# 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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### Universal formulation of central-moments-based lattice Boltzmann method with external forcing for the simulation of multiphysics phenomena

Physics of Fluids, Volume 31, Issue 11, November 2019.

The cascaded or central-moments-based lattice Boltzmann method (CM-LBM) is a robust alternative to the more conventional Bhatnagar-Gross-Krook-LBM for the simulation of high-Reynolds number flows. Unfortunately, its original formulation makes its extension to a broader range of physics quite difficult. In addition, it relies on CMs that are derived in an ad hoc manner, i.e., by mimicking those of the Maxwell-Boltzmann distribution to ensure their Galilean invariance a posteriori. This work aims at tackling both issues by deriving Galilean invariant CMs in a systematic and a priori manner, thanks to the Hermite polynomial expansion framework. More specifically, the proposed formalism fully takes advantage of the D3Q27 discretization by relying on the corresponding set of 27 Hermite polynomials (up to the sixth-order) for the derivation of both the discrete equilibrium state and the forcing term in an a priori manner. Furthermore, while keeping the numerical properties of the original CM-LBM, this work leads to a compact and simple algorithm, representing a universal methodology based on CMs and external forcing within the lattice Boltzmann framework. To support these statements, mathematical derivations and a comparative study with four other forcing schemes are provided. The universal nature of the proposed methodology is eventually proved through the simulation of single phase, multiphase (using both pseudopotential and color-gradient formulations), and magnetohydrodynamic flows.

The cascaded or central-moments-based lattice Boltzmann method (CM-LBM) is a robust alternative to the more conventional Bhatnagar-Gross-Krook-LBM for the simulation of high-Reynolds number flows. Unfortunately, its original formulation makes its extension to a broader range of physics quite difficult. In addition, it relies on CMs that are derived in an ad hoc manner, i.e., by mimicking those of the Maxwell-Boltzmann distribution to ensure their Galilean invariance a posteriori. This work aims at tackling both issues by deriving Galilean invariant CMs in a systematic and a priori manner, thanks to the Hermite polynomial expansion framework. More specifically, the proposed formalism fully takes advantage of the D3Q27 discretization by relying on the corresponding set of 27 Hermite polynomials (up to the sixth-order) for the derivation of both the discrete equilibrium state and the forcing term in an a priori manner. Furthermore, while keeping the numerical properties of the original CM-LBM, this work leads to a compact and simple algorithm, representing a universal methodology based on CMs and external forcing within the lattice Boltzmann framework. To support these statements, mathematical derivations and a comparative study with four other forcing schemes are provided. The universal nature of the proposed methodology is eventually proved through the simulation of single phase, multiphase (using both pseudopotential and color-gradient formulations), and magnetohydrodynamic flows.

Categories: Latest papers in fluid mechanics

### The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

Physics of Fluids, Volume 31, Issue 11, November 2019.

The stress singularity for Phan-Thien–Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like [math], where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like [math] and dominates the polymer stresses, which are like [math] for PTT and [math] for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.

The stress singularity for Phan-Thien–Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like [math], where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like [math] and dominates the polymer stresses, which are like [math] for PTT and [math] for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.

Categories: Latest papers in fluid mechanics

### Direct numerical simulations of multi-mode immiscible Rayleigh-Taylor instability with high Reynolds numbers

Physics of Fluids, Volume 31, Issue 11, November 2019.

In this paper, we conduct the high-resolution direct numerical simulations of multimode immiscible Rayleigh-Taylor instability (RTI) with a low Atwood number (At = 0.1) using an improved phase field lattice Boltzmann method. The effect of the Reynolds number on the evolutional interfacial dynamics and bubble/spike amplitudes is first investigated by considering its wide range, from 100 up to a high value of 30 000. The numerical results show that, for sufficiently large Reynolds numbers, a sequence of distinguishing stages in the immiscible RTI can be observed, which includes the linear growth, saturated velocity growth, and chaotic development stages. At the late stage, the RTI induces a complex topology structure of the interface and a mass of dissociative drops can be significantly observed in the system. The accelerations of the bubble and spike front are also measured, and it is reported that their normalized values at the late time are, respectively, approximate to the constant values of around 0.025 and 0.027, exhibiting a terminally quadratic growth. As the Reynolds number is reduced to small ones, the multiple disturbances of the RTI are found to merge into a larger one at the initial stage. Then, the evolutional interfaces display the patterns familiar from the single-mode RTI. The phase interfaces in the whole process become very smooth without the appearance of the breakup phenomenon, and the spike and bubble velocities at the late time approach constant values. Furthermore, we also analyze the effects of the initial conditions in terms of the perturbation wavelength and amplitude, and it is found that the instability undergoes a faster growth at the intermediate stage for a larger wavelength, while the late-time bubble and spike growth rates are insensitive to the changes of the initially perturbed wavelength and amplitude.

