# Latest papers in fluid mechanics

### Pattern method for higher harmonics of first normal stress difference from molecular orientation in oscillatory shear flow

This study examines the simplest relevant molecular model of a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow: rigid dumbbells suspended in a Newtonian solvent. For such suspensions, the viscoelastic response of the polymeric liquid depends exclusively on the dynamics of dumbbell orientation. Previously, the explicit analytical expressions of the zeroth, second, and fourth harmonics of the alternating first normal stress difference response in LAOS have been derived. In this paper, we correct and extend these expressions by seeking an understanding of the next higher harmonic. Specifically, this paper continues a series of studies that shed light on molecular theory as a useful approach in investigating the response of polymeric liquids to oscillatory shear. Following the general method of Bird and Armstrong [“Time-dependent flows of dilute solutions of rodlike macromolecules,” J. Chem. Phys. 56, 3680 (1972)], we derive the expression of the first normal stress coefficient up to and including the sixth harmonic. Our analysis relies on the extension of the orientation distribution function to the sixth power of the shear rate. Our expression is the only one to have been derived from a molecular theory for a sixth harmonic and thus provides the first glimpse of the molecular origins of a first normal stress difference higher than the fourth.

### Universal scaling parameter for a counter jet drag reduction technique in supersonic flows

The reduction in aerodynamic drag by injecting a gaseous jet from the nose of a blunt body into a supersonic stream is investigated numerically. The penetration of the jet into the supersonic flow modifies the shock structure around the body and creates a low pressure recirculation zone, thereby decreasing the wave drag significantly. Combining various theoretical estimates of different flow features and numerical simulations, we identify a universal parameter, called the jet to freestream momentum ratio (RmA), which uniquely governs the drag on the blunt body. The momentum ratio fundamentally decides the penetration of the jet as well as the extent of low pressure envelope around the body. In addition, various influencing parameters reported in the literature are reviewed for different steady jet flow conditions. Furthermore, their limitations in regulating the flowfield are explained by correlating the facts with the jet to freestream momentum ratio. We perform the simulations for various combinations of physical and flow parameters of the jet and the freestream to show a universal dependence of drag on the momentum ratio.

### Shape oscillations of a viscoelastic droplet suspended in a viscoelastic host liquid

Author(s): Fang Li, Xie-Yuan Yin, and Xie-Zhen Yin

The small-amplitude oscillation of a liquid droplet suspended in an immiscible host liquid is studied, where both liquids are assumed viscoelastic. An analytical characteristic equation is derived, and the damping rate and angular frequency that describe the droplet oscillation are numerically solved. The effect of the properties of the host liquid, including its density, viscosity and elasticity, on the viscoelastic droplet oscillation is examined for the quadrupole mode. The host liquid is found to make the droplet oscillation more complex and the mechanisms behind that are discussed.

[Phys. Rev. Fluids 5, 033610] Published Thu Mar 19, 2020

### Triad resonant instability of horizontally periodic internal modes

Author(s): Bruce R. Sutherland and Riley Jefferson

Nonuniform stratification and vertical confinement significantly restricts the development of triad resonant instability for low mode internal gravity waves with and without background rotation.

[Phys. Rev. Fluids 5, 034801] Published Thu Mar 19, 2020

### Randomized resolvent analysis

Author(s): Jean Hélder Marques Ribeiro, Chi-An Yeh, and Kunihiko Taira

Randomized numerical algebra is incorporated into resolvent analysis to reduce large-scale resolvent operators to their low-rank approximations. The key to finding the resolvent modes accurately is to weigh the random test matrix using insights from the base flow. Turbulent flow over a NACA0012 airfoil at Re = 23,000 is used to demonstrate significant speedup and memory savings to accurately find principal resolvent modes.

