Fluctuating Velocity Gradient Research Articles

Dissolution and evolution effects on pressure fluctuations in a space micropump

The purpose of this paper is to reveal the dynamic mass transfer effects between gas and liquid on pressure fluctuations in a space micropump using our proposed computational model of gas–liquid mass transfer. Complex dissolution and evolution processes were applied to achieve accurate dynamic gas–liquid mass transfer predictions in the micropump. The validation of experiments was conducted by measuring the performance characteristics of the micropump, and the mass transfer model was verified by a dissolved oxygen concentration experiment in plug discharge flow. Based on this, four conditions including unsteady single-phase, two-phase without mass transfer, dissolution, and coexisting dissolved–released flows calculations are discussed to clarify the frequency contents, generation reasons, and propagation law. Combined with entropy production analysis, the pressure fluctuation influenced by the dissolution and evolution is illustrated. The results exhibit that the evolution of the gas is located on the head of the long blade suction surface in the impeller. When a unidirectional gas-to-liquid dissolution process occurs, the fluctuating amplitude of the characteristic dominant 5 fn is significantly weakened; otherwise, when dissolution and evolution coexist, the amplitude is significantly promoted as the released gas increase the flow instability. In addition, the distributions of local high entropy production induced by mean and fluctuating velocity gradients overlap that of large mass transfer rates and high amplitude of the mixing frequency in the volute, exhibiting the relationship between entropy production, mass transfer, and flow instability. The current study provides a guidance that the dissolved gases’ concentration must be controlled strictly to avoid the evolution of gas for the safety and stability of the space hydraulic system.

Stochastic model for the alignment and tumbling of rigid fibers in two-dimensional turbulent shear flow

Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analytically. To reproduce the single-time orientation statistics obtained numerically, it is found that one has to properly account for the combined effect of the mean shear, for anisotropic velocity gradient fluctuations, and for the presence of persistent rotating structures in the flow that bias Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibres angular increment. Still, it does not reproduce an intricate behaviour observed in numerics for intermediate times: the unfolded angle is there very similar to a L\'evy walk with distributions of increments displaying intermediate power-law tails.

Velocity gradient statistics in turbulent shear flow: an extension of Kolmogorov's local equilibrium theory

This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation$\left \langle \epsilon \right \rangle$per unit mass, kinematic viscosity$\nu$and parameter$\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$, provided that$\gamma$is small in an appropriate sense, where$S$is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small$\gamma$, and the anisotropy of moments decreases approximately in proportion to$\gamma$. This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number$Re_\tau$up to$\approx 8000$. In the TCF, there is a range$y$where$\gamma$scales approximately$\propto y^ {-1/2}$, and the anisotropy of the moments of the gradients decreases with$y$nearly in proportion to$y^ {-1/2}$, where$y$is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.

A comparison of different methods for estimating turbulent dissipation rate in under-resolved flow fields from synthetic PIV images

The turbulent energy dissipation rate is an important parameter that determines the transfer rates of mass, heat and momentum in chemical engineering industry. Especially in a multiphase reactor, the local instantaneous turbulent dissipation rate affects the breakage and coalescence of bubbles or droplets. To directly determine the local turbulent dissipation rate, fluctuating velocity gradients have to be measured down to the Kolmogorov length scale. This paper compares the advantages and disadvantages of three different methods, namely, correlation method, optical flow method and hybrid method, in the evaluation of velocity fields, which are subsequently used to calculate the turbulent dissipation rate. An instantaneous flow field of homogeneous isotropic turbulence from DNS results is used as a benchmark. The velocity field and turbulent dissipation rate field obtained by these three methods are compared to the DNS results. It is shown that the hybrid method performs better in both velocity field evaluation and local turbulent dissipation rate estimation compared to the other two methods. This is because the hybrid method combines the advantages of the correlation method in achieving a stable averaged velocity field and the optical flow method in achieving pixel-level resolution measurement of the flow field.