In this paper, we conduct the high-resolution direct numerical simulations of multimode immiscible Rayleigh-Taylor instability (RTI) with a low Atwood number (At = 0.1) using an improved phase field lattice Boltzmann method. The effect of the Reynolds number on the evolutional interfacial dynamics and bubble/spike amplitudes is first investigated by considering its wide range, from 100 up to a high value of 30 000. The numerical results show that, for sufficiently large Reynolds numbers, a sequence of distinguishing stages in the immiscible RTI can be observed, which includes the linear growth, saturated velocity growth, and chaotic development stages. At the late stage, the RTI induces a complex topology structure of the interface and a mass of dissociative drops can be significantly observed in the system. The accelerations of the bubble and spike front are also measured, and it is reported that their normalized values at the late time are, respectively, approximate to the constant values of around 0.025 and 0.027, exhibiting a terminally quadratic growth. As the Reynolds number is reduced to small ones, the multiple disturbances of the RTI are found to merge into a larger one at the initial stage. Then, the evolutional interfaces display the patterns familiar from the single-mode RTI. The phase interfaces in the whole process become very smooth without the appearance of the breakup phenomenon, and the spike and bubble velocities at the late time approach constant values. Furthermore, we also analyze the effects of the initial conditions in terms of the perturbation wavelength and amplitude, and it is found that the instability undergoes a faster growth at the intermediate stage for a larger wavelength, while the late-time bubble and spike growth rates are insensitive to the changes of the initially perturbed wavelength and amplitude.

Categories: Latest papers in fluid mechanics

### Wake transitions and laminar separation bubble in the flow past an Eppler 61 airfoil

Physics of Fluids, Volume 31, Issue 11, November 2019.

The flow past an Eppler 61 airfoil at 10° angle to the free-stream is investigated numerically for 100 ≤ Re ≤ 87 000. Vortex shedding is observed beyond Re ∼ 600 and three-dimensionality sets in at Re ∼ 1280.9 via the mode C instability and hairpin vortex structures that grow with an increase in Re. At larger Re, the shear layer vortices, arising from the instability of the separated shear layer, interact with the flow close to the airfoil and cause it to reattach. A Laminar Separation Bubble (LSB) forms at Re ∼ 20 000 and beyond. The airfoil experiences a very significant increase in the lift and a decrease in the drag at the formation of LSB. Although the flow is three-dimensional, the primary mechanism of the formation of LSB appears to be two-dimensional. The length of the LSB decreases with an increase in Re. The variation of the shear layer and primary wake frequency, with Re, is studied. Both follow the power law. However, the variation before and after the formation of LSB is quite different. Unlike the primary wake frequency, the shear layer frequency suffers a jump at the formation of LSB. Wake formation length (Lf), estimated via the spatial distribution of Reynolds stress, exhibits spanwise periodicity at the onset of three-dimensionality.

The flow past an Eppler 61 airfoil at 10° angle to the free-stream is investigated numerically for 100 ≤ Re ≤ 87 000. Vortex shedding is observed beyond Re ∼ 600 and three-dimensionality sets in at Re ∼ 1280.9 via the mode C instability and hairpin vortex structures that grow with an increase in Re. At larger Re, the shear layer vortices, arising from the instability of the separated shear layer, interact with the flow close to the airfoil and cause it to reattach. A Laminar Separation Bubble (LSB) forms at Re ∼ 20 000 and beyond. The airfoil experiences a very significant increase in the lift and a decrease in the drag at the formation of LSB. Although the flow is three-dimensional, the primary mechanism of the formation of LSB appears to be two-dimensional. The length of the LSB decreases with an increase in Re. The variation of the shear layer and primary wake frequency, with Re, is studied. Both follow the power law. However, the variation before and after the formation of LSB is quite different. Unlike the primary wake frequency, the shear layer frequency suffers a jump at the formation of LSB. Wake formation length (Lf), estimated via the spatial distribution of Reynolds stress, exhibits spanwise periodicity at the onset of three-dimensionality.

Categories: Latest papers in fluid mechanics

### Numerical investigations on stability of the spatially oscillating planar two-phase liquid jet in a quiescent atmosphere

Physics of Fluids, Volume 31, Issue 11, November 2019.

