# Latest papers in fluid mechanics

### Coalescence of drops on the free-surface of a liquid pool at elevated temperatures

The coalescence dynamics of ethanol drops injected from a needle on the free-surface of an ethanol pool maintained at a higher temperature than the drop is experimentally studied using a high-speed imaging system. The drop is always kept at 25 °C, and the temperature of the ethanol pool is varied using a heater. The coalescence behavior depends on the size of the drop, the height of the needle tip from the free-surface, and the temperature of the ethanol pool. A parametric study is carried out by varying these parameters. The drop exhibits a residence period at low impact velocity, when it floats on the free-surface before the coalescence begins. Subsequently, the complete coalescence and partial coalescence dynamics are observed for different sets of parameters considered. It is found that increasing the temperature of the ethanol pool reduces the residence time of the drop. This phenomenon is explained by analyzing the forces acting on the drop and the capillary waves generated due to the temperature gradient between the drop and the ethanol pool. During partial coalescence, we also observed that the diameter of the daughter droplet decreases as the size of the primary drop and pool temperature are increased. As expected, due to the gravity effect, increasing the size of the drop also decreases the residence time. A regime map designating the complete coalescence and partial coalescence dynamics is plotted in the pool temperature and drop impact height space.

### Freezing process of ferrofluid droplets: Numerical and scaling analyses

Author(s): Wen-Zhen Fang, Hui Zhang, Chao-Yang Zhang, and Chun Yang

Direct numerical simulations, using the lattice Boltzmann method, and scaling analyses present the underlying physics for the freezing process of ferrofluid droplets under magnetic squeeze and lift conditions. The effects of magnetization and volume expansion upon freezing on the shape profile of droplets are included in the numerical models. The morphology evolution of ferrofluid droplets and its influences on the droplet freezing time are presented.

[Phys. Rev. Fluids 5, 053601] Published Wed May 06, 2020

### Biaxial extensional viscous dissipation in sheets expansion formed by impact of drops of Newtonian and non-Newtonian fluids

Author(s): Ameur Louhichi, Carole-Ann Charles, Ty Phou, Dimitris Vlassopoulos, Laurence Ramos, and Christian Ligoure

Freely expanding liquid sheets of Newtonian fluids and non-Newtonian fluids are produced by the impact of a drop on a repellent surface that suppresses shear viscous dissipation. For each sample, an effective biaxial extensional Ohnesorge number is identified that controls entirely the maximum expansion factor.

[Phys. Rev. Fluids 5, 053602] Published Wed May 06, 2020

### Mean transport of inertial particles in viscous streaming flows

Author(s): Mathieu Le Provost and Jeff D. Eldredge

A body oscillating in a fluid can produce a steady viscous streaming flow due to mean nonlinear interactions of oscillatory flow components. Effects of the fast-scale oscillations on the slow mean transport are difficult to resolve when calculating inertial particle transport. Using asymptotic analysis and generalized Lagrangian mean theory, we develop mean transport equations for the long-time particle trajectories, skipping over the fast scales without ignoring their effect. Calculations with these transport equations are several orders of magnitude faster than was previously possible.

[Phys. Rev. Fluids 5, 054302] Published Wed May 06, 2020

### Breakdown characteristics of carbon dioxide–ethane azeotropic mixtures near the critical point

The properties of traditional dielectric media have been a major limiting factor impacting the design and operation of many applications spanning from particle accelerators over x-ray radiography and radiotherapy to electrical power systems. Supercritical fluids (SCFs) combine the properties of high dielectric strength, low viscosity, and excellent heat transfer capability. Here, we show, for the first time, the anomalous breakdown strength characteristics of SCF mixtures, such as carbon dioxide (CO2) and ethane (C2H6) mixtures and their azeotropic mixture under supercritical conditions. Our experiments suggest that the dielectric behavior deviates significantly from the established theory of gas discharge known by the work of Townsend and Paschen. Our results reveal that not only pure substances such as CO2 exhibit a discontinuity of the dielectric strength near the critical point, but the supercritical mixture also manifests a discontinuity. The effect of random particle clustering in the pure substance and the mixture is observed, which impacts the mean free path of electrons. We present the measured breakdown voltage in a 0.1 mm gap with a uniform electric field over a wide range of mixture ratios and fluid densities and use a mathematical model by Stanley to show the density fluctuations that peak at around the critical point. By adjusting the mixing ratio, we prove that the mixture forms a useful combination of dielectric strengths and critical points and broadens the applicability of SC mixtures for a variety of purposes.

