# Physics of Fluids

Table of Contents for Physics of Fluids. List of articles from both the latest and ahead of print issues.

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### Pulse propagation in gravity currents

Physics of Fluids, Volume 32, Issue 1, January 2020.

Real world gravity current flows rarely exist as a single discrete event, but are instead made up of multiple surges. This paper examines the propagation of surges as pulses in gravity currents. Using theoretical shallow-water modeling, we analyze the structure of pulsed flows created by the sequential release of two lock-boxes. The first release creates a gravity current, while the second creates a pulse that eventually propagates to the head of the first current. Two parameters determine the flow structure: the densimetric Froude number at the head of the current, Fr, and a dimensionless time between releases, tre. The shallow-water model enables the flow behavior to be mapped in (Fr, tre) space. Pulse speed depends on three critical characteristic curves: two that derive from the first release and correspond to a wavelike disturbance which reflects between the head of the current and the back of the lock-box and a third that originates from the second release and represents the region of the flow affected by the finite supply of source material. Pulses have non-negative acceleration until they intersect the third characteristic, after which they decelerate. Variations in pulse speed affect energy transfer and dissipation. Critically for lahars, landslides, and avalanches, pulsed flows may change from erosional to depositional, further affecting their dynamics. Gravity current hazard prediction models for such surge-prone flows may underpredict risk if they neglect internal flow dynamics.

Real world gravity current flows rarely exist as a single discrete event, but are instead made up of multiple surges. This paper examines the propagation of surges as pulses in gravity currents. Using theoretical shallow-water modeling, we analyze the structure of pulsed flows created by the sequential release of two lock-boxes. The first release creates a gravity current, while the second creates a pulse that eventually propagates to the head of the first current. Two parameters determine the flow structure: the densimetric Froude number at the head of the current, Fr, and a dimensionless time between releases, tre. The shallow-water model enables the flow behavior to be mapped in (Fr, tre) space. Pulse speed depends on three critical characteristic curves: two that derive from the first release and correspond to a wavelike disturbance which reflects between the head of the current and the back of the lock-box and a third that originates from the second release and represents the region of the flow affected by the finite supply of source material. Pulses have non-negative acceleration until they intersect the third characteristic, after which they decelerate. Variations in pulse speed affect energy transfer and dissipation. Critically for lahars, landslides, and avalanches, pulsed flows may change from erosional to depositional, further affecting their dynamics. Gravity current hazard prediction models for such surge-prone flows may underpredict risk if they neglect internal flow dynamics.

Categories: Latest papers in fluid mechanics

### Imaging fluid injections into soft biological tissue to extract permeability model parameters

Physics of Fluids, Volume 32, Issue 1, January 2020.

One of the most common health care procedures is injecting fluids, in the form of drugs and vaccines, into our bodies, and hollow microneedles are emerging medical devices that deliver such fluids into the skin. Fluid injection into the skin through microneedles is advantageous because of improved patient compliance and the dose sparing effect for vaccines. Since skin tissue is a deformable porous medium, injecting fluid into the skin involves a coupled interaction between the injected fluid flow and the deformation of the soft porous matrix of skin tissue. Here, we introduce a semiempirical model that describes the fluid transport through skin tissue based on experimental data and constitutive equations of flow through biological tissue. Our model assumes that fluid flows radially outward and tissue deformation varies spherically from the microneedle tip. The permeability of tissue, assumed to be initially homogeneous, varies as a function of volumetric strain in the tissue based on a two-parameter exponential relationship. The model is optimized to extract two macroscopic parameters, k0 and m, for each of the seven experiments on excised porcine skin, using a radial form of Darcy’s law, the two-parameter exponential dependence of permeability on strain, and the experimental data on fluid flow recorded by a flow sensor and tissue deformation captured in real time using optical coherence tomography. The fluid flow estimated by the permeability model with optimized macroscopic parameters matches closely with the recorded flow rate, thus validating our semiempirical model.

One of the most common health care procedures is injecting fluids, in the form of drugs and vaccines, into our bodies, and hollow microneedles are emerging medical devices that deliver such fluids into the skin. Fluid injection into the skin through microneedles is advantageous because of improved patient compliance and the dose sparing effect for vaccines. Since skin tissue is a deformable porous medium, injecting fluid into the skin involves a coupled interaction between the injected fluid flow and the deformation of the soft porous matrix of skin tissue. Here, we introduce a semiempirical model that describes the fluid transport through skin tissue based on experimental data and constitutive equations of flow through biological tissue. Our model assumes that fluid flows radially outward and tissue deformation varies spherically from the microneedle tip. The permeability of tissue, assumed to be initially homogeneous, varies as a function of volumetric strain in the tissue based on a two-parameter exponential relationship. The model is optimized to extract two macroscopic parameters, k0 and m, for each of the seven experiments on excised porcine skin, using a radial form of Darcy’s law, the two-parameter exponential dependence of permeability on strain, and the experimental data on fluid flow recorded by a flow sensor and tissue deformation captured in real time using optical coherence tomography. The fluid flow estimated by the permeability model with optimized macroscopic parameters matches closely with the recorded flow rate, thus validating our semiempirical model.

Categories: Latest papers in fluid mechanics

### Mathematical modelling of mass transfer of paramagnetic ions through an inert membrane by the transient magnetic concentration gradient force

Physics of Fluids, Volume 32, Issue 1, January 2020.

The objective of this work is to suggest a mathematical model for mass-transfer of a paramagnetic electrolyte, nickel(ii)chloride solution, through an inert, thin membrane from one chamber to another under the influence of magnetic fields which are applied perpendicular to the membrane. The model is based on the magnetic concentration gradient force, the Fick’s law of diffusion, and the Hagen-Poiseuille law for paramagnetic ion transport in the membrane. The magnetic concentration gradient force is found to be elusive and points in the direction of the magnetic field, in our case, the direction of the Fick diffusion flux. The reason is the gradient of the magnetic volume susceptibility for the electrolyte in the membrane, which decreases in the direction of the magnetic field. This is in accordance with the variable-reluctance principle. Mass balances for transport of Ni ions in distilled water through the membrane are derived and governed by a partial differential equation in one-dimensional space and time with specified initial and boundary conditions. The associated flux is superimposed on the pure Fick diffusion flux. The total flux is described by a nonlinear partial differential equation, which has not previously been used to describe transfer phenomena in paramagnetic solutions in magnetic fields. The simulated results were compared with experimental results and coincide approximately in all points for unstirred solutions. In stirred solutions, where the mass transfer coefficient at the membrane inlet approaches infinity if the mixing is ideal, no experimental or simulated effect was observed of the magnetic field.

