# Latest papers in fluid mechanics

### Fast estimation of internal flowfields in scramjet intakes via reduced-order modeling and machine learning

The interface between fluid mechanics and machine learning has ushered in a new avenue of scientific inquiry for complex fluid flow problems. This paper presents the development of a reduced-order predictive framework for the fast and accurate estimation of internal flowfields in two classes of scramjet intakes for hypersonic airbreathing propulsion. Proper orthogonal decomposition is employed as a reduced-order model while the moving least squares-based regression model and the multilayer perceptron-based neural network technique are employed. The samples required for the training process are generated using a sampling strategy, such as Latin hypercube sampling, or obtained as an outcome of multi-objective optimization. The study explores the flowfield estimation capability of this framework for the two test cases, each representing a unique type of scramjet intake. The importance of tuning the user-defined parameters as well as the use of multiple reduced-order bases instead of a global basis are highlighted. It is also demonstrated that the bias involved in the generation of input samples in an optimization problem can potentially be utilized to build a reduced-order predictive framework while using only a moderate number of training samples. This offers the potential to significantly reduce the computational time involved in expensive optimization problems, especially those relying on a population-based approach to identify global optimal solutions.

### Hydrodynamics of a fish-like body undulation mechanism: Scaling laws and regimes for vortex wake modes

A comprehensive two-dimensional numerical investigation has been undertaken to calculate the energetic cost of propulsion and the various flow transitions of a fish-like body undulation mechanism based on a National Advisory Committee for Aeronautics 0012 hydrofoil. This covers a wide range of Strouhal [math] and Reynolds [math] numbers from simulations based on a level-set function immersed-interface method. It is found that the time-averaged thrust coefficient displays a quadratic relationship with increasing St, and increases significantly with Re. Additionally, the time-averaged input power coefficient exhibits a cubic dependence with increasing St but is independent of Re. Both St dependences agree with those previously observed experimentally and numerically for an oscillating foil; however, for similar ranges of governing parameters, comparisons suggest that the body undulation mechanism possesses a higher propulsive efficiency. The [math] scaling for the drag-to-thrust transition is consistent with that found for a wide variety of fish and birds in nature. Interestingly, for cases with an undulation wave-speed below the free-stream speed, the time-averaged drag coefficient is found to be higher than that of a stationary hydrofoil at the same Re. Furthermore, the time-averaged input power coefficient is negative, indicating the potential for the undulation mechanism to extract energy from the free-stream. Eight different wake patterns/transitions are documented for the parameter space; these have been assembled into a wake-regime parameter-space map. The present findings should aid in predicting and understanding different hydrodynamic forces and wake patterns for undulating kinematics.

### Image-clustering analysis of the wave–structure interaction processes under breaking and non-breaking waves

This contribution presents the effectiveness and the potentialities of a consolidated technique—the video-cluster analysis—to the study of turbulent flow and breaking waves, in order to demonstrate its suitability as a low-cost, non-intrusive method to derive quantitative key parameters describing the wave–structure interaction processes at coastal defense structures. For this purpose, a new methodology, consisting of a series of pre- and post-processing techniques developed to optimize the automatic detection of clusters in video imagery, was designed to process the video-records of experiments of wave run-up and wave overtopping at sea-dikes subjected to irregular waves. The results of the cluster analysis were elaborated to reconstruct the instantaneous profiles of the free-surface elevations across the structure crest and derive simultaneous information on overtopping volumes, discharges, depths, and velocities and to get spatial-time maps of the concentration of the air entrapped in the liquid phase. The accuracy of the methodology is demonstrated by comparing the quantities derived from the cluster analysis to laboratory measurements performed with resistive gauges and acoustic Doppler profilers. The novelty of the work is either represented by the results of the application of the cluster-analysis and by the procedures of optimizations, whose ensemble may establish a best practice and represent a guideline for other applications.

### Experimental investigation of droplet breakup of oxide-forming liquid metals

This work investigates the breakup of liquid metal droplets experimentally using a shock tube. Droplets of Field's metal melt are produced and their flow-induced deformation and rupture are captured by a high-speed camera. Results are compared to previous data on Galinstan droplet breakup using image sequences and deformation data. Regarding differences, we find that Field's metal droplets show slightly larger deformations and breakup into a larger number of smaller fragments, especially at low Weber numbers. We expect this to be an effect of different oxidation rates. However, both oxidizing metals show a very similar behavior with respect to the breakup morphology, transition between modes, and the timing of the deformation across the investigated Weber number range of 10–100. In addition, core features that distinguish the breakup of metals from that of conventional, water-like liquids are confirmed. Based on the similarities, we propose that the findings can be generalized to also represent other oxide-forming metals. Weber number-dependent fits are presented for the initial deformation time, the time of the onset of breakup, and the maximum cross-stream diameter. In addition, we provide an empirical fit for the time-dependent cross-stream deformation and evaluate it against experimental data and models from the literature. The fits can be used directly in numerical studies or help improve current breakup models.

