# 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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### Effect of non-parallel mean flow on the acoustic spectrum of heated supersonic jets: Explanation of “jet quietening”

Physics of Fluids, Volume 31, Issue 10, October 2019.

Noise measurements of heated axisymmetric jets at a fixed supersonic acoustic Mach number indicate that the acoustic spectrum reduces when the temperature ratio increases. The “spectral quietening” effect has been observed both experimentally and computationally using large Eddy simulations. It was explained by Afsar, Goldstein, and Fagan [AIAA J. 49, 2522 (2011)] through the cancellation introduced by the enthalpy flux/momentum flux coupling term using the generalized acoustic analogy formulation. But the parallel flow assumption is known to give inaccurate predictions at high jet speeds. In this paper, we therefore extend the nonparallel flow asymptotic theory of Goldstein, Sescu, and Afsar [J. Fluid Mech. 695, 199 (2012)] for the vector Green’s function of the adjoint linearized Euler equations in the analogy. Using the steady Reynolds averaged Navier Stokes calculation for the jet mean flow, we find that the coupling term propagator is positive-definite and asymptotically subdominant at low frequencies corresponding to the peak jet noise when nonparallel flow effects are taken into account and self-consistent approximations for the turbulence structure are made. We then assess the validity of the non-parallel flow-based acoustic analogy model by computing the overall sound pressure level (OASPL) at various observation angles. Interpretation of the latter allows a more rational explanation of the quietening effect. In general, our noise predictions are in very good agreement with acoustic data beyond the peak frequency.

Noise measurements of heated axisymmetric jets at a fixed supersonic acoustic Mach number indicate that the acoustic spectrum reduces when the temperature ratio increases. The “spectral quietening” effect has been observed both experimentally and computationally using large Eddy simulations. It was explained by Afsar, Goldstein, and Fagan [AIAA J. 49, 2522 (2011)] through the cancellation introduced by the enthalpy flux/momentum flux coupling term using the generalized acoustic analogy formulation. But the parallel flow assumption is known to give inaccurate predictions at high jet speeds. In this paper, we therefore extend the nonparallel flow asymptotic theory of Goldstein, Sescu, and Afsar [J. Fluid Mech. 695, 199 (2012)] for the vector Green’s function of the adjoint linearized Euler equations in the analogy. Using the steady Reynolds averaged Navier Stokes calculation for the jet mean flow, we find that the coupling term propagator is positive-definite and asymptotically subdominant at low frequencies corresponding to the peak jet noise when nonparallel flow effects are taken into account and self-consistent approximations for the turbulence structure are made. We then assess the validity of the non-parallel flow-based acoustic analogy model by computing the overall sound pressure level (OASPL) at various observation angles. Interpretation of the latter allows a more rational explanation of the quietening effect. In general, our noise predictions are in very good agreement with acoustic data beyond the peak frequency.

Categories: Latest papers in fluid mechanics

### Inertial focusing of a neutrally buoyant particle in stratified flows

Physics of Fluids, Volume 31, Issue 10, October 2019.

Particles in microfluidic channels experience two dominant lift forces in the direction transverse to the flow—the shear gradient lift force and the wall lift force. These forces contribute to the lift experienced by the particle and cause their cross stream migration until they attain an equilibrium position where the net lift force in the transverse direction is zero. Stratified coflow of two liquids with different viscosities is a stable flow-regime observed under some operating conditions. The presence of the second fluid alters the shear gradient induced lift force and the wall force acting on the particle at each point, changing the final equilibrium position. These positions can be tuned and controlled by altering the viscosity or the flow rates of the two fluids so that the particles focus in one fluid. A numerical method based on the combined Immersed Boundary-Lattice Boltzmann Method is used to study inertial focusing of neutrally buoyant particles in stratified Couette flows and pressure driven flows. We analyze how different factors such as the Reynolds number, flow rate ratio, viscosity ratio of the fluids, and particle size affect the particle migration in two-dimensional (2D) and three-dimensional (3D) geometries. Our study shows that in Couette flows, the particle focuses in the low viscosity fluid when the interface is at the center. We also found that a critical viscosity ratio exists beyond which particle focusing in low viscous fluid is guaranteed, for a given flow rate ratio in pressure driven flows.

Particles in microfluidic channels experience two dominant lift forces in the direction transverse to the flow—the shear gradient lift force and the wall lift force. These forces contribute to the lift experienced by the particle and cause their cross stream migration until they attain an equilibrium position where the net lift force in the transverse direction is zero. Stratified coflow of two liquids with different viscosities is a stable flow-regime observed under some operating conditions. The presence of the second fluid alters the shear gradient induced lift force and the wall force acting on the particle at each point, changing the final equilibrium position. These positions can be tuned and controlled by altering the viscosity or the flow rates of the two fluids so that the particles focus in one fluid. A numerical method based on the combined Immersed Boundary-Lattice Boltzmann Method is used to study inertial focusing of neutrally buoyant particles in stratified Couette flows and pressure driven flows. We analyze how different factors such as the Reynolds number, flow rate ratio, viscosity ratio of the fluids, and particle size affect the particle migration in two-dimensional (2D) and three-dimensional (3D) geometries. Our study shows that in Couette flows, the particle focuses in the low viscosity fluid when the interface is at the center. We also found that a critical viscosity ratio exists beyond which particle focusing in low viscous fluid is guaranteed, for a given flow rate ratio in pressure driven flows.

Categories: Latest papers in fluid mechanics

### Influence of convection regimes of two-layer thermal convection with large viscosity contrast on the thermal and mechanical states at the interface of the two layers: Implications for dynamics in the present-day and past Earth

Physics of Fluids, Volume 31, Issue 10, October 2019.

