# Latest papers in fluid mechanics

### Investigation of the structures in the unstable rotating-cone boundary layer

Author(s): K. Kato, T. Kawata, P. H. Alfredsson, and R. J. Lingwood

Experiments show that crossflow vortices in the boundary layer on a rotating wide cone start to meander before breakdown. The structural development of the overturning process of these vortices, where high-momentum upwelling of the vortices leads to transition to turbulence, is described in detail.

[Phys. Rev. Fluids 4, 053903] Published Thu May 30, 2019

### Coexistence of multiple long-time solutions for two-dimensional laminar flow past a linearly sprung circular cylinder with a rotational nonlinear energy sink

Author(s): Antoine B. Blanchard, Lawrence A. Bergman, Alexander F. Vakakis, and Arne J. Pearlstein

Phase diagram showing the variation of the number of long-time solutions with Re and a dimensionless measure of spring stiffness. Each combination of symbol shape and color indicates a different set of such solutions, with the pentagons corresponding to three unsteady solutions and one steady one.

[Phys. Rev. Fluids 4, 054401] Published Thu May 30, 2019

### Refinement of the logarithmic law of the wall

Author(s): F. Laadhari

A refinement of the logarithmic law for the mean velocity in canonical wall-bounded turbulent flows is presented. The relevant length scale in the overlap region is found to be based on the weighted mean velocity gradient instead of the classical length scale based on the wall friction velocity.

[Phys. Rev. Fluids 4, 054605] Published Thu May 30, 2019

### Publisher’s Note: “An experimental study on the effect of swirl number on pollutant formation in propane bluff-body stabilized swirl diffusion flames” [Phys. Fluids 31, 055105 (2019)]

### Hydrodynamic interaction for rigid dumbbell suspensions in steady shear flow

From kinetic molecular theory, we can attribute the rheological behaviors of polymeric liquids to macromolecular orientation. The simplest model to capture the orientation of macromolecules is the rigid dumbbell. For a suspension of rigid dumbbells, subject to any shear flow, for instance, we must first solve the diffusion equation for the orientation distribution function. From this distribution, we then calculate the first and second normal stress differences. To get reasonable results for the normal stress differences in steady shear flow, one must account for hydrodynamic interaction between the dumbbell beads. However, for the power series expansions for these normal stress differences, three series arise. The coefficients for two of these series, (ck, dk), are not known, not even approximately, beyond the second power of the shear rate. Analytical work on many viscoelastic material functions in shear flow must be checked for consistency, in their steady shear flow limits, against these normal stress difference power series expansions. For instance, for large-amplitude oscillatory shear flow, we must recover the power series expansions in the limits of low frequency. In this work, for (ck, dk), we arrive at the exact expressions for the first 18 of these coefficients.

### Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence

The effect of internal-kinetic energy exchange on transient spectral energy transfer in compressible turbulence is investigated. We derive the spectral evolution equations for kinetic energy and pressure fields to highlight the key mechanisms that affect the turbulence spectral evolution. Direct numerical simulations of decaying isotropic turbulence are performed from solenoidal, dilatational, and mixed velocity initial conditions. It is shown that internal-kinetic energy exchange arising due to pressure-dilatation renders the dilatational kinetic energy amplitudes at large scales of motion to be oscillatory. The oscillatory behavior of amplitude diminishes with increasing wavenumber. The dilatational energy spectrum also exhibits a wider range of scales due to its inherent tendency to form shocks. The findings are expected to lead to an improved understanding of energy dynamics in high-speed compressible flows.

### Empirical scaling analysis of supersonic jet control using steady fluidic injection

Following our previous work [Kumar et al., “Fluidic injectors for supersonic jet control,” Phys. Fluids 30(12), 126101 (2018)], we experimentally investigate the effect of fluidic injection on the mixing enhancement of a Mach 2.0 jet. The mass flow rate ratio Cm of the injectors to that of the main jet and the expansion ratio pe/pa (where pe and pa are the nozzle exit and atmospheric pressures, respectively) to understand the mixing capability at design and off-design conditions are examined in detail. Extensive Pitot pressure measurements are performed along the jet centerline, and the jet stream has been visualized using the shadowgraph technique in the orthogonal planes of the manipulated jet. The mixing capability of the manipulated jet quantified based on the reduction in supersonic core length [math] exhibits a strong dependence on Cm and pe/pa. Empirical scaling analysis of the jet control reveals that the relationship [math] = f1(Cm, pe, pa, D, d) may be reduced to [math] = f2(ξ), where f1 and f2 are different functions and the scaling factor ξ = [math], where MR is the momentum ratio of the injector to the main jet, D and d are the nozzle exit diameter and the injector exit diameter, respectively. The scaling parameter [math] = f2(ξ) provides important insights into the jet control physics.

### Retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid

Retarded or frequency-dependent hydrodynamic interactions are relevant for velocity relaxation of colloidal particles immersed in a fluid, sufficiently close that their flow patterns interfere. The interactions are also important for periodic motions, such as occur in swimming. Analytic expressions are derived for the set of scalar mobility functions of a pair of spheres. Mutual hydrodynamic interactions are evaluated in one-propagator approximation, characterized by a single Green function acting between the two spheres. Self-mobility functions are evaluated in a two-propagator approximation, characterized by a single reflection between the two spheres. The approximations should yield accurate results for intermediate and long distances between the spheres. Both translations and rotations are considered. For motions perpendicular to the line of centers, there is a translation-rotation coupling. Extensive use is made of Faxén theorems, which yield the hydrodynamic force and torque acting on a sphere in an incident oscillating flow. The derived results are important for the study of velocity relaxation of two interacting spheres immersed in a fluid and for the study of swimming of assemblies of spheres.

### The influence of dipolar particle interactions on the magnetization and the rotational viscosity of ferrofluids

The effect of the dipolar particle interactions on the behavior of ferrofluids under a shear flow is not yet well understood. The equilibrium magnetization in the absence of flow is studied in Paper I [A. P. Rosa, G. C. Abade, and F. R. Cunha, “Computer simulation of equilibrium magnetization and microstructure in magnetic fluids,” Phys. Fluids 29(9), 092006 (2017)]. In this paper, we present the results of magnetization and rheology in terms of a rotational viscosity obtained by applying Brownian dynamics simulations for a periodic magnetic suspension, where the many body long-range dipole-dipole interactions are calculated by the Ewald summation technique. The dependence of these macroscopic properties on the dipolar interactions is explored in ferrofluids undergoing both weak and strong shear flows in the presence of a uniform magnetic field. Through the simulations, the suspension microstructure is also analyzed in order to characterize the interplay between the structure and the investigated macroscopic properties. We show that for weak shear flows the dipole-dipole interactions produces a magnetization increasing. In contrast, a decrease in the ferrofluid magnetization with the shear rate is substantially intensified as the dipolar interactions are accounted for. Therefore, for strong shear flows, the dipolar interactions always have an effect of decreasing magnetization. In addition, while the dipolar particle interactions produce an increase in the rotational viscosity for weak flows, variations in the same property are not perceptible under the condition of strong flows. The numerical simulations show chain-structure formation oriented in the direction of the magnetic field (i.e., perpendicular to the direction of the shear) for weak flows, which explains the remarkable increasing of the suspension rotational viscosity as a function of the applied magnetic field and of the dipolar interactions parameters. A detailed comparison shows that our simulation results of magnetization and the rotational viscosity are in excellent agreement with approximate theoretical predictions reported in the literature for the case of noninteracting particles.

### Taylor-Couette flow of shear-thinning fluids

The flow between two concentric cylinders, one of which is rotating (Taylor-Couette flow), has been the focus of extensive research, due to the number of flow instabilities that may occur and its use in various industrial applications. We examine Taylor-Couette flow of Newtonian and shear-thinning fluids (solutions of xanthan gum in water/glycerol) using a combination of particle-image velocimetry and flow visualization for a wide range of Reynolds number, spanning the circular Couette flow, Taylor vortex flow, and wavy vortex flow regimes. Shear thinning is associated with an increase in the axial wavelength and has a nonmonotonic effect on the critical Reynolds number for transition to Taylor vortex flow and wavy vortex flow. The magnitude of vorticity and the strength of the radial jets transporting fluid away from the inner cylinder (“outward jets”) are both reduced in shear-thinning fluids relative to the Newtonian case; the vorticity in the shear-thinning fluids also tends to concentrate at the edges of vortices, rather than in the cores. In the wavy vortex flow regime for Newtonian fluids, the amplitudes of the waves at the “inward jets” (moving toward the inner cylinder) are low compared to those at the outward jets. However, for the shear-thinning fluids, the amplitudes of the waves at both the inward and outward jets tend to be significantly larger. Finally, shear thinning is associated with greater variations in time and space: we observe slow drifts in the axial positions of vortices and spatial variations in the amplitudes of the wavy instability, which are absent in Newtonian fluids.

