# Latest papers in fluid mechanics

### Revisiting [math]-law for the evaporation of dilute droplets

In a wide range of applications, the estimate of droplet evaporation time is based on the classical [math]-law, which, assuming a fast mixing and fixed environmental properties, states that the droplet surface decreases linearly with time at a determined rate. However, in many cases the predicted evaporation rate is overestimated. In this Letter, we propose a revision of the [math]-law capable of accurately determining droplet evaporation rate in dilute conditions by a proper estimate of the asymptotic droplet properties. Besides a discussion of the main assumptions, we tested the proposed model against data from direct numerical simulations finding an excellent agreement for predicted droplet evaporation time in dilute turbulent jet-sprays.

### Bifurcation analysis for a flow of viscoelastic fluid due to peristaltic activity

In this article, bifurcation analysis is performed to study the qualitative nature of stagnation points and various flow regions for a peristaltic transport of viscoelastic fluid through an axisymmetric tube. The rheological behavior of viscoelastic fluid is characterized by the simplified Phan–Than–Tanner fluid model. An analytic solution in a wave frame is obtained subject to the low Reynolds number and long wavelength approximations. The stagnation points and their bifurcations (critical conditions) are explored by developing a system of autonomous differential equations. The dynamical system theory is employed to examine the nature and bifurcations of obtained stagnation points. The ranges of various flow phenomena and their bifurcations are scrutinized graphically through global bifurcation diagrams. This analysis reveals that the bifurcation in the flow is manifested at large flow rate for high extensional parameter and Weissenberg number. Backward flow phenomenon enhances and trapping diminishes with an increase in the Weissenberg number. At the end, the results of present analysis are verified by making a comparison with the existing literature.

### A high-fidelity numerical study on the propulsive performance of pitching flexible plates

In this paper, we numerically investigate the propulsive performance of three-dimensional pitching flexible plates with varying flexibility and trailing edge shapes. We employ our recently developed body-conforming fluid-structure interaction solver for our high-fidelity numerical study. To eliminate the effect of other geometric parameters, only the trailing edge angle is varied from [math] (concave plate), [math] (rectangular plate) to [math] (convex plate) while maintaining the constant area of the flexible plate. For a wide range of flexibility, three distinctive flapping motion regimes are classified based on the variation of the flapping dynamics: (i) low bending stiffness [math], (ii) moderate bending stiffness [math] near resonance, and (iii) high bending stiffness [math]. We examine the impact of the frequency ratio [math] defined as the ratio of the natural frequency of the flexible plate to the actuated pitching frequency. Through our numerical simulations, we find that the global maximum mean thrust occurs near [math] corresponding to the resonance condition. However, the optimal propulsive efficiency is achieved around [math] = 1.54 instead of the resonance condition. While the convex plate with low and high bending stiffness values shows the best performance, the rectangular plate with moderate [math] is the most efficient propulsion configuration. To examine the flow features and the correlated structural motions, we employ the sparsity-promoting dynamic mode decomposition. We find that the passive deformation induced by the flexibility effect can help in redistributing the pressure gradient, thus, improving the efficiency and the thrust production. A momentum-based thrust evaluation approach is adopted to link the temporal and spatial evolution of the vortical structures with the time-dependent thrust. When the vortices detach from the trailing edge, the instantaneous thrust shows the largest values due to the strong momentum change and convection process. Moderate flexibility and convex shape help to transfer momentum to the fluid, thereby improving the thrust generation and promoting the transition from drag to thrust. The increase in the trailing edge angle can broaden the range of flexibility that produces positive mean thrust. The role of added mass effect on the thrust generation is quantified for different pitching plates and the bending stiffness. These findings are of great significance to the optimal design of propulsion systems with flexible wings.

