# Physics of Fluids

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

*Updated:*2 days 19 hours ago

### Likelihood of survival of coronavirus in a respiratory droplet deposited on a solid surface

Physics of Fluids, Volume 32, Issue 6, June 2020.

We predict and analyze the drying time of respiratory droplets from a COVID-19 infected subject, which is a crucial time to infect another subject. Drying of the droplet is predicted by using a diffusion-limited evaporation model for a sessile droplet placed on a partially wetted surface with a pinned contact line. The variation in droplet volume, contact angle, ambient temperature, and humidity are considered. We analyze the chances of the survival of the virus present in the droplet based on the lifetime of the droplets under several conditions and find that the chances of the survival of the virus are strongly affected by each of these parameters. The magnitude of shear stress inside the droplet computed using the model is not large enough to obliterate the virus. We also explore the relationship between the drying time of a droplet and the growth rate of the spread of COVID-19 in five different cities and find that they are weakly correlated.

We predict and analyze the drying time of respiratory droplets from a COVID-19 infected subject, which is a crucial time to infect another subject. Drying of the droplet is predicted by using a diffusion-limited evaporation model for a sessile droplet placed on a partially wetted surface with a pinned contact line. The variation in droplet volume, contact angle, ambient temperature, and humidity are considered. We analyze the chances of the survival of the virus present in the droplet based on the lifetime of the droplets under several conditions and find that the chances of the survival of the virus are strongly affected by each of these parameters. The magnitude of shear stress inside the droplet computed using the model is not large enough to obliterate the virus. We also explore the relationship between the drying time of a droplet and the growth rate of the spread of COVID-19 in five different cities and find that they are weakly correlated.

Categories: Latest papers in fluid mechanics

### Erratum: “A vortex identification method based on local fluid rotation” [Phys. Fluids 32, 015104 (2020)]

Physics of Fluids, Volume 32, Issue 6, June 2020.

Categories: Latest papers in fluid mechanics

### Field-induced shaping of sessile paramagnetic drops

Physics of Fluids, Volume 32, Issue 6, June 2020.

We use the electromagnetic stress tensor to describe the elongation of paramagnetic drops in uniform magnetic fields. This approach implies a linear relationship between the shape of the drops and the square of the applied field, which we confirm experimentally. We show that this effect scales with the volume and susceptibility of the drops. By using this unified electromagnetic approach, we highlight the potential applications of combining electric and magnetic techniques for controlled shaping of drops in liquid displays, liquid lenses, and chemical mixing of drops in microfluidics.

We use the electromagnetic stress tensor to describe the elongation of paramagnetic drops in uniform magnetic fields. This approach implies a linear relationship between the shape of the drops and the square of the applied field, which we confirm experimentally. We show that this effect scales with the volume and susceptibility of the drops. By using this unified electromagnetic approach, we highlight the potential applications of combining electric and magnetic techniques for controlled shaping of drops in liquid displays, liquid lenses, and chemical mixing of drops in microfluidics.

Categories: Latest papers in fluid mechanics

### Spherical and cylindrical shocks in a non-ideal dusty gas with magnetic field under the action of heat conduction and radiation heat flux

Physics of Fluids, Volume 32, Issue 6, June 2020.

In this article, the propagation of spherical or cylindrical shock waves in a mixture of small solid particles of microsize and a non-ideal gas with conductive as well as radiative heat fluxes are studied under the influence of an azimuthal or axial magnetic field. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow conditions are maintained and the moving piston continuously supplies the variable energy input. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. Heat conduction is expressed in terms of Fourier’s law, and the radiation is considered to be of diffusion type for an optically thick gray gas model. The thermal conductivity and the absorption coefficient are assumed to vary with temperature and density. Numerical calculations have been performed to obtain the flow profiles of variables. The effects of different values of the non-idealness parameter, the strength of the magnetic field, the mass concentration, the ratio of the density of solid particles to the initial density of the gas, the piston velocity index, and the adiabatic index are shown in detail. It is interesting to note that in the presence of an azimuthal magnetic field, the pressure and density vanish at the piston, and hence, a vacuum is formed at the center of symmetry, which is in excellent agreement with the laboratory condition to produce the shock wave.