This paper reports on results for the thermal and mechanical states at the interface of two-layer thermal convection in two-dimensional (2-D) spherical geometry solved by numerical calculations. The two-layer system was composed of a highly viscous layer (HVL) and a low-viscosity layer (LVL) underneath. The two end-member convection regimes were studied by varying two free model parameters, which control the degree of layering in HVL convection and separate the HVL into the upper and lower parts. One of the regimes was a nearly whole-layer convection regime in which the upwelling and downwelling plumes easily penetrated into another layer in the HVL, while the other was a so-called hybrid convection regime, which represented a transitional regime between the whole-layer convection and the double-layer convection. The spatiotemporal analyses of convection behavior showed that the lateral scale of HVL convection and the resultant lateral scale of thermal heterogeneity beneath the HVL–LVL interface tended to be larger in the hybrid convection regime than those in the whole-layer convection regime. On the other hand, the fluctuation of shear-stress at the HVL–LVL interface was more time-dependent in the hybrid convection regime, whereas the mechanical heterogeneity near the HVL–LVL interface was larger in the whole-layer convection regime. The present results on the differences in the scale of dynamically determined thermal and mechanical states beneath the HVL–LVL interface between the two end-member convection regimes may apply to issues on the relationship between thermal and mechanical conditions at the Earth’s core–mantle boundary and the strength of the geomagnetic field.

This paper reports on results for the thermal and mechanical states at the interface of two-layer thermal convection in two-dimensional (2-D) spherical geometry solved by numerical calculations. The two-layer system was composed of a highly viscous layer (HVL) and a low-viscosity layer (LVL) underneath. The two end-member convection regimes were studied by varying two free model parameters, which control the degree of layering in HVL convection and separate the HVL into the upper and lower parts. One of the regimes was a nearly whole-layer convection regime in which the upwelling and downwelling plumes easily penetrated into another layer in the HVL, while the other was a so-called hybrid convection regime, which represented a transitional regime between the whole-layer convection and the double-layer convection. The spatiotemporal analyses of convection behavior showed that the lateral scale of HVL convection and the resultant lateral scale of thermal heterogeneity beneath the HVL–LVL interface tended to be larger in the hybrid convection regime than those in the whole-layer convection regime. On the other hand, the fluctuation of shear-stress at the HVL–LVL interface was more time-dependent in the hybrid convection regime, whereas the mechanical heterogeneity near the HVL–LVL interface was larger in the whole-layer convection regime. The present results on the differences in the scale of dynamically determined thermal and mechanical states beneath the HVL–LVL interface between the two end-member convection regimes may apply to issues on the relationship between thermal and mechanical conditions at the Earth’s core–mantle boundary and the strength of the geomagnetic field.

Categories: Latest papers in fluid mechanics

### Numerical investigation of toroidal bubble dynamics in a compressible fluid based on boundary integral method

Physics of Fluids, Volume 31, Issue 10, October 2019.

The external liquid compressibility cannot be ignored because the speed of the bubble jet emerging at the end of bubble collapse reaches hundreds of meters per second. Additionally, when the bubble jet penetrates the surface of a bubble, a toroidal bubble forms and the singly connected flow domain changes to a doubly connected topology. As the Biot–Savart law is based on the assumption of incompressibility, the vortex ring model is very difficult to extend to compressible fluids. This paper describes the use of the boundary integral method to establish a numerical model of a toroidal bubble, considering the external liquid compressibility and the internal gas wave effect. A cut is introduced into the fluid domain so that it can be considered as singly connected, with the discontinuity of velocity across this cut equal to the circulation of the flow. Furthermore, the initial bubble condition is calculated by the volume acceleration model. The numerical model is validated through comparisons with experimental data from underwater explosions. The numerical results are found to correlate well with the experimental results. Then, the influence of buoyancy parameters and the internal gas wave effect on toroidal bubble dynamics in a gravitational field is investigated.

The external liquid compressibility cannot be ignored because the speed of the bubble jet emerging at the end of bubble collapse reaches hundreds of meters per second. Additionally, when the bubble jet penetrates the surface of a bubble, a toroidal bubble forms and the singly connected flow domain changes to a doubly connected topology. As the Biot–Savart law is based on the assumption of incompressibility, the vortex ring model is very difficult to extend to compressible fluids. This paper describes the use of the boundary integral method to establish a numerical model of a toroidal bubble, considering the external liquid compressibility and the internal gas wave effect. A cut is introduced into the fluid domain so that it can be considered as singly connected, with the discontinuity of velocity across this cut equal to the circulation of the flow. Furthermore, the initial bubble condition is calculated by the volume acceleration model. The numerical model is validated through comparisons with experimental data from underwater explosions. The numerical results are found to correlate well with the experimental results. Then, the influence of buoyancy parameters and the internal gas wave effect on toroidal bubble dynamics in a gravitational field is investigated.

Categories: Latest papers in fluid mechanics

### Shear thickening of dense suspensions: The role of friction

Physics of Fluids, Volume 31, Issue 10, October 2019.

Shear thickening of particle suspensions is caused by a transition between lubricated and frictional contacts between the particles. Using three-dimensional (3D) numerical simulations, we study how the interparticle friction coefficient (μm) influences the effective macroscopic friction coefficient (μ) and hence the microstructure and rheology of dense shear thickening suspensions. We propose expressions for μ in terms of distance to jamming for varying shear stresses and μm values. We find μ to be rather insensitive to interparticle friction, which is perhaps surprising but agrees with recent theory and experiments. Unifying behaviors were observed between the average coordination numbers of particles across a wide range of viscous numbers and μm values.

Shear thickening of particle suspensions is caused by a transition between lubricated and frictional contacts between the particles. Using three-dimensional (3D) numerical simulations, we study how the interparticle friction coefficient (μm) influences the effective macroscopic friction coefficient (μ) and hence the microstructure and rheology of dense shear thickening suspensions. We propose expressions for μ in terms of distance to jamming for varying shear stresses and μm values. We find μ to be rather insensitive to interparticle friction, which is perhaps surprising but agrees with recent theory and experiments. Unifying behaviors were observed between the average coordination numbers of particles across a wide range of viscous numbers and μm values.

Categories: Latest papers in fluid mechanics

### Experimental study of elevation- and depression-type internal solitary waves generated by gravity collapse

Physics of Fluids, Volume 31, Issue 10, October 2019.