The objective of this work is to suggest a mathematical model for mass-transfer of a paramagnetic electrolyte, nickel(ii)chloride solution, through an inert, thin membrane from one chamber to another under the influence of magnetic fields which are applied perpendicular to the membrane. The model is based on the magnetic concentration gradient force, the Fick’s law of diffusion, and the Hagen-Poiseuille law for paramagnetic ion transport in the membrane. The magnetic concentration gradient force is found to be elusive and points in the direction of the magnetic field, in our case, the direction of the Fick diffusion flux. The reason is the gradient of the magnetic volume susceptibility for the electrolyte in the membrane, which decreases in the direction of the magnetic field. This is in accordance with the variable-reluctance principle. Mass balances for transport of Ni ions in distilled water through the membrane are derived and governed by a partial differential equation in one-dimensional space and time with specified initial and boundary conditions. The associated flux is superimposed on the pure Fick diffusion flux. The total flux is described by a nonlinear partial differential equation, which has not previously been used to describe transfer phenomena in paramagnetic solutions in magnetic fields. The simulated results were compared with experimental results and coincide approximately in all points for unstirred solutions. In stirred solutions, where the mass transfer coefficient at the membrane inlet approaches infinity if the mixing is ideal, no experimental or simulated effect was observed of the magnetic field.

Categories: Latest papers in fluid mechanics

### Interface-resolved numerical simulations of particle-laden turbulent channel flows with spanwise rotation

Physics of Fluids, Volume 32, Issue 1, January 2020.

Interface-resolved simulations of particle-laden turbulent channel flows with spanwise rotation at a Reynolds number of 180 and different rotation numbers ranging from 0.1 to 1.0 are performed with a fictitious domain method. The difficulty of the centrifugal force on the particles not satisfying the periodic boundary condition is circumvented by the feature of the fictitious domain formulation for the neutrally buoyant case, where the centrifugal force in the particle motion equation vanishes, and by only considering a low rotation number of 0.1 and setting the rotation center to be far away from the channel for the non-unity density ratio case. Our results show that the heavy particles (i.e., the particle density being larger than the fluid density) migrate towards the pressure wall, whereas the light particles migrate towards the suction wall. For the density ratio being unity, the particle concentration is higher near the pressure wall than near the suction wall, and we attribute the reason to the effects of the mean secondary flow structure (i.e., the Taylor–Görtler vortices), since similar particle concentration distribution and secondary flow structure are observed in a rotating laminar channel flow. The mean velocities of heavy particles are smaller in the pressure-side half channel except the near-wall region, and larger in the suction-side half channel, compared to the fluid mean velocity; the opposite occurs for the light particle case. The addition of the finite-size particles increases the flow drag. The flow drag is not sensitive to the density ratio for the light particles and increases with increasing density ratio for the heavy particles. The effects of the particles on the fluid root-mean-square velocities of the rotating turbulent channel flow are generally similar to the non-rotating channel case, but become more complicated because of the asymmetric turbulence intensity and particle concentration distribution near two walls caused by the channel rotation.

Interface-resolved simulations of particle-laden turbulent channel flows with spanwise rotation at a Reynolds number of 180 and different rotation numbers ranging from 0.1 to 1.0 are performed with a fictitious domain method. The difficulty of the centrifugal force on the particles not satisfying the periodic boundary condition is circumvented by the feature of the fictitious domain formulation for the neutrally buoyant case, where the centrifugal force in the particle motion equation vanishes, and by only considering a low rotation number of 0.1 and setting the rotation center to be far away from the channel for the non-unity density ratio case. Our results show that the heavy particles (i.e., the particle density being larger than the fluid density) migrate towards the pressure wall, whereas the light particles migrate towards the suction wall. For the density ratio being unity, the particle concentration is higher near the pressure wall than near the suction wall, and we attribute the reason to the effects of the mean secondary flow structure (i.e., the Taylor–Görtler vortices), since similar particle concentration distribution and secondary flow structure are observed in a rotating laminar channel flow. The mean velocities of heavy particles are smaller in the pressure-side half channel except the near-wall region, and larger in the suction-side half channel, compared to the fluid mean velocity; the opposite occurs for the light particle case. The addition of the finite-size particles increases the flow drag. The flow drag is not sensitive to the density ratio for the light particles and increases with increasing density ratio for the heavy particles. The effects of the particles on the fluid root-mean-square velocities of the rotating turbulent channel flow are generally similar to the non-rotating channel case, but become more complicated because of the asymmetric turbulence intensity and particle concentration distribution near two walls caused by the channel rotation.

Categories: Latest papers in fluid mechanics

### A numerical investigation of puddle jumping

Physics of Fluids, Volume 32, Issue 1, January 2020.

The nearly step reduction in gravity arising in routine drop tower tests leads to numerous interesting large-length-scale capillary flow phenomena. For example, a liquid puddle at equilibrium on a hydrophobic substrate is observed to spontaneously jump from the substrate during such tests. Implementing a modified version of the open-source Gerris code, we numerically investigate such a puddle jump phenomenon for a variety of water puddles on flat substrates. We quantify a range of puddle jump characteristics including jump time, jump velocity, and free puddle oscillation modes for an unearthly range of drop volumes between 0.001 ml and 15 ml and substrate contact angles between 60° and 175°. A numerical regime map is constructed identifying no jump, standard jump, bubble ingestion, geyser formation, drop fission, and satellite puddle jump regimes. Favorable agreement is found between the simulations, experiments, simple theoretical models, and scaling laws.