In this article, the propagation of spherical or cylindrical shock waves in a mixture of small solid particles of microsize and a non-ideal gas with conductive as well as radiative heat fluxes are studied under the influence of an azimuthal or axial magnetic field. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow conditions are maintained and the moving piston continuously supplies the variable energy input. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. Heat conduction is expressed in terms of Fourier’s law, and the radiation is considered to be of diffusion type for an optically thick gray gas model. The thermal conductivity and the absorption coefficient are assumed to vary with temperature and density. Numerical calculations have been performed to obtain the flow profiles of variables. The effects of different values of the non-idealness parameter, the strength of the magnetic field, the mass concentration, the ratio of the density of solid particles to the initial density of the gas, the piston velocity index, and the adiabatic index are shown in detail. It is interesting to note that in the presence of an azimuthal magnetic field, the pressure and density vanish at the piston, and hence, a vacuum is formed at the center of symmetry, which is in excellent agreement with the laboratory condition to produce the shock wave.

Categories: Latest papers in fluid mechanics

### Investigating the effect of reagent parameters on the efficiency of cell lysis within droplets

Physics of Fluids, Volume 32, Issue 6, June 2020.

Cell lysis is an essential primary step in cell assays. In the process of cell lysis, the cell membrane is destroyed and the substances inside the cell are extracted. By utilizing a droplet-based microfluidic platform for cell lysis, the mixer unit that is required for mixing lysis reagents with the cells can be excluded, and thus, the complexity of the fabrication process is reduced. In addition, lysing the cells within the droplets will prevent the cells from exposure to the channel walls, and as a result, cleanliness of the samples and the device is maintained. In this study, cell lysis within the droplets and the parameters affecting the efficiency of this process are investigated using a computational fluid dynamics model. Both the cell solution and the lysis reagents are encapsulated within a droplet and the lysis procedure is simulated inside the droplet. It is known that the secondary flows generated inside the droplet facilitate the mixing process. In this study, we used this effect to improve the efficiency of cell lysis in droplet and the improvement is shown to be attributed to activating an advection mechanism besides the diffusion mechanism inside the droplet. It is also shown that increasing the concentration of the lysis reagents does not have a significant effect on the efficiency of the cell lysis. The effect of the volume fraction of the lysis reagents is also studied, which is shown to be an effective factor in controlling the efficiency of the cell lysis. The lysis procedure is simulated with lysis reagent volume fractions of 50%, 66%, 80%, 90%, and 97%. The lysis efficiency is found to be 38.45%, 45.3%, 57.6%, 82.4%, and 100%, respectively, while the droplet travels through a 2 mm-long microchannel within 0.25 s. This study shows that the droplet microfluidic platform is a powerful tool for performing fast and reliable cell lysis.

Cell lysis is an essential primary step in cell assays. In the process of cell lysis, the cell membrane is destroyed and the substances inside the cell are extracted. By utilizing a droplet-based microfluidic platform for cell lysis, the mixer unit that is required for mixing lysis reagents with the cells can be excluded, and thus, the complexity of the fabrication process is reduced. In addition, lysing the cells within the droplets will prevent the cells from exposure to the channel walls, and as a result, cleanliness of the samples and the device is maintained. In this study, cell lysis within the droplets and the parameters affecting the efficiency of this process are investigated using a computational fluid dynamics model. Both the cell solution and the lysis reagents are encapsulated within a droplet and the lysis procedure is simulated inside the droplet. It is known that the secondary flows generated inside the droplet facilitate the mixing process. In this study, we used this effect to improve the efficiency of cell lysis in droplet and the improvement is shown to be attributed to activating an advection mechanism besides the diffusion mechanism inside the droplet. It is also shown that increasing the concentration of the lysis reagents does not have a significant effect on the efficiency of the cell lysis. The effect of the volume fraction of the lysis reagents is also studied, which is shown to be an effective factor in controlling the efficiency of the cell lysis. The lysis procedure is simulated with lysis reagent volume fractions of 50%, 66%, 80%, 90%, and 97%. The lysis efficiency is found to be 38.45%, 45.3%, 57.6%, 82.4%, and 100%, respectively, while the droplet travels through a 2 mm-long microchannel within 0.25 s. This study shows that the droplet microfluidic platform is a powerful tool for performing fast and reliable cell lysis.

Categories: Latest papers in fluid mechanics

### Steady-state modeling of extrusion cast film process, neck-in phenomenon, and related experimental research: A review

Physics of Fluids, Volume 32, Issue 6, June 2020.