### Ceiling effects on the aerodynamics of a flapping wing at hovering condition

The ceiling effect on the aerodynamics of a hovering flapping wing is investigated by solving the three-dimensional incompressible Navier-Stokes equations. Computations have been carried out for some parameters including the distance between the wing and the ceiling, the Reynolds number, the stroke amplitude, and the mid-stroke angle of incidence. The ceiling effect on the force production and vortical structures around the wing is analyzed. It is shown that the ceiling effect increases the aerodynamic forces. This improvement in force production in the ceiling effect is caused by the increments both in the relative velocity of oncoming flow and the effective angle of attack of the wing. The underlying mechanism is that the presence of the ceiling acts as a mirror as if there exists a mirroring leading-edge vortex (LEV). This mirroring LEV not only increases the relative velocity of the oncoming flow ahead of the wing but also produces an upwash to the oncoming flow, hence increasing the effective angle of attack of the wing.

### On the role of vortical structures in aerodynamic performance of a hovering mosquito

Mosquitoes have slimmer wings, higher flapping frequencies, and much lower amplitudes than most other insects. These unique features signify special aerodynamic mechanisms. Besides the leading-edge vortex, which is one of the most common mechanisms of flapping-wing flight, mosquitoes have two distinctive mechanisms: trailing-edge vortex and rotational drag. In this study, the three-dimensional flow field around a hovering mosquito is simulated by using the immersed boundary method. The numerical results agree well with previous experimental data. Mechanisms unique to mosquitoes are identified from the instantaneous pressure and vorticity fields. The flow domains, containing several vortical structures produced by the flapping wings, are divided into different regions for quantitatively analyzing the contribution of vortical structures to the lift. Advection of the trailing-edge vortex and production of the leading-edge vortex each contribute peaks in lift. Passive deformation of the wings is also important, as it stabilizes delayed stall and decreases by 26% the maximum aerodynamic power required for hovering flight. In addition, the lift coefficient and power economy are improved as the Reynolds number increases, which explains the better ability of larger mosquitoes to seek and feed on hosts from the aerodynamic point of view.

### Numerical simulation of the interaction of two shear layers in double backward-facing steps

In this paper, Navier-Stokes equations were solved with high-order accurate schemes to investigate the basic structure and regularity of the flow field during the interaction of a supersonic jet and a codirectional supersonic incoming flow. A double backward-facing step model was proposed to investigate the interaction between the jet/supersonic incoming flow shear layers. The two shear layers interact to produce a secondary jet. The secondary jet produced by the action has a unique periodicity that is related to the overall oscillation of the shear layer. The secondary jet is generated when the horizontal angle of the jet shear layer reaches a certain value. This paper focused on the analysis and discussion of the periodicity of the secondary jet. When the aspect ratio is different, the period of the secondary jet changes significantly. However, when the static pressure ratio is different, the period of the secondary jet does not change much.

### Rapidly rotating Rayleigh-Bénard convection with a tilted axis

Author(s): L. Novi, J. von Hardenberg, D. W. Hughes, A. Provenzale, and E. A. Spiegel

We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector. The angle ϕ between the rotation axis a...

[Phys. Rev. E 99, 053116] Published Tue May 28, 2019

### Scaling of Reynolds stresses in a differentially heated vertical channel

Author(s): Tie Wei

Reexamination of some direct numerical simulation data for a turbulent differentially heated vertical channel finds that the components of the Reynolds stress scale with the product of the friction velocity and the maximum mean flow.

[Phys. Rev. Fluids 4, 051501(R)] Published Tue May 28, 2019

### Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria

Author(s): Peter B. Weichman

Equilibrium properties of axisymmetric flow in cylindrical (Taylor-Couette) geometries are studied using the methods of statistical mechanics. The system is constrained by an infinite number of conservation laws, leading to an intricate interplay between the toroidal (σ) and poloidal (ξ) flow fields.