### Prediction of drag coefficient and ultimate settling velocity for high-density spherical particles in a cylindrical pipe

In this paper, we build a model for calculating the drag coefficient and ultimate settling velocity of high-density and large-diameter solid particles in cylindrical pipes. The settling process of particles in a cylindrical pipe was recorded by using a high-speed camera. Based on the analysis of the experimental data, a gap area ratio was introduced into the model in order to predict the resistance coefficient and ultimate settling velocity of the particles settling in the open space, enabling the model to predict the resistance coefficient and ultimate settling velocity of spherical particles settling in cylindrical pipes. The values of the ultimate settling velocity and the drag coefficient that were obtained with the proposed formulas were validated using experimental data and computational fluid dynamics results. The model was found to be accurate and reliable.

### Erratum: “The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids” [Phys. Fluids 31, 113102 (2019)]

### Cluster-based Markov model to understand the transition dynamics of a supersonic mixing layer

The cluster-based Markov model (CMM) is performed on a numerically simulated supersonic mixing layer at Re = 10 400 to extract physical mechanisms. The high-dimensional state space of the supersonic mixing layer is automatically partitioned into ten relatively homogeneous clusters with representative states called centroids via the cluster analysis. The transition dynamics is conceptualized as a Markov model between centroids using the cluster transition matrix from a probabilistic point of view. A comprehensive analysis of CMM’s outcomes reveals two flow regimes: the single/double-vortex interaction (SDV) and multiple-vortex interaction (MV). The SDV regime plays the dominant role in the supersonic mixing layer, although any single centroid from the MV group carries much larger energy than that from the SDV group. More complicated patterns of vortex are well captured in an intelligent way associated with triple-vortex, quadruple-vortex, and even quintuple-vortex interaction. These vortex formations transport much more energy than the double-vortex pairing/merging. The CMM reveals a complicated set of dynamics that intermittently appear in the two regimes. The inner-circulation transition inside the SDV regime is the most probable route in the supersonic mixing layer. The MV regime can only be accessed from the SDV regime; meanwhile, it inclines to move back to the SDV regime. The transitions linking two regimes undergo large energy fluctuations. The predicted distribution of future cluster probability converges to a unique stationary distribution, which approximates the statistical probability distribution of the dataset.

### Numerical investigation of aero-optical effects of flow past a flat-windowed cylindrical turret

Improved delayed detached eddy simulation is performed to investigate aero-optical distortions induced by Mach 0.5 flow over a cylindrical turret with a flat window at a Reynolds number of Re = 5.9 × 105 based on the turret radius. Optical wave-front distortions and associated far-field intensities at elevation angles of 90°, 100°, and 120° are calculated using the geometric ray-tracing method and Fourier optics theory, respectively. The time-averaged properties of the flow fields and the optical distortions are compared with the experimental results, and reasonable agreements are obtained. It is shown that large-scale coherent structures significantly increase the optical distortions and, consequently, degrade the optical performance in terms of the far-field intensity level, while the optical effects of the attached turbulent boundary layers and separation bubbles are trivial and negligible. Very little difference is observed in both instantaneous and statistical optical results calculated with and without defocus and astigmatism components, indicating the adequacy of the removing piston and tilt components for aiding in the design of adaptive optical systems. Both co-flow and blowing jet fluid control methods are introduced with a steady mass flow to alleviate wave-front distortions, and preliminary simulations demonstrate the practicability of these fluid control methods in suppressing the optical distortions. It is shown that both the active control methods are competent to reduce the overall wave-front root-mean-square of the optical path difference.

### Mean velocity and temperature scaling for near-wall turbulence with heat transfer at supercritical pressure

Compared with conventional gaseous and liquid fluids, fluids operating at supercritical pressure undergo drastic variations in thermophysical properties within a small temperature range across the pseudo-critical point. Therefore, the effect of these variations on flow and heat transfer must be studied. This paper presents direct numerical simulations (DNSs) of the turbulent heat transfer of CO2 at supercritical pressure in a fully developed channel flow between two isothermal walls. The thermophysical property tables generated from the REFPROF 9.1 database were used in this DNS. The velocity and temperature scaling and the analogy between momentum and scalar transport are comprehensively explored by using stress balance and semi-local methods. The results show that at small temperature differences, the velocity transformation developed by Trettel and Larsson [“Mean velocity scaling for compressible wall turbulence with heat transfer,” Phys. Fluids 28, 026102 (2016)] with a semi-local coordinate provides a good description of the near-wall turbulence of supercritical fluids. Upon including how large specific-heat variations affect temperature transformation, the logarithmic region of the cooled wall becomes consistent, as does the heated wall in a certain temperature range. In addition, in near-wall turbulence with small temperature differences at supercritical pressure, momentum transport is highly analogous to scalar transport.