This review provides the current state of knowledge of steady-state modeling of the extrusion cast film process used to produce flat polymer films, as well as related experimental research with a particular focus on the flow instability neck-in. All kinematic models used (i.e., 1-, 1.5-, 2-, and 3-dimensional models) together with the utilized constitutive equations, boundary conditions, simplified assumptions, and numerical methods are carefully summarized. The effect of draw ratio, Deborah number (i.e., melt relaxation time related to experimental time), film cooling, second to first normal stress difference ratio at the die exit, uniaxial extensional strain hardening, and planar-to-uniaxial extensional viscosity ratio on the neck-in is discussed.

This review provides the current state of knowledge of steady-state modeling of the extrusion cast film process used to produce flat polymer films, as well as related experimental research with a particular focus on the flow instability neck-in. All kinematic models used (i.e., 1-, 1.5-, 2-, and 3-dimensional models) together with the utilized constitutive equations, boundary conditions, simplified assumptions, and numerical methods are carefully summarized. The effect of draw ratio, Deborah number (i.e., melt relaxation time related to experimental time), film cooling, second to first normal stress difference ratio at the die exit, uniaxial extensional strain hardening, and planar-to-uniaxial extensional viscosity ratio on the neck-in is discussed.

Categories: Latest papers in fluid mechanics

### Thermal convection studies in liquid metal flow inside a horizontal duct under the influence of transverse magnetic field

Physics of Fluids, Volume 32, Issue 6, June 2020.

An experiment is conducted to study the effect of magnetic field on heat transfer in a magnetohydrodynamic flow of molten lead–lithium, in a stainless steel thin-wall duct, with a uniform surface heat flux at its bottom wall. A novel technique of measuring fluid temperature inside the duct is applied to map the temperature profile in a flow cross section, at both ends of the heated section, using a 4 × 4 array of 16 equidistantly placed thermocouples. Surface temperature, as well as electric potential, is measured at seven different locations along the heated section. Based on the temperature profiles obtained at the outlet of the heated section, various flow regimes have been identified over the experimentally investigated flow parameters. Distinctively, three flow regimes have been recognized depending on the dominance of buoyancy force, electromagnetic force, and inertial force. In order to better understand the experimental data, numerical simulations are performed using COMSOL. In the buoyancy force dominated flow regime, a quasi-two-dimensional turbulence flow is predominant, determining the overall heat transfer mechanism. In the electromagnetic force dominated regime, the perturbation due to buoyancy force is suppressed by the magnetic field. Finally, in the inertial force dominated regime, the electromagnetic force and buoyancy force do not play a significant role in determining the heat transfer mechanism. The transition between observed flow regimes has been identified in terms of Grashof, Reynolds, and Hartmann numbers, and the Nusselt number has been calculated for quantitative comparison of heat transfer in these flow regimes.

An experiment is conducted to study the effect of magnetic field on heat transfer in a magnetohydrodynamic flow of molten lead–lithium, in a stainless steel thin-wall duct, with a uniform surface heat flux at its bottom wall. A novel technique of measuring fluid temperature inside the duct is applied to map the temperature profile in a flow cross section, at both ends of the heated section, using a 4 × 4 array of 16 equidistantly placed thermocouples. Surface temperature, as well as electric potential, is measured at seven different locations along the heated section. Based on the temperature profiles obtained at the outlet of the heated section, various flow regimes have been identified over the experimentally investigated flow parameters. Distinctively, three flow regimes have been recognized depending on the dominance of buoyancy force, electromagnetic force, and inertial force. In order to better understand the experimental data, numerical simulations are performed using COMSOL. In the buoyancy force dominated flow regime, a quasi-two-dimensional turbulence flow is predominant, determining the overall heat transfer mechanism. In the electromagnetic force dominated regime, the perturbation due to buoyancy force is suppressed by the magnetic field. Finally, in the inertial force dominated regime, the electromagnetic force and buoyancy force do not play a significant role in determining the heat transfer mechanism. The transition between observed flow regimes has been identified in terms of Grashof, Reynolds, and Hartmann numbers, and the Nusselt number has been calculated for quantitative comparison of heat transfer in these flow regimes.

Categories: Latest papers in fluid mechanics

### The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence

Physics of Fluids, Volume 32, Issue 6, June 2020.

The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.

The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.

Categories: Latest papers in fluid mechanics

### Acoustic-wave-induced cooling in onset of hypersonic turbulence

Physics of Fluids, Volume 32, Issue 6, June 2020.