The liquid jet when perturbed sinusoidally will lead to instability under certain conditions. Understanding the causes and consequences of such a behavior is still obscure. Hence, numerical investigations are reported in the present study for a two phase spatially oscillating planar jet in a quiescent air. Simulations are performed by solving the Navier-Stokes equations and using the volume of fluid method to track the air-water interface. It is demonstrated that an increase in amplitude of oscillation is caused due to the formation of a low pressure region created by the vortical structures in air near the leading edge of the jet when deflected. This two way coupling between air and water is analyzed with the help of enstrophy, divergence of the Lamb vector, and vortex forces. It is found through a parametric study that surface tension and viscosity stabilize the perturbations in an oscillating planar jet. On the other hand, an increase in Froude number (Fr) initially leads to an augmentation of perturbation amplitude and later causes its damping when surface tension forces become dominant. The numerical analysis for different inlet velocity profiles establishes that the jet is more stable when subjected to a parabolic inlet velocity profile as compared to a uniform profile due to lower relative velocity at the interface. The present work also reveals the role of capillary instability in addition to Kelvin-Helmholtz and Rayleigh-Taylor instabilities that induce primary breakup in the jet.

The liquid jet when perturbed sinusoidally will lead to instability under certain conditions. Understanding the causes and consequences of such a behavior is still obscure. Hence, numerical investigations are reported in the present study for a two phase spatially oscillating planar jet in a quiescent air. Simulations are performed by solving the Navier-Stokes equations and using the volume of fluid method to track the air-water interface. It is demonstrated that an increase in amplitude of oscillation is caused due to the formation of a low pressure region created by the vortical structures in air near the leading edge of the jet when deflected. This two way coupling between air and water is analyzed with the help of enstrophy, divergence of the Lamb vector, and vortex forces. It is found through a parametric study that surface tension and viscosity stabilize the perturbations in an oscillating planar jet. On the other hand, an increase in Froude number (Fr) initially leads to an augmentation of perturbation amplitude and later causes its damping when surface tension forces become dominant. The numerical analysis for different inlet velocity profiles establishes that the jet is more stable when subjected to a parabolic inlet velocity profile as compared to a uniform profile due to lower relative velocity at the interface. The present work also reveals the role of capillary instability in addition to Kelvin-Helmholtz and Rayleigh-Taylor instabilities that induce primary breakup in the jet.

Categories: Latest papers in fluid mechanics

### A dynamic wall model for large eddy simulation of turbulent flow over complex/moving boundaries based on the immersed boundary method

Physics of Fluids, Volume 31, Issue 11, November 2019.

A hybrid immersed boundary/wall-model approach for large eddy simulation is developed for turbulent flows with complex/moving boundaries. The filtered Navier-Stokes equations are solved on a regular Eulerian mesh, with the no-slip condition on the wall imposed through the continuous forcing of the immersed boundary (IB) method. To implement the wall model, the thin boundary layer equation is solved on an embedded mesh refined along the wall-normal direction and a dynamic matching procedure is adopted. Near-wall subgrid-scale viscosity is further modified by taking into account the influence of IB forcing. The proposed method is tested on several numerical examples, including turbulent channel flow, turbulent flow over periodic hills, and turbulent channel flow with a traveling wavy wall. The mean velocity profile and turbulent fluctuations are reasonably well predicted in the canonical channel flow, as well as in flows with a complex/moving boundary and large flow separation.

A hybrid immersed boundary/wall-model approach for large eddy simulation is developed for turbulent flows with complex/moving boundaries. The filtered Navier-Stokes equations are solved on a regular Eulerian mesh, with the no-slip condition on the wall imposed through the continuous forcing of the immersed boundary (IB) method. To implement the wall model, the thin boundary layer equation is solved on an embedded mesh refined along the wall-normal direction and a dynamic matching procedure is adopted. Near-wall subgrid-scale viscosity is further modified by taking into account the influence of IB forcing. The proposed method is tested on several numerical examples, including turbulent channel flow, turbulent flow over periodic hills, and turbulent channel flow with a traveling wavy wall. The mean velocity profile and turbulent fluctuations are reasonably well predicted in the canonical channel flow, as well as in flows with a complex/moving boundary and large flow separation.

Categories: Latest papers in fluid mechanics

### Internal flow during mixing induced in acoustically levitated droplets by mode oscillations

Physics of Fluids, Volume 31, Issue 11, November 2019.

In this paper, we describe a mixing method with mode oscillation on the internal flow field of a levitated droplet. The effect of internal flow on the mixing performance of droplets acoustically levitated via ultrasonic phased arrays remains unclear. To better understand the mixing mechanism of a levitated droplet, clarifying the effect of the internal flow field on droplet mixing from mode oscillation during acoustic levitation is necessary. We used a 50 wt. % glycerol aqueous solution with 6th mode oscillation. We applied particle image velocimetry (PIV) to study the internal flow fields under interfacial oscillation. The PIV results indicated that the visualized flow field enhanced mixing performance with increasing Reynolds number. We demonstrated the nonlinear characteristics of droplet mixing compared to potential flow. The nonlinearity of the droplet oscillation was driven by the nonlinear acoustic field exerted on the levitated droplet. Mode oscillation on the droplet surface induced a pressure gradient and caused internal flow in the droplet. The pressure gradient in the droplet from the interfacial oscillation was quantitatively analyzed. Pressure induced by the interfacial oscillation, which can be roughly ten times larger than the hydrostatic pressure in the droplet, drastically enhanced the mixing performance in the droplet. Our experimental findings provide deeper physical insights into noncontact fluid manipulation for potential lab-in-a-drop applications.