### Robust deep learning for emulating turbulent viscosities

From the simplest models to complex deep neural networks, modeling turbulence with machine learning techniques still offers multiple challenges. In this context, the present contribution proposes a robust strategy using patch-based training to learn turbulent viscosity from flow velocities and demonstrates its efficient use on the Spalart–Allmaras turbulence model. Training datasets are generated for flow past two-dimensional obstacles at high-Reynolds numbers and used to train an auto-encoder type convolutional neural network with local patch inputs. Compared to a standard training technique, patch-based learning not only yields increased accuracy but also reduces the computational cost required for training.

### A generally applicable hybrid unsteady Reynolds-averaged Navier–Stokes closure scaled by turbulent structures

This work demonstrates a strategy for hybrid turbulence modeling that relies on parameters identifying flow structures to regulate the model's level of scale resolution, independent of the computational grid and user input. The approach can be classified as second-generation unsteady Reynolds-averaged Navier–Stokes (URANS), where it is assumed that increased scale resolution inside rapidly deformed turbulence regions can consistently reduce modeling error compared to basic URANS closures. The methodology selects flow structures by evaluating the second invariant of the velocity gradient tensor in the resolved field. The functions used for this purpose are similar to techniques applied in topology studies to identify coherent structures. The proposed formulation extends a baseline nonlinear eddy-viscosity URANS model and achieves completeness by means of a differential Lagrangian operator that approximates a locally computed average. The model addresses the lack of general applicability deriving from globally filtering at small scales by reverting to the baseline URANS in flow locations with low acceleration, in which the URANS solution achieves best accuracy. Three flow test cases are presented, demonstrating substantial accuracy enhancement over the baseline URANS on the same grid sizes. Results obtained with this new closure demonstrate robust applicability to internal flows, showing large-eddy simulation (LES)-like statistics on coarse RANS computational grids. The observed increase in computational cost compared to the baseline URANS is only 3% to 24%, which represents almost two orders of magnitude reduction from LES.

### Asymmetric invasion in anisotropic porous media

Author(s): Dario Maggiolo, Francesco Picano, and Federico Toschi

We report and discuss, by means of pore-scale numerical simulations, the possibility of achieving a directional-dependent two-phase flow behavior during the process of invasion of a viscous fluid into anisotropic porous media with controlled design. By customising the pore-scale morphology and heter...

[Phys. Rev. E 104, 045103] Published Thu Oct 14, 2021

### Bifurcations of a plane parallel flow with Kolmogorov forcing

Author(s): Kannabiran Seshasayanan, Vassilios Dallas, and Stephan Fauve

We study the primary bifurcations of a plane parallel flow in a channel with Kolmogorov forcing. We find a new type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold. We call this a stationary drift bifurcation. The laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast to the case of doubly periodic boundary conditions for which the primary bifurcation is stationary.

[Phys. Rev. Fluids 6, 103902] Published Thu Oct 14, 2021

### Logarithmic energy profile of the streamwise velocity for wall-attached eddies along the spanwise direction in turbulent boundary layer

The present work explores the spanwise logarithmic decay of the turbulence intensity for wall-attached eddies per Townsend's attached eddy hypothesis. Within the dataset spanning a friction Reynolds number range [math], the coherence between the turbulence in the logarithmic region along the spanwise direction and that at a near-wall reference location is used to assess the scale-dependent coherence. Linear coherence spectrum analysis is applied as a filter to separate the coherent and incoherent portions. After this separation procedure, the turbulence intensity decay for wall-attached eddies in the spanwise direction is described in a log-linear manner, which also identifies how the scaling parameter increases with the Reynolds number. This variation is parametrized and consequently can be used to improve existing near-wall models.