[Phys. Rev. Fluids 4, 054703] Published Tue May 28, 2019

### Active control of vortex shedding from a blunt trailing edge using oscillating piezoelectric flaps

Author(s): Bradley Gibeau, Charles Robert Koch, and Sina Ghaemi

Active control of vortex shedding from a blunt trailing edge is achieved experimentally using piezoelectric actuators. The system can suppress and amplify the vortex shedding pattern in the wake, as well as force near-wake symmetry. An application of closed-loop control is also demonstrated.

[Phys. Rev. Fluids 4, 054704] Published Tue May 28, 2019

### Experimental investigation of the impact and freezing processes of a hot water droplet on an ice surface

Water droplet freezing on an ice surface is a common phenomenon and poses hazards to a lot of applications, including wind turbines, aircraft, and power transmission lines. Since the water droplet temperature is critical, many studies have been carried out to understand the influence of the water droplet temperature on both the impact and freezing processes of droplets on different surfaces. However, the past research studies mainly focused on supercooled water droplets, not on hot water droplets. For applications such as hot-water ice-drilling, the understanding of freezing of hot water droplets on an ice surface is necessary. In the present study, we report the detailed dynamic motions of a hot water droplet impacting on an ice surface. The impact and freezing processes of the hot water droplet on the ice surface are recorded by two cameras. The effects of the water droplet temperature and the ice surface temperature on the impact and freezing processes of the water droplet were experimentally investigated. The results showed that, at the same ice surface temperature, the increase of the water droplet temperature resulted in the increase of the maximum spreading factor, the reduction of the height of the ice bead, and the slight increase of the freezing time. In addition, during the droplet spreading process, the experimental results of the normalized contact diameter fitted well with the exponential model and the water droplet temperature was found to have an apparent influence on the lamella thickness.

### Dynamic mode decomposition with exogenous input for data-driven modeling of unsteady flows

This work proposes a data-driven reduced-order modeling algorithm for complex, high-dimensional, and unsteady fluid systems with exogenous input and control. This algorithm is a variant of dynamic mode decomposition (DMD), which is an equation-free method for identifying coherent structures and modeling complex flow dynamics. Compared with existing methods, the proposed method improves the capability of predicting the flow evolution near the unstable equilibrium state. The method is achieved by two steps. First, the system matrix without input is identified by standard DMD to represent the intrinsic flow dynamics. Second, the input term, represented by a state space equation, is identified through existing methods for DMD with control effects. The whole system with input is described by the superposition of both the system matrix and the input term. The proposed method is validated by one simple two-dimensional dynamic system and two test cases of unsteady flow, including flow past a circular cylinder at Reynolds number 45 and flow past a NACA0012 airfoil at an angle of attack 25°. Results indicate that the proposed method gives more accurate description on the flow evolution with or without external forcing.

### Electric field mediated squeezing to bending transitions of interfacial instabilities for digitization and mixing of two-phase microflows

Electric field mediated instabilities in a tri-layer oil-water flow inside a microchannel have been explored with the help of the analytical models and computational fluid dynamic simulations. The twin oil-water interfaces undergo either in-phase bending or antiphase squeezing mode of deformation when a direct current (DC) electric field is applied locally inside the channel. The selection of modes largely depends on the magnitudes of the electric field intensity and oil-water interfacial tension. The instability modes grow to form an array of miniaturized oil-droplets with a significantly higher surface to volume ratio. While squeezing mode leads to a time-periodic dripping of droplets at relatively lower field intensities, the bending mode develops into a whiplash ejection of miniaturized droplets at higher field intensities. Subsequently, a transition from purely laminar to chaotic flow is observed, resembling the von Kármán vortex street from a flow past immersed body, suitable for augmented heat, mass, and momentum transport inside a microfluidic channel. Under these conditions, the simulations also reveal the formation of multiple microvortices inside and outside the droplets, which helps in increase in the local Reynolds number for a better mixing efficiency in such microflows. Use of alternating current electric field instead of DC is also found to create on-demand flow features in a time-periodic manner following the mode selection. The amplitude, frequency, and waveform of such electric field is found to generate miniaturized oil-droplets along with the formation of an array of flow features, namely, thread, slugs, plugs, among others.