### Scaling patch analysis of turbulent planar plume

Proper scaling in turbulent planar plumes is investigated here using a scaling patch approach. Based on the scaled boundary conditions, a proper velocity scale for the mean axial flow is the plume centerline velocity [math], and a proper temperature scale for the temperature excess is [math], where Tctr is the plume centerline temperature and [math] is the ambient fluid temperature. By seeking an admissible scaling, a key concept in the scaling patch approach, for the mean continuity, mean momentum, and mean energy equations, respectively, the following is found: (1) a proper scale for the mean transverse flow is [math], where [math] is the growth rate of the plume width. (2) A proper scale for the Reynolds shear stress is [math], a mix of the scales for the mean axial and transverse flows. (3) A proper scale for the turbulent heat flux is [math], a mix of the scales for the mean transverse flow and mean temperature excess. The mean transverse flow thus plays a critical role in the scaling of turbulent planar plumes. Approximate functions are developed for the scaled mean transverse flow, Reynolds shear stress, and turbulent temperature flux, and are found to agree favorably with experimental and numerical simulation data. The integral analysis of the mean momentum equation yields a Richardson number Ri, which remains invariant in the axial direction. The Richardson number is defined as [math], where g is the gravitational acceleration, β is the thermal expansion coefficient, and δt is the plume half-width based on the mean temperature profile. This Richardson number arises directly from the scaling patch analysis of the mean momentum equation, including both the streamwise and transverse velocity scales.

### Behavior of synchronous and asynchronous spatially oscillating planar liquid jets in tandem

Understanding the behavior of oscillating liquid jets in tandem is vital for improving the efficacy of numerous industrial applications. An interpretation of their behavior in the terms of development, associated instability, and interactions when used in tandem remains unclear. Therefore, the present study reports the numerical investigations on spatially oscillating liquid jets in tandem. Numerical simulations are carried out by solving Navier–Stokes equations coupled with volume of fluid method to track the air–water interface. The development of the tandem jets, growth in amplitude of oscillation, and interaction between the coherent structures is analyzed for both synchronous and asynchronous liquid jets in tandem. Moreover, the effect of nozzle spacing on these parameters is also reported in the present study. It is demonstrated that a decrease in the nozzle spacing destabilizes these jets and promotes an early merging between them. This decrease in nozzle spacing also improves the stream-wise entrainment of the surrounding fluid. Furthermore, synchronous jets are found to be more stable as compared to asynchronous jets owing to a relatively constant spacing between the two jets. Asynchronous jets provide better fluid entrainment than synchronous jets owing to their higher amplitude of oscillation and stronger jet front interactions. Moreover, it is demonstrated that these interactions at the jet front give rise to a staggered vortex front in asynchronous jets, whereas the vortex front remains symmetric in synchronous jets.

### Velocity and density measurements in forced fountains with negative buoyancy

In fluid mechanics, fountains take place when a source fluid is driven by its own momentum into a surrounding ambient fluid, and it is counterbalanced by buoyancy. These phenomena are largely encountered in nature and human activities. Despite the numerous studies on the subject, few experimental data are available about the internal structure of turbulent fountains. Here, we present a set of laboratory experiments with the aim to (i) get direct velocity and density measurements of fountains in a controlled environment and (ii) obtain insights about the basic physics of the phenomenon. The results concern the characteristics of the mean and turbulent flow: we report the analysis of the turbulent kinetic energy, the velocity skewness, and the Reynolds stresses, including a quadrant analysis of the fluctuating velocities. For some tests, the correlation between density and vertical velocity is investigated for both mean and fluctuating values. We have quantified the momentum transport, which is mainly out-downward at the nozzle axis with peaks at the mean rise height, where also maximum levels of the buoyancy and mass fluxes are present. The ability of acoustic Doppler current profilers to identify the rise height of the fountain and to measure the velocity field is also discussed.

### Air filament contraction

We investigate numerically the contraction dynamics of a long air filament surrounded by liquid for a range of Ohnesorge numbers Oh. The contraction velocity rises to a maximum value Umax and then decreases due to the hydrodynamic drag force from the liquid medium. Umax follows a capillary-inertial scaling for low Oh while it shifts to a capillary-viscous scaling with increasing Oh. Our simulations reveal that contracting air filaments always first rupture via end-pinching mechanism before the Rayleigh–Plateau instability can develop.