Numerical investigation of transient liquid nitrogen cavitating flows with special emphasis on force evolution and entropy features

This study aims to investigate the impacts of thermal effect on cavitation, entropy features and interactions among vorticity, force coefficients and entropy production rate (EPR) in a liquid nitrogen cavitating flow. Numerical results demonstrate that the increase in free-stream temperature intensifies thermal effect and weakens cavitation intensity. The EPR by fluctuating velocity gradient (ṠD'''') dominates the evolution of EPR, and the cavity shedding or collapse increases the velocity gradient, thereby increasing ṠD''''. The variation trend of EPR by wall region (ṠW) is consistent with ṠD''''. By contrast, EPR by temperature gradient (ṠC'''',ṠC¯''') always distributes on the liquid–vapour interface and increases with the development of the attached sheet cavity. Higher temperature acts as a disincentive to each item of EPR. The stable development of negative sheet vorticity at the leading edge increases the force coefficients nearly linearly. The counter clockwise vortex forms at the hydrofoil tail because the small-scale cloud cavity decreases the lift coefficient sharply. As the instability of attached sheet cavity is enhanced by re-entrant jet, the large vortex is dispersed into several clockwise vortexes, and positive vorticity occurs inside the cavity. Consequently, the force coefficients present a downward trend overall with intensive fluctuation. The distribution of EPR is similar to that of vorticity, and the opposite trend of entropy production integral and lift coefficient illustrates that the force can be used as a representation of energy loss to a certain extent.

Extreme velocity gradients in turbulent flows

Fully turbulent flows are characterized by intermittent formation of very localized and intense velocity gradients. These gradients can be orders of magnitude larger than their typical value and lead to many unique properties of turbulence. Using direct numerical simulations of the Navier–Stokes equations with unprecedented small-scale resolution, we characterize such extreme events over a significant range of turbulence intensities, parameterized by the Taylor-scale Reynolds number (). Remarkably, we find the strongest velocity gradients to empirically scale as , with , where is the Kolmogorov time scale (with its inverse, , being the rms of velocity gradient fluctuations). Additionally, we observe velocity increments across very small distances , where η is the Kolmogorov length scale, to be as large as the rms of the velocity fluctuations. Both observations suggest that the smallest length scale in the flow behaves as , with , which is at odds with predictions from existing phenomenological theories. We find that extreme gradients are arranged in vortex tubes, such that strain conditioned on vorticity grows on average slower than vorticity, approximately as a power law with an exponent , which weakly increases with . Using scaling arguments, we get , which suggests that β would also slowly increase with . We conjecture that approaching the mathematical limit of infinite , strain and vorticity would scale similarly resulting in and hence extreme events occurring at a scale corresponding to .

Onset of intermittency in stochastic Burgers hydrodynamics.

We study the onset of intermittency in stochastic Burgers hydrodynamics, as characterized by the statistical behavior of negative velocity gradient fluctuations. The analysis is based on the response functional formalism, where specific velocity configurations-the viscous instantons-are assumed to play a dominant role in modeling the left tails of velocity gradient probability distribution functions. We find, as expected on general grounds, that the field-theoretical approach becomes meaningful in practice only if the effects of fluctuations around instantons are taken into account. Working with a systematic cumulant expansion, it turns out that the integration of fluctuations yields, in leading perturbative order, to an effective description of the Burgers stochastic dynamics given by the renormalization of its associated heat kernel propagator and the external force-force correlation function.

Application of optical flow methods to aerated skimming flows above triangular and trapezoidal step cavities

ABSTRACTStepped chutes are built to provide safe flood passage in dams. The steps are associated with strong turbulence generation, which allow air to be entrained from the free surface. The present work adopts a non-intrusive local optical flow technique to examine the effects of partial cavity blockage on the mean and turbulent properties in aerated skimming flows in two large-size stepped chute models. The partial cavity blockage was found to be associated with decreased mean velocities in the overflow. The step edge was identified as the most significant source of turbulent production. General increases in turbulence levels and large scale turbulent motions were identified for the partially blocked cavities, which could be linked to a reduction in mutual sheltering between adjacent roughness elements. It was implied that the cavity blockage might have some effects on the velocity gradient fluctuations.