The nearly step reduction in gravity arising in routine drop tower tests leads to numerous interesting large-length-scale capillary flow phenomena. For example, a liquid puddle at equilibrium on a hydrophobic substrate is observed to spontaneously jump from the substrate during such tests. Implementing a modified version of the open-source Gerris code, we numerically investigate such a puddle jump phenomenon for a variety of water puddles on flat substrates. We quantify a range of puddle jump characteristics including jump time, jump velocity, and free puddle oscillation modes for an unearthly range of drop volumes between 0.001 ml and 15 ml and substrate contact angles between 60° and 175°. A numerical regime map is constructed identifying no jump, standard jump, bubble ingestion, geyser formation, drop fission, and satellite puddle jump regimes. Favorable agreement is found between the simulations, experiments, simple theoretical models, and scaling laws.

Categories: Latest papers in fluid mechanics

### Immersed boundary–simplified thermal lattice Boltzmann method for incompressible thermal flows

Physics of Fluids, Volume 32, Issue 1, January 2020.

An immersed boundary–simplified lattice Boltzmann method (IB-STLBM) is proposed in this paper for the simulation of incompressible thermal flows with immersed objects. The fractional step technique is adopted to resolve the problem in two successive steps. In the predictor step, the simplified thermal lattice Boltzmann method (STLBM) is utilized to resolve the intermediate flow variables without considering the immersed objects. The STLBM is advantageous over the conventional thermal lattice Boltzmann method (TLBM) in memory cost, boundary treatment, and numerical stability. In the corrector step, the boundary condition-enforced immersed boundary method (IBM) is used to give correction values of velocity and temperature for accurate interpretation of the Dirichlet boundary conditions on the surface of the immersed objects. Based on the present IBM, some novel strategies can be applied in the evaluation of hydrodynamic forces or thermal parameters of the immersed objects. Five numerical examples are presented for comprehensive validation of the accuracy and robustness of IB-STLBM in various two- and three-dimensional thermal flow problems.

An immersed boundary–simplified lattice Boltzmann method (IB-STLBM) is proposed in this paper for the simulation of incompressible thermal flows with immersed objects. The fractional step technique is adopted to resolve the problem in two successive steps. In the predictor step, the simplified thermal lattice Boltzmann method (STLBM) is utilized to resolve the intermediate flow variables without considering the immersed objects. The STLBM is advantageous over the conventional thermal lattice Boltzmann method (TLBM) in memory cost, boundary treatment, and numerical stability. In the corrector step, the boundary condition-enforced immersed boundary method (IBM) is used to give correction values of velocity and temperature for accurate interpretation of the Dirichlet boundary conditions on the surface of the immersed objects. Based on the present IBM, some novel strategies can be applied in the evaluation of hydrodynamic forces or thermal parameters of the immersed objects. Five numerical examples are presented for comprehensive validation of the accuracy and robustness of IB-STLBM in various two- and three-dimensional thermal flow problems.

Categories: Latest papers in fluid mechanics

### Self-similar dynamics of radiative blast waves

Physics of Fluids, Volume 32, Issue 1, January 2020.

The self-similar study of cooling blast waves (BWs) is performed for the case of a homogeneous self-similar cooling of the gas. This analysis is crucial to better understand its internal structure and global evolution when the BW loses a significant amount of energy due to cooling processes. The evolution of the shock front radius Rsh follows the law Rsh(t) ∝ tα where the decelerating parameter α covers the range 1/4 ≤ α ≤ 2/5 depending on the magnitude of the cooling rate. When the cooling is negligible, α = 2/5, and we recover the analytical solution of Sedov-Taylor (ST) where the total BW energy is conserved. For the internal structure of the cooling BW, we demonstrate that there exist two types of solutions. The first type is the ST-type solution, which is smooth until the center of the BW and only exists for 1/4 < α′ ≤ α ≤ 2/5, where α′ is a specific value of α. This special solution is determined through an eigenvalue problem. The second type is a shell-type solution where a thin cooled shell is bounded by a contact discontinuity separating the shell from a hot rarefied interior bubble where the pressure is homogeneous. The shell becomes thinner and denser when the cooling rate increases. For a strong enough cooling rate, the density inside the shell can diverge at the contact discontinuity while the temperature goes to zero.

The self-similar study of cooling blast waves (BWs) is performed for the case of a homogeneous self-similar cooling of the gas. This analysis is crucial to better understand its internal structure and global evolution when the BW loses a significant amount of energy due to cooling processes. The evolution of the shock front radius Rsh follows the law Rsh(t) ∝ tα where the decelerating parameter α covers the range 1/4 ≤ α ≤ 2/5 depending on the magnitude of the cooling rate. When the cooling is negligible, α = 2/5, and we recover the analytical solution of Sedov-Taylor (ST) where the total BW energy is conserved. For the internal structure of the cooling BW, we demonstrate that there exist two types of solutions. The first type is the ST-type solution, which is smooth until the center of the BW and only exists for 1/4 < α′ ≤ α ≤ 2/5, where α′ is a specific value of α. This special solution is determined through an eigenvalue problem. The second type is a shell-type solution where a thin cooled shell is bounded by a contact discontinuity separating the shell from a hot rarefied interior bubble where the pressure is homogeneous. The shell becomes thinner and denser when the cooling rate increases. For a strong enough cooling rate, the density inside the shell can diverge at the contact discontinuity while the temperature goes to zero.

Categories: Latest papers in fluid mechanics

### Experimental investigation of vortex shedding past a circular cylinder in the high subcritical regime

Physics of Fluids, Volume 32, Issue 1, January 2020.

Vortex shedding in the near wake of a circular cylinder is investigated using surface pressure measurements and two component Particle Image Velocimetry (2C PIV) for 1.49 × 105 ≤ Re ≤ 5 × 105. Space-time distribution of surface pressure shows that regular vortex shedding is interspersed with bursts of weakened activity. Its occurrence increases with an increase in Re. As a result, the rms of the lift coefficient decreases significantly in the subcritical regime with an increase in Re. Proper Orthogonal Decomposition (POD) of the surface pressure data and the 2C PIV data at the midspan of the cylinder shows that most of the energy is contained within the antisymmetric (AS) and symmetric (S) modes. The AS mode is responsible for the regular von Karman vortex shedding, while the S mode is related to intermittent expansion and contraction of the vortex formation region. The energy of the AS mode decreases at a faster rate as compared to that of the S mode with an increase in Re. The S mode is the most dominant mode beyond Re ∼ 3.2 × 105. In the critical regime, the POD modes are modified due to the presence of the intermittent Laminar Separation Bubble (LSB). 2C PIV at the midspan of the cylinder reveals that the weakening of the AS mode is accompanied by an increase in the formation length, Lf. (Lf/d) increases from 1.4 in the low subcritical to 2.0 in the high subcritical regime, where d is the diameter of the cylinder. The weakening of the AS mode and increase in Lf/d collectively lead to a significant decrease in fluctuating lift with an increase in Re. 2C PIV of a spanwise section shows that weakening of vortex shedding is nearly uniform along the span of the cylinder.

