# Physics of Fluids

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

*Updated:*1 day 6 hours ago

### Mesh-free peridynamic coupled simulation of impacting collapse of a granular column with various heights

Physics of Fluids, Volume 33, Issue 10, October 2021.

In this study, a coupled model of Peridynamics into the mesh-free method is extended to simulate the impacting collapse of a granular column with various suspended heights. Experiments on the impacting collapse were conducted to validate the numerical model. It is found that the simulated free surface profiles have a good agreement with the experimental measurements. The numerical model is also validated by simulating the granular column collapse. The simulated kinetic energy and potential energy agree well with the discrete element method (DEM) simulation results in the literature. In the validations, different numerical particle distances are implemented to discrete the fluid and a good numerical convergence is achieved for the numerical method. After model validation, the energy variations in the impacting collapse of the granular column with different suspended heights are analyzed, which include evolutions of the potential energy, horizontal kinetic energy, and vertical kinetic energy. In the analysis, the relative total energy dissipation rate for the flow with various suspended heights is calculated. The analysis shows that the suspended height in the impacting collapse of the granular column can affect the energy dissipation significantly.

In this study, a coupled model of Peridynamics into the mesh-free method is extended to simulate the impacting collapse of a granular column with various suspended heights. Experiments on the impacting collapse were conducted to validate the numerical model. It is found that the simulated free surface profiles have a good agreement with the experimental measurements. The numerical model is also validated by simulating the granular column collapse. The simulated kinetic energy and potential energy agree well with the discrete element method (DEM) simulation results in the literature. In the validations, different numerical particle distances are implemented to discrete the fluid and a good numerical convergence is achieved for the numerical method. After model validation, the energy variations in the impacting collapse of the granular column with different suspended heights are analyzed, which include evolutions of the potential energy, horizontal kinetic energy, and vertical kinetic energy. In the analysis, the relative total energy dissipation rate for the flow with various suspended heights is calculated. The analysis shows that the suspended height in the impacting collapse of the granular column can affect the energy dissipation significantly.

Categories: Latest papers in fluid mechanics

### Rotational and reflectional equivariant convolutional neural network for data-limited applications: Multiphase flow demonstration

Physics of Fluids, Volume 33, Issue 10, October 2021.

This article deals with approximating steady-state particle-resolved fluid flow around a fixed particle of interest under the influence of randomly distributed stationary particles in a dispersed multiphase setup using convolutional neural network (CNN). The considered problem involves rotational symmetry about the mean velocity (streamwise) direction. Thus, this work enforces this symmetry using SE(3)-equivariant, special Euclidean group of dimension 3, CNN architecture, which is translation and three-dimensional rotation equivariant. This study mainly explores the generalization capabilities and benefits of a SE(3)-equivariant network. Accurate synthetic flow fields for Reynolds number and particle volume fraction combinations spanning over a range of [86.22, 172.96] and [0.11, 0.45], respectively, are produced with careful application of symmetry-aware data-driven approach.

This article deals with approximating steady-state particle-resolved fluid flow around a fixed particle of interest under the influence of randomly distributed stationary particles in a dispersed multiphase setup using convolutional neural network (CNN). The considered problem involves rotational symmetry about the mean velocity (streamwise) direction. Thus, this work enforces this symmetry using SE(3)-equivariant, special Euclidean group of dimension 3, CNN architecture, which is translation and three-dimensional rotation equivariant. This study mainly explores the generalization capabilities and benefits of a SE(3)-equivariant network. Accurate synthetic flow fields for Reynolds number and particle volume fraction combinations spanning over a range of [86.22, 172.96] and [0.11, 0.45], respectively, are produced with careful application of symmetry-aware data-driven approach.

Categories: Latest papers in fluid mechanics

### Vortex criteria can be objectivized by unsteadiness minimization

Physics of Fluids, Volume 33, Issue 10, October 2021.

Reference frame optimization is a generic framework to calculate a spatially varying observer field that views an unsteady fluid flow in a reference frame that is as-steady-as-possible. In this paper, we show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer. To check objectivity, the optimized velocity vectors and the coordinates in which they are defined must both be connected by an Euclidean transformation. In this paper, we show that a recent publication applied this definition incorrectly, falsely concluding that reference frame optimizations are not objective. Furthermore, we prove the objectivity of the variational formulation of the reference frame optimization that was recently proposed and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization.

Reference frame optimization is a generic framework to calculate a spatially varying observer field that views an unsteady fluid flow in a reference frame that is as-steady-as-possible. In this paper, we show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer. To check objectivity, the optimized velocity vectors and the coordinates in which they are defined must both be connected by an Euclidean transformation. In this paper, we show that a recent publication applied this definition incorrectly, falsely concluding that reference frame optimizations are not objective. Furthermore, we prove the objectivity of the variational formulation of the reference frame optimization that was recently proposed and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization.

Categories: Latest papers in fluid mechanics

### Striped patterns in radially driven suspensions with open boundaries

Physics of Fluids, Volume 33, Issue 10, October 2021.

We study the motion of radially driven fluid–immersed particles in a novel Hele–Shaw cell with open boundaries. The initially uniform suspension forms a striped pattern within a specific range of horizontal oscillation frequencies and for sufficiently large amplitudes. We observe that the initial coarsening dynamics of the stripes gradually slows down and the pattern reaches a steady state after a few minutes. The distance between the stripes in the steady state exhibits an exponentially saturating increase with increased oscillation amplitude or frequency. The width of the stripes decreases as a power-law with the frequency, while its amplitude dependence follows a logistic function. We propose a mechanism—based on the interplay between shear stress, hydrodynamic interactions, and frictional forces—to link the structural characteristics of the stripes to the properties of the oscillatory external drive.