Reduced power consumption in stirred vessels by means of fractal impellers

Earlier studies have shown that the power consumption of an unbaffled stirred vessel decreases significantly when the regular blades are replaced by fractal ones. In this paper, the physical explanation for this reduction is investigated using Direct Numerical Simulations at Re = 1600. The gaps around the fractal blade perimeter create jets that penetrate inside the recirculation zone in the wake and break up the trailing vortices into smaller ones. This affects the time‐average recirculation pattern on the suction side. The volume of the separation region is 7% smaller in the wake of the fractal blades. The lower torque of the fractal impeller is equivalent to a decreased transport of angular momentum; this difference stems from the reduced turbulent transport induced by the smaller trailing vortices. The major difference in the turbulent dissipation is seen in the vicinity of trailing vortices, due to fluctuations of velocity gradients at relatively low frequencies. © 2018 American Institute of Chemical Engineers AIChE J, 64: 1485–1499, 2018

A multifractal model for the velocity gradient dynamics in turbulent flows

We develop a stochastic model for the velocity gradient dynamics along a Lagrangian trajectory in isotropic and homogeneous turbulent flows. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the recent fluid deformation (RFD) approximation that was shown to give a realistic picture of small-scale statistics of turbulence only up to moderate Reynolds numbers. As a second step, we constrain the dynamics, in the spirit of Girimaji & Pope (Phys. Fluids A, vol. 2, 1990, p. 242), in order to impose a peculiar statistical structure to the dissipation seen by the Lagrangian particle. This probabilistic closure uses as a building block a random field that fulfils the statistical description of the intermittency, i.e. multifractal, phenomenon. To do so, we define and generalize to a statistically stationary framework a proposition made by Schmitt (Eur. Phys. J. B, vol. 34, 2003, p. 85). These considerations lead us to propose a nonlinear and non-Markovian closed dynamics for the elements of the velocity gradient tensor. We numerically integrate this dynamics and observe that a stationary regime is indeed reached, in which (i) the gradient variance is proportional to the Reynolds number, (ii) gradients are typically correlated over the (small) Kolmogorov time scale and gradient norms over the (large) integral time scale, (iii) the joint probability distribution function of the two non-vanishing invariants$Q$and$R$reproduces the characteristic teardrop shape, (iv) vorticity becomes preferentially aligned with the intermediate eigendirection of the deformation tensor and (v) gradients are strongly non-Gaussian and intermittent, a behaviour that we quantify by appropriate high-order moments. Additionally, we examine the problem of rotation rate statistics of (axisymmetric) anisotropic particles as observed in direct numerical simulations. Although our realistic picture of velocity gradient fluctuations leads to better results when compared to the former RFD approximation, it is still unable to provide an accurate description for the rotation rate variance of oblate spheroids.

Deposition fraction of ellipsoidal fibers in a model of human nasal cavity for laminar and turbulent flows

In this study, the deposition fraction of ellipsoidal fibers in a realistic model of a human nasal cavity was evaluated for laminar and turbulent airflow conditions. The mean flow field was simulated by solving the discretized continuity and Navier-Stokes equations using the ANSYS-Fluent software. For ellipsoidal particle trajectory analysis, several user defined functions (UDFs) were developed and coupled to the ANSYS-Fluent discrete phase model (DPM). The presented formulation accounted for the coupled translation and rotational motions of ellipsoidal fibers and included the stochastic modeling of turbulence velocity and velocity gradient fluctuations. Particular attention was given to the proper modeling of turbulence fluctuation fields especially in the near wall region for accurate evaluation of fiber deposition rate. The predicted fiber deposition fractions in the nasal passage were compared with the available experimental and numerical results for both laminar and turbulent flows and good agreement was observed.