[Phys. Rev. Fluids 5, 033902] Published Wed Mar 18, 2020

### Tip-vortex flow characteristics investigation of a novel bird-like morphing discrete wing structure

A bird-like morphing discrete wing, inspired by primary feathers of birds’ wings, was designed to control the wing-tip vortex strength. The influence of both the morphing process and discrete (non-continuous) surface feature for the bird-like wing structure on the tip-vortex flow characteristics was investigated in detail at Re = 87 000. The results reveal that the morphing process of the bird-like discrete wing structure can achieve the effective control of the core vortex strength by changing the flow structures around the tip-vortex core center(s). The induced drag yielded by the bird-like morphing wing structure is tightly related to its vorticity distribution in the near-wake region. Moreover, compared with the fully extended fixed-wing model with a continuous surface structure, the bird-like discrete wing model with the fully extended morphing state can suppress the core vortex strength by destroying the tip-vortex merging process. Meanwhile, the core vortex strength of the fully extended discrete wing model decays more sharply with the increase in x/c. The maximum proportions of the induced drag relative to the total drag for both the discrete and continuous wing models with the fully extended shape are 14.33% and 19.97%, respectively. However, the fully folding process of the bird-like wing structure significantly weakens the induced-drag reduction effect of the discrete surface structure. The maximum proportions of the induced drag relative to the total drag for both the discrete and continuous wing models with the fully folded shape are 17.59% and 18.41%, respectively.

### Numerical simulations of inert and reactive highly underexpanded jets

In this study, the high-resolution numerical simulations of the two-dimensional (2D) multi-component inert and reactive highly underexpanded jets are conducted to quantify the influences of the injected gas mixture properties on the flow structure. First, the gas mixture with the specified species mass fractions is imposed to exhaust into the quiescent air with a Mach number of 1.0, of which the specific heat ratios (γe) range from 1.3 to 1.6. Our results indicate that the larger γe yields a relatively shorter and thinner jet core under the same inlet pressure ratio due to the sound speed increasing. Next, we focus on the chemical reaction effects on the jets with a premixed hydrogen–air mixture injection. The results reveal that the shock-induced combustion develops into a detonation, inducing numerous vortices behind the combustion wave, while the combustion in the mixing layer cannot be preserved due to the instability of the supersonic shearing. During the detonation process, the increasing pressure accompanied by the heat release forces the Riemann wave to move upstream compared with the inert one. The violent detonation periodically propagates between the two jet triple points. The detonation collision leads to the intersection of their slip lines, which causes distinct vortex formation. In addition, the main frequencies, corresponding to the Riemann wave movement, the oscillation of the shock-induced ignition positions, the periodical propagation of the detonation, and the collision of the detonation triple points, are explored to explain the unsteady process of the reactive highly underexpanded jet.

### Numerical simulations of inert and reactive highly underexpanded jets

In this study, the high-resolution numerical simulations of the two-dimensional (2D) multi-component inert and reactive highly underexpanded jets are conducted to quantify the influences of the injected gas mixture properties on the flow structure. First, the gas mixture with the specified species mass fractions is imposed to exhaust into the quiescent air with a Mach number of 1.0, of which the specific heat ratios (γe) range from 1.3 to 1.6. Our results indicate that the larger γe yields a relatively shorter and thinner jet core under the same inlet pressure ratio due to the sound speed increasing. Next, we focus on the chemical reaction effects on the jets with a premixed hydrogen–air mixture injection. The results reveal that the shock-induced combustion develops into a detonation, inducing numerous vortices behind the combustion wave, while the combustion in the mixing layer cannot be preserved due to the instability of the supersonic shearing. During the detonation process, the increasing pressure accompanied by the heat release forces the Riemann wave to move upstream compared with the inert one. The violent detonation periodically propagates between the two jet triple points. The detonation collision leads to the intersection of their slip lines, which causes distinct vortex formation. In addition, the main frequencies, corresponding to the Riemann wave movement, the oscillation of the shock-induced ignition positions, the periodical propagation of the detonation, and the collision of the detonation triple points, are explored to explain the unsteady process of the reactive highly underexpanded jet.