We report a newly identified aerodynamic cooling mechanism in the onset of hypersonic wall-bounded turbulence. We first experimentally investigated a flared cone with a smooth surface in a Ma 6 wind tunnel using fast-response pressure sensors, Rayleigh scattering flow visualization, and infrared thermography, which confirmed a cooled region (denoted as CS) downstream of a highly heated region (denoted as HS) on the model, as shown by Franko and Lele [J. Fluid Mech. 730, 491–532 (2013)] and Sivasubramanian and Fasel [J. Fluid Mech. 768, 175–218 (2015)]. We then performed calculations based on both linear stability theory and direct numerical simulations to understand this mechanism. We found that the phase difference ϕpθ between the periodic pressure and dilatation waves plays an important role in the interchange between thermal and mechanical energy in a hypersonic wall-bounded flow. Using porous steel to modify the model surface’s sound admittance, we experimentally show that it is possible to modify the cosine value of ϕpθ to be negative near the wall and thus reduce the temperature growth. These results can provide insight into the thermal protection design of future hypersonic vehicles.

We report a newly identified aerodynamic cooling mechanism in the onset of hypersonic wall-bounded turbulence. We first experimentally investigated a flared cone with a smooth surface in a Ma 6 wind tunnel using fast-response pressure sensors, Rayleigh scattering flow visualization, and infrared thermography, which confirmed a cooled region (denoted as CS) downstream of a highly heated region (denoted as HS) on the model, as shown by Franko and Lele [J. Fluid Mech. 730, 491–532 (2013)] and Sivasubramanian and Fasel [J. Fluid Mech. 768, 175–218 (2015)]. We then performed calculations based on both linear stability theory and direct numerical simulations to understand this mechanism. We found that the phase difference ϕpθ between the periodic pressure and dilatation waves plays an important role in the interchange between thermal and mechanical energy in a hypersonic wall-bounded flow. Using porous steel to modify the model surface’s sound admittance, we experimentally show that it is possible to modify the cosine value of ϕpθ to be negative near the wall and thus reduce the temperature growth. These results can provide insight into the thermal protection design of future hypersonic vehicles.

Categories: Latest papers in fluid mechanics

### Unsteady characteristics of compressible reattaching shear layers

Physics of Fluids, Volume 32, Issue 6, June 2020.

Compressible reattaching flows occur in many aerospace applications and are characterized by high aerothermal loads at reattachment and a broad range of characteristic time scales. The flowfield in this study involves a separated shear layer reattaching onto a 20° ramp at a freestream unit Reynolds number of 6.7 × 107 m−1 and Mach number of 2.9. Delayed detached eddy simulations were carried out using OVERFLOW 2.2K by Leger et al. [“Detached-eddy simulation of a supersonic reattaching shear layer,” AIAA J. 55, 3722–3733 (2017)], and the corresponding results were analyzed to determine the unsteady features of this flowfield using statistical techniques. The simulations were run for long integration times, which ensured sufficient temporal resolution of low-frequency unsteadiness in the range of [math]. The mean flow data highlighted essential flow components such as an expansion fan at the separation point, a large recirculation vortex, and a reattachment shock. Fourier analysis of wall pressure data revealed several high energy frequency bands, which appeared to correspond to separation bubble breathing, shear-layer flapping, and shedding of vortices from the recirculation zone. The spectra also highlighted the possible presence of Rossiter modes, suggesting a feedback mechanism through the recirculation zone. Correlations in the shear-layer and recirculation zone confirmed the presence of large-scale turbulence structures, with an increase in length and time scales downstream. The spectra of reattachment shock location suggested a broadband nature of the oscillations. The role of upstream events on the same was investigated by examining coherence, conditional averages, and correlations. A similar exercise was carried out to investigate the nature of unsteadiness at the mean reattachment location.