### Higher-order harmonic induced wave resonance for two side-by-side boxes in close proximity

Wave resonance in a narrow gap formed by two boxes in the side-by-side arrangement is investigated by employing a numerical wave flume based on the OpenFOAM® package. In the present study, the main focus lies in the exploration of the nonlinear resonant behavior induced by the second- and third-order harmonic components which occur around half and one third of the resonant frequencies. A wide range of incident wave frequencies is considered, by which the higher-order harmonic induced wave resonance at lower wave frequencies is discussed. By the harmonic analysis, it is revealed that at the resonant frequency the first-order harmonic component dominates the wave resonance and the second-order effect is minor, which coincides with the conclusion drawn in the previous study. However, around half or one third of the resonant frequencies, the first-order harmonic component is in good agreement with the linear potential flow solutions, but the second- or third-order harmonic component is significant, causing the corresponding higher-order harmonic induced wave resonance. In addition, the numerical results show that the second- and third-order harmonic induced wave resonances only influence the wave properties inside the narrow gap, such as the wave response in the narrow gap and the horizontal wave forces on the boxes. The wave responses around the two-box system, the vertical wave forces on the boxes and the reflection and transmission coefficients are hardly affected by the wave resonance around the corresponding half and one third of the resonant frequencies. The higher-order harmonic induced normalized wave elevation in the narrow gap increases with the increase in incident wave height, implying the increased influence of the free surface nonlinearity.

### Impact dynamics of a charged droplet onto different substrates

Impact behaviors of an electrically charged water droplet on different solid substrates and subsequent dynamic mechanisms were experimentally investigated in this study. Droplets were generated from a metal capillary by a syringe pump with a constant diameter of about 2.2 mm. The capillary was directly connected to a high voltage direct current power supply, while a lower counter ring electrode was grounded. A high-speed camera was utilized to visualize the droplet impact morphology. The influences of the droplet charge density, substrate wettability, and surface temperature were analyzed. The results showed that the impact on hydrophilic surfaces exhibited a greater spreading diameter but a smaller recoiling height than that on a hydrophobic surface, which was attributed to the increased viscous dissipation on the substrate. In addition, compared with a neutral droplet, the maximum spreading diameter of a charged droplet was found to be improved by about 8.4%, where the enhancing effects were proportional to the droplet charge ratio. This was due to the weakening effects of the Coulomb repulsion on the liquid surface tension. Moreover, the impact of charged droplets on a hot copper substrate in three different boiling regimes, called convection, nucleate and film boiling, was also discussed. Finally, a model of the maximum spreading ratio of a charged droplet based on the Weber number, charge ratio, and wettability was established. This study demonstrated that the free charges in a droplet was able to influence its impact behaviors, which would hold great promise for some related technologies.

### Effect of heating on topology of vortex breakdown in Vogel–Escudier flow

This paper examines the effect of unstable thermal stratification on vortex breakdown in Vogel–Escudier flow. A three-dimensional direct numerical simulation of Navier–Stokes and energy equations are used to simulate a flow inside a cylindrical container generated by rotating the top lid. The top and bottom are kept at two constant temperatures such that unstable stratification is maintained. The rotation speed is related to the Reynolds number (Re), and buoyancy is linked to the Rayleigh number (Ra). The streamline and vertical velocity contour plots indicate different regimes of the flow depending on the Re and Ra. The convection dominated (CD) regime has a characteristic large-scale circulation similar to the Rayleigh–Bénard convection, and the rotation dominated (RD) regime has a central axial vortex and breakdowns. A transitional regime between RD and CD regimes is also identified from energy consideration. The influence of Ra on a vortex breakdown bubble and its relation to azimuthal vorticity is investigated in detail. Consistent with the literature on Vogel–Escudier flow, the azimuthal vorticity is shown to be essential for the breakdown in the presence of buoyancy as well. In the low Re limits, the energy of flow tends to be associated with the r–z plane velocity field, while at large Re, the energy is associated with the out-of-the-plane velocity field. Thermal plumes align along the axis for large rotations and are affected by the vortex breakdown bubble. The velocity perturbation structures and plumes show a remarkable distinction between rotation and convection-dominated regimes in the topology.

### Capillary flow of magnetic fluids with effect of hydrodynamic dispersion

A theoretical study is conducted on the influence of a shear-induced dispersion on the rheological response of a magnetic suspension. A capillary geometry is considered, in which a dilute ferrofluid flows under the action of a longitudinal applied magnetic field. The shear-induced dispersion is assumed to arise either due to particle roughness or non-sphericity (i.e., shape anisotropy). A new asymptotic solution for a suspension of rough spheres in the limit of weak flows is developed. The numerical results indicate that the dispersive flux by shear rate gradient produces a particle migration toward the center of the tube. In the case of smooth prolate spheroidal particles, the shape anisotropy may either intensify or reduce the viscous dissipation according to the non-dimensional physical parameters. For weak applied fields and weak shear rates, the relative viscosity presented a rising dependence with the aspect ratio. In contrast, at strong flows and/or large applied fields, the net result was a relative viscosity reduction in comparison with a suspension of spheres. The results provide useful insights into the rheology of ferrofluids in quadratic flows, especially to suspensions designed for biomedical applications, such as hyperthermia and magnetic drug targeting in the blood vessels.