We study the motion of radially driven fluid–immersed particles in a novel Hele–Shaw cell with open boundaries. The initially uniform suspension forms a striped pattern within a specific range of horizontal oscillation frequencies and for sufficiently large amplitudes. We observe that the initial coarsening dynamics of the stripes gradually slows down and the pattern reaches a steady state after a few minutes. The distance between the stripes in the steady state exhibits an exponentially saturating increase with increased oscillation amplitude or frequency. The width of the stripes decreases as a power-law with the frequency, while its amplitude dependence follows a logistic function. We propose a mechanism—based on the interplay between shear stress, hydrodynamic interactions, and frictional forces—to link the structural characteristics of the stripes to the properties of the oscillatory external drive.

Categories: Latest papers in fluid mechanics

### Kinetic theory of polydisperse gas–solid flow: Navier–Stokes transport coefficients

Physics of Fluids, Volume 33, Issue 10, October 2021.

The particulate phase stress and solid–solid drag force in the multifluid modeling of polydisperse gas–solid flows are usually closed using kinetic theory. This research aims to establish the hydrodynamic equations and constitutive relations of the multifluid model for polydisperse systems via species kinetic theory, in which the non-equipartition of energy and interphase slip velocity between different species are considered. Whereas previous studies have used approximations, such as Taylor series expansions, to simplify the calculation of collision integrals, the present study, for the first time, solves the collision integrals analytically without any approximations to obtain accurate constitutive relations. Explicit expressions for the constitutive laws are obtained, including the particle stress tensor, solid–solid drag force, heat flux, and energy dissipation rate up to the Navier–Stokes order. The present study offers more complete and mathematically rigorous constitutive laws for the multifluid modeling of polydisperse gas–solid flows.

The particulate phase stress and solid–solid drag force in the multifluid modeling of polydisperse gas–solid flows are usually closed using kinetic theory. This research aims to establish the hydrodynamic equations and constitutive relations of the multifluid model for polydisperse systems via species kinetic theory, in which the non-equipartition of energy and interphase slip velocity between different species are considered. Whereas previous studies have used approximations, such as Taylor series expansions, to simplify the calculation of collision integrals, the present study, for the first time, solves the collision integrals analytically without any approximations to obtain accurate constitutive relations. Explicit expressions for the constitutive laws are obtained, including the particle stress tensor, solid–solid drag force, heat flux, and energy dissipation rate up to the Navier–Stokes order. The present study offers more complete and mathematically rigorous constitutive laws for the multifluid modeling of polydisperse gas–solid flows.

Categories: Latest papers in fluid mechanics

### Prediction of airway deformation effect on pulmonary air-particle dynamics: A numerical study

Physics of Fluids, Volume 33, Issue 10, October 2021.

Most existing whole lung models neglect the airway deformation kinematics and assume the lung airways are static. However, neglecting the airway deformation effect on pulmonary air-particle flow dynamics significantly limits the modeling capability under disease-specific lung conditions. Therefore, a novel elastic truncated whole-lung (TWL) modeling framework has been developed to simulate the disease-specific airway deformation kinematics simultaneously with pulmonary air-particle flow dynamics using one-way coupled Euler–Lagrange method plus the dynamic mesh method. Specifically, the deformation kinematics of the elastic TWL model was calibrated with clinical data and pulmonary function test results for both healthy lung and lungs with chronic obstructive pulmonary diseases (COPDs). The transport dynamics of spherical sub micrometer and micrometer particles were investigated. Results show that noticeable differences in air-particle flow predictions between static and elastic lung models can be found, which demonstrates the necessity to model airway deformation kinematics in whole-lung models. The elastic TWL model predicted lower deposition fraction in mouth-throat regions and higher deposition fraction in lower airways. The effect of disease-specific airway deformation kinematics on particle transport and deposition in the whole lung was investigated, with a focus on the targeted drug delivery efficiency in small airways from generation (G8) to alveoli as the designated lung sites for COPD treatment using inhalation therapy. Simulation results indicate that with the exacerbation of COPD disease conditions, the highest delivery efficiency of the inhaled drug particles decreases which indicates that delivering aerosolized medications to small airways to treat COPD is more challenging for patients with severe disease conditions.

Most existing whole lung models neglect the airway deformation kinematics and assume the lung airways are static. However, neglecting the airway deformation effect on pulmonary air-particle flow dynamics significantly limits the modeling capability under disease-specific lung conditions. Therefore, a novel elastic truncated whole-lung (TWL) modeling framework has been developed to simulate the disease-specific airway deformation kinematics simultaneously with pulmonary air-particle flow dynamics using one-way coupled Euler–Lagrange method plus the dynamic mesh method. Specifically, the deformation kinematics of the elastic TWL model was calibrated with clinical data and pulmonary function test results for both healthy lung and lungs with chronic obstructive pulmonary diseases (COPDs). The transport dynamics of spherical sub micrometer and micrometer particles were investigated. Results show that noticeable differences in air-particle flow predictions between static and elastic lung models can be found, which demonstrates the necessity to model airway deformation kinematics in whole-lung models. The elastic TWL model predicted lower deposition fraction in mouth-throat regions and higher deposition fraction in lower airways. The effect of disease-specific airway deformation kinematics on particle transport and deposition in the whole lung was investigated, with a focus on the targeted drug delivery efficiency in small airways from generation (G8) to alveoli as the designated lung sites for COPD treatment using inhalation therapy. Simulation results indicate that with the exacerbation of COPD disease conditions, the highest delivery efficiency of the inhaled drug particles decreases which indicates that delivering aerosolized medications to small airways to treat COPD is more challenging for patients with severe disease conditions.

Categories: Latest papers in fluid mechanics

### On independent degrees of freedom of turbulent mixing: The one-dimensional formulation

Physics of Fluids, Volume 33, Issue 10, October 2021.