Compressible reattaching flows occur in many aerospace applications and are characterized by high aerothermal loads at reattachment and a broad range of characteristic time scales. The flowfield in this study involves a separated shear layer reattaching onto a 20° ramp at a freestream unit Reynolds number of 6.7 × 107 m−1 and Mach number of 2.9. Delayed detached eddy simulations were carried out using OVERFLOW 2.2K by Leger et al. [“Detached-eddy simulation of a supersonic reattaching shear layer,” AIAA J. 55, 3722–3733 (2017)], and the corresponding results were analyzed to determine the unsteady features of this flowfield using statistical techniques. The simulations were run for long integration times, which ensured sufficient temporal resolution of low-frequency unsteadiness in the range of [math]. The mean flow data highlighted essential flow components such as an expansion fan at the separation point, a large recirculation vortex, and a reattachment shock. Fourier analysis of wall pressure data revealed several high energy frequency bands, which appeared to correspond to separation bubble breathing, shear-layer flapping, and shedding of vortices from the recirculation zone. The spectra also highlighted the possible presence of Rossiter modes, suggesting a feedback mechanism through the recirculation zone. Correlations in the shear-layer and recirculation zone confirmed the presence of large-scale turbulence structures, with an increase in length and time scales downstream. The spectra of reattachment shock location suggested a broadband nature of the oscillations. The role of upstream events on the same was investigated by examining coherence, conditional averages, and correlations. A similar exercise was carried out to investigate the nature of unsteadiness at the mean reattachment location.

Categories: Latest papers in fluid mechanics

### Publisher’s Note: “New similarity reductions and exact solutions for helically symmetric viscous flows” [Phys. Fluids 32, 053604 (2020)]

Physics of Fluids, Volume 32, Issue 6, June 2020.

Categories: Latest papers in fluid mechanics

### Optimization of knee joint maximum angle on dolphin kick

Physics of Fluids, Volume 32, Issue 6, June 2020.

The dolphin kick stroke is seen as an important technique used in setting many swimming records because the stroke is used in various races and it has an influence on them. The propulsion of the dolphin kick is determined by a combination of the angles and motions of various joints. We evaluated the knee maximum angle, which is known as one of the most influential factors for propulsion. An evaluation was conducted by computation using the unstructured moving grid finite volume method, which is a highly reproducible approach in the body fitted coordinate system. Furthermore, the approach can prevent accumulation of errors caused by the traveling of grid points. A swimmer model was created using the video footage of a swimmer. To express a traveling swimmer in a pool, the moving computational domain method was adopted. In this method, the motion of the swimmer model itself creates flows around the model instead of putting the model in a uniform flow. Thus, we can calculate acceleration/deceleration and rotational motion of the swimmer model. In this paper, the relationship between the knee joint maximum angle and the ring vortex created by a kick-down motion is also mentioned.

The dolphin kick stroke is seen as an important technique used in setting many swimming records because the stroke is used in various races and it has an influence on them. The propulsion of the dolphin kick is determined by a combination of the angles and motions of various joints. We evaluated the knee maximum angle, which is known as one of the most influential factors for propulsion. An evaluation was conducted by computation using the unstructured moving grid finite volume method, which is a highly reproducible approach in the body fitted coordinate system. Furthermore, the approach can prevent accumulation of errors caused by the traveling of grid points. A swimmer model was created using the video footage of a swimmer. To express a traveling swimmer in a pool, the moving computational domain method was adopted. In this method, the motion of the swimmer model itself creates flows around the model instead of putting the model in a uniform flow. Thus, we can calculate acceleration/deceleration and rotational motion of the swimmer model. In this paper, the relationship between the knee joint maximum angle and the ring vortex created by a kick-down motion is also mentioned.

Categories: Latest papers in fluid mechanics

### The singular hydrodynamic interactions between two spheres in Stokes flow

Physics of Fluids, Volume 32, Issue 6, June 2020.

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

Categories: Latest papers in fluid mechanics

### Non-linear instability analysis of the three-dimensional Navier–Stokes equations: Taylor–Green vortex problem

Physics of Fluids, Volume 32, Issue 6, June 2020.

The three-dimensional (3D) Taylor–Green vortex (TGV) flow is one of the simplest systems to study the generation of different scales of vortices due to the growth of disturbances via effects of different physical mechanisms, including vortex-stretching as an additional source for instability, showing not only the creation of turbulence but also turbulent decay. The strong anisotropic and well-organized flow becomes unstable at early time due to transfer of energy to small scales. The analysis of instability of the periodic 3D TGV flow for the Reynolds number of Re = 2000 is reported here. The direct numerical simulation of the periodic 3D TGV flow is carried out using high accuracy numerical methods for the (vector-potential, vorticity)-formulation, which exactly satisfy the solenoidality condition for vector-potential and vorticity in the computational domain. The evolution of disturbances is examined using the instability theories of the disturbance mechanical energy of the Navier–Stokes equation and the role of rotationality by the disturbance enstrophy transport equation (DETE), which is derived from the enstrophy transport equation. The 3D TGV flow exhibits a tornado-type structure at the center of the domain at intermediate stages of transition to turbulence, which is analyzed using the vortex-identification method of λ2-criteria and the DETE method, as described by Sengupta et al. [“Tracking disturbances in transitional and turbulent flows: Coherent structures,” Phys. Fluids 31(12), 124106 (2019)]. Here, it is observed that the coherent structure is diffused in the λ2-contours. Third generation vortex-identification methods are analyzed for capturing the transient, rotating vortex. The combination of new Omega- and the Liutex/Rortex-methods, as reviewed by C. Liu et al. [“Third generation of vortex identification methods: Omega and Liutex/Rortex based systems,” J. Hydrodyn. 31(2), 205–223 (2019)], captures the evolution of the transient vortex, but the structure identified by these methods appears to be diffused, while the DETE method clearly captures the vortex geometry and highlights the formation of the vortex at early times to aid in predicting the flow evolution.

