▪ Abstract  Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of Rλ ≈ 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgrid-scale model. It is based on the ‘resolved stress’ tensor Lij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during large-eddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between Lij and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lij and the strain-rate tensor, is introduced. A ‘dynamic’ version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

Better understanding of particle-particle and particle-fluid interactions requires accurate 3D measurements of particle distributions and motions. We introduce the application of in-line digital holographic microscopy as a viable tool for measuring distributions of dense micrometer (3.2 microm) and submicrometer (0.75 microm) particles in a liquid solution with large depths of 1-10 mm. By recording a magnified hologram, we obtain a depth of field of approximately 1000 times the object diameter and a reduced depth of focus of approximately 10 particle diameters, both representing substantial improvements compared to a conventional microscope and in-line holography. Quantitative information on depth of field, depth of focus, and axial resolution is provided. We demonstrate that digital holographic microscopy can resolve the locations of several thousand particles and can measure their motions and trajectories using cinematographic holography. A sample trajectory and detailed morphological information of a free-swimming copepod nauplius are presented.

Three-dimensional velocity distributions of a turbulent flow in the core region of a square duct at ReH = 1.2 × 105 are measured using holographic particle image velocimetry (HPIV). Spatial filtering of the 136 × 130 × 128 velocity vector maps enables calculation of subgrid-scale (SGS) stresses and parameters based on the filtered velocity gradients, such as the filtered strain-rate tensor and vorticity vector. Probability density functions (p.d.f.) of scalar parameters characterizing eigenvalue structures confirm that the most probable strain-rate topology is axisymmetric extension, and show that the most probable SGS stress state is axisymmetric contraction. Conditional sampling shows that high positive SGS dissipation occurs preferentially in regions with these preferred strain-rate and stress topologies. High negative SGS dissipation (backscatter) occurs preferentially in regions of axisymmetric contracting SGS stress topology, but is not associated with any particular strain-rate topology. The nonlinear model produces the same trends but tends to overpredict the likelihood of the preferred stress state.Joint p.d.f.s of relative angles are used to investigate the alignments of the SGS stress eigenvectors relative to the vorticity and eigenvectors associated with eddy viscosity and similarity/nonlinear models. The results show that the most extensive SGS stress eigenvector is preferentially aligned at 32° to the most contracting strain-rate eigenvector. This alignment trend persists, with some variations in angle and peak probability, during conditional samplings based on the SGS dissipation rate, vorticity and strain-rate magnitudes. The relative alignment of the other two stress and strain-rate eigenvectors has a bimodal behaviour with the most contracting and intermediate stress eigenvectors ‘switching places’: from being aligned at 32° to the most extensive strain-rate eigenvector to being parallel to the intermediate strain-rate eigenvector. Conditional sampling shows that one of the alignment configurations occurs preferentially in regions of high vorticity magnitude, whereas the other one dominates in regions where the filtered strain-rate tensor has axisymmetric contracting topology. Analysis of DNS data for isotropic turbulence at lower Re shows similar trends.Conversely, the measured stress eigenvectors are preferentially aligned with those of the nonlinear model. This alignment persists in various regions of the flow (high vorticity, specific flow topologies, etc). Furthermore, the alignment between the strain-rate and nonlinear model tensors also exhibits a bimodal behaviour, but the alignment angle of both configurations is 42°. Implications of alignment trends on SGS dissipation are explored and conditions for high backscatter are identified based on the orientation of the stress eigenvectors. Several dynamical and kinematical arguments are presented that may explain some of the observed preferred alignments among tensors. These arguments motivate further analysis of the mixed model, which shows good alignment properties owing to the dominance of the Leonard stress on the alignments. Nevertheless, the data also show that the mixed model produces some unrealistic features in probability distributions of SGS dissipation, and unphysical eigenvector alignments in selected subregions of the flow.

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The quantification of three-dimensional (3D) flow structures and particle dynamics is crucial for unveiling complex interactions in turbulent flows. This review summarizes recent advances in volumetric particle detection and 3D flow velocimetry involving holography. We introduce the fundamental principle of holography and discuss the debilitating depth-of-focus problem, along with methods that have been implemented to circumvent it. The focus of this review is on recent advances in the development of in-line digital holography in general, and digital holographic microscopy in particular. A mathematical background for the numerical reconstruction of digital holograms is followed by a summary of recently introduced 3D particle tracking and velocity measurement techniques. The review concludes with sample applications, including 3D velocity measurements that fully resolve the flow in the inner part of a turbulent boundary layer, the diffusion of oil droplets in high-Reynolds number turbulence, and predator-prey interactions among swimming microorganisms in dense suspensions, as well as oceanic and atmospheric field experiments.

The forces that act on bubbles as they are entrained by a vortex are measured using particle image velocimetry. Triple exposure images are used to measure the velocity and acceleration of the fluid and the bubbles simultaneously. Distinction between phases is achieved by using fluorescent particles as liquid flow tracers. The buoyancy, pressure, and inertia forces are computed from the data, while the drag and the lift forces are determined from a force balance on each bubble. It is found that in the present range of bubble diameters, 500 pm

Turbulence characteristics in the coastal ocean bottom boundary layer are measured using a submersible Particle Image Velocimetry (PIV) system with a sample area of 20 ϫ 20 cm 2. Measurements are performed in the New York Bight at elevations ranging from 10 cm to about 1.4 m above the seafloor. Recorded data for each elevation consists of 130 s of image pairs recorded at 1 Hz. After processing, the data at each elevation consist of 130 instantaneous spatial velocity distributions within the sample area. The vertical distribution of mean velocity indicates the presence of large-scale shear even at the highest measurement station. The flow also undergoes variations at timescales longer than the present data series. Spatial spectra of the energy and dissipation are calculated from individual vector maps. The data extend well beyond the peak in the dissipation spectrum and demonstrate that the turbulence is clearly anisotropic even in the dissipation range. The vector maps are also patched together to generate extended velocity distributions using the Taylor hypothesis. Spectra calculated from the extended data cover about three decades in wavenumber space. For the overlapping range the extended spectra show small differences from those determined using the instantaneous distributions. Use of the Taylor hypothesis causes ''contamination'' of the extended spectra with surface waves. Nevetheless, the results still indicate that the turbulence is also anisotropic at low wavenumbers (energy containing eddies). The vertical component of velocity fluctuations at energy containing scales is significantly damped as the bottom is approached, while the horizontal component maintains a similar energy level at all elevations. Different methods of estimating the turbulent energy dissipation are compared. Several of these methods are possible only with 2D data, such as that provided by PIV, including a ''direct'' method, which is based on measured components of the deformation tensor. Estimates based on assumptions of isotropy are typically larger than those based on the direct method (using available velocity gradients and least number of assumptions), but the differences vary from 30% to 100%.