In the present theoretical work, spatially locked, predominantly one-dimensional (1D) turbulent eddies hosting [math] fluid parcels that exchange chaotically their positions [math] are approached as discretized, one-dimensional, “generic” rearrangements (permutations) that comprise assemblages, [math], of minor, “mixing” rearrangements, [math], satisfying three topological–kinematical criteria that outline their mixing extent. In turn, the criteria lead to the derivation of two theorems of mixing that help count the number of all possible mixing rearrangements. The “universal” set of all generic rearrangements, [math], is organized into subsets characterized by the same domain structure, [math], that determines the size and location of a characteristic, minor mixing eddy [math] within the major, generic one, [math] Under the guidance of the first of the two aforementioned theorems of mixing, there can be gathered all pairwise disjoint, domain-structured subsets that add up to the universal set. Then, a class of “independent degrees of freedom of turbulent mixing” has been assembled, a new functional tool in the probability theory of one-dimensional turbulent mixing. The theorem-dictated condition for making up a class of independent degrees of freedom of turbulent mixing is that the characteristic, minor, mixing domains [math] of the participating subsets are all sharing one at least common point of the generic domain.

In the present theoretical work, spatially locked, predominantly one-dimensional (1D) turbulent eddies hosting [math] fluid parcels that exchange chaotically their positions [math] are approached as discretized, one-dimensional, “generic” rearrangements (permutations) that comprise assemblages, [math], of minor, “mixing” rearrangements, [math], satisfying three topological–kinematical criteria that outline their mixing extent. In turn, the criteria lead to the derivation of two theorems of mixing that help count the number of all possible mixing rearrangements. The “universal” set of all generic rearrangements, [math], is organized into subsets characterized by the same domain structure, [math], that determines the size and location of a characteristic, minor mixing eddy [math] within the major, generic one, [math] Under the guidance of the first of the two aforementioned theorems of mixing, there can be gathered all pairwise disjoint, domain-structured subsets that add up to the universal set. Then, a class of “independent degrees of freedom of turbulent mixing” has been assembled, a new functional tool in the probability theory of one-dimensional turbulent mixing. The theorem-dictated condition for making up a class of independent degrees of freedom of turbulent mixing is that the characteristic, minor, mixing domains [math] of the participating subsets are all sharing one at least common point of the generic domain.

Categories: Latest papers in fluid mechanics

### Boundary-layer instability measurements on a cone at freestream Mach 3.5

Physics of Fluids, Volume 33, Issue 10, October 2021.

An experimental study was conducted in the NASA (National Aeronautics and Space Administration) Langley Supersonic Low-Disturbance Tunnel to investigate naturally occurring instabilities in a supersonic boundary layer on a 7° half-angle cone at nominal freestream conditions: Mach 3.5, total temperature of 299.8 K, and unit Reynolds numbers (millions per m) of 9.89, 13.85, 21.77, and 25.73. Instability measurements were acquired under noisy-flow and quiet-flow conditions. Pitot-pressure and calibrated hot-wire measurements were obtained using a model-integrated traverse system to document the model flow field. In noisy-flow conditions, growth rates and mode shapes achieved good agreement between the measured results and linear stability theory (LST). The corresponding N factor at transition from LST is [math]. Under quiet-flow conditions, the most unstable first-mode instabilities as predicted by LST were measured, but this mode was not the dominant instability measured in the boundary layer. Instead, the dominant instabilities were less-amplified, low-frequency disturbances predicted by LST, and grew according to linear theory. These low-frequency unstable disturbances were initiated by freestream acoustic disturbances through a receptivity process believed to occur near the branch I location of the cone. Under quiet-flow conditions, the boundary layer remained laminar up to the last measurement station for the largest unit Reynolds number, implying a transition N factor of N > 8.5.

An experimental study was conducted in the NASA (National Aeronautics and Space Administration) Langley Supersonic Low-Disturbance Tunnel to investigate naturally occurring instabilities in a supersonic boundary layer on a 7° half-angle cone at nominal freestream conditions: Mach 3.5, total temperature of 299.8 K, and unit Reynolds numbers (millions per m) of 9.89, 13.85, 21.77, and 25.73. Instability measurements were acquired under noisy-flow and quiet-flow conditions. Pitot-pressure and calibrated hot-wire measurements were obtained using a model-integrated traverse system to document the model flow field. In noisy-flow conditions, growth rates and mode shapes achieved good agreement between the measured results and linear stability theory (LST). The corresponding N factor at transition from LST is [math]. Under quiet-flow conditions, the most unstable first-mode instabilities as predicted by LST were measured, but this mode was not the dominant instability measured in the boundary layer. Instead, the dominant instabilities were less-amplified, low-frequency disturbances predicted by LST, and grew according to linear theory. These low-frequency unstable disturbances were initiated by freestream acoustic disturbances through a receptivity process believed to occur near the branch I location of the cone. Under quiet-flow conditions, the boundary layer remained laminar up to the last measurement station for the largest unit Reynolds number, implying a transition N factor of N > 8.5.

Categories: Latest papers in fluid mechanics

### The function of the alula with different geometric parameters on the flapping wing

Physics of Fluids, Volume 33, Issue 10, October 2021.

Birds in nature have the ability to maintain high aerodynamic efficiency in complex flight conditions. This agility stems from the multi-degree-of-freedom flapping motion and specialized feather systems that evolved over millions of years. The leading-edge alula is considered a typical feather system that can enhance the flight envelope and capabilities of birds at low speed and high incidence. Previous studies usually adopted a static model, ignoring unsteady effects caused by flapping motion. Thus, we numerically investigated the function of the alula with different geometric parameters on the flapping wing in this paper. The alula has both the slot effect and vortex generator effect during the flapping motion, whereas the effect that plays a main role in lift enhancement changes as time varies. At the beginning of the upstroke, the slot effect plays the main role. At mid-time of the upstroke, the vortex generator effect plays the main role. Different geometric parameters have different influences on these two effects. The dimensionless spanwise location affects the strength of both the ATEV (alula trailing edge vortex) and ASV (alula streamwise vortex). The relative angle affects mainly the strength of the ATEV, whereas the deflection angle affects mainly the strength of the ASV. The optimal geometric parameters to obtain maximum lift enhancement are a dimensionless spanwise location of 0.5, a relative angle of 0°, and a deflection angle of 10°, with a lift enhancement of 5.5% compared to the baseline wing.