Vortex shedding in the near wake of a circular cylinder is investigated using surface pressure measurements and two component Particle Image Velocimetry (2C PIV) for 1.49 × 105 ≤ Re ≤ 5 × 105. Space-time distribution of surface pressure shows that regular vortex shedding is interspersed with bursts of weakened activity. Its occurrence increases with an increase in Re. As a result, the rms of the lift coefficient decreases significantly in the subcritical regime with an increase in Re. Proper Orthogonal Decomposition (POD) of the surface pressure data and the 2C PIV data at the midspan of the cylinder shows that most of the energy is contained within the antisymmetric (AS) and symmetric (S) modes. The AS mode is responsible for the regular von Karman vortex shedding, while the S mode is related to intermittent expansion and contraction of the vortex formation region. The energy of the AS mode decreases at a faster rate as compared to that of the S mode with an increase in Re. The S mode is the most dominant mode beyond Re ∼ 3.2 × 105. In the critical regime, the POD modes are modified due to the presence of the intermittent Laminar Separation Bubble (LSB). 2C PIV at the midspan of the cylinder reveals that the weakening of the AS mode is accompanied by an increase in the formation length, Lf. (Lf/d) increases from 1.4 in the low subcritical to 2.0 in the high subcritical regime, where d is the diameter of the cylinder. The weakening of the AS mode and increase in Lf/d collectively lead to a significant decrease in fluctuating lift with an increase in Re. 2C PIV of a spanwise section shows that weakening of vortex shedding is nearly uniform along the span of the cylinder.

Categories: Latest papers in fluid mechanics

### Transition to chaos in electro-thermo-convection of a dielectric liquid in a square cavity

Physics of Fluids, Volume 32, Issue 1, January 2020.

The transition process from laminar to chaotic flow in electro-thermal convection of a dielectric liquid is numerically investigated using a unified lattice Boltzmann method. The liquid is confined in a closed square cavity, and free charges are introduced into the system through a strong unipolar injection mechanism. Three cases with different Rayleigh numbers are considered. With the increase of electric Rayleigh number, various complicated dynamical behaviors are observed and three diverse transition routes to chaos are identified, namely, the quasi-periodic sequence involving four incommensurable frequencies, the intermittency sequence, and the alternating periodic-chaotic sequence. Numerical results are illustrated using time histories, Fourier frequency spectra, and phase portraits. The chaotic behavior is quantitatively analyzed through the calculation of fractal dimension and Lyapunov exponent. Typical flow patterns for both steady-state regime and periodic regime are also presented and discussed.

The transition process from laminar to chaotic flow in electro-thermal convection of a dielectric liquid is numerically investigated using a unified lattice Boltzmann method. The liquid is confined in a closed square cavity, and free charges are introduced into the system through a strong unipolar injection mechanism. Three cases with different Rayleigh numbers are considered. With the increase of electric Rayleigh number, various complicated dynamical behaviors are observed and three diverse transition routes to chaos are identified, namely, the quasi-periodic sequence involving four incommensurable frequencies, the intermittency sequence, and the alternating periodic-chaotic sequence. Numerical results are illustrated using time histories, Fourier frequency spectra, and phase portraits. The chaotic behavior is quantitatively analyzed through the calculation of fractal dimension and Lyapunov exponent. Typical flow patterns for both steady-state regime and periodic regime are also presented and discussed.

Categories: Latest papers in fluid mechanics

### Microfluidic shear rheology and wall-slip of viscoelastic fluids using holography-based flow kinematics

Physics of Fluids, Volume 32, Issue 1, January 2020.

In this study, we report microfluidic shear rheology and wall-slip using the 3D-resolved flow kinematics obtained from digital holography microscopy (DHM). We computationally reconstruct the recorded holograms to visualize the tracer imbued flow volume in linear microchannels, followed by the implementation of particle tracking velocimetry (PTV) to quantitate spatially resolved velocity fields in 3D. In order to select optimal parameters for DHM-PTV characterization of viscoelastic fluids, we studied the effect of the hologram recording distance, seeding density, and particle size. Using the optimal parameters, we show quantitative characterization of the shear rheology from the velocity fields without any a priori assumptions of wall boundary conditions or constitutive equation. The viscosity vs shear rate data for Newtonian and polyethylene oxide (PEO) solutions could be measured in the range of ≈0.05 to 20 000 s−1 with just three input pressures using sample volumes as low as 20 µl. These data from holographic shear rheometry were found to be in good agreement with computational fluid dynamics simulations and macrorheometry. With respect to the wall-slip, we find that highly viscoelastic PEO solutions can show slip lengths in the order of few microns. Finally, we discuss holographic visualization of particle migration in microfluidic flows, which can limit flow field access, whereas at the same time provide a fingerprint of the suspending fluid rheology.

In this study, we report microfluidic shear rheology and wall-slip using the 3D-resolved flow kinematics obtained from digital holography microscopy (DHM). We computationally reconstruct the recorded holograms to visualize the tracer imbued flow volume in linear microchannels, followed by the implementation of particle tracking velocimetry (PTV) to quantitate spatially resolved velocity fields in 3D. In order to select optimal parameters for DHM-PTV characterization of viscoelastic fluids, we studied the effect of the hologram recording distance, seeding density, and particle size. Using the optimal parameters, we show quantitative characterization of the shear rheology from the velocity fields without any a priori assumptions of wall boundary conditions or constitutive equation. The viscosity vs shear rate data for Newtonian and polyethylene oxide (PEO) solutions could be measured in the range of ≈0.05 to 20 000 s−1 with just three input pressures using sample volumes as low as 20 µl. These data from holographic shear rheometry were found to be in good agreement with computational fluid dynamics simulations and macrorheometry. With respect to the wall-slip, we find that highly viscoelastic PEO solutions can show slip lengths in the order of few microns. Finally, we discuss holographic visualization of particle migration in microfluidic flows, which can limit flow field access, whereas at the same time provide a fingerprint of the suspending fluid rheology.