### On exactly self-similar turbulent boundary layers and single-layer scaling

Turbulent boundary layers are traditionally thought to obey two layer scaling. However, it was shown by Dixit and Ramesh [J. Fluid Mech. 615, 445–475 (2008)] that the so-called sink flow turbulent boundary layer obeys a Blasius-like scaling all through the boundary layer. We make this single layer scaling as an operational definition for perfectly equilibrium flow and derive equations for a general class of equilibrium flows with permeable walls allowing suction and blowing of fluid. Using the above definition, constraints required to be satisfied by such equilibrium turbulent boundary layers are derived and discussed in detail. Using direct numerical simulations (DNS), we demonstrate the existence of such turbulent boundary layers, and various properties of these boundary layers are studied in detail. In particular, it is shown that the mean velocity profiles and other turbulent statistics display excellent similarity, and various boundary layer parameters in such boundary layers vary systematically with the magnitude of blowing/suction parameter. By extension, it is also demonstrated by DNS that higher order moments (up to third order moments) also display streamwise self-similarity. We believe that this is an important result where self-similarity is demonstrated for the first time for higher moments of fluctuating velocity. The non-universality of the log-law coefficients in favorable pressure gradient flows is already known [S. A. Dixit and O. N. Ramesh, J. Fluid Mech. 615, 445–475 (2008) and Chauhan et al., in Proceedings of 11th EUROMECH European Turbulence Conference on Turbulence, Porto, Portugal (Springer Proceedings in Physics, 2007), pp. 239–241]. We demonstrate their non-universal nature in equilibrium boundary layers with suction and blowing using the present direct numerical simulations. Remarkably, the ratio of Reynolds shear stress ([math]) to streamwise ([math]) and spanwise turbulent intensity ([math]) in these boundary layers is constant over the log-region in the mean velocity profile even though [math], and [math] by themselves show variation with [math] in the log region, indicative of the absence of inactive motion in these flows. It is shown that the dominance of active motion and more importantly the absence of inactive motion leads to a universality of scaling that manifests as a single layer Blasius scaling with self-similarity in such flows. Equivalently, it could also be seen as a single layer scaling with all variables all through the boundary layer. This would mean that the friction velocity scales the entire boundary layer, which could be taken to suggest equivalence of attached eddies with the active component of motion.

### Development of an immersed boundary-multiphase lattice Boltzmann flux solver with high density ratio for contact line dynamics

Interaction between a two-phase fluid and a structure involving contact line dynamics is a common phenomenon. In this paper, we aim to develop a fluid–solid coupling model that can study contact line dynamics in the case of a high density ratio between the two fluids. The fluids are treated using a multiphase lattice Boltzmann flux solver (MLBFS) that uses the cell-centered finite volume method to obtain macroscopic flow variables, and the interface fluxes are reconstructed locally by the standard lattice Boltzmann method (LBM) solutions. This approach retains the advantages of the original LBM while being more flexible in handling nonuniform grids and external force terms. The immersed boundary method (IBM) is an effective method for processing structural information, and here, the implicit boundary-condition-enforced IBM is used to accurately satisfy the Dirichlet boundary condition (no-slip boundary). Moreover, the Neumann boundary condition is deemed to represent the contribution from the structure boundary flux and is incorporated into the IB-MLBFS. The developed IB-MLBFS is verified by several test cases, including contact line motion of a two-phase fluid along a circular cylinder and droplet spreading on a flat plate, where both equilibrium results and dynamic process are correctly reproduced for different density ratios and wettability conditions. Furthermore, based on the IB-MLBFS established here, the contact line dynamics of a two-phase fluid between two square cylinders or two circular cylinders is studied. The effects of distance, structure size, and wettability on the interface state and the contact angle are studied in detail. The robustness of the proposed model is verified.