Birds in nature have the ability to maintain high aerodynamic efficiency in complex flight conditions. This agility stems from the multi-degree-of-freedom flapping motion and specialized feather systems that evolved over millions of years. The leading-edge alula is considered a typical feather system that can enhance the flight envelope and capabilities of birds at low speed and high incidence. Previous studies usually adopted a static model, ignoring unsteady effects caused by flapping motion. Thus, we numerically investigated the function of the alula with different geometric parameters on the flapping wing in this paper. The alula has both the slot effect and vortex generator effect during the flapping motion, whereas the effect that plays a main role in lift enhancement changes as time varies. At the beginning of the upstroke, the slot effect plays the main role. At mid-time of the upstroke, the vortex generator effect plays the main role. Different geometric parameters have different influences on these two effects. The dimensionless spanwise location affects the strength of both the ATEV (alula trailing edge vortex) and ASV (alula streamwise vortex). The relative angle affects mainly the strength of the ATEV, whereas the deflection angle affects mainly the strength of the ASV. The optimal geometric parameters to obtain maximum lift enhancement are a dimensionless spanwise location of 0.5, a relative angle of 0°, and a deflection angle of 10°, with a lift enhancement of 5.5% compared to the baseline wing.

Categories: Latest papers in fluid mechanics

### A unified multi-phase and multi-material formulation for combustion modeling

Physics of Fluids, Volume 33, Issue 10, October 2021.

The motivation of this work is to produce an integrated formulation for material response (e.g., elastoplastic, viscous, viscoplastic) due to detonation wave loading. Here, we focus on elastoplastic structural response. In particular, we want to capture miscible and immiscible behavior within condensed-phase explosives arising from the co-existence of a reactive carrier mixture of miscible materials and several material interfaces due to the presence of immiscible impurities such as particles or cavities. The dynamic and thermodynamic evolution of the explosive is communicated to one or more inert confiners through their shared interfaces, which may undergo severe topological change. We also wish to consider elastic and plastic structural response of the confiners rather than make a hydrodynamic assumption for their behavior. The previous work by these authors has met these requirements by means of the simultaneous solution of appropriate systems of equations for the behavior of the condensed-phase explosive and the elastoplastic behavior of the confiners. To that end, both systems were written in the same mathematical form as a system of inhomogeneous hyperbolic partial differential equations (PDEs), which were solved on the same discrete space using the same algorithms, as opposed to coupling fluid and solid algorithms (co-simulation). In the present work, we employ a single system of PDEs proposed by Peshkov and Romenski [Peshkov and Romenski, “A hyperbolic model for viscous Newtonian flows,” Continuum Mech. Thermodyn. 28, 85 (2016)], which is able to account for different states of matter by means of generalizing the concept of distortion tensors beyond solids. We amalgamate that formulation with a single system of PDEs, which meets the requirement of co-existing miscible and immiscible explosive mixtures. We present the mathematical derivation and construct appropriate algorithms for its solution. The resulting model is validated against exact solutions for several one-dimensional use-cases, including mechanically and thermally induced, inviscid, and viscous detonations. Results indicate that the model can accurately simulate a very broad range of problems involving the nonlinear interaction between reactive and inert materials within a single framework.

The motivation of this work is to produce an integrated formulation for material response (e.g., elastoplastic, viscous, viscoplastic) due to detonation wave loading. Here, we focus on elastoplastic structural response. In particular, we want to capture miscible and immiscible behavior within condensed-phase explosives arising from the co-existence of a reactive carrier mixture of miscible materials and several material interfaces due to the presence of immiscible impurities such as particles or cavities. The dynamic and thermodynamic evolution of the explosive is communicated to one or more inert confiners through their shared interfaces, which may undergo severe topological change. We also wish to consider elastic and plastic structural response of the confiners rather than make a hydrodynamic assumption for their behavior. The previous work by these authors has met these requirements by means of the simultaneous solution of appropriate systems of equations for the behavior of the condensed-phase explosive and the elastoplastic behavior of the confiners. To that end, both systems were written in the same mathematical form as a system of inhomogeneous hyperbolic partial differential equations (PDEs), which were solved on the same discrete space using the same algorithms, as opposed to coupling fluid and solid algorithms (co-simulation). In the present work, we employ a single system of PDEs proposed by Peshkov and Romenski [Peshkov and Romenski, “A hyperbolic model for viscous Newtonian flows,” Continuum Mech. Thermodyn. 28, 85 (2016)], which is able to account for different states of matter by means of generalizing the concept of distortion tensors beyond solids. We amalgamate that formulation with a single system of PDEs, which meets the requirement of co-existing miscible and immiscible explosive mixtures. We present the mathematical derivation and construct appropriate algorithms for its solution. The resulting model is validated against exact solutions for several one-dimensional use-cases, including mechanically and thermally induced, inviscid, and viscous detonations. Results indicate that the model can accurately simulate a very broad range of problems involving the nonlinear interaction between reactive and inert materials within a single framework.

Categories: Latest papers in fluid mechanics

### The feedback loops of discrete tones in under-expanded impinging jets

Physics of Fluids, Volume 33, Issue 10, October 2021.