Gravity collapse in density stratified fluids can excite internal solitary waves (ISWs), which have been studied by creating initial steplike rectangular disturbances in two-layer fluid systems in stratified fluid flumes. The observation of the elevation- and depression-type ISWs has brought detailed insights into the conditions that generate stable propagation of ISWs and the properties thereof. In this work, the results of such experiments are compared with the wave profile, phase velocity, characteristic frequency, and induced flow velocity calculated by using nonlinear theories. In addition, we analyze the capacity of nonlinear theories to accurately predict the characteristics of stable ISWs generated by gravity collapse. The results show that ISWs are generated by the evolution of the propagating vortex that develops from the vertical shear movement in the mixed-density region of the fluid. The length L and depth D of the initial steplike disturbance and the upper- and lower-layer thicknesses h1 and h2 together determine the amplitude, number, and propagation state of the ISWs that are generated. For small-amplitude elevation- and depression-type ISWs (|a/H| < 0.04, where a and H are the amplitude and total depth of the stratified fluid, respectively), the generated ISWs are consistent with the Korteweg de Vries (KdV) theory. Large-amplitude ISWs, however, tend to be consistent with the extended KdV (eKdV) or Miyata–Choi–Camassa (MCC) theories. For larger (smaller) differences |h2 − h1|, the MCC (eKdV) theory provides the best prediction of large-amplitude ISWs.

Gravity collapse in density stratified fluids can excite internal solitary waves (ISWs), which have been studied by creating initial steplike rectangular disturbances in two-layer fluid systems in stratified fluid flumes. The observation of the elevation- and depression-type ISWs has brought detailed insights into the conditions that generate stable propagation of ISWs and the properties thereof. In this work, the results of such experiments are compared with the wave profile, phase velocity, characteristic frequency, and induced flow velocity calculated by using nonlinear theories. In addition, we analyze the capacity of nonlinear theories to accurately predict the characteristics of stable ISWs generated by gravity collapse. The results show that ISWs are generated by the evolution of the propagating vortex that develops from the vertical shear movement in the mixed-density region of the fluid. The length L and depth D of the initial steplike disturbance and the upper- and lower-layer thicknesses h1 and h2 together determine the amplitude, number, and propagation state of the ISWs that are generated. For small-amplitude elevation- and depression-type ISWs (|a/H| < 0.04, where a and H are the amplitude and total depth of the stratified fluid, respectively), the generated ISWs are consistent with the Korteweg de Vries (KdV) theory. Large-amplitude ISWs, however, tend to be consistent with the extended KdV (eKdV) or Miyata–Choi–Camassa (MCC) theories. For larger (smaller) differences |h2 − h1|, the MCC (eKdV) theory provides the best prediction of large-amplitude ISWs.

Categories: Latest papers in fluid mechanics

### Small-angle light scattering in large-amplitude oscillatory shear

Physics of Fluids, Volume 31, Issue 10, October 2019.

We explore wormlike micellar orientation during oscillatory shear using small-angle light scattering. Previous oscillatory-shear light scattering measurements focused on phase separation in polymeric solutions undergoing shear and none on wormlike micelles. We correlate light scattering videos of wormlike micelles undergoing oscillatory shear with molecular orientation. Specifically, we compare our orientation measurements with the predictions of rigid dumbbell theory. We find that “tulip” shaped scattering patterns caused by micellar orientation are only partially captured by the predicted scattering generated by rigid dumbbell theory. Additionally, we confirm that rigid dumbbell theory cannot describe the “butterfly” shaped scattering patterns arising from concentration fluctuations during micelle breakup. We successfully create a theory to describe both orientation and concentration fluctuation scattering by combining rigid rod Rayleigh-Debye scattering theory with flow induced Helfand-Fredrickson scattering theory.

We explore wormlike micellar orientation during oscillatory shear using small-angle light scattering. Previous oscillatory-shear light scattering measurements focused on phase separation in polymeric solutions undergoing shear and none on wormlike micelles. We correlate light scattering videos of wormlike micelles undergoing oscillatory shear with molecular orientation. Specifically, we compare our orientation measurements with the predictions of rigid dumbbell theory. We find that “tulip” shaped scattering patterns caused by micellar orientation are only partially captured by the predicted scattering generated by rigid dumbbell theory. Additionally, we confirm that rigid dumbbell theory cannot describe the “butterfly” shaped scattering patterns arising from concentration fluctuations during micelle breakup. We successfully create a theory to describe both orientation and concentration fluctuation scattering by combining rigid rod Rayleigh-Debye scattering theory with flow induced Helfand-Fredrickson scattering theory.

Categories: Latest papers in fluid mechanics

### Computational fluid radiative dynamics of the Galileo Jupiter entry

Physics of Fluids, Volume 31, Issue 10, October 2019.

On December 7th, 1995, the Galileo descent probe entered Jupiter’s atmosphere at a relative velocity of 47.4 km s−1. Flight data revealed an unforeseen recession profile: while the stagnation region had been significantly oversized, the shoulder almost completely ablated. In an attempt to understand why numerical predictions diverge from the flight data, several sensitivity studies were performed at the 180 km altitude point. The inaccuracy of the Wilke/Blottner/Eucken model at temperatures above 5000 K was confirmed. When applied to Galileo’s entry, it predicts a narrower shock with higher peak temperatures compared to the Gupta/Yos model. The effects of He and H2 line-by-line radiation were studied. Inclusion of these systems increased radiative heating by 9% at the stagnation point, even when precursor heating is unaccounted for. Otherwise, the internal excitation of H2 due to absorption of radiation originating from the highly emitting shock layer promotes H2 emission before dissociation occurs at the shock, yielding 196% higher radiative heat fluxes. This emphasizes the importance of H2 radiation not only on the recession experienced by Galileo but also for future entries in gas giants. Accordingly, thermal nonequilibrium resulted in 25% lower radiative heating when compared to an equilibrium solution, contrary to previous investigations that neglected H2. Ablation products absorption was shown to counteract the increased emission due to precursor heating of H2. However, the ablation layer temperature must be accurately predicted using a material-response code coupled to the flowfield since radiative heating has been shown to significantly depend on this energy-exchange interaction. Finally, the tangent-slab and ray-tracing models agreed to within 12%.

