatic reduction of the Reynolds shear stress. However, only a mild reduction of the drag is observed because the fibres generate considerable shear stress components at the wall at the configuration chosen.