The three-dimensional (3D) Taylor–Green vortex (TGV) flow is one of the simplest systems to study the generation of different scales of vortices due to the growth of disturbances via effects of different physical mechanisms, including vortex-stretching as an additional source for instability, showing not only the creation of turbulence but also turbulent decay. The strong anisotropic and well-organized flow becomes unstable at early time due to transfer of energy to small scales. The analysis of instability of the periodic 3D TGV flow for the Reynolds number of Re = 2000 is reported here. The direct numerical simulation of the periodic 3D TGV flow is carried out using high accuracy numerical methods for the (vector-potential, vorticity)-formulation, which exactly satisfy the solenoidality condition for vector-potential and vorticity in the computational domain. The evolution of disturbances is examined using the instability theories of the disturbance mechanical energy of the Navier–Stokes equation and the role of rotationality by the disturbance enstrophy transport equation (DETE), which is derived from the enstrophy transport equation. The 3D TGV flow exhibits a tornado-type structure at the center of the domain at intermediate stages of transition to turbulence, which is analyzed using the vortex-identification method of λ2-criteria and the DETE method, as described by Sengupta et al. [“Tracking disturbances in transitional and turbulent flows: Coherent structures,” Phys. Fluids 31(12), 124106 (2019)]. Here, it is observed that the coherent structure is diffused in the λ2-contours. Third generation vortex-identification methods are analyzed for capturing the transient, rotating vortex. The combination of new Omega- and the Liutex/Rortex-methods, as reviewed by C. Liu et al. [“Third generation of vortex identification methods: Omega and Liutex/Rortex based systems,” J. Hydrodyn. 31(2), 205–223 (2019)], captures the evolution of the transient vortex, but the structure identified by these methods appears to be diffused, while the DETE method clearly captures the vortex geometry and highlights the formation of the vortex at early times to aid in predicting the flow evolution.

Categories: Latest papers in fluid mechanics

### A charged finitely extensible dumbbell model: Explaining rheology of dilute polyelectrolyte solutions

Physics of Fluids, Volume 32, Issue 6, June 2020.

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

Categories: Latest papers in fluid mechanics

### Theory-based Reynolds-averaged Navier–Stokes equations with large eddy simulation capability for separated turbulent flow simulations

Physics of Fluids, Volume 32, Issue 6, June 2020.

The prediction and investigation of very high Reynolds number turbulent wall flows pose a significant challenge: experimental studies and large eddy simulation (LES) are often inapplicable to these flows, and Reynolds-averaged Navier–Stokes (RANS) methods often fail to characterize the essential flow characteristics, in particular, for separated flows. These facts explain the need for the development of hybrid RANS-LES methods. The predominant approach to deal with this question is the combination of RANS and LES equation elements. This often implies shortcomings in simulations: the lack of control of modeled and resolved motions, which are involved in hybrid simulations, can lead to inconsistencies and imbalances. A novel approach based on a theoretical solution to the latter problem (referred to as continuous eddy simulation method) is investigated here via simulations of periodic hill flows (involving flow separation and reattachment) for a range of very high Reynolds numbers. We study the mechanism and simulation performance of these new hybrid methods. The results presented demonstrate their excellent performance and advantages to differently designed hybrid methods. We also consider the reliability of flow predictions for which data for model validation are unavailable. Criteria for the reliability of such hybrid simulations are suggested. It is shown that the new hybrid method satisfy these criteria for reliable flow predictions. The results indicate the existence of an asymptotic flow regime far above Reynolds numbers that can be realized in experimental studies and resolved LES.