### Theory for undercompressive shocks in tears of wine

Author(s): Yonatan Dukler, Hangjie Ji, Claudia Falcon, and Andrea L. Bertozzi

Tears of wine, in which a thin layer of a water-ethanol mixture travel up an inclined surface against gravity and then fall down in the form of tears, have been observed in wine glasses for centuries. It has been modeled with a conservation law with a nonconvex flux and higher order diffusion due to the bulk surface tension. The resulting nonclassical “undercompressive” shock solutions are the main driver of the destabilizing front forming the “wine tears”. Prior modeling did not address the wine tears but rather the behavior of the film at earlier stages and the behavior of the meniscus.

[Phys. Rev. Fluids 5, 034002] Published Tue Mar 17, 2020

### Unsteady behaviors of separated flow over a finite blunt plate at different inclination angles

This study comprises an extensive analysis of unsteady behaviors of a separated flow over a finite blunt plate at three different inclination angles, θ = 0°, 3°, and 6°. It was found that these three distinctly different flow patterns resulted from increasing inclination angles: reattachment, intermittent reattachment, and non-reattachment. The separated flow fields and wall-pressure fluctuations were experimentally measured with planar particle image velocimetry (PIV) operating at 1 Hz and a microphone array sampling at 5 kHz, respectively. Flow patterns were discussed in terms of the time-averaged flow fields and distributions of the statistical quantities (i.e., the reverse-flow intermittency, or velocity fluctuation intensity). A slender separation bubble formed in the front area of the plate (0 < x/D < 4.6) in the system where θ = 0° and then it enlarged to the whole surface of the plate in the system where θ = 3°. In contrast, in the system where θ = 6°, the plate surface was entirely engulfed by a large recirculation zone extending to the near-wake region. In the wall-pressure fluctuation analysis, two characteristic frequencies, St = 0.036 and 0.107, could be readily identified in all three systems; these corresponded to the flapping of separation bubble and the shedding of large-scale vortical structures, respectively. In addition, in the system where θ = 0°, wall-pressure fluctuations of the Karman vortex were detected at St = 0.154 but were suppressed in the systems where θ = 3° and 6° due to the extensive interaction between the shedding of large-scale vortical structures and the unsteady wake. Subsequently, a field-programmable gate array taking full advantage of dynamic mode decomposition (DMD) on the wall-pressure fluctuations was constructed, and a real-time conditional signal corresponding to individual unsteady behavior was generated to fire the phase-locking PIV measurement. High-resolution spatiotemporal evolutions of dominant flow behaviors (i.e., enlargement-and-shrinkage motion of the separation bubble and shedding motion of the large-scale vortical structures) were determined. In the system where θ = 6°, a separation bubble enlarged and shrank dramatically together with the shedding of large-scale vortical structures, leading to a large recirculation zone over the blunt plate, distinct from the behavior in the systems where θ = 0° and 3°. Finally, a joint dominant mode analysis of flow structures and wall-pressure fluctuations was further evaluated, which delineated the complex unsteady processes in separated flow clearly and provided more information and references for other studies on wind engineering, fluid-induced structure vibrations, and acoustic emissions.

### Linear global stability of liquid metal mixed convection in a horizontal bottom-heating duct under strong transverse magnetic field