### Direct experimental observations of the impact of viscosity contrast on convective mixing in a three-dimensional porous medium

Analog fluids have been widely used to mimic the convective mixing of carbon dioxide into brine in the study of geological carbon storage. Although these fluid systems had many characteristics of the real system, the viscosity contrast between the resident fluid and the invading front was significantly different and largely overlooked. We used x-ray computed tomography to image convective mixing in a three-dimensional porous medium formed of glass beads and compared two invading fluids that had a viscosity 3.5× and 16× that of the resident fluid. The macroscopic behavior such as the dissolution rate and onset time scaled well with the viscosity contrast. However, with a more viscous invading fluid, fundamentally different plume structures and final mixing state were observed due in large part to greater dispersion.

### A flexion-based approach for the simulation of turbulent flows

Turbulent flows at high Reynolds numbers are dominated by vortex filaments and/or sheets with sharp gradients in the vorticity field near the boundaries of the vortical structures. Numerical simulations of high Reynolds number flows are computationally demanding due to the fine grid required to accurately resolve these sharp gradient regions. In this paper, an alternative approach is proposed to improve the computational efficiency of Navier–Stokes solvers by reformulating the momentum equations as a set of equations for the time-dependent evolution of the flexion field. The flexion vector represents the curl of the vorticity field and is better able to resolve nonlinear effects in regions with large vorticity gradients. The improved resolution capabilities of the flexion-based approach are illustrated through the pseudospectral computations of the rollup of a perturbed 2D shear layer and the transition to a turbulence/viscous decay of the three-dimensional (3D) Taylor–Green vortex. The flexion-based formulation also provides further insight into the dynamics of turbulence through the evolution of the mean-square flexion or palinstrophy. Analysis of data from the Taylor–Green vortex simulations shows that the observed rapid growth of small-scale features and palinstrophy in 3D turbulent flows is primarily associated with flexion amplification by the curl of the vortex stretching vector. Consequently, we hypothesize that the primary physical mechanism responsible for energy cascade from large to small scales is the curl of the vortex stretching vector of interacting vortex tubes, as opposed to the stretching of individual vortex tubes.

### Baroclinic toroidal quasi-geostrophic vortices

We investigate the stability and nonlinear evolution of two tori of opposite-signed uniform potential vorticity, located one above the other, in a three-dimensional, continuously stratified, quasi-geostrophic flow. We focus on the formation of hetons as a result of the destabilization of the tori of potential vorticity. Hetons are pairs of vortices of opposite signs lying at different depths capable of transporting heat, momentum, and mass over large distances. Particular attention is paid to the condition under which the hetons move away from their region of formation. We show that their formation and evolution depend on the aspect ratio of the tori, as well as the vertical gap separating them. The aspect ratio of a torus is the ratio of its major (center line) radius to its minor (cross-sectional) radius. Pairs of thin opposite-signed potential vorticity tori self-organize into a large number of hetons. On the other hand, increasing the vertical gap between the tori decreases the coupling between the opposite-signed vortices forming the hetons. This results in a more convoluted dynamics where the vortices remain near the center of the domain.

### A fractional subgrid-scale model for turbulent flows: Theoretical formulation and a priori study

Coherent structures/motions in turbulence inherently give rise to intermittent signals with sharp peaks, heavy-skirt, and skewed distributions of velocity increments, highlighting the non-Gaussian nature of turbulence. This suggests that the spatial nonlocal interactions cannot be ruled out of the turbulence physics. Furthermore, filtering the Navier–Stokes equations in the large eddy simulation of turbulent flows would further enhance the existing nonlocality, emerging in the corresponding subgrid scale fluid motions. This urges the development of new nonlocal closure models, which respect the corresponding non-Gaussian statistics of the subgrid stochastic motions. To this end and starting from the filtered Boltzmann equation, we model the corresponding equilibrium distribution function with a Lévy-stable distribution, leading to the proposed fractional-order modeling of subgrid-scale stresses. We approximate the filtered equilibrium distribution function with a power-law term and derive the corresponding filtered Navier–Stokes equations. Subsequently in our functional modeling, the divergence of subgrid-scale stresses emerges as a single-parameter fractional Laplacian, (−Δ)α(·), α ∈ (0, 1], of the filtered velocity field. The only model parameter, i.e., the fractional exponent, appears to be strictly dependent on the filter-width and the flow Reynolds number. We furthermore explore the main physical and mathematical properties of the proposed model under a set of mild conditions. Finally, the introduced model undergoes a priori evaluations based on the direct numerical simulation database of forced and decaying homogeneous isotropic turbulent flows at relatively high and moderate Reynolds numbers, respectively. Such analysis provides a comparative study of predictability and performance of the proposed fractional model.