On December 7th, 1995, the Galileo descent probe entered Jupiter’s atmosphere at a relative velocity of 47.4 km s−1. Flight data revealed an unforeseen recession profile: while the stagnation region had been significantly oversized, the shoulder almost completely ablated. In an attempt to understand why numerical predictions diverge from the flight data, several sensitivity studies were performed at the 180 km altitude point. The inaccuracy of the Wilke/Blottner/Eucken model at temperatures above 5000 K was confirmed. When applied to Galileo’s entry, it predicts a narrower shock with higher peak temperatures compared to the Gupta/Yos model. The effects of He and H2 line-by-line radiation were studied. Inclusion of these systems increased radiative heating by 9% at the stagnation point, even when precursor heating is unaccounted for. Otherwise, the internal excitation of H2 due to absorption of radiation originating from the highly emitting shock layer promotes H2 emission before dissociation occurs at the shock, yielding 196% higher radiative heat fluxes. This emphasizes the importance of H2 radiation not only on the recession experienced by Galileo but also for future entries in gas giants. Accordingly, thermal nonequilibrium resulted in 25% lower radiative heating when compared to an equilibrium solution, contrary to previous investigations that neglected H2. Ablation products absorption was shown to counteract the increased emission due to precursor heating of H2. However, the ablation layer temperature must be accurately predicted using a material-response code coupled to the flowfield since radiative heating has been shown to significantly depend on this energy-exchange interaction. Finally, the tangent-slab and ray-tracing models agreed to within 12%.

Categories: Latest papers in fluid mechanics

### Path to turbulence in a transitional asymmetric planar wake

Physics of Fluids, Volume 31, Issue 10, October 2019.

We report on a transitional, high-resolution direct numerical simulation of a temporally developing planar asymmetric wake at Re = 4000 based on the mass flux deficit. The asymmetric wake is formed by a Blasius and a fully turbulent boundary layer on either side of an infinitely thin splitter plate. Such a setup has direct relevance in low-Reynolds number aeronautics where pressure gradients on an airfoil can relaminarize transitional wall-bounded flows, thus generating a half-laminar/half-turbulent wake. The spreading and normalized turbulence intensity of the asymmetric wake are lower than the initially fully laminar wake but greater than the initially turbulent wake. In the far-field, the flow reaches a fully symmetric and nearly self-similar state with a high level of structural organization, originating from the transition of the laminar side. The structures are generated by the mutual interaction of the turbulent/laminar half-wakes. A forcing from the turbulent side accelerates the development of spanwise-organized structures on the laminar side, which evolve and develop a high-level of spanwise coherence. Unlike the classical transitioning wakes, the pairing of the roller is bypassed. Instead, the spanwise-aligned bulges appear from the initially turbulent half-wake. Under the local shear of the Blasius boundary layer, these bulges undergo a “kinking-and-stretching” mechanism similar to that of the mixing layer. The spanwise organization of the structures is maintained far downstream.

We report on a transitional, high-resolution direct numerical simulation of a temporally developing planar asymmetric wake at Re = 4000 based on the mass flux deficit. The asymmetric wake is formed by a Blasius and a fully turbulent boundary layer on either side of an infinitely thin splitter plate. Such a setup has direct relevance in low-Reynolds number aeronautics where pressure gradients on an airfoil can relaminarize transitional wall-bounded flows, thus generating a half-laminar/half-turbulent wake. The spreading and normalized turbulence intensity of the asymmetric wake are lower than the initially fully laminar wake but greater than the initially turbulent wake. In the far-field, the flow reaches a fully symmetric and nearly self-similar state with a high level of structural organization, originating from the transition of the laminar side. The structures are generated by the mutual interaction of the turbulent/laminar half-wakes. A forcing from the turbulent side accelerates the development of spanwise-organized structures on the laminar side, which evolve and develop a high-level of spanwise coherence. Unlike the classical transitioning wakes, the pairing of the roller is bypassed. Instead, the spanwise-aligned bulges appear from the initially turbulent half-wake. Under the local shear of the Blasius boundary layer, these bulges undergo a “kinking-and-stretching” mechanism similar to that of the mixing layer. The spanwise organization of the structures is maintained far downstream.

Categories: Latest papers in fluid mechanics

### Experimental study of forward-facing cavity with energy deposition in hypersonic flow conditions

Physics of Fluids, Volume 31, Issue 10, October 2019.

Controlling nose surface peak heat fluxes is crucial to the design of hypersonic vehicles. In this study, we report a novel technique of convective heat flux reduction on the nose surface of a spherically blunted cone by employing a forward-facing cavity combined with heat energy deposition inside the cavity. The heat deposition is achieved by the exothermic reaction of a chromium film coated on the cavity surface. The experiments are performed in hypersonic shock tunnels using air as the test gas at free stream stagnation enthalpy conditions of 2.2 ± 0.08 MJ/kg (H1), 3.2 ± 0.018 MJ/kg (H2), and 5.4 ± 0.02 MJ/kg (H3), for a geometry of cavity length to diameter ratio of 1. Schlieren images of the flow are captured using a high-speed camera and a high-power pulsed diode laser light source. The surface heat flux measurements were performed using calibrated platinum thin film sensors. We observed that the heat deposition altered the cavity flow field significantly by lowering the flow oscillation frequency and increasing the shock standoff distances with higher oscillation amplitudes. The overall surface mean heat flux reduction is increased from ≈13% to ≈49% compared to the blunt body geometry, whose nose radius is 30 mm and enhanced with reference to the cavity without heat deposition from ≈15% to ≈35%. Chromium film surface reactions are studied using X-ray Photoelectron spectroscopy, and the results confirm that the exothermic surface reactions of the Cr thin film are attributed to the formation of oxides and nitrides of Cr.