Two-dimensional steady-state solutions of liquid metal mixed convection in a horizontal bottom-heating duct under a strong magnetic field are first computed numerically by the Newton iteration method along with the spatial discretization of the Taylor–Hood finite element. Two branches of steady solutions with symmetrical rolls and a pair of asymmetrical solutions with a single roll are identified and can be regarded as the base flow for linear global stability analysis. The symmetrical steady solution for the first branch has a nearly uniform distribution for the temperature field in the transverse direction, while the second branch occurs at much larger Grashof numbers and the temperature field becomes nonuniform transversely. The linear stability analysis is performed for a fixed Reynolds number and Prandtl number with Re = 5000 and Pr = 0.0321. For the symmetrical rolls of the first branch, with an increase in the Grashof number, two-dimensional stationary instabilities first occur at small Hartmann numbers, while three-dimensional oscillatory instabilities first appear at moderate or large Hartmann numbers. From the further study of the two-dimensional instabilities, it is revealed that the asymmetrical solution is actually bifurcated supercritically from the symmetrical solution at a two-dimensional critical Grashof number. In addition, the critical curve of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode shows that there exists a minimum critical Grashof number, which occurs at a moderate Hartmann number. The critical curves of the one-roll asymmetrical solution are also exhibited and determined by two three-dimensional oscillatory unstable modes. It is revealed that there exists a minimum Hartmann number below which the asymmetrical steady-state can always remain stable for all Grashof numbers (5 × 105–107). The energy analyses at the oscillatory critical thresholds with different Hartmann numbers are performed to exhibit that buoyancy is the dominant destabilizing term, and the magnetic force is always the main stabilization term for both symmetrical and asymmetrical solutions. In addition, both streamwise and cross-sectional shears of the basic flow are important for the determination of the linear stability boundary of the asymmetrical solution.

### Hole expansion from a bubble at a liquid surface

For millimetre to micron sized bubbles, floating at the free surface of different low viscosity fluids with different surface tensions, and then collapsing, we study the ensuing expansion of the outer radius of the hole (ro) at the free surface, as well as its velocity of expansion (uo). Since the thin film cap of the bubble disintegrates before the hole in it reaches the static rim, the hole expansion at intermediate times occurs as if it initiates at the bubble’s static rim of radius Rr; the evolution of ro then results to be a strong function of gravity, since Rr depends strongly on the bubble radius R. A scaling analysis, which includes the increase in the tip radius due to mass accumulation and the resulting change in the retraction force, along with the gravity effects by considering the hole radius in excess of its initial static radius, re = ro − Rr, results in a novel scaling law [math], where [math] is the capillary time scale; this scaling law is shown to capture the evolution of the hole radii in the present study. The dimensionless velocities of hole expansion, namely, the Weber numbers of hole expansion, [math], scale as [math], independent of gravity effects, matching the observations. We also show that these Weber numbers, which reduce with time, begin with a constant initial Weber number of 64, while the viscous limit of the present phenomena occurs when the bubble Ohnesorge number [math].

### Dynamo action in sliding plates of anisotropic electrical conductivity

Author(s): T. Alboussière, K. Drif, and F. Plunian

With materials of anisotropic electrical conductivity, it is possible to generate a dynamo with a simple velocity field, of the type precluded by Cowling's theorems with isotropic materials. Following a previous study by Ruderman and Ruzmaikin [M. S. Ruderman and A. A. Ruzmaikin, Magnetic field gene...

[Phys. Rev. E 101, 033107] Published Mon Mar 16, 2020

### Formation of vase-shaped drops

Author(s): Martin Coux, Pierre Chantelot, Lucie Domino, Christophe Clanet, Antonin Eddi, and David Quéré

Beautiful, elusive shapes are obtained when a water droplet deposited on a nonwetting substrate is subjected to a strong vertical impulse. Drops are highly reshaped to form truncated cones that eventually collapse. The evolution of the geometrical features of these so-called “vase-shaped droplets” is reported and discussed.

[Phys. Rev. Fluids 5, 033609] Published Mon Mar 16, 2020

### Bifurcations in the dynamics of a dipolar spheroid in a shear flow subjected to an external field

Author(s): V. Kumaran

The dynamical phase behavior of a spheroid with a magnetic dipole rotating in a shear flow and external field is investigated theoretically. Depending on the ratio of the torques due to the shear flow, a sequence of bifurcations leads to a rich complexity in the phase portraits in orientation space. We also find that the dynamics of an ideal thin rod could be very different from that of a high aspect ratio spheroid.