The upstream-propagating waves of feedback loops in under-expanded impinging jets are investigated through several typical flow conditions. Schlieren image capturing and far-field acoustic measurement are applied to the jet. The flow structures related to the discrete tones are extracted through dynamic mode decomposition (DMD), and the standing waves formed by the resonance are identified from the normalized amplitude fields of the chosen DMD modes. Spatial Fourier transforms are applied along the axial direction on the chosen DMD modes to obtain the spatial wavenumber spectra, from which the two wave disturbances caused by the interaction between Kelvin–Helmholtz (K–H) waves and shock cells can be identified. The upstream and downstream-propagating components of the feedback loop are rebuilt, respectively, via bandpass filtering. The wavenumbers of the upstream-propagating components in the feedback loop and the wavenumbers of disturbance waves are compared along the jet shear layer; the results suggest more than one closure mechanism of the feedback loop. For the impinging tones which have the same frequencies as the A1 and B modes in free jet, the closure mechanism of the feedback loop is the Mach radiation generated by the interaction between K–H waves and shock cells. For the other tones, their frequencies are consistent well with the allowable frequency ranges of the neutral wave mode. The combination of the neutral wave mode with the classical feedback loop model also gives strong evidence that the upstream-propagating waves for these tones may be the jets' inherent neutral wave.

The upstream-propagating waves of feedback loops in under-expanded impinging jets are investigated through several typical flow conditions. Schlieren image capturing and far-field acoustic measurement are applied to the jet. The flow structures related to the discrete tones are extracted through dynamic mode decomposition (DMD), and the standing waves formed by the resonance are identified from the normalized amplitude fields of the chosen DMD modes. Spatial Fourier transforms are applied along the axial direction on the chosen DMD modes to obtain the spatial wavenumber spectra, from which the two wave disturbances caused by the interaction between Kelvin–Helmholtz (K–H) waves and shock cells can be identified. The upstream and downstream-propagating components of the feedback loop are rebuilt, respectively, via bandpass filtering. The wavenumbers of the upstream-propagating components in the feedback loop and the wavenumbers of disturbance waves are compared along the jet shear layer; the results suggest more than one closure mechanism of the feedback loop. For the impinging tones which have the same frequencies as the A1 and B modes in free jet, the closure mechanism of the feedback loop is the Mach radiation generated by the interaction between K–H waves and shock cells. For the other tones, their frequencies are consistent well with the allowable frequency ranges of the neutral wave mode. The combination of the neutral wave mode with the classical feedback loop model also gives strong evidence that the upstream-propagating waves for these tones may be the jets' inherent neutral wave.

Categories: Latest papers in fluid mechanics

### A lattice Boltzmann method for single- and two-phase models of nanofluids: Newtonian and non-Newtonian nanofluids

Physics of Fluids, Volume 33, Issue 10, October 2021.

Nanofluids play an important role in many different industries for an improvement of heat transfer. The modeling and simulation of such fluids is developing continuously. Two important models for studying nanofluids are mixture (or single-phase) and two-phase (or Buongiorno) forms, which have been examined in various ways. Non-Newtonian behavior of nanofluids (shear-thinning and viscoplasticity) has been observed in experimental tests and simulated in several studies. However, a lattice Boltzmann method (LBM), which can employ either model depending on the particular non-Newtonian constitutive equation, has not been considered to date within the suite of available numerical methods. Here, we propose a comprehensive LBM to simulate both Newtonian and non-Newtonian nanofluids. The approach has the potential to incorporate any format of extra tensor directly and is independent to the relaxation time; the upshot is that our method is appropriate for studying non-Newtonian nanofluids. The derivations for both models are presented and discussed in some detail. To evaluate the proposed method, it was compared with previous studies into a benchmark problem, natural convection in a square enclosure filled with Newtonian nanofluids and non-Newtonian fluids. Then, the applied macroscopic and LBM equations, using the power-law and viscoplastic models, for the benchmark are derived and the results are presented.

Nanofluids play an important role in many different industries for an improvement of heat transfer. The modeling and simulation of such fluids is developing continuously. Two important models for studying nanofluids are mixture (or single-phase) and two-phase (or Buongiorno) forms, which have been examined in various ways. Non-Newtonian behavior of nanofluids (shear-thinning and viscoplasticity) has been observed in experimental tests and simulated in several studies. However, a lattice Boltzmann method (LBM), which can employ either model depending on the particular non-Newtonian constitutive equation, has not been considered to date within the suite of available numerical methods. Here, we propose a comprehensive LBM to simulate both Newtonian and non-Newtonian nanofluids. The approach has the potential to incorporate any format of extra tensor directly and is independent to the relaxation time; the upshot is that our method is appropriate for studying non-Newtonian nanofluids. The derivations for both models are presented and discussed in some detail. To evaluate the proposed method, it was compared with previous studies into a benchmark problem, natural convection in a square enclosure filled with Newtonian nanofluids and non-Newtonian fluids. Then, the applied macroscopic and LBM equations, using the power-law and viscoplastic models, for the benchmark are derived and the results are presented.

Categories: Latest papers in fluid mechanics

### Reynolds number effect on statistics of turbulent flows over periodic hills

Physics of Fluids, Volume 33, Issue 10, October 2021.

The wall-resolved large-eddy simulations of turbulent flows over periodic hills are carried out to study the Reynolds number effect on flow statistics. Five different Reynolds numbers ranging from 2800 to 37 000 are considered. The present simulations are validated by comparing the time-averaged flow statistics with those from the literature. The Reynolds number effect is first examined on the skin friction and pressure coefficients, the isosurfaces of [math] and Q criteria, and the vertical profiles of flow statistics. The results show that (1) at most locations the magnitude of friction coefficient decreases with the increase in Reynolds number, while the pressure coefficient varies in the opposite direction; (2) smaller turbulence structures arise at higher Reynolds numbers; and (3) the mean velocities and Reynolds stresses in general exhibit asymptotic behaviors with the increase in Reynolds number. The statistical properties of turbulence structures are further examined via the probability density function and time correlation of velocity fluctuations. At last, the dynamics in the separation bubble is investigated by examining the flow statistics and the budget equation of mean kinetic energy (MKE) on the coordinate with its origin fixed at the recirculation center, and the power spectral density of the velocity fluctuations. Similarities are in general observed for the mean velocities, Reynolds stresses, and the MKE budget in the rear part of the separation bubble. The mean convection term and turbulence convection term are observed playing a key role on the decrease in bubble size with the increase in Reynolds number.