Categories: Latest papers in fluid mechanics

### Central uprising sheet in simultaneous and near-simultaneous impact of two high kinetic energy droplets onto dry surface and thin liquid film

Physics of Fluids, Volume 32, Issue 1, January 2020.

Droplet impact on both dry and wet surfaces is present in several applications, and often multiple droplets, instead of one single droplet, are involved. This paper focuses on the problem of two-droplet impingement on dry and wet surfaces with two Weber numbers (We) of 115 and 230, corresponding to two Reynolds numbers (Re) of 6100 and 8620, respectively. We study impact dynamics phenomena, compare simultaneous and time-delayed impact dynamics of two droplets, and investigate the time evolution of a central uprising sheet formed between the two droplets impinged on dry or wet surfaces, a problem that has been barely studied. A central uprising sheet forms between two impinging droplets at sufficiently high Re and We and short droplet to droplet spacing (high kinetic energy at the point of spread contact). Three different shapes for the central uprising sheet are observed for two droplet impact on a dry surface with various time delays: ordered two-dimensional (2D) semilunar shape (vertical and inclined), curved or C-shaped three-dimensional (3D) shape, and irregular splash. Our experiments show that the central uprising sheet undergoes splashing under conditions not predicted by existing correlations; also, during the early formation of the central uprising sheet, the effect of gravity force on the sheet evolution is negligible. Mixing and surface waves are also studied, taking advantage of liquids with three different colors.

Droplet impact on both dry and wet surfaces is present in several applications, and often multiple droplets, instead of one single droplet, are involved. This paper focuses on the problem of two-droplet impingement on dry and wet surfaces with two Weber numbers (We) of 115 and 230, corresponding to two Reynolds numbers (Re) of 6100 and 8620, respectively. We study impact dynamics phenomena, compare simultaneous and time-delayed impact dynamics of two droplets, and investigate the time evolution of a central uprising sheet formed between the two droplets impinged on dry or wet surfaces, a problem that has been barely studied. A central uprising sheet forms between two impinging droplets at sufficiently high Re and We and short droplet to droplet spacing (high kinetic energy at the point of spread contact). Three different shapes for the central uprising sheet are observed for two droplet impact on a dry surface with various time delays: ordered two-dimensional (2D) semilunar shape (vertical and inclined), curved or C-shaped three-dimensional (3D) shape, and irregular splash. Our experiments show that the central uprising sheet undergoes splashing under conditions not predicted by existing correlations; also, during the early formation of the central uprising sheet, the effect of gravity force on the sheet evolution is negligible. Mixing and surface waves are also studied, taking advantage of liquids with three different colors.

Categories: Latest papers in fluid mechanics

### Pattern method for higher harmonics from macromolecular orientation in oscillatory shear flow

Physics of Fluids, Volume 32, Issue 1, January 2020.

For a suspension of rigid dumbbells, in any simple shear flow, we must first solve the diffusion equation for the orientation distribution function by a power series expansion in the shear rate. Our recent work has uncovered the pattern in the coefficients of this power series [L. M. Jbara and A. J. Giacomin, “Orientation distribution function pattern for rigid dumbbell suspensions in any simple shear flow,” Macromol. Theory Simul. 28, 1800046-1–1800046-16 (2019)]. Specifically, we have here used this pattern on large-amplitude oscillatory shear (LAOS) flow, for which we have extended the orientation distribution function to the 6th power of the shear rate. In this letter, we embed this extension into the Giesekus expression for the extra stress tensor to arrive at the alternant shear stress response, up to and including the seventh harmonic. We thus demonstrate that the pattern method for macromolecular orientation now allows our harmonic analysis to penetrate the shear stress response to oscillatory shear flow far more deeply than ever.

For a suspension of rigid dumbbells, in any simple shear flow, we must first solve the diffusion equation for the orientation distribution function by a power series expansion in the shear rate. Our recent work has uncovered the pattern in the coefficients of this power series [L. M. Jbara and A. J. Giacomin, “Orientation distribution function pattern for rigid dumbbell suspensions in any simple shear flow,” Macromol. Theory Simul. 28, 1800046-1–1800046-16 (2019)]. Specifically, we have here used this pattern on large-amplitude oscillatory shear (LAOS) flow, for which we have extended the orientation distribution function to the 6th power of the shear rate. In this letter, we embed this extension into the Giesekus expression for the extra stress tensor to arrive at the alternant shear stress response, up to and including the seventh harmonic. We thus demonstrate that the pattern method for macromolecular orientation now allows our harmonic analysis to penetrate the shear stress response to oscillatory shear flow far more deeply than ever.

Categories: Latest papers in fluid mechanics

### Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data

Physics of Fluids, Volume 32, Issue 1, January 2020.

We put forth a modular approach for distilling hidden flow physics from discrete and sparse observations. To address functional expressiblity, a key limitation of the black-box machine learning methods, we have exploited the use of symbolic regression as a principle for identifying relations and operators that are related to the underlying processes. This approach combines evolutionary computation with feature engineering to provide a tool for discovering hidden parameterizations embedded in the trajectory of fluid flows in the Eulerian frame of reference. Our approach in this study mainly involves gene expression programming (GEP) and sequential threshold ridge regression (STRidge) algorithms. We demonstrate our results in three different applications: (i) equation discovery, (ii) truncation error analysis, and (iii) hidden physics discovery, for which we include both predicting unknown source terms from a set of sparse observations and discovering subgrid scale closure models. We illustrate that both GEP and STRidge algorithms are able to distill the Smagorinsky model from an array of tailored features in solving the Kraichnan turbulence problem. Our results demonstrate the huge potential of these techniques in complex physics problems, and reveal the importance of feature selection and feature engineering in model discovery approaches.