### Dynamic mode decomposition and Koopman spectral analysis of boundary layer separation-induced transition

In the present work, dynamic mode decomposition (DMD) and Koopman spectral analysis are applied to flat plate particle image velocimetry experimental data. Experiments concerning separated-flow transition process were carried out in a test section allowing the variation of the Reynolds number (Re), the adverse pressure gradient (APG) and the free-stream turbulence intensity (Tu). The analysis accounts for two different Re numbers, two different Tu levels, and a fixed APG condition inducing flow separation, as it may occur in low pressure turbine-like conditions. For every flow condition, instantaneous velocity fields clearly show the formation of Kelvin–Helmholtz (KH) vortices induced by the KH instability. The most effective definition of the observable matrix for Koopman analysis able to characterize these vortices is addressed first for a reference Tu and Re number condition. Successively, the robustness of DMD and Koopman modal decomposition has been examined for different Tu levels and Re numbers. On a short time trace (10 KH periods), the Koopman analysis is shown to identify the proper KH vortex shedding frequency and wavelength for every condition tested, while DMD fails especially with low Tu and high Re. To validate the results on a longer time trace, a statistical analysis of the dominant unstable eigenvalues captured by the two procedures is successively performed considering several temporal blocks for different inflow conditions. Overall, the Koopman analysis always performs better than DMD since it finds a larger number of unstable eigenvalues at the KH instability frequency and wavelength.

### A novel local-variable-based Reynolds-averaged Navier–Stokes closure model for bypass and laminar separation induced transition

A one-equation Reynolds-averaged Navier–Stokes closure model is established for bypass transition in this paper. A new local indicator is proposed to describe the variation of turbulence intensities and pressure gradients. Based on this new indicator, a novel and efficient transition criterion is formulated. For laminar separation bubble induced transition, a reasonable modified intermittency factor is developed to complete the reattachment process and control the size of separation bubbles. Incorporated with Menter's [math] shear stress transport turbulence model, the new transition-turbulence model is built for a high turbulence intensity environment. Several classical flow cases, including the ERCOFTAC (European Research Community on Flow, Turbulence and Combustion) series flat plates with various pressure gradients, the Pratt and Whitney low pressure turbine cascade, and a highly loaded linear compressor cascade, are all employed for the model verifications. Decent agreement with the experimental data and direct numerical simulation data can be obtained in a wide range of incoming flow conditions.

### Squirming in density-stratified fluids

In this work, we study the axisymmetric motion of a prototypical swimming organism, a spherical squirmer, in a linearly density-stratified fluid. We assume the inertia is negligible, the stratification is weak, and the swimmer is oriented either vertically upward or downward. The swimmer acts as a settling particle or a neutrally buoyant organism depending on the relative magnitude of its density and the ambient fluid density. While the stratification reduces the speed of a settling particle, it increases (respectively, reduces) the speed of a neutrally buoyant puller (respectively, pusher). The stratification only affects the flow far from the swimmer and this far-field flow due to a settling particle (respectively, neutrally buoyant organism) at low advection of density is same as the flow due to a point-force (respectively, force-dipole) placed in a stratified fluid. The swimmer mixes the surrounding fluid. The mixing caused by a settling particle scales as the ratio of the particle size to the stratification length scale but the mixing due to a neutrally buoyant organism scales as the ratio of the buoyancy to the viscous forces. The latter mixing estimate is much larger than the previous estimate based on the force-dipole model of an organism but still negligible compared to the oceanic mixing. These results are useful to understand the stratification effects on the motility of the organism and the induced mixing.

### Parameter optimization of open-loop control of a circular cylinder by simplified reinforcement learning

Open-loop control is commonly considered an efficient approach in flow control, in which the search for control parameters with excellent performance is mostly carried out by grid search, leading to an extremely tedious process of parameter optimization. With extensive applications of machine learning, reinforcement learning (RL) has emerged as a powerful tool to achieve optimal strategies, which constructively leads to the result that parameter optimization can be performed by RL. In this paper, we provide the concept of simplified RL formally and show the corresponding properties. In order to implement simplified RL for flow control, a high-order numerical approach is coupled with simplified RL to develop a new framework for parameter optimization and determination. In order to validate the performance of the framework, flows past a rotary oscillating circular cylinder at low Reynolds number Re = 200 (defined as [math], where [math] is the free-stream velocity and ν is the kinematic viscosity) are investigated by varying the parameters of rotary amplitude and frequency individually or simultaneously. By numerical investigations, a satisfactory drag reduction effect is achieved, which demonstrates the capability of the framework to perform parameter optimization in terms of open-loop control.