Small-scale anisotropy in turbulent boundary layers

In a channel flow, the velocity fluctuations are inhomogeneous and anisotropic. Yet, the small-scale properties of the flow are expected to behave in an isotropic manner in the very-large-Reynolds-number limit. We consider the statistical properties of small-scale velocity fluctuations in a turbulent channel flow at moderately high Reynolds number ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$), using the Johns Hopkins University Turbulence Database. Away from the wall, in the logarithmic layer, the skewness of the normal derivative of the streamwise velocity fluctuation is approximately constant, of order 1, while the Reynolds number based on the Taylor scale is$R_{\unicode[STIX]{x1D706}}\approx 150$. This defines a small-scale anisotropy that is stronger than in turbulent homogeneous shear flows at comparable values of$R_{\unicode[STIX]{x1D706}}$. In contrast, the vorticity–strain correlations that characterize homogeneous isotropic turbulence are nearly unchanged in channel flow even though they do vary with distance from the wall with an exponent that can be inferred from the local dissipation. Our results demonstrate that the statistical properties of the fluctuating velocity gradient in turbulent channel flow are characterized, on one hand, by observables that are insensitive to the anisotropy, and behave as in homogeneous isotropic flows, and on the other hand by quantities that are much more sensitive to the anisotropy. How this seemingly contradictory situation emerges from the simultaneous action of the flux of energy to small scales and the transport of momentum away from the wall remains to be elucidated.

Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932

The invariants of the velocity gradient tensor,$R$and$Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results are analysed at different scales. We show that the$R$–$Q$plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the instantaneous behaviour of the flow since they are the result of an averaging process where the mean is 3–5 times smaller than the corresponding standard deviation. The orbital periods in the$R$–$Q$plane are shown to be independent of the intensity of the events, and of the same order of magnitude as those in the enstrophy/enstrophy-production and strain/strain-production planes. Our final goal is to test whether the dynamics of the flow is self-similar in the inertial range, and the answer turns out to be that it is not. The mean shear is found to be responsible for the absence of self-similarity and progressively controls the dynamics of the eddies observed as the filter width increases. However, a self-similar behaviour emerges when the calculations are repeated for the fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms of vortex stretching is considered by computing the alignment of the vorticity at a given scale with the strain at a different one. These results generally support a non-negligible role of the phenomenological energy-cascade model formulated in terms of vortex stretching.

Assessment of subgrid-scale modeling for large-eddy simulation of a spatially-evolving compressible turbulent boundary layer

Abstract The performance of three standard subgrid-scale (SGS) models, namely the wall-adapting local eddy-viscosity (WALE) model, the Dynamic Smagorinsky model (DSM) and the Coherent Structures model (CSM), are investigated in the case of a spatially-evolving supersonic turbulent boundary layer (STBL) over a flat plate at M ∞ = 2 and Reθ ≈ 2600. A high-order split-centered scheme is used to discretize the convective fluxes of the Navier–Stokes equations, and is found to be highly effective to overcome the dissipative character of the standard shock-capturing WENO scheme. The consistency and the accuracy of the simulations are evaluated using direct numerical simulations taken from the literature. It is demonstrated that all SGS models require a comparable minimum grid refinement in order to capture accurately the near-wall turbulence. Overall, the models exhibit correct behavior when predicting the dynamic properties, but show different performances for the temperature distribution in the near-wall region even for cases with satisfactory energy resolution of more than 80%. For a well-resolved LES, the SGS dissipation due to the fluctuating velocity gradients is found to dominate the total SGS dissipation.

Zooplankton in flowing water near benthic communities encounter rapidly fluctuating velocity gradients and accelerations

The fine-scale temporal patterns of water velocities, accelerations, and velocity gradients encountered by individual zooplankters carried in ambient flow can affect their dispersal, behavior, and interaction with other organisms, but have not yet been measured in realistic flow environments. We focused on zooplankton in wavy turbulent boundary layer flow near benthic communities because such flow affects important processes, including larval settlement and prey capture by benthic zooplanktivores. Flow across fouling communities measured in the field was mimicked in a wave flume, where time-varying velocity fields over biofouled surfaces were quantified using particle image velocimetry (PIV). Trajectories of simulated zooplankters seeded into these flow fields were followed to quantify temporal patterns of velocity gradients and accelerations that individuals encountered. We found that such zooplankters are not subjected to steady velocities or velocity gradients, but rather encounter rapidly fluctuating accelerations and velocity gradients with peaks reaching several orders of magnitude above mean values and lasting fractions of a second, much shorter than the wave period. We calculated the proportion of time zooplankters spent affected (e.g., being damaged, changing behavior) by accelerations or velocity gradients and found that a small increase in mean velocity can cause a much larger increase in time affected. Animal reaction threshold and reaction time also changed the fraction of time they were affected by the flow. Using different PIV spatial resolutions showed that inter-vector spacing should be ≤0.5 Kolmogorov length (smallest eddy scale) to accurately capture velocity gradients along trajectories, but coarser resolutions (≤2–6 × Kolmogorov length) are sufficient for velocities, accelerations, and zooplankton trajectories.