The wall-resolved large-eddy simulations of turbulent flows over periodic hills are carried out to study the Reynolds number effect on flow statistics. Five different Reynolds numbers ranging from 2800 to 37 000 are considered. The present simulations are validated by comparing the time-averaged flow statistics with those from the literature. The Reynolds number effect is first examined on the skin friction and pressure coefficients, the isosurfaces of [math] and Q criteria, and the vertical profiles of flow statistics. The results show that (1) at most locations the magnitude of friction coefficient decreases with the increase in Reynolds number, while the pressure coefficient varies in the opposite direction; (2) smaller turbulence structures arise at higher Reynolds numbers; and (3) the mean velocities and Reynolds stresses in general exhibit asymptotic behaviors with the increase in Reynolds number. The statistical properties of turbulence structures are further examined via the probability density function and time correlation of velocity fluctuations. At last, the dynamics in the separation bubble is investigated by examining the flow statistics and the budget equation of mean kinetic energy (MKE) on the coordinate with its origin fixed at the recirculation center, and the power spectral density of the velocity fluctuations. Similarities are in general observed for the mean velocities, Reynolds stresses, and the MKE budget in the rear part of the separation bubble. The mean convection term and turbulence convection term are observed playing a key role on the decrease in bubble size with the increase in Reynolds number.

Categories: Latest papers in fluid mechanics

### Patterns of interfacial flow around a lubricated rolling point contact region

Physics of Fluids, Volume 33, Issue 10, October 2021.

It is of great importance to develop an in-depth understanding of interfacial flow around a lubricated rolling point contact region (RPCR). Consideration of the flow patterns around an RPCR will be important for the lubrication, cooling, and cleaning of machine parts, such as rolling bearings and gearboxes. In this study, an experiment using laser-induced fluorescence and a simulation using computational fluid dynamics of interfacial flow around an oil-lubricated ball-on-disk RPCR are presented. The results show good agreement with each other, and the flow patterns are clearly classified. The forming mechanisms are analyzed in terms of force competition caused by inertia, viscosity, pressure gradient, and air–oil surface tension. Quantitative criteria are proposed to evaluate the behaviors at the air–oil interface and the transition of flow patterns. Analyses have shown that the competition between these forces drives the motion of the air–oil interface, and the formation of flow patterns can be considered a self-adjusting process for the air–oil interface toward the equilibrium positions of the forces. High surface tension is beneficial for maintaining interface stability and can prevent the meniscus from rupturing at an outlet and concaving at an inlet. High capillary numbers may increase the risk for outlet meniscus rupture and the degree of concavity of the inlet meniscus.

It is of great importance to develop an in-depth understanding of interfacial flow around a lubricated rolling point contact region (RPCR). Consideration of the flow patterns around an RPCR will be important for the lubrication, cooling, and cleaning of machine parts, such as rolling bearings and gearboxes. In this study, an experiment using laser-induced fluorescence and a simulation using computational fluid dynamics of interfacial flow around an oil-lubricated ball-on-disk RPCR are presented. The results show good agreement with each other, and the flow patterns are clearly classified. The forming mechanisms are analyzed in terms of force competition caused by inertia, viscosity, pressure gradient, and air–oil surface tension. Quantitative criteria are proposed to evaluate the behaviors at the air–oil interface and the transition of flow patterns. Analyses have shown that the competition between these forces drives the motion of the air–oil interface, and the formation of flow patterns can be considered a self-adjusting process for the air–oil interface toward the equilibrium positions of the forces. High surface tension is beneficial for maintaining interface stability and can prevent the meniscus from rupturing at an outlet and concaving at an inlet. High capillary numbers may increase the risk for outlet meniscus rupture and the degree of concavity of the inlet meniscus.

Categories: Latest papers in fluid mechanics

### Unified prediction of turbulent mixing induced by interfacial instabilities via Besnard−Harlow−Rauenzahn-2 model

Physics of Fluids, Volume 33, Issue 10, October 2021.

Turbulent mixing induced by interfacial instabilities, such as Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz (KH) instabilities, widely exist in natural phenomena and engineering applications. On the one hand, the Reynolds-averaged Navier–Stokes (RANS) method, mainly involving physical model and model coefficients, is still the most viable approach in application. On the other hand, predicting different mixing problems with the same physical model and model coefficients—defined as “unified prediction” in this paper—is the basis for practice because (1) different instabilities usually exist simultaneously in a flow system and are coupled to each other; (2) mixing processes involve a wide range of parameters (e.g., time-dependent density ratio and acceleration history, etc.). However, few models can achieve such a unified prediction. Recently, we proposed a RANS route to realize this unified prediction by setting model coefficients to match the given physical model. This study attempts to apply this to the widely used BHR2 model to achieve unified predictions of different turbulent mixing problems, including basic problems (i.e., classical RT, RM, and KH mixing) and complex problems (i.e., re-shocked RM, tilted-RT, and spherical implosion mixing). Good agreement between experiments, large-eddy simulations, and RANS results were obtained. The temporal evolution of mixing width and spatial profiles of important physical quantities are presented. Based on our achievements of the k – L and [math] models for unified predictions, the success of BHR2 model further confirms that our RANS route is robust for different turbulent mixing models and may be expanded to other fields.