[Phys. Rev. Fluids 5, 033701] Published Mon Mar 16, 2020

### Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow

Author(s): Aditya S. Khair and Jason K. Kabarowski

The cross-streamline migration of a spherical particle undergoing electrophoresis in weakly inertial or viscoelastic simple shear flow is quantified via asymptotic analysis. The findings for the migration direction and speed are in reasonable agreement with previous experimental studies on migration of electrophoretic colloids in Poiseuille microchannel flow.

[Phys. Rev. Fluids 5, 033702] Published Mon Mar 16, 2020

### Improvement of the parabolized stability equation to predict the linear evolution of disturbances in three-dimensional boundary layers based on ray tracing theory

Author(s): Runjie Song, Lei Zhao, and Zhangfeng Huang

A new method to predict the linear evolution of disturbances in three-dimensional inhomogeneous boundary layers is proposed, named RTPSE, in which the line-marching parabolized stability equation (PSE) is improved by applying ray tracing (RT) theory. Results show that RTPSE can accurately predict the spanwise wave number and amplitude ratio for both stationary and traveling crossflow waves, while the traditional PSE could not.

[Phys. Rev. Fluids 5, 033901] Published Mon Mar 16, 2020

### Collapse of a bubble injected side-by-side with another bubble into an incipiently fluidized bed: A CFD-DEM study

Author(s): A. Padash and C. M. Boyce

Computational fluid dynamics–discrete element method simulation results demonstrate the bubble collapse phenomenon observed in a prior experimental study when two bubbles are injected side-by-side into an incipiently fluidized bed. Results confirm that one of the bubbles collapses when there is a slight size difference between the two bubbles and its extent is beyond a critical value. The collapse occurs because of a preferential gas channeling toward the larger bubble, which leaves the smaller bubble without sufficient gas flow to support its shape.

[Phys. Rev. Fluids 5, 034304] Published Mon Mar 16, 2020

### Can preferential concentration of finite-size particles in plane Couette turbulence be reproduced with the aid of equilibrium solutions?

Author(s): Tiago Pestana, Markus Uhlmann, and Genta Kawahara

Fluid-particle interaction is studied in plane Couette flow by considering a nontrivial equilibrium solution that features exact coherent structures representative of wall-bounded shear flows. Despite the advantage of a much reduced complexity in comparison with a true turbulent flow, this strategy is shown to reproduce the phenomena of particle preferential concentration. The proposed approach is expected to be fruitful in future studies on various aspects of particulate flow, and perspectives for future works are also discussed.

[Phys. Rev. Fluids 5, 034305] Published Mon Mar 16, 2020

### Unified prediction of reshocked Richtmyer–Meshkov mixing with K-L model

Hydrodynamic instabilities, including Rayleigh–Taylor, Richtmyer–Meshkov (RM), and Kelvin–Helmholtz, induced turbulent mixing broadly occur in both natural phenomena, such as supernova explosions, and high-energy-density applications, such as inertial confinement fusion. Reshocked RM mixing is the most fundamental physical process that is closely related to practical problems, as it involves three classical instabilities. In complex applications, the Reynolds-averaged Navier–Stokes model analysis continues to play a major role. However, there are very few turbulence models that facilitate unified predictions of the outcome of reshocked RM mixing experiments under different physical conditions. Thus, we aim to achieve this objective using the K-L model based on three considerations: deviatoric shear stress is considered when constructing Reynolds stress tensor; the model coefficients used are derived based on a new systematic procedure; the performance of different numerical schemes are studied to ensure high resolution but basically no numerical oscillation. Consequently, a unified prediction is obtained for the first time for a series of reshocked RM mixing experiments under incident shock Mach numbers Ma = 1.2–1.98, Atwood numbers At = ±0.67, and test section lengths 8 cm ≤ δ ≤ 110 cm. The results reveal the feasibility of demonstrating different reshocked RM processes using a single model, without adjusting the model coefficients, which sheds light on the further application of the present model to practical engineering, such as inertial confinement fusion.