We put forth a modular approach for distilling hidden flow physics from discrete and sparse observations. To address functional expressiblity, a key limitation of the black-box machine learning methods, we have exploited the use of symbolic regression as a principle for identifying relations and operators that are related to the underlying processes. This approach combines evolutionary computation with feature engineering to provide a tool for discovering hidden parameterizations embedded in the trajectory of fluid flows in the Eulerian frame of reference. Our approach in this study mainly involves gene expression programming (GEP) and sequential threshold ridge regression (STRidge) algorithms. We demonstrate our results in three different applications: (i) equation discovery, (ii) truncation error analysis, and (iii) hidden physics discovery, for which we include both predicting unknown source terms from a set of sparse observations and discovering subgrid scale closure models. We illustrate that both GEP and STRidge algorithms are able to distill the Smagorinsky model from an array of tailored features in solving the Kraichnan turbulence problem. Our results demonstrate the huge potential of these techniques in complex physics problems, and reveal the importance of feature selection and feature engineering in model discovery approaches.

Categories: Latest papers in fluid mechanics

### Droplet mobilization at the walls of a microfluidic channel

Physics of Fluids, Volume 32, Issue 1, January 2020.

The mechanism of dynamic wetting and the fluid dynamics during the onset of droplet mobilization driven by a microchannel flow are not clearly understood. In this work, we use microparticle tracking velocimetry to visualize the velocity distribution inside the droplet both prior to and during mobilization. Time-averaged and instantaneous velocity vectors are determined using fluorescent microscopy for various capillary numbers. A circulating flow exists inside the droplet at a subcritical capillary number, in which case the droplet is pinned to the channel walls. When the capillary number exceeds a critical value, droplet mobilization occurs, and this process can be divided into two stages. In the first stage, the location of the internal circulation vortex center moves to the rear of the droplet and the droplet deforms, but the contact lines at the top walls remain fixed. In the second stage, the droplet rolls along the solid wall, with fixed contact angles keeping the vortex center in the rear part of the droplet. The critical capillary number for the droplet mobilization is larger for the droplet fluid with a larger viscosity. A force-balance model of the droplet, considering the effect of fluid properties, is formulated to explain the experimental trends of advancing and receding contact angles with the capillary number. Numerical simulations on internal circulations for the pinned droplet indicate that the reversed flow rate, when normalized by the inlet flow rate and the kinematic viscosity ratio of the wetting and nonwetting phases, is independent of the capillary number and the droplet composition.

The mechanism of dynamic wetting and the fluid dynamics during the onset of droplet mobilization driven by a microchannel flow are not clearly understood. In this work, we use microparticle tracking velocimetry to visualize the velocity distribution inside the droplet both prior to and during mobilization. Time-averaged and instantaneous velocity vectors are determined using fluorescent microscopy for various capillary numbers. A circulating flow exists inside the droplet at a subcritical capillary number, in which case the droplet is pinned to the channel walls. When the capillary number exceeds a critical value, droplet mobilization occurs, and this process can be divided into two stages. In the first stage, the location of the internal circulation vortex center moves to the rear of the droplet and the droplet deforms, but the contact lines at the top walls remain fixed. In the second stage, the droplet rolls along the solid wall, with fixed contact angles keeping the vortex center in the rear part of the droplet. The critical capillary number for the droplet mobilization is larger for the droplet fluid with a larger viscosity. A force-balance model of the droplet, considering the effect of fluid properties, is formulated to explain the experimental trends of advancing and receding contact angles with the capillary number. Numerical simulations on internal circulations for the pinned droplet indicate that the reversed flow rate, when normalized by the inlet flow rate and the kinematic viscosity ratio of the wetting and nonwetting phases, is independent of the capillary number and the droplet composition.

Categories: Latest papers in fluid mechanics

### Spreading and penetration of a micro-sized water droplet impacting onto oil layers

Physics of Fluids, Volume 32, Issue 1, January 2020.

This paper describes a theoretical and numerical investigation of the impact dynamics and outcomes of a microsized water droplet falling onto an oil layer. The shape of the water droplet floating on the oil layer is predicted theoretically to understand the balancing of the three interfacial tensions. Direct numerical simulations coupled with a three-phase volume-of-fluid method are performed on an axisymmetric model, considering the balancing and motion of the triple-line. The effects of the impact velocity, viscosity ratio of oil and water, height of the oil layer, and the combination of the three interfacial tensions on the impact dynamics and outcomes are systematically studied. Regime diagrams of the nonpenetration and penetration outcomes are obtained under different combinations of the flow and physical parameters. It is found that the balance among the three interfacial tensions is well maintained at the triple-line due to the low capillary number. The maximum horizontal spreading of the water droplet is proportional to the square root of the Weber number when the impact velocity is low. Moreover, the maximum penetration for high impact velocities is independent of the spreading parameter. To understand the lower transition between nonpenetration and penetration, the critical penetration distance at which the triple-line is about to collapse is obtained from simulation results as a function of the spreading parameter, and these indicate weak dependence on the viscosity ratio. A semiempirical model is used to predict the boundary of lower transitions, and these are in good agreement with the simulations results.

This paper describes a theoretical and numerical investigation of the impact dynamics and outcomes of a microsized water droplet falling onto an oil layer. The shape of the water droplet floating on the oil layer is predicted theoretically to understand the balancing of the three interfacial tensions. Direct numerical simulations coupled with a three-phase volume-of-fluid method are performed on an axisymmetric model, considering the balancing and motion of the triple-line. The effects of the impact velocity, viscosity ratio of oil and water, height of the oil layer, and the combination of the three interfacial tensions on the impact dynamics and outcomes are systematically studied. Regime diagrams of the nonpenetration and penetration outcomes are obtained under different combinations of the flow and physical parameters. It is found that the balance among the three interfacial tensions is well maintained at the triple-line due to the low capillary number. The maximum horizontal spreading of the water droplet is proportional to the square root of the Weber number when the impact velocity is low. Moreover, the maximum penetration for high impact velocities is independent of the spreading parameter. To understand the lower transition between nonpenetration and penetration, the critical penetration distance at which the triple-line is about to collapse is obtained from simulation results as a function of the spreading parameter, and these indicate weak dependence on the viscosity ratio. A semiempirical model is used to predict the boundary of lower transitions, and these are in good agreement with the simulations results.

Categories: Latest papers in fluid mechanics

### Experimental investigation of turbulent flow in a rotating straight channel with continuous ribs

Physics of Fluids, Volume 32, Issue 1, January 2020.