Turbulent mixing induced by interfacial instabilities, such as Rayleigh–Taylor (RT), Richtmyer–Meshkov (RM), and Kelvin–Helmholtz (KH) instabilities, widely exist in natural phenomena and engineering applications. On the one hand, the Reynolds-averaged Navier–Stokes (RANS) method, mainly involving physical model and model coefficients, is still the most viable approach in application. On the other hand, predicting different mixing problems with the same physical model and model coefficients—defined as “unified prediction” in this paper—is the basis for practice because (1) different instabilities usually exist simultaneously in a flow system and are coupled to each other; (2) mixing processes involve a wide range of parameters (e.g., time-dependent density ratio and acceleration history, etc.). However, few models can achieve such a unified prediction. Recently, we proposed a RANS route to realize this unified prediction by setting model coefficients to match the given physical model. This study attempts to apply this to the widely used BHR2 model to achieve unified predictions of different turbulent mixing problems, including basic problems (i.e., classical RT, RM, and KH mixing) and complex problems (i.e., re-shocked RM, tilted-RT, and spherical implosion mixing). Good agreement between experiments, large-eddy simulations, and RANS results were obtained. The temporal evolution of mixing width and spatial profiles of important physical quantities are presented. Based on our achievements of the k – L and [math] models for unified predictions, the success of BHR2 model further confirms that our RANS route is robust for different turbulent mixing models and may be expanded to other fields.

Categories: Latest papers in fluid mechanics

### Rigid fiber motion in slightly non-Newtonian viscoelastic fluids

Physics of Fluids, Volume 33, Issue 10, October 2021.

The perturbation technique based on the retardation-motion expansion is a simple method to obtain flow solutions at low Weissenberg number. In this context, this perturbation analysis is used to develop simple expressions for the motion of fibers suspended in viscoelastic fluids. In particular, the suspending fluid is characterized by a second-order fluid, Giesekus and PPT (Phan–Thien–Tanner) models, and their derivatives, such as the upper and lower convected Maxwell models. The first-order perturbation results in a similar effective velocity gradient that is exploited to express the translation and rotational motion of a single fiber and the associated extra stress tensor. In terms of a parameter related to the various viscoelastic fluid models, it is found that a fiber aligns along the vorticity direction when subjected to a shear flow. However, when a lower convected Maxwell model is considered, the elongated particle orients in the flow direction, as basically predicted by the Jeffery solution for a Newtonian suspending fluid. Furthermore, the conservation equation for particle concentration leads to particle migration in a pressure-driven flow channel and good agreement is observed with experimental data.

The perturbation technique based on the retardation-motion expansion is a simple method to obtain flow solutions at low Weissenberg number. In this context, this perturbation analysis is used to develop simple expressions for the motion of fibers suspended in viscoelastic fluids. In particular, the suspending fluid is characterized by a second-order fluid, Giesekus and PPT (Phan–Thien–Tanner) models, and their derivatives, such as the upper and lower convected Maxwell models. The first-order perturbation results in a similar effective velocity gradient that is exploited to express the translation and rotational motion of a single fiber and the associated extra stress tensor. In terms of a parameter related to the various viscoelastic fluid models, it is found that a fiber aligns along the vorticity direction when subjected to a shear flow. However, when a lower convected Maxwell model is considered, the elongated particle orients in the flow direction, as basically predicted by the Jeffery solution for a Newtonian suspending fluid. Furthermore, the conservation equation for particle concentration leads to particle migration in a pressure-driven flow channel and good agreement is observed with experimental data.

Categories: Latest papers in fluid mechanics

### Diffusive instabilities of baroclinic lenticular vortices

Physics of Fluids, Volume 33, Issue 10, October 2021.

We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying the algebraic Bilharz criterion to the latter, we establish that the stability conditions are reduced to three inequalities that define the stability domain in the space of parameters. The main destabilization mechanism is either monotonic or oscillatory axisymmetric instability depending on the Schmidt number (Sc), vortex Rossby number, and the difference between radial and axial density gradients as well as the difference between epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of monotonic and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all Sc > 0.

We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying the algebraic Bilharz criterion to the latter, we establish that the stability conditions are reduced to three inequalities that define the stability domain in the space of parameters. The main destabilization mechanism is either monotonic or oscillatory axisymmetric instability depending on the Schmidt number (Sc), vortex Rossby number, and the difference between radial and axial density gradients as well as the difference between epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of monotonic and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all Sc > 0.

Categories: Latest papers in fluid mechanics

### The effects of roughness levels on the instability of the boundary-layer flow over a rotating disk with an enforced axial flow

Physics of Fluids, Volume 33, Issue 10, October 2021.

This paper investigates the effects of surface roughness on the convective stability behavior of boundary-layer flow over a rotating disk. An enforced axial flow and the Miklavčič and Wang (MW) model of roughness are applied to this flow. The effects of both anisotropic and isotropic surface roughness on the distinct instability properties of the boundary-layer flow over a rotating disk will also be examined for this model. It is possible to implement these types of roughness on this geometric shape while considering an axial flow. This approach requires a modification for the no-slip condition and that the current boundary conditions are partial-slip conditions. The Navier–Stokes equations are used to obtain the steady mean-flow system, and linear stability equations are then formulated to obtain neutral stability curves while investigating the convective instability behavior for stationary modes. The stability analysis results are then confirmed by the linear convective growth rates for stationary disturbances and the energy analysis. The stability characteristics of the inviscid type I (or cross-flow) instability and the viscous type II instability are examined over a rough, rotating disk within the boundary layer at all axial flow rates considered. Our findings indicate that the radial grooves have a strong destabilizing effect on the type II mode as the axial flow is increased, whereas the concentric grooves and isotropic surface roughness stabilize the boundary-layer flow for the type I mode. It is worth noting that the flows over a concentrically grooved disk with increasing enforced axial flow strength are the most stable for the inviscid type I instability.