### Influence of upstream disturbances on the primary and secondary instabilities in a supersonic separated flow over a backward-facing step

The development of primary and secondary instabilities is investigated numerically for a supersonic transitional flow over a backward-facing step at Ma = 1.7 and [math]. Oblique Tollmien–Schlichting (T–S) waves with properties according to linear stability theory (LST) are introduced at the domain inlet with zero, low, or high amplitude (cases ZA, LA, and HA). A well-resolved large eddy simulation (LES) is carried out for the three cases to characterize the transition process from laminar to turbulent flow. The results for the HA case show a rapid transition due to the high initial disturbance level such that the non-linear interactions already occur upstream of the step, before the Kelvin–Helmholtz (K–H) instability could get involved. In contrast, cases ZA and LA share a similar transition road map in which transition occurs in the separated shear flow behind the step. Case LA is analyzed in detail based on the results from LST and LES to scrutinize the evolution of T–S, K–H, and secondary instabilities, as well as their interactions. Upstream of the step, the linear growth of the oblique T–S waves is the prevailing instability. Both T–S and K–H modes act as the primary mode within a short distance behind the step and undergo a quasi-linear growth with a weak coupling. Upon pairing of the large K–H vortices, subharmonic waves are produced, and secondary instabilities begin to dominate the transition. Simultaneously, the growth of T–S waves is retarded by the slow resonance between subharmonic K–H and secondary instabilities. The vortex breakdown and reattachment downstream further contribute to the development of the turbulent boundary layer.

### Theoretical analysis on the atomization characteristics of liquid nitrogen jets

Atomization with cryogenic fluids is complicated due to the combined effects of both the mechanical breakup and thermodynamic flashing. However, the weight of the two effects on the atomization has not been determined yet. This paper proposes a hybrid mathematical model to evaluate the weight of the thermodynamic effect and aerodynamic effect. The entire process of jet spray breakup, including the primary breakup and secondary atomization, is described in the model. A linear stability analysis is carried out using the dispersion equations for the primary breakup. In the secondary atomization, the Taylor analogy breakup and bubble micro-explosion models are employed to study aerodynamic and thermodynamic effects on the atomization, respectively. The homogeneous nucleation is implemented in the bubble micro-explosion models. A criterion is obtained which can be evaluated to predict the dominant effect. Accordingly, the influence of factors such as ambient pressure and other physical parameters on the atomization process is analyzed. The results reveal that, under low ambient pressure, thermodynamic flashing is stronger. Besides, the mechanical action is restrained when the surface tension is increased.

### Integral-equation approach to resonances in circular two-layer flows around an island with bottom topography

This paper presents an integral-equation approach to the linear instability problem of two-layer quasi-geostrophic flows around circular islands with radial offshore bottom slope. The flows are composed of concentric uniform potential-vorticity (PV) rings in each layer, with the PV of each ring being opposite in sign. The study extends an earlier similar barotropic model and focuses on the degree to which the topographic waves resonate with the deformation waves at the rings’ peripheries. The integral approach poses the instability problem in a physically elucidating way, whereby the resonating waves in the system are directly identified. Four types of instabilities are identified: instability caused by the resonance of waves at the liquid contours at the edge of each PV ring, instability caused by the resonance of the wave at the upper-layer contour and the topographic waves outside the lower-layer contour, a similar resonance of the lower-layer contour with the topographic waves, and a resonance between one of the eigenmodes of the contour subsystem with the topographic waves. The three latter resonances lead to critical layer instabilities and can be identified as resonances between the contour waves and a collection of singular topographic modes with a critical layer. The PV perturbations in the outer region can be represented asymptotically (far from the origin) as a combination of barotropic and baroclinic modes. Usually, the asymptotically barotropic mode is the mode in resonance with the contours, but, for small growth rates, the asymptotically baroclinic mode may be the dominant mode.