### Convergent Richtmyer–Meshkov instability on a light gas layer with perturbed inner and outer surfaces

The instability of an annular helium gas layer surrounded by air with sinusoidal inner and outer interfaces, formed by a novel soap-film technique, impacted by a cylindrically convergent shock is experimentally studied in a semi-annular shock tube. Detailed evolution of the interfaces and wave patterns is captured by a high-speed Schlieren system. The focus is placed on the influences of layer thickness and phase difference between the inner and outer interfaces on the instability development. It is found that the larger the layer thickness, the quicker the early stage development of the outer interface. This is because the layer thickness affects the arrival time of the reflected shock (RS) at the outer interface and further determines the direction of baroclinic vorticity deposited on the outer interface by RS; namely, RS inhibits or promotes the instability growth depending on the layer thickness. It is also found that phase difference between the inner and outer perturbations produces a negligible (an evident) influence on the early stage (late-stage) instability growth at the outer interface, whereas a considerable (weak) influence on the early stage (late-stage) instability growth at the inner interface. This finding suggests that the early stage development of the outer (inner) interface can be modulated by changing the layer thickness (perturbation phase difference). Empirical coefficient in the Charakhch'an model [J. Appl. Mech. 41, 23–31 (2000)] is calculated to be [math] by comparing the prediction with the experimental results. The model with [math] gives a reasonable prediction of the post-reshock growth rate for all the cases considered in this work.

### The influence of micro vortex generator on inception cavitation

The objective of the present paper is to investigate the influence of the micro vortex generator (mVG) on the inception cavitation number and mode around a National Advisory Committee for Aeronautics 66 hydrofoil. Two different sets of mVG with varying position are employed in this paper, i.e., the mVG-1 (located upstream of the laminar separation point of the baseline hydrofoil) and the mVG-2 (located in the laminar separation zone of the baseline hydrofoil). A high-speed camera is applied to visualize the inception cavitating structures, and numerical simulation is assisted to the effect of mVG. The results indicate that compared to the baseline hydrofoil, the mVG-1 can promote the earlier inception cavitation while the mVG-2 delays the inception, especially for the cases with smaller angle of attack (α = 4°–8°). For the mVG-1 hydrofoil, there are two reasons to be responsible for this phenomenon. One is that the fingerlike vortex at the rear of mVG-1 induces the fingerlike vortex cavitation earlier. The other is that the mVG-1 increases the length of the laminar separation bubble (LSB) by comparison with the baseline hydrofoil, thus causing a cavitation due to the laminar boundary layer separation. For the mVG-2 hydrofoil, it is located at the high-pressure zone of leading edge and reduces the length of the LSB. More precisely, the fingerlike vortex in the high-pressure zone is not enough to induce a fingerlike vortex cavitation, and the smaller length of the LSB than that of the baseline hydrofoil suppressing the cavitation at some angles of attack.

### On a vertical chain of small bubbles ascending in a viscoelastic fluid

Recently, our direct numerical simulations [Yuan et al., “Hydrodynamic interaction and coalescence of two inline bubbles rising in a viscoelastic liquid,” Phys. Fluids 33, 083102 (2021)] indicated that a stable chain can be formed for a pair of bubbles rising in a viscoelastic liquid, consistent with experimental observations. Motivated by the fact that the flow in bubble chains is still poorly understood, this Letter extends the investigations to multiple small bubbles ascending in a vertical file in a viscoelastic medium with different configurations. With an increasing bubble number, it is found that the rising velocity of the bubble group increases and the vertical chain of bubbles becomes unstable due to the distinct oscillation of the uppermost bubble. The terminal separation distance between two adjacent bubbles decreases in the upward direction, diminished by the neighborhood rising bubbles due to increasing loading. By probing the polymeric stresses and deformation, our results demonstrated that the accumulation of viscoelastic normal stresses promotes the aggregation of rising bubbles, while the successive chain of bubbles is stable because of the near-field repulsion induced by the non-monotonic polymer stretching among the bubble chain. In addition, the large bubble deformation appears to enhance the accumulative polymeric normal stress effect, and the bubbles can form more stable vertical chains at increasing initial spacing. Our findings provide insights into the mechanism of bubbles clustering in viscoelastic fluids, as chaining of bubbles is believed to be more prevailing in highly elastic flows.