The prediction and investigation of very high Reynolds number turbulent wall flows pose a significant challenge: experimental studies and large eddy simulation (LES) are often inapplicable to these flows, and Reynolds-averaged Navier–Stokes (RANS) methods often fail to characterize the essential flow characteristics, in particular, for separated flows. These facts explain the need for the development of hybrid RANS-LES methods. The predominant approach to deal with this question is the combination of RANS and LES equation elements. This often implies shortcomings in simulations: the lack of control of modeled and resolved motions, which are involved in hybrid simulations, can lead to inconsistencies and imbalances. A novel approach based on a theoretical solution to the latter problem (referred to as continuous eddy simulation method) is investigated here via simulations of periodic hill flows (involving flow separation and reattachment) for a range of very high Reynolds numbers. We study the mechanism and simulation performance of these new hybrid methods. The results presented demonstrate their excellent performance and advantages to differently designed hybrid methods. We also consider the reliability of flow predictions for which data for model validation are unavailable. Criteria for the reliability of such hybrid simulations are suggested. It is shown that the new hybrid method satisfy these criteria for reliable flow predictions. The results indicate the existence of an asymptotic flow regime far above Reynolds numbers that can be realized in experimental studies and resolved LES.

Categories: Latest papers in fluid mechanics

### Combined wicking and evaporation of NaCl solution with efflorescence formation: The efflorescence exclusion zone

Physics of Fluids, Volume 32, Issue 6, June 2020.

An experiment combining wicking and evaporation of a NaCl solution and leading to the formation of salt efflorescence is presented. The experiment shows that efflorescence develops over the porous medium surface exposed to evaporation except in the bottom region of the sample. This region remains free of efflorescence and is called the exclusion zone. It is shown that the exclusion zone extent depends on the solute concentration in the bottom reservoir. A model is developed, and it helps understand the exclusion phenomenon. The arch shape of the exclusion zone upper boundary is explained and modeled. The study is also seen as a successful test for the model of efflorescence growth driven by evaporation and salt precipitation presented in a previous study. The modeling approach is expected to help develop better models of salt transport with crystallization at the surface of porous media in relation with soil salinization issues or the salt weathering of porous materials.

An experiment combining wicking and evaporation of a NaCl solution and leading to the formation of salt efflorescence is presented. The experiment shows that efflorescence develops over the porous medium surface exposed to evaporation except in the bottom region of the sample. This region remains free of efflorescence and is called the exclusion zone. It is shown that the exclusion zone extent depends on the solute concentration in the bottom reservoir. A model is developed, and it helps understand the exclusion phenomenon. The arch shape of the exclusion zone upper boundary is explained and modeled. The study is also seen as a successful test for the model of efflorescence growth driven by evaporation and salt precipitation presented in a previous study. The modeling approach is expected to help develop better models of salt transport with crystallization at the surface of porous media in relation with soil salinization issues or the salt weathering of porous materials.

Categories: Latest papers in fluid mechanics

### Effect of spanwise domain size on direct numerical simulations of airfoil noise during flow separation and stall

Physics of Fluids, Volume 32, Issue 6, June 2020.

It is well established that a large spanwise domain size is required for accurate numerical simulations of flow past an airfoil in stall. A number of numerical experiments support this conclusion with regard to aerodynamic and turbulence statistics. However, very little has been reported concerning the effect of the span length on aeroacoustic results. In this paper, a detailed investigation is carried out into the influence of spanwise domain length on the prediction of airfoil stall noise when spanwise periodic boundary conditions are applied. This study is based on direct numerical simulations of an NACA0012 airfoil at Re∞ = 50 000 and M∞ = 0.4 at near- and full-stall conditions. There are three main findings in this paper. First, the far-field acoustics are found to be highly sensitive to the choice of spanwise domain length. In the full-stall case, a span length equal to 20% of the airfoil chord over-predicts the radiated noise by more than 10 dB at low-to-medium frequencies relative to a case with one chord length in span. Discrepancies are found to occur for acoustic wavelengths shorter than the spanwise domain size. Under near-stall conditions, the changes caused by the small spanwise domain are noticeably milder. Second, the lower noise predictions from the large span simulation at low frequencies are attributed to the spanwise breakup of large scale flow structures and reduced spanwise coherence near the trailing edge. Third, a more destructive source phase relationship is observed with a large span for medium frequencies, which was inhibited by the periodic forcing in the small span case.