### Shear-induced viscosity stratified flow past a pair of heated side-by-side square cylinders in a confined domain

In this article, investigations have been carried out to decipher the effect of thermal buoyancy in a viscosity stratified flow field for a shear-thinning fluid flowing past a pair of heated side-by-side square cylinders, which is an extension part of our recent study [Sanyal, A. and Dhiman, A., “Wake interactions in a fluid flow past a pair of side-by-side square cylinders in presence of mixed convection,” Phys. Fluids 29, 103602 (2017)]. It is found that the leading-edge flow-separations from the square cylinders influence the near-wake structures and vortex shedding patterns in the presence of shear-thinning effects, which is otherwise missing for Newtonian fluid flow at Reynolds number Re = 40 and Richardson number Ri = 1. The distribution of wall-viscosity η along the inner surfaces of the side-by-side square cylinders, at different values of transverse spacings s/d and flow-behavior indices n, hints at large dependency on the inflections in the velocity profile within the gap-flow region. Under thermal buoyancy-driven mild shear-thinning flow conditions (n = 0.6 and 0.8), the gap-flow characteristics have been classified into “pressure-driven” and “momentum-driven” flow regimes, which provides a good explanation for the aberrations noted in the distribution pattern of η. The root-mean-square fluctuations of the velocity-magnitude and vortex shedding phenomenon are found to reciprocate a consistent flow physics associated with a shear-thinning flow at near and far-field downstream. The single body deflected type flow is primarily seen under predominant shear-thinning flow conditions (n = 0.4), compared to chaotic or quasi-periodic flow under mild shear-thinning conditions. Besides, the evolution of non-linear dynamics-based flow regimes (classified with respect to s/d using power spectrum density analysis) at different values of n and s/d is thoroughly summarized. The time-variant fluctuations of lift and drag force parameters are also found to be unified through cause and effects.

### Numerical investigation of lift enhancement in flapping hover flight

The focus of this work is the study of lift enhancement in flapping hover flight using numerical simulations. An idealized set of kinematics for a NACA0012 airfoil consisting of sequential translations and rotations is considered for this purpose, such that the Cl response can be demarcated into translational and rotational parts, which facilitates comparison of forces attributed to translation and rotation. Additionally, comparisons with pure translation and pure rotation are done to isolate the effect of wing–wake interactions. The investigation reveals that the majority of lift is produced in the translational phase. The wing–wake interactions affect the translational phase of the response more than the rotational phase. However, the rotation rate determines the extent of influence of wing–wake interactions on the translational lift response. The effect of different durations of overlap between the translational and rotational motions is also assessed based on the Cl time histories and mean Cl, and the study reveals that an optimum duration of overlap can maximize the lift. An immersed-boundary method with integrated surface-load reconstruction capabilities is used for the computations presented here. The reconstruction of the surface stresses and their integration are carried out with the framework of a parallel solver. The method is validated for a flow past a NACA0012 airfoil executing a non-periodic plunge motion and a non-periodic pitch/plunge motion and a flow around an elliptic airfoil executing a flapping motion.

### Large-amplitude oscillatory shear flow loops for long-chain branching from general rigid bead-rod theory

General rigid bead-rod theory [O. Hassager, “Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60, 4001–4008 (1974)] explains polymer viscoelasticity from macromolecular orientation. By means of this theory, we relate the complex viscosity of polymeric liquids to the architecture of axisymmetric branched macromolecules. In this work, we explore how adding long-chain branching to polymers affects the shapes of large-amplitude oscillatory shear (LAOS) flow loops. By loops, we mean plots of the alternant part of the shear stress response vs the cosinusoidal shear rate. We choose LAOS for its ability to amplify subtle differences in small-amplitude oscillatory shear flow at a high Weissenberg number. When non-dimensionalized with the product of the zero-shear viscosity and the shear rate amplitude, the loop shapes depend on the sole dimensionless architectural parameter, the macromolecular lopsidedness of the long-chain branched macromolecule. In this work, in this way, we compare and contrast the loop shapes of macromolecular chains that are straight with those branched. Specifically, we explore symmetric branch multiplicity, branch functionality, branch length, branch position, branch distribution, and multiple branch asymmetry. We find that adding branching collapses and distorts the loops. We then find that so long as branch length, branch position, and branch distribution are held constant and so long as the branching is symmetric about the center of mass, the peak shear stress increases with branch multiplicity. We also find that branch functionality hardly affects the loops. The structural details explored in this paper have yet to be explored in the laboratory.