Controlling nose surface peak heat fluxes is crucial to the design of hypersonic vehicles. In this study, we report a novel technique of convective heat flux reduction on the nose surface of a spherically blunted cone by employing a forward-facing cavity combined with heat energy deposition inside the cavity. The heat deposition is achieved by the exothermic reaction of a chromium film coated on the cavity surface. The experiments are performed in hypersonic shock tunnels using air as the test gas at free stream stagnation enthalpy conditions of 2.2 ± 0.08 MJ/kg (H1), 3.2 ± 0.018 MJ/kg (H2), and 5.4 ± 0.02 MJ/kg (H3), for a geometry of cavity length to diameter ratio of 1. Schlieren images of the flow are captured using a high-speed camera and a high-power pulsed diode laser light source. The surface heat flux measurements were performed using calibrated platinum thin film sensors. We observed that the heat deposition altered the cavity flow field significantly by lowering the flow oscillation frequency and increasing the shock standoff distances with higher oscillation amplitudes. The overall surface mean heat flux reduction is increased from ≈13% to ≈49% compared to the blunt body geometry, whose nose radius is 30 mm and enhanced with reference to the cavity without heat deposition from ≈15% to ≈35%. Chromium film surface reactions are studied using X-ray Photoelectron spectroscopy, and the results confirm that the exothermic surface reactions of the Cr thin film are attributed to the formation of oxides and nitrides of Cr.

Categories: Latest papers in fluid mechanics

### Laser induced cavitation: Plasma generation and breakdown shockwave

Physics of Fluids, Volume 31, Issue 10, October 2019.

Laser induced cavitation is one of the effective techniques to generate controlled cavitation bubbles, both for basic study and for applications in different fields of engineering and medicine. Unfortunately, control of bubble formation and symmetry is hardly achieved due to a series of concurrent causes. In particular, the need to focus the laser beam at the bubble formation spot leads, in general, to a conical region proximal to the light source where conditions are met for plasma breakdown. A finite sized region then exists where the electric field may fluctuate depending on several disturbing agents, leading to possible plasma fragmentation and plasma intensity variation. Such irregularities may induce asymmetry in the successive bubble dynamics, a mostly undesired effect if reproducible conditions are sought for. In the present paper, the structure of the breakdown plasma and the ensuing bubble dynamics are analyzed by means of high speed imaging and intensity measurements of the shockwave system launched at breakdown. It is found that the parameters of the system can be tuned to optimize repeatability and sphericity. In particular, symmetric rebound dynamics is achieved almost deterministically when a pointlike plasma is generated at the breakdown threshold energy. Spherical symmetry is also favored by a large focusing angle combined with a relatively large pulse energy, a process which, however, retains a significant level of stochasticity. Outside these special conditions, the elongated and often fragmented conical plasma shape is found to be correlated with anisotropic and multiple breakdown shockwave emission.

Laser induced cavitation is one of the effective techniques to generate controlled cavitation bubbles, both for basic study and for applications in different fields of engineering and medicine. Unfortunately, control of bubble formation and symmetry is hardly achieved due to a series of concurrent causes. In particular, the need to focus the laser beam at the bubble formation spot leads, in general, to a conical region proximal to the light source where conditions are met for plasma breakdown. A finite sized region then exists where the electric field may fluctuate depending on several disturbing agents, leading to possible plasma fragmentation and plasma intensity variation. Such irregularities may induce asymmetry in the successive bubble dynamics, a mostly undesired effect if reproducible conditions are sought for. In the present paper, the structure of the breakdown plasma and the ensuing bubble dynamics are analyzed by means of high speed imaging and intensity measurements of the shockwave system launched at breakdown. It is found that the parameters of the system can be tuned to optimize repeatability and sphericity. In particular, symmetric rebound dynamics is achieved almost deterministically when a pointlike plasma is generated at the breakdown threshold energy. Spherical symmetry is also favored by a large focusing angle combined with a relatively large pulse energy, a process which, however, retains a significant level of stochasticity. Outside these special conditions, the elongated and often fragmented conical plasma shape is found to be correlated with anisotropic and multiple breakdown shockwave emission.

Categories: Latest papers in fluid mechanics

### Separation of particle-laden gravity currents down a slope in linearly stratified environments

Physics of Fluids, Volume 31, Issue 10, October 2019.

Particle-laden gravity currents down a slope in stratified fluid are important processes in lake, estuary, and ocean environments. By conducting direct numerical simulations, this study investigates the detailed dynamic features of lock-exchange particle-laden gravity currents down a slope in linearly stratified environments. The front velocity, separation depth, water entrainment ratio, and energy budget are quantitatively analyzed. This evolutionary process can be divided into three stages, i.e., the acceleration stage, deceleration stage, and separation stage, if the relative stratification parameter is larger than unity. At the acceleration stage, as the collapse of the dense fluid leads to fast entrainment of ambient water into the current, the entrainment ratios have large values, while the settling velocity and the ambient stratification are shown to have less impact on both the entrainment ratios and the front velocity. At the deceleration stage, a larger slope angle, a weaker ambient stratification, and a smaller settling velocity bring a greater front velocity. At the separation stage, the head of the current leaves the slope and intrudes into the environment; meanwhile, the dense fluid at the body of the current also intrudes into the ambient water because the density contrast has largely been reduced due to water entrainment, particle settling, and the density increase in the ambient fluid. A predictive model is developed to determine the separation depth by considering the presence of particles. The fingerlike horizontal intrusions enhance the entrainment effect between the current and the ambient water. A stronger ambient stratification suppresses the conversion of the potential energy to the kinetic energy, while a larger settling velocity accelerates the conversion of the kinetic energy to the dissipated energy.

Particle-laden gravity currents down a slope in stratified fluid are important processes in lake, estuary, and ocean environments. By conducting direct numerical simulations, this study investigates the detailed dynamic features of lock-exchange particle-laden gravity currents down a slope in linearly stratified environments. The front velocity, separation depth, water entrainment ratio, and energy budget are quantitatively analyzed. This evolutionary process can be divided into three stages, i.e., the acceleration stage, deceleration stage, and separation stage, if the relative stratification parameter is larger than unity. At the acceleration stage, as the collapse of the dense fluid leads to fast entrainment of ambient water into the current, the entrainment ratios have large values, while the settling velocity and the ambient stratification are shown to have less impact on both the entrainment ratios and the front velocity. At the deceleration stage, a larger slope angle, a weaker ambient stratification, and a smaller settling velocity bring a greater front velocity. At the separation stage, the head of the current leaves the slope and intrudes into the environment; meanwhile, the dense fluid at the body of the current also intrudes into the ambient water because the density contrast has largely been reduced due to water entrainment, particle settling, and the density increase in the ambient fluid. A predictive model is developed to determine the separation depth by considering the presence of particles. The fingerlike horizontal intrusions enhance the entrainment effect between the current and the ambient water. A stronger ambient stratification suppresses the conversion of the potential energy to the kinetic energy, while a larger settling velocity accelerates the conversion of the kinetic energy to the dissipated energy.