It is well established that a large spanwise domain size is required for accurate numerical simulations of flow past an airfoil in stall. A number of numerical experiments support this conclusion with regard to aerodynamic and turbulence statistics. However, very little has been reported concerning the effect of the span length on aeroacoustic results. In this paper, a detailed investigation is carried out into the influence of spanwise domain length on the prediction of airfoil stall noise when spanwise periodic boundary conditions are applied. This study is based on direct numerical simulations of an NACA0012 airfoil at Re∞ = 50 000 and M∞ = 0.4 at near- and full-stall conditions. There are three main findings in this paper. First, the far-field acoustics are found to be highly sensitive to the choice of spanwise domain length. In the full-stall case, a span length equal to 20% of the airfoil chord over-predicts the radiated noise by more than 10 dB at low-to-medium frequencies relative to a case with one chord length in span. Discrepancies are found to occur for acoustic wavelengths shorter than the spanwise domain size. Under near-stall conditions, the changes caused by the small spanwise domain are noticeably milder. Second, the lower noise predictions from the large span simulation at low frequencies are attributed to the spanwise breakup of large scale flow structures and reduced spanwise coherence near the trailing edge. Third, a more destructive source phase relationship is observed with a large span for medium frequencies, which was inhibited by the periodic forcing in the small span case.

Categories: Latest papers in fluid mechanics

### Eulerian and Lagrangian analysis of coherent structures in separated shear flow by time-resolved particle image velocimetry

Physics of Fluids, Volume 32, Issue 6, June 2020.

We investigate the turbulent shear flow that separates from a two-dimensional backward-facing step. We aim to analyze the unsteady separated and reattaching shear flow in both the Eulerian and Lagrangian frameworks in order to provide complementary insight into the self-sustaining coherent structures and Lagrangian transport of the entrainment process. The Reynolds number is Reh = 1.0 × 103, based on the incoming free-stream velocity and step height. The separated and reattaching shear flow as well as the recirculation region beneath is measured by time-resolved planar particle image velocimetry. As a result, time sequences of velocity vector fields in a horizontal–vertical plane in the center of the step model are obtained. In the Eulerian approach, a set of temporally orthogonal dynamic modes are extracted, and each one represents a single-frequency vortex pattern that neutrally evolves in time. The self-sustaining coherent structures are represented by reduced-order reconstruction of the identified high-amplitude dynamic modes, showing the basic unsteady flapping motion of the shear layer and the vortex rolling-up, pairing, and shedding processes superimposed on it. On the other hand, trajectories of passive fluid tracers depict the Lagrangian fluid transport by the entrainment process in the separated shear flow and identify the time-dependent vortex rolling-up process as well as complex vortex interactions. The contours of the finite-time Lyapunov exponent reveal the unsteady Lagrangian coherent structures that significantly shape the vortex patterns and contribute substantial parts to the fluid entrainment in the shear flow.

We investigate the turbulent shear flow that separates from a two-dimensional backward-facing step. We aim to analyze the unsteady separated and reattaching shear flow in both the Eulerian and Lagrangian frameworks in order to provide complementary insight into the self-sustaining coherent structures and Lagrangian transport of the entrainment process. The Reynolds number is Reh = 1.0 × 103, based on the incoming free-stream velocity and step height. The separated and reattaching shear flow as well as the recirculation region beneath is measured by time-resolved planar particle image velocimetry. As a result, time sequences of velocity vector fields in a horizontal–vertical plane in the center of the step model are obtained. In the Eulerian approach, a set of temporally orthogonal dynamic modes are extracted, and each one represents a single-frequency vortex pattern that neutrally evolves in time. The self-sustaining coherent structures are represented by reduced-order reconstruction of the identified high-amplitude dynamic modes, showing the basic unsteady flapping motion of the shear layer and the vortex rolling-up, pairing, and shedding processes superimposed on it. On the other hand, trajectories of passive fluid tracers depict the Lagrangian fluid transport by the entrainment process in the separated shear flow and identify the time-dependent vortex rolling-up process as well as complex vortex interactions. The contours of the finite-time Lyapunov exponent reveal the unsteady Lagrangian coherent structures that significantly shape the vortex patterns and contribute substantial parts to the fluid entrainment in the shear flow.

Categories: Latest papers in fluid mechanics

### On a non-linear droplet oscillation theory via the unified method

Physics of Fluids, Volume 32, Issue 6, June 2020.

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

Categories: Latest papers in fluid mechanics