Evolution of the velocity-gradient tensor in a spatially developing turbulent flow

AbstractAn experimental study of turbulence generated by a low-blockage space-filling fractal square grid was performed using cinematographic stereoscopic particle image velocimetry in a water tunnel. All fluctuating velocity gradients were measured and their statistics were computed at three different stations along the streamwise direction downstream of the grid: in the production region, at the location of peak turbulence intensity and in the non-equilibrium decay region. The usual signatures of these statistics are only found in the decay region, where a well-defined$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2/3$power-law dependence of the second-order structure function on two-point distance is also present. However, this$2/3$exponent is well defined over a wide range of scales even at the peak location, where the statistics of the fluctuating velocity-gradient tensor are very unusual. There, as at the production region station, the$Q\text {--}R$teardrop shape is not yet fully developed, vortex stretching only slightly dominates over compression and they both fluctuate very widely, reaching very high low-probability values. In these two stations, there is also only marginal preference between sheet-like and tube-like velocity-gradient structures as seen by the sign of the second eigenvalue of the strain-rate tensor. Yet, there are subregions of the flow in the production region where the$2/3$exponent is present and where the$Q\text {--}R$teardrop shape is as undeveloped as for the entire data set at this station.

Energy Dissipation Rates of Newtonian and Non‐Newtonian Fluids in a Stirred Vessel

AbstractEnergy dissipation rates of water and glycerol as Newtonian fluids and carboxyl methyl carbonate solution as non‐Newtonian fluid in a stirred vessel are investigated by 2D particle image velocimetry and compared. Mean velocity profiles reflect the Reynolds (Re) number similarity of two flow fields with different rheological properties, but the root mean square velocity profiles differ in rheology at the same Re‐number. Energy dissipation rates are estimated by direct calculation of fluctuating velocity gradients. The varying energy dissipation rates of Newtonian and non‐Newtonian fluids result from the difference in fluid rheology and apparent viscosity distribution which decides largely the flow pattern, circulation intensity, and rate of turbulence generation.

Statistics of spatial derivatives of velocity and pressure in turbulent channel flow

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.

On the Relation between Small-scale Intermittency and Shocks in Turbulent Flows

High Reynolds number turbulence is characterized by extreme fluctuations of velocity gradients which can interact with shock waves in compressible flows. While these processes are traditionally thought to happen at very disparate range of scales, both turbulence gradients as well as shock gradients become stronger as the Reynolds number increases. Our interest here is to in- vestigate their relation in the high-Reynolds number limit. Our conclusion is that for intermittent turbulence with inertial range scaling exponents which grow more slowly than linear at asymptotically high orders, small-scale intermittency produces gradients which are commensurate with shocks. This result is interpreted in the context of shock-turbulence interactions where intermittency appears to be responsible, in part, for the holes observed in shocks from simulations and experiments. This effect is aided by the correlation between strong gradients and flow retardation ahead of the shock which is observed from analysis of our direct numerical simulation database of incompressible and compressible turbulence.

Effects of forcing geometry on two-dimensional weak turbulence

Using high-resolution particle tracking velocimetry, we study the effects of the forcing geometry on the statistics of an electromagnetically stirred thin-layer flow. We consider two forcing arrangements: one that produces a lattice of vortices as a base flow, and one that produces an array of shear bands. We find that the vortex flow drives stronger fluctuating kinetic energy while the shear-band flow leads to more intense fluctuating velocity gradients. We explain our results by considering the spectral flow of energy in the system. Our results have implications for the design of two-dimensional flow experiments.