Categories: Latest papers in fluid mechanics

### Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Physics of Fluids, Volume 31, Issue 10, October 2019.

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Categories: Latest papers in fluid mechanics

### A smoothed particle hydrodynamics (SPH) formulation of a two-phase mixture model and its application to turbulent sediment transport

Physics of Fluids, Volume 31, Issue 10, October 2019.

A Smoothed Particle Hydrodynamics (SPH) formulation and implementation of the classical two-phase mixture model are reported, with a particular focus on the turbulent sediment transport and the sediment disturbances generated by moving equipment operating near or on the seabed. In the mixture model, the fluid-particle system is considered to be an equivalent medium whose evolution is described by a set of equations for the mixture continuity and momentum conservation, with the particle volume fraction being tracked by a transport equation. The governing equations are adapted to a Lagrangian, weakly-compressible SPH framework, the turbulence is modeled by a Reynolds-averaged Navier-Stokes approach, and adaptive boundary conditions for shear stress and turbulent quantities are implemented to account for laminar or turbulent local flow conditions. The complex rheological behavior of clay sediment/water mixtures is modeled using a volume fraction, shear rate-dependent viscosity which accounts for the existence of a yield stress. Hence, the proposed work encompasses several challenging modeling aspects: turbulence, non-Newtonian fluid behavior, sediment transport, and fluid-structure interactions. It is then illustrated on diverse cases of interest: a fluid-particle mixture column release, its subsequent turbulent transport and return to a hydrostatic equilibrium, the settling of particle clouds and two cases of particle-driven gravity currents, and their comparisons with available results. Finally, SPH simulation results for the disturbance of a bed of clay sediment/water mixture induced by a moving plate are reported and compared with experiments performed in our laboratory. The proposed SPH two-phase mixture model agrees well with the existing results considered in this study.

A Smoothed Particle Hydrodynamics (SPH) formulation and implementation of the classical two-phase mixture model are reported, with a particular focus on the turbulent sediment transport and the sediment disturbances generated by moving equipment operating near or on the seabed. In the mixture model, the fluid-particle system is considered to be an equivalent medium whose evolution is described by a set of equations for the mixture continuity and momentum conservation, with the particle volume fraction being tracked by a transport equation. The governing equations are adapted to a Lagrangian, weakly-compressible SPH framework, the turbulence is modeled by a Reynolds-averaged Navier-Stokes approach, and adaptive boundary conditions for shear stress and turbulent quantities are implemented to account for laminar or turbulent local flow conditions. The complex rheological behavior of clay sediment/water mixtures is modeled using a volume fraction, shear rate-dependent viscosity which accounts for the existence of a yield stress. Hence, the proposed work encompasses several challenging modeling aspects: turbulence, non-Newtonian fluid behavior, sediment transport, and fluid-structure interactions. It is then illustrated on diverse cases of interest: a fluid-particle mixture column release, its subsequent turbulent transport and return to a hydrostatic equilibrium, the settling of particle clouds and two cases of particle-driven gravity currents, and their comparisons with available results. Finally, SPH simulation results for the disturbance of a bed of clay sediment/water mixture induced by a moving plate are reported and compared with experiments performed in our laboratory. The proposed SPH two-phase mixture model agrees well with the existing results considered in this study.

Categories: Latest papers in fluid mechanics

### Separated boundary layer transition under pressure gradient in the presence of free-stream turbulence

Physics of Fluids, Volume 31, Issue 10, October 2019.

Large-eddy simulation (LES) has been carried out to investigate the transition process of a separated boundary layer on a flat plate. A streamwise pressure distribution is imposed to mimic the suction surface of a low-pressure turbine blade, and the free-stream turbulence intensity at the plate leading edge is 2.9%. A dynamic subgrid scale model is employed in the study, and the current LES results compare well with available experimental data and previous LES results. The transition process has been thoroughly analyzed, and streamwise streaky structures, known as the Klebanoff streaks, have been observed much further upstream of the separation. However, transition occurs in the separated shear layer and is caused by two mechanisms: streamwise streaks and the inviscid K-H instability. Analysis suggests that streamwise streaks play a dominant role in the transition process as those streaks severely disrupt and break up the K-H rolls once they are formed, leading to significant three-dimensional (3D) motions very rapidly. It is also demonstrated in the present study that the usual secondary instability stage under low free-stream turbulence intensity where coherent two-dimensional (2D) spanwise rolls get distorted gradually and eventually broken up into 3D structures has been bypassed.

Large-eddy simulation (LES) has been carried out to investigate the transition process of a separated boundary layer on a flat plate. A streamwise pressure distribution is imposed to mimic the suction surface of a low-pressure turbine blade, and the free-stream turbulence intensity at the plate leading edge is 2.9%. A dynamic subgrid scale model is employed in the study, and the current LES results compare well with available experimental data and previous LES results. The transition process has been thoroughly analyzed, and streamwise streaky structures, known as the Klebanoff streaks, have been observed much further upstream of the separation. However, transition occurs in the separated shear layer and is caused by two mechanisms: streamwise streaks and the inviscid K-H instability. Analysis suggests that streamwise streaks play a dominant role in the transition process as those streaks severely disrupt and break up the K-H rolls once they are formed, leading to significant three-dimensional (3D) motions very rapidly. It is also demonstrated in the present study that the usual secondary instability stage under low free-stream turbulence intensity where coherent two-dimensional (2D) spanwise rolls get distorted gradually and eventually broken up into 3D structures has been bypassed.