We experimentally study the combined effects of continuous ribs and rotations constructed in a square duct on the turbulent flows and flow separation. The ribs obstruct the channel by 10% of its height and are arranged in three different pitch-to-height ratios (P/e) of 10, 12, and 15. The Reynolds number (Re = ρU0D/μ) is fixed at 10 000, and the rotation number (Ro = ΩD/U0) varies from 0 to 0.52. A time-resolved particle image velocimetry system is applied to provide insights into the main flow and turbulence mechanism. Results show that rotation significantly changes main flow and turbulent characteristics. In particular, a main flow phenomenon has been found: on account of the secondary flow near the ribs, velocity profile deflects to the leading side under a low rotation number, and when Ro rises to 0.48 (critical value), velocity profile deflects to the trailing side. It gives an insight into main flow in a ribbed channel. Reattachment law has been investigated, which can optimize heat transfer by optimize rib arrangement. A proper orthogonal decomposition analysis is also considered to identify the spatial characteristics of the superimposed flow fields. Based on the experimental data, the existence of ribs with different P/e ratios and Coriolis forces play significant roles in rib-generated vortices as well as their turbulent activities.

We experimentally study the combined effects of continuous ribs and rotations constructed in a square duct on the turbulent flows and flow separation. The ribs obstruct the channel by 10% of its height and are arranged in three different pitch-to-height ratios (P/e) of 10, 12, and 15. The Reynolds number (Re = ρU0D/μ) is fixed at 10 000, and the rotation number (Ro = ΩD/U0) varies from 0 to 0.52. A time-resolved particle image velocimetry system is applied to provide insights into the main flow and turbulence mechanism. Results show that rotation significantly changes main flow and turbulent characteristics. In particular, a main flow phenomenon has been found: on account of the secondary flow near the ribs, velocity profile deflects to the leading side under a low rotation number, and when Ro rises to 0.48 (critical value), velocity profile deflects to the trailing side. It gives an insight into main flow in a ribbed channel. Reattachment law has been investigated, which can optimize heat transfer by optimize rib arrangement. A proper orthogonal decomposition analysis is also considered to identify the spatial characteristics of the superimposed flow fields. Based on the experimental data, the existence of ribs with different P/e ratios and Coriolis forces play significant roles in rib-generated vortices as well as their turbulent activities.

Categories: Latest papers in fluid mechanics

### Small droplet bouncing on a deep pool

Physics of Fluids, Volume 32, Issue 1, January 2020.

Droplet bouncing on liquid surfaces frequently occurs for low-Weber-number impacts. Previous studies typically used large droplets with oscillation initiated by their creation process but without determining the effects of these oscillations. Here, we use small droplets, providing the means to reduce oscillations to show that the probability of the droplet bounce does not depend on the droplet oscillations. The time from the moment of contact to the maximum penetration depth was found to be independent of the Weber number for droplets of fixed diameter but increased with an increase in diameter. Both the maximum penetration depth and the maximum rebound height increased monotonically with the Weber number. A simple model predicting the maximum penetration depth was proposed and validated through comparison with experimental data.

Droplet bouncing on liquid surfaces frequently occurs for low-Weber-number impacts. Previous studies typically used large droplets with oscillation initiated by their creation process but without determining the effects of these oscillations. Here, we use small droplets, providing the means to reduce oscillations to show that the probability of the droplet bounce does not depend on the droplet oscillations. The time from the moment of contact to the maximum penetration depth was found to be independent of the Weber number for droplets of fixed diameter but increased with an increase in diameter. Both the maximum penetration depth and the maximum rebound height increased monotonically with the Weber number. A simple model predicting the maximum penetration depth was proposed and validated through comparison with experimental data.

Categories: Latest papers in fluid mechanics

### Accelerating viscous flow past a wedge

Physics of Fluids, Volume 32, Issue 1, January 2020.

This paper presents direct numerical simulations of accelerated viscous flow past an infinite wedge with a focus on the effect of the accelerating rate on vortex dynamics and material transport near the wedge tip. The wedge angle ranges π/3 ≤ βπ ≤ 5π/6 and the acceleration rate is 0 ≤ p ≤ 1. Since the wedge is infinite, the inviscid self-similar analysis predicts that the solution of one viscosity at a time is the same as the solution of a different viscosity at a scaled time. This theoretical prediction is numerically verified in the current work. Evolution of the vorticity, streamlines, and streaklines is shown in great detail. In the vorticity field, hierarchical vortex separations at the wedge tip are observed, which associate with multiple circulating regions in the streamlines. We also compare the solution at varying acceleration rates and wedge angles and explore the scaling laws in the solution. Streaklines are used to illustrate material transport in the fluid flow. The streaklines form two spirals, a major one that corresponds to the starting vortex and a second one that appears near the wedge tip. The formation of both the major and secondary spirals are investigated and diagnosed using the strain rate in the fluid flow.

This paper presents direct numerical simulations of accelerated viscous flow past an infinite wedge with a focus on the effect of the accelerating rate on vortex dynamics and material transport near the wedge tip. The wedge angle ranges π/3 ≤ βπ ≤ 5π/6 and the acceleration rate is 0 ≤ p ≤ 1. Since the wedge is infinite, the inviscid self-similar analysis predicts that the solution of one viscosity at a time is the same as the solution of a different viscosity at a scaled time. This theoretical prediction is numerically verified in the current work. Evolution of the vorticity, streamlines, and streaklines is shown in great detail. In the vorticity field, hierarchical vortex separations at the wedge tip are observed, which associate with multiple circulating regions in the streamlines. We also compare the solution at varying acceleration rates and wedge angles and explore the scaling laws in the solution. Streaklines are used to illustrate material transport in the fluid flow. The streaklines form two spirals, a major one that corresponds to the starting vortex and a second one that appears near the wedge tip. The formation of both the major and secondary spirals are investigated and diagnosed using the strain rate in the fluid flow.

Categories: Latest papers in fluid mechanics

### A theoretical model of thermo-dynamic interpretation for the Pettit effect and associated heat transport processes

Physics of Fluids, Volume 32, Issue 1, January 2020.