This paper investigates the effects of surface roughness on the convective stability behavior of boundary-layer flow over a rotating disk. An enforced axial flow and the Miklavčič and Wang (MW) model of roughness are applied to this flow. The effects of both anisotropic and isotropic surface roughness on the distinct instability properties of the boundary-layer flow over a rotating disk will also be examined for this model. It is possible to implement these types of roughness on this geometric shape while considering an axial flow. This approach requires a modification for the no-slip condition and that the current boundary conditions are partial-slip conditions. The Navier–Stokes equations are used to obtain the steady mean-flow system, and linear stability equations are then formulated to obtain neutral stability curves while investigating the convective instability behavior for stationary modes. The stability analysis results are then confirmed by the linear convective growth rates for stationary disturbances and the energy analysis. The stability characteristics of the inviscid type I (or cross-flow) instability and the viscous type II instability are examined over a rough, rotating disk within the boundary layer at all axial flow rates considered. Our findings indicate that the radial grooves have a strong destabilizing effect on the type II mode as the axial flow is increased, whereas the concentric grooves and isotropic surface roughness stabilize the boundary-layer flow for the type I mode. It is worth noting that the flows over a concentrically grooved disk with increasing enforced axial flow strength are the most stable for the inviscid type I instability.

Categories: Latest papers in fluid mechanics

### Electric-field induced phase transitions in capillary electrophoretic systems

Physics of Fluids, Volume 33, Issue 10, October 2021.

The movement of particles in a capillary electrophoretic system under electroosmotic flow was modeled using Monte Carlo simulation with the Metropolis algorithm. Two different cases with repulsive and attractive interactions between molecules were taken into consideration. Simulation was done using a spin-like system, where the interactions between the nearest and second closest neighbors were considered in two separate steps of the modeling study. A total of 20 different cases with different rates of interactions for both repulsive and attractive interactions were modeled. The movement of the particles through the capillary is defined as current. At a low interaction level between molecules, a regular electroosmotic flow is obtained; on the other hand, with increasing interactions between molecules, the current shows a phase transition behavior. The results also show that a modular electroosmotic flow can be obtained for separations by tuning the ratio between molecular interactions and electric field strength.

The movement of particles in a capillary electrophoretic system under electroosmotic flow was modeled using Monte Carlo simulation with the Metropolis algorithm. Two different cases with repulsive and attractive interactions between molecules were taken into consideration. Simulation was done using a spin-like system, where the interactions between the nearest and second closest neighbors were considered in two separate steps of the modeling study. A total of 20 different cases with different rates of interactions for both repulsive and attractive interactions were modeled. The movement of the particles through the capillary is defined as current. At a low interaction level between molecules, a regular electroosmotic flow is obtained; on the other hand, with increasing interactions between molecules, the current shows a phase transition behavior. The results also show that a modular electroosmotic flow can be obtained for separations by tuning the ratio between molecular interactions and electric field strength.

Categories: Latest papers in fluid mechanics

### Pore-resolved volume-of-fluid simulations of two-phase flow in porous media: Pore-scale flow mechanisms and regime map

Physics of Fluids, Volume 33, Issue 10, October 2021.

Two-phase flow through porous media is important to the development of secondary and tertiary oil recovery. In the present work, we have simulated oil recovery through a pore-resolved three-dimensional medium using volume-of-fluid method. The effects of wettability and interfacial tension (IFT) on two-phase flow mechanisms are investigated using pore-scale events, oil-phase morphology, forces acting on oil ganglia surfaces, and oil recovery curves, for Capillary numbers (Ca) in the range of 1.2 × 10−3 to 6 × 10−1. We found that the two-phase flow through oil-wet medium is governed by pore-by-pore filling mechanism dominated by the Haines-jumps. At low Ca values, a change in the wettability from oil- to neutrally wet resulted into the change of pore-by-pore filling mechanism to co-operative pore filling and as the medium wettability changes from the neutrally to the weakly water-wet, the corner flow events begin to emerge. At low Ca values, the invasion through weakly water-wet porous medium is dominated by co-operative filling and results into an increased oil recovery, whereas the two-phase flow through strongly water-wet medium is governed by corner flow events resulting in a low oil recovery. The corner flow events are found to be a function of not only the medium wettability, but also of Ca and are a characteristic of controlled imbibition. Further, we show that a substantial decrease in the IFT results in a fingerlike invasion at pore-scale, irrespective of the medium wettability. Finally, a two-phase flow regime map is proposed in terms of Ca and contact angle based on the two-phase interface morphology.

Two-phase flow through porous media is important to the development of secondary and tertiary oil recovery. In the present work, we have simulated oil recovery through a pore-resolved three-dimensional medium using volume-of-fluid method. The effects of wettability and interfacial tension (IFT) on two-phase flow mechanisms are investigated using pore-scale events, oil-phase morphology, forces acting on oil ganglia surfaces, and oil recovery curves, for Capillary numbers (Ca) in the range of 1.2 × 10−3 to 6 × 10−1. We found that the two-phase flow through oil-wet medium is governed by pore-by-pore filling mechanism dominated by the Haines-jumps. At low Ca values, a change in the wettability from oil- to neutrally wet resulted into the change of pore-by-pore filling mechanism to co-operative pore filling and as the medium wettability changes from the neutrally to the weakly water-wet, the corner flow events begin to emerge. At low Ca values, the invasion through weakly water-wet porous medium is dominated by co-operative filling and results into an increased oil recovery, whereas the two-phase flow through strongly water-wet medium is governed by corner flow events resulting in a low oil recovery. The corner flow events are found to be a function of not only the medium wettability, but also of Ca and are a characteristic of controlled imbibition. Further, we show that a substantial decrease in the IFT results in a fingerlike invasion at pore-scale, irrespective of the medium wettability. Finally, a two-phase flow regime map is proposed in terms of Ca and contact angle based on the two-phase interface morphology.

Categories: Latest papers in fluid mechanics