Categories: Latest papers in fluid mechanics

### Three-dimensional rotation of paramagnetic and ferromagnetic prolate spheroids in simple shear and uniform magnetic field

Physics of Fluids, Volume 31, Issue 10, October 2019.

We examine a time-dependent, three-dimensional rotation of magnetic ellipsoidal particles in a two-dimensional, simple shear flow and a uniform magnetic field. We consider that the particles have paramagnetic and ferromagnetic properties, and we compare their rotational dynamics due to the strengths and directions of the applied uniform magnetic field. We determine the critical magnetic field strength that can pin the particles’ rotations. Above the critical field strength, the particles’ stable steady angles were determined. In a weak magnetic regime (below the critical field strength), a paramagnetic particle’s polar angle will oscillate toward the magnetic field plane while its azimuthal angle will execute periodic rotations. A ferromagnetic particle’s rotation depends on its initial angles and the magnetic field strength and direction. Even when it is exposed to a critical magnetic field strength, its rotational dynamics will either be pinned in or out of the magnetic field plane. In a weak magnetic regime, a ferromagnetic particle will either execute out-of-plane rotations or will oscillate toward the magnetic field plane and perform periodic rotations. For both particles, we analytically determine the peaks and troughs of their oscillations and study their time-dependent rotations through analytical and numerical analyses.

We examine a time-dependent, three-dimensional rotation of magnetic ellipsoidal particles in a two-dimensional, simple shear flow and a uniform magnetic field. We consider that the particles have paramagnetic and ferromagnetic properties, and we compare their rotational dynamics due to the strengths and directions of the applied uniform magnetic field. We determine the critical magnetic field strength that can pin the particles’ rotations. Above the critical field strength, the particles’ stable steady angles were determined. In a weak magnetic regime (below the critical field strength), a paramagnetic particle’s polar angle will oscillate toward the magnetic field plane while its azimuthal angle will execute periodic rotations. A ferromagnetic particle’s rotation depends on its initial angles and the magnetic field strength and direction. Even when it is exposed to a critical magnetic field strength, its rotational dynamics will either be pinned in or out of the magnetic field plane. In a weak magnetic regime, a ferromagnetic particle will either execute out-of-plane rotations or will oscillate toward the magnetic field plane and perform periodic rotations. For both particles, we analytically determine the peaks and troughs of their oscillations and study their time-dependent rotations through analytical and numerical analyses.

Categories: Latest papers in fluid mechanics

### A generalized minimal residual method-based immersed boundary-lattice Boltzmann flux solver coupled with finite element method for non-linear fluid-structure interaction problems

Physics of Fluids, Volume 31, Issue 10, October 2019.

A generalized minimal residual method (GMRES) based immersed boundary-lattice Boltzmann flux solver (IB-LBFS) coupled with the finite element method (FEM) is presented in this paper for nonlinear fluid-structure interaction (FSI) problems. This approach effectively combines LBFS for the simulation of the flow field, the total Lagrangian FEM for the evaluation of nonlinear structural deformations, and the immersed boundary method (IBM) for the exchange of information on the fluid-solid interface and implementation of boundary conditions. Both the multidirect forcing and the implicit IBM are considered to examine their effects on numerical accuracy and efficiency. Through numerical simulations on flow past a cylinder, it is shown that the implicit IBM with the GMRES for the linear equation system is more efficient and accurate, which justify the conventional misunderstanding that implicit IBM is always less efficient than explicit methods. Numerical simulations on the lid-driven cavity flow in an inclined cavity, incompressible flows of a uniformly accelerated vertical plate, and the flow induced vibrations of a beam attached behind a cylinder in a channel are also successfully carried out and the obtained results are in good agreement with the published data, which verify the reliability and flexibility of the proposed solver for simulating nonlinear FSI problems. After that, the external flows past two hyperelastic cylinder-beam structures at the Reynolds number of 40–300 are studied and three different modes of static, linear, and nonlinear deformations of the beam are obtained, demonstrating its capability of simulating flows with nonlinear FSI problems with multiple deformable objects.

A generalized minimal residual method (GMRES) based immersed boundary-lattice Boltzmann flux solver (IB-LBFS) coupled with the finite element method (FEM) is presented in this paper for nonlinear fluid-structure interaction (FSI) problems. This approach effectively combines LBFS for the simulation of the flow field, the total Lagrangian FEM for the evaluation of nonlinear structural deformations, and the immersed boundary method (IBM) for the exchange of information on the fluid-solid interface and implementation of boundary conditions. Both the multidirect forcing and the implicit IBM are considered to examine their effects on numerical accuracy and efficiency. Through numerical simulations on flow past a cylinder, it is shown that the implicit IBM with the GMRES for the linear equation system is more efficient and accurate, which justify the conventional misunderstanding that implicit IBM is always less efficient than explicit methods. Numerical simulations on the lid-driven cavity flow in an inclined cavity, incompressible flows of a uniformly accelerated vertical plate, and the flow induced vibrations of a beam attached behind a cylinder in a channel are also successfully carried out and the obtained results are in good agreement with the published data, which verify the reliability and flexibility of the proposed solver for simulating nonlinear FSI problems. After that, the external flows past two hyperelastic cylinder-beam structures at the Reynolds number of 40–300 are studied and three different modes of static, linear, and nonlinear deformations of the beam are obtained, demonstrating its capability of simulating flows with nonlinear FSI problems with multiple deformable objects.

Categories: Latest papers in fluid mechanics

### Wake adjustment and vortex-induced vibration of a circular cylinder with a C-shaped plate at a low Reynolds number of 100

Physics of Fluids, Volume 31, Issue 10, October 2019.