Based on several fundamental preassumptions, a one-dimensional convection-diffusion equation for heat transport inside liquid wedges causing the Pettit effect is proposed. With a hot or cold thermode placed at the middle of the liquid wedge, the average temperature of the liquid wedge determined from the convection-diffusion equation proposed shows a maximum, which corresponds to a particular liquid flow rate. The state achieved at this maximum temperature is believed to be the most stable for its minimum interfacial energy. The theory suggests a thermodynamic mechanism, which drives the liquid to flow in directions corresponding to those observed in experiments. It is believed that this work improves the thermodynamic interpretation proposed previously since the new-form convection-diffusion equation is more rigorously deduced and is thus more accurate. In addition, the work also presents a detailed theoretical analysis for heat transport. The results show that, in practical situations, the manifested heat-transport behaviors of a liquid wedge are governed by conductive heat transfer because convective heat flow is self-balancing due to the restriction by the law of mass conservation. Meanwhile, based on the asymmetric features of the conductive heat flows transiting within two different halves of the liquid wedge, a closed-loop formed by connecting a hot-thermode-driven liquid wedge and a cold-thermode-driven liquid wedge is proposed such that a hot thermode-cold thermode loop can lead to controllable heat transfer with which targeted heating or cooling may be realized. The effect may reveal the technical principles upon which novel small-size thermal engines, pumps, heaters, and coolers can be built.

Based on several fundamental preassumptions, a one-dimensional convection-diffusion equation for heat transport inside liquid wedges causing the Pettit effect is proposed. With a hot or cold thermode placed at the middle of the liquid wedge, the average temperature of the liquid wedge determined from the convection-diffusion equation proposed shows a maximum, which corresponds to a particular liquid flow rate. The state achieved at this maximum temperature is believed to be the most stable for its minimum interfacial energy. The theory suggests a thermodynamic mechanism, which drives the liquid to flow in directions corresponding to those observed in experiments. It is believed that this work improves the thermodynamic interpretation proposed previously since the new-form convection-diffusion equation is more rigorously deduced and is thus more accurate. In addition, the work also presents a detailed theoretical analysis for heat transport. The results show that, in practical situations, the manifested heat-transport behaviors of a liquid wedge are governed by conductive heat transfer because convective heat flow is self-balancing due to the restriction by the law of mass conservation. Meanwhile, based on the asymmetric features of the conductive heat flows transiting within two different halves of the liquid wedge, a closed-loop formed by connecting a hot-thermode-driven liquid wedge and a cold-thermode-driven liquid wedge is proposed such that a hot thermode-cold thermode loop can lead to controllable heat transfer with which targeted heating or cooling may be realized. The effect may reveal the technical principles upon which novel small-size thermal engines, pumps, heaters, and coolers can be built.

Categories: Latest papers in fluid mechanics

### Numerical investigation of flow-induced vibrations of two cylinders in tandem arrangement with full wake interference

Physics of Fluids, Volume 32, Issue 1, January 2020.

This paper presents a numerical investigation on flow-induced vibration (FIV) of two elastically mounted cylinders in a tandem arrangement at subcritical Reynolds numbers. The tandem spacing between the cylinder centers is set at four cylinder diameters, placing the FIV problem within the full wake interference regime. A fluid-structure interaction numerical methodology based on a two-dimensional discrete vortex method is developed and applied to the FIV simulation of two cylinders. To investigate the effect of the upstream wake frequency and Reynolds number on the FIV response of the downstream cylinder separately, two experimental cases are designed. In one case, the FIV response is investigated with the Reynolds number varying and the upstream wake frequency fixed. In another case, the Reynolds number is fixed and the upstream wake frequency varies. In both cases, the FIV response of the upstream cylinder roughly resembles the lower branch of the typical vortex-induced vibration response of the single cylinder with the upstream reduced velocity varying from 6 to 9. For the downstream cylinder, the FIV response is characterized by two frequency branches: a dominant frequency branch associated with the wake interference mechanism and a secondary frequency branch associated with the upstream vortex shedding mechanism. It is found that the variation of the vortex shedding frequency in the upstream wake has little effect on the amplitude and dominant frequency of the downstream FIV response but directly causes the variation of the secondary frequency as the secondary frequency strictly follows the upstream vortex shedding frequency. The FIV response amplitude and dominant frequency of the downstream cylinder shows a strong dependence on the Reynolds number. The wake pattern of FIV shows that the FIV vortex shedding of each cylinder is directly related to the motion of itself but slightly modified by the wake of other cylinders for the full wake interference regime. The upstream wake vortices interfering with the downstream motion induce a flow in favor of the downstream FIV response.

This paper presents a numerical investigation on flow-induced vibration (FIV) of two elastically mounted cylinders in a tandem arrangement at subcritical Reynolds numbers. The tandem spacing between the cylinder centers is set at four cylinder diameters, placing the FIV problem within the full wake interference regime. A fluid-structure interaction numerical methodology based on a two-dimensional discrete vortex method is developed and applied to the FIV simulation of two cylinders. To investigate the effect of the upstream wake frequency and Reynolds number on the FIV response of the downstream cylinder separately, two experimental cases are designed. In one case, the FIV response is investigated with the Reynolds number varying and the upstream wake frequency fixed. In another case, the Reynolds number is fixed and the upstream wake frequency varies. In both cases, the FIV response of the upstream cylinder roughly resembles the lower branch of the typical vortex-induced vibration response of the single cylinder with the upstream reduced velocity varying from 6 to 9. For the downstream cylinder, the FIV response is characterized by two frequency branches: a dominant frequency branch associated with the wake interference mechanism and a secondary frequency branch associated with the upstream vortex shedding mechanism. It is found that the variation of the vortex shedding frequency in the upstream wake has little effect on the amplitude and dominant frequency of the downstream FIV response but directly causes the variation of the secondary frequency as the secondary frequency strictly follows the upstream vortex shedding frequency. The FIV response amplitude and dominant frequency of the downstream cylinder shows a strong dependence on the Reynolds number. The wake pattern of FIV shows that the FIV vortex shedding of each cylinder is directly related to the motion of itself but slightly modified by the wake of other cylinders for the full wake interference regime. The upstream wake vortices interfering with the downstream motion induce a flow in favor of the downstream FIV response.

Categories: Latest papers in fluid mechanics