### A high-order implicit least square-based finite difference-finite volume method for incompressible flows on unstructured grids

In this study, a high-order implicit least squares-based finite difference-finite volume (ILSFD-FV) method with a lattice Boltzmann flux solver is presented for the simulation of two-dimensional incompressible flows on unstructured grids. In this method, a high-order polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by the least squares-based finite difference scheme. The volume integral of the high-order polynomial over the control cell results in a pre-multiplied coefficient matrix in the time-dependent term. This makes the high-order method be implicit in nature. With this feature, a high-order implicit Runge–Kutta time integration scheme, namely, the explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) scheme, is applied to obtain the time-accurate solutions for flow problems. The non-linear system of equations arising from each ESDIRK stage except for the first explicit stage is solved by a dual time stepping approach. A matrix-free lower-upper symmetric Gauss–Seidel solver is then used to efficiently march the solution in the pseudo time. The present high-order ILSFD-FV method is verified and validated by both steady and unsteady 2D incompressible flow problems. Numerical results indicate that the developed implicit method outperforms its explicit counterpart in terms of the convergence property and computational efficiency. The speedup ratio of the computational effort is about 3–22.

### Topological constraints in the reconnection of vortex braids

We study the relaxation of a topologically nontrivial vortex braid with zero net helicity in a barotropic fluid. The aim is to investigate the extent to which the topology of the vorticity field—characterized by braided vorticity field lines—determines the dynamics, particularly the asymptotic behavior under vortex reconnection in evolution at high Reynolds numbers (25 000). Analogous to the evolution of braided magnetic fields in plasma, we find that the relaxation of our vortex braid leads to a simplification of the topology into large-scale regions of opposite swirl, consistent with an inverse cascade of the helicity. The change of topology is facilitated by a cascade of vortex reconnection events. During this process, the existence of regions of positive and negative kinetic helicities imposes a lower bound for the kinetic energy. For the enstrophy, we derive analytically a lower bound given by the presence of unsigned kinetic helicity, which we confirm in our numerical experiments.

### Quantitative prediction of flow dynamics and mechanical retention of surface-altered red blood cells through a splenic slit

Normal red blood cells (RBCs) have remarkable properties of deformability, which enable them to squeeze through tiny splenic inter-endothelial slits (IESs) without any damage. Decreased surface-area-to-volume (SA/V) ratio through the loss of membrane surface is a key determinant of splenic entrapment of surface-altered RBCs due to cell aging or disease. Here, we investigate the flow dynamics and mechanical retention of the surface-altered RBCs with different extents of surface area loss, using a multiscale RBC (MS-RBC) model implemented in dissipative particle dynamics (DPD). We show that the DPD-based MS-RBC simulations can accurately reproduce the ex vivo experimentally measured rate of RBC mechanical retention when we take into account the distribution of RBC surface area (i.e., the size difference within the RBC population). We also examine the cumulative effect of the cell surface area loss on the traversal dynamics of the surface-altered RBCs, where we found that the final values of cell surface area (or the SA/V ratio) play a key role in determining the RBC traversal dynamics, regardless of the loss pathway of cell surface area. Taken together, these simulation results have implications for understanding the sensitivity of the splenic IESs to retain and clear the surface-altered RBCs with increased surface area loss, providing an insight into the fundamental flow dynamics and mechanical clearance of the surface-altered RBCs by the human spleen.

### Erratum: “Frameworks for investigation of nonlinear dynamics: Experimental study of the turbulent jet” [Phys. Fluids 32, 085112 (2020)]

### Drag analysis with a self-propelled flexible swimmer

Author(s): David Gross, Yann Roux, Christophe Raufaste, and Argentina Médéric

Fish swim by undulating their body to ensure propulsion. In a steady state, thrust is balanced by the total drag force, for which the dominant terms depend on the Reynolds number and the flow regime. In this article we propose a set of simple scaling laws to determine the contribution of each mechanism to the drag exerted on the swimmer.