The vortex-induced vibration (VIV) of a circular cylinder with a C-shaped plate arranged in its wake at a low Reynolds number of 100 is numerically investigated in this work using the direct numerical simulation. Four typical streamwise spacing ratios of 1.5, 3, 4.5, and 6 are examined in the computations that were carried out for the range of reduced velocities (Ur = 2–12). In terms of shear layer reattachment, wake interference, and vortex shedding, five flow regimes are identified, i.e., the extended-body regime, the front-face reattachment regime, the shear-layer combination regime, the one-row co-shedding regime, and the two-row co-shedding regime. The wake regime is sensitive to the spacing ratio and the reduced velocity. The switching of the flow regime occurs at the transition between the initial VIV branch and the lower VIV branch, accompanying a phase jump of 180°. Furthermore, the shift of the wake regime leads to the prominent fluctuation of the response amplitude. Among the five regimes, the two-row co-shedding regime has the maximum wake width, resulting in the maximum amplitude. In contrast, the shear layers are elongated in the extended-body regime and hence the prolongation of the vortex formation length, contributing to the suppression of VIV. The best suppression is achieved by placing the C-shaped plate behind the cylinder with a spacing of 1.5D, and the reductions in the lift force and the cross-flow amplitude reach 85.5% and 94.5%, respectively.

The vortex-induced vibration (VIV) of a circular cylinder with a C-shaped plate arranged in its wake at a low Reynolds number of 100 is numerically investigated in this work using the direct numerical simulation. Four typical streamwise spacing ratios of 1.5, 3, 4.5, and 6 are examined in the computations that were carried out for the range of reduced velocities (Ur = 2–12). In terms of shear layer reattachment, wake interference, and vortex shedding, five flow regimes are identified, i.e., the extended-body regime, the front-face reattachment regime, the shear-layer combination regime, the one-row co-shedding regime, and the two-row co-shedding regime. The wake regime is sensitive to the spacing ratio and the reduced velocity. The switching of the flow regime occurs at the transition between the initial VIV branch and the lower VIV branch, accompanying a phase jump of 180°. Furthermore, the shift of the wake regime leads to the prominent fluctuation of the response amplitude. Among the five regimes, the two-row co-shedding regime has the maximum wake width, resulting in the maximum amplitude. In contrast, the shear layers are elongated in the extended-body regime and hence the prolongation of the vortex formation length, contributing to the suppression of VIV. The best suppression is achieved by placing the C-shaped plate behind the cylinder with a spacing of 1.5D, and the reductions in the lift force and the cross-flow amplitude reach 85.5% and 94.5%, respectively.

Categories: Latest papers in fluid mechanics

### Edge flame propagation statistics in igniting monodisperse droplet-laden mixtures

Physics of Fluids, Volume 31, Issue 10, October 2019.

The effects of droplet diameter, overall (i.e., liquid+gaseous phases) equivalence ratio, and turbulence intensity on the edge flame propagation statistics for localized forced ignition of uniformly dispersed n-heptane droplet-laden mixtures under homogeneous isotropic decaying turbulence have been analyzed based on direct numerical simulations data. It has been found that the edge flame structure becomes increasingly prominent for large overall equivalence ratios and droplet diameters. Although the mean edge flame speed has been found to be positive and its most probable value remains comparable to the theoretical value for laminar edge flames in purely gaseous mixtures, the mean values have been found to decrease and the probabilities of finding locally negative edge flame speeds have been found to increase with increasing turbulence intensity. The marginal probability density function and curvature and strain rate dependences of the edge flame speed have been found to be principally governed by the displacement speed of the fuel mass fraction isosurface intersecting the stoichiometric mixture fraction isosurface. The displacement speed of the stoichiometric mixture fraction isosurface has also been found to influence the local scalar gradient dependences of the edge flame speed in this configuration, especially for large droplets. The displacement speed of the fuel mass fraction isosurface Sd has been found to be principally governed by leading order contributions of the reaction and molecular diffusion components and the evaporation contribution remains weak in comparison to these leading order contributors. The local edge flame speed exhibits nonlinear curvature and strain rate dependences and its variation with the magnitudes of both fuel mass fraction and mixture fraction gradients has been found to be nonmonotonic for all cases considered here. The correlations of the edge flame speed with curvature, strain rate, and scalar gradient have been found to be qualitatively similar to the corresponding statistics reported in the existing literature for edge flames in purely gaseous mixtures. Additionally, the curvature and tangential strain rate dependences of the edge flame speed have been found to be dependent on the droplet size and overall equivalence ratio, and these dependences become weak for cases with large droplets.

The effects of droplet diameter, overall (i.e., liquid+gaseous phases) equivalence ratio, and turbulence intensity on the edge flame propagation statistics for localized forced ignition of uniformly dispersed n-heptane droplet-laden mixtures under homogeneous isotropic decaying turbulence have been analyzed based on direct numerical simulations data. It has been found that the edge flame structure becomes increasingly prominent for large overall equivalence ratios and droplet diameters. Although the mean edge flame speed has been found to be positive and its most probable value remains comparable to the theoretical value for laminar edge flames in purely gaseous mixtures, the mean values have been found to decrease and the probabilities of finding locally negative edge flame speeds have been found to increase with increasing turbulence intensity. The marginal probability density function and curvature and strain rate dependences of the edge flame speed have been found to be principally governed by the displacement speed of the fuel mass fraction isosurface intersecting the stoichiometric mixture fraction isosurface. The displacement speed of the stoichiometric mixture fraction isosurface has also been found to influence the local scalar gradient dependences of the edge flame speed in this configuration, especially for large droplets. The displacement speed of the fuel mass fraction isosurface Sd has been found to be principally governed by leading order contributions of the reaction and molecular diffusion components and the evaporation contribution remains weak in comparison to these leading order contributors. The local edge flame speed exhibits nonlinear curvature and strain rate dependences and its variation with the magnitudes of both fuel mass fraction and mixture fraction gradients has been found to be nonmonotonic for all cases considered here. The correlations of the edge flame speed with curvature, strain rate, and scalar gradient have been found to be qualitatively similar to the corresponding statistics reported in the existing literature for edge flames in purely gaseous mixtures. Additionally, the curvature and tangential strain rate dependences of the edge flame speed have been found to be dependent on the droplet size and overall equivalence ratio, and these dependences become weak for cases with large droplets.

Categories: Latest papers in fluid mechanics

### Response to “Comment on ‘A periodic grain consolidation model of porous media’” [Phys. Fluids 31, 109101 (2019)]

Physics of Fluids, Volume 31, Issue 10, October 2019.

Categories: Latest papers in fluid mechanics