[Phys. Rev. Fluids 6, 053101] Published Mon May 03, 2021

### Scaling laws for extremely strong thermals

Author(s): Alex Skvortsov, Timothy C. DuBois, Milan Jamriska, and Martin Kocan

Since the seminal results of Batchelor, Morton and Turner who laid the foundations of classical convective plume theory, it has been recognized that convective thermals exhibit remarkable scaling properties. In this paper, the authors identify dynamics of strong thermals which can be drastically different from weak Boussinesq-type thermals. In general, the evolution of thermals is affected by the interplay of two processes: entrainment flux caused by thermal expansion and solid-body acceleration of the thermal centroid. As a result, depending on the density contrast between the thermal and the ambient environment, scaling laws are modified with different power-law exponents.

[Phys. Rev. Fluids 6, 053501] Published Mon May 03, 2021

### Erratum: Material transport in the left ventricle with aortic valve regurgitation [Phys. Rev. Fluids <b>3</b>, 113101 (2018)]

Author(s): Giuseppe Di Labbio, Jérôme Vétel, and Lyes Kadem

[Phys. Rev. Fluids 6, 059901] Published Mon May 03, 2021

### Reversed role of liquid viscosity on drop splash

The influence of liquid viscosity on the splashing behavior of a drop impacting a smooth surface is investigated. It is found that the liquid viscosity plays a reversed role on the drop splash, and the liquid viscosity promotes the drop splash in the low-viscosity cases but suppresses the drop splash in the high-viscosity cases. Here, we focus on the dynamics of the lifting lamella during the splashing behavior and demonstrate that the increase in liquid viscosity could contribute to the lower splashing angle αs. Furthermore, an empirical relationship of the splashing angle is obtained according to the experimental results. The reversed role of liquid viscosity is reflected in the positive effect both on the aerodynamic force and viscous force. For low-Oh cases, the increase in aerodynamic force predominates and promotes the drop splash. For high-Oh cases, the increase in viscous force predominates and suppresses the drop splash. We eventually put forward a splash criterion on smooth surfaces, which could successfully characterize the splashing behavior in this work and previous studies, and indicate the reversed role of liquid viscosity on drop splash.

### Optic nerve microcirculation: Fluid flow and electrodiffusion

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system (CNS) are important in a wide range of biological and medical processes [C. Nicholson and S. Hrabětová, “Brain extracellular space: The final frontier of neuroscience,” Biophys. J. 113(10), 2133 (2017)]. However, the exact driving mechanisms are often not known. In this paper, we investigate flows induced by action potentials in an optic nerve as a prototype of the CNS. Different from traditional fluid dynamics problems, flows in biological tissues such as the CNS are coupled with ion transport. It is driven by osmosis created by the concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group [R. K. Orkand, J. G. Nicholls, and S. W. Kuffler, “Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia,” J. Neurophysiol. 29(4), 788 (1966)]. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.

### Stochastic modeling for subgrid-scale particle dispersion in large-eddy simulation of inhomogeneous turbulence

We consider Lagrangian modeling of heavy particle motion in inhomogeneous turbulence. The dynamics of point particles is one-way coupled to the large-eddy simulation (LES) of fluid flow. To account for the effect of non-resolved (subgrid) flow scales on particle motion, we propose a model for the fluid velocity along the particle trajectories. The model, based on a stochastic diffusion process, accounts for turbulence anisotropy and utilizes the statistical estimates of subgrid quantities: the velocity components (the r.m.s. and covariance) and the Lagrangian time scales. The turbulent channel flow case is taken for validation. First, we discuss the outcome of an a priori LES study. Then, the proposed subgrid dispersion model is tested in a true LES computation. The resulting velocity statistics, particle concentration profiles, and the deposition velocity are compared against available reference data from direct numerical simulations.

### Thixotropy in viscoplastic drop impact on thin films

Author(s): Samya Sen, Anthony G. Morales, and Randy H. Ewoldt

We report the first-ever experimental study of thixotropic aging in viscoplastic drop impact. A new dimensionless group is proposed and validated. The results will be useful in predicting splash behavior in a variety of applications from spray coating to fire suppression.

[Phys. Rev. Fluids 6, 043301] Published Fri Apr 30, 2021