In this paper, we describe a mixing method with mode oscillation on the internal flow field of a levitated droplet. The effect of internal flow on the mixing performance of droplets acoustically levitated via ultrasonic phased arrays remains unclear. To better understand the mixing mechanism of a levitated droplet, clarifying the effect of the internal flow field on droplet mixing from mode oscillation during acoustic levitation is necessary. We used a 50 wt. % glycerol aqueous solution with 6th mode oscillation. We applied particle image velocimetry (PIV) to study the internal flow fields under interfacial oscillation. The PIV results indicated that the visualized flow field enhanced mixing performance with increasing Reynolds number. We demonstrated the nonlinear characteristics of droplet mixing compared to potential flow. The nonlinearity of the droplet oscillation was driven by the nonlinear acoustic field exerted on the levitated droplet. Mode oscillation on the droplet surface induced a pressure gradient and caused internal flow in the droplet. The pressure gradient in the droplet from the interfacial oscillation was quantitatively analyzed. Pressure induced by the interfacial oscillation, which can be roughly ten times larger than the hydrostatic pressure in the droplet, drastically enhanced the mixing performance in the droplet. Our experimental findings provide deeper physical insights into noncontact fluid manipulation for potential lab-in-a-drop applications.

Categories: Latest papers in fluid mechanics

### Review of transport processes and particle self-assembly in acoustically levitated nanofluid droplets

Physics of Fluids, Volume 31, Issue 11, November 2019.

Acoustic levitation has been the cornerstone of many interesting studies across multiple application domains ranging from biomedical engineering to spray drying. In the sphere of colloidal or nanofluid droplets, acoustic levitation allows researchers to probe deep into the physical mechanisms concerning stability, heat and mass transfer processes, and subsequent particle self-assembly. It also offers a plethora of opportunities to custom engineer the transport mechanisms, thereby enabling unique morphological features of the dried precipitate. The high degree of spatial control in a levitator and ease of experimental diagnostics ensure one to study any such transport process in great detail. In this review, we have systematically elucidated three important paradigms in acoustic levitation of nanofluid droplets. First, we have provided a detailed understanding of the fluid mechanics of the process by delving into the pressure and velocity fields the droplet encounters. We have provided descriptions about the key nondimensional number responsible for successful levitation of the droplet. Second, we have studied the transport processes in nanofluid droplets and investigated the important transport mechanisms that are affected by flow and the acoustic field of the levitator. In particular, we look into the heat and mass transfer limitation for particle laden droplets. Third, we have analyzed the particle self-assembly and formation of nanoporous viscoelastic shell. Subsequently, we provided detailed insights into the morphological transitions of the shell through buckling and cavity ingression. We also showcase how the morphology of the shell can be controlled using differential heating and doping. Finally, we conclude by showcasing some unique application context-like photonic crystal behavior that can emerge from unique particle assembly in acoustic levitation.

Acoustic levitation has been the cornerstone of many interesting studies across multiple application domains ranging from biomedical engineering to spray drying. In the sphere of colloidal or nanofluid droplets, acoustic levitation allows researchers to probe deep into the physical mechanisms concerning stability, heat and mass transfer processes, and subsequent particle self-assembly. It also offers a plethora of opportunities to custom engineer the transport mechanisms, thereby enabling unique morphological features of the dried precipitate. The high degree of spatial control in a levitator and ease of experimental diagnostics ensure one to study any such transport process in great detail. In this review, we have systematically elucidated three important paradigms in acoustic levitation of nanofluid droplets. First, we have provided a detailed understanding of the fluid mechanics of the process by delving into the pressure and velocity fields the droplet encounters. We have provided descriptions about the key nondimensional number responsible for successful levitation of the droplet. Second, we have studied the transport processes in nanofluid droplets and investigated the important transport mechanisms that are affected by flow and the acoustic field of the levitator. In particular, we look into the heat and mass transfer limitation for particle laden droplets. Third, we have analyzed the particle self-assembly and formation of nanoporous viscoelastic shell. Subsequently, we provided detailed insights into the morphological transitions of the shell through buckling and cavity ingression. We also showcase how the morphology of the shell can be controlled using differential heating and doping. Finally, we conclude by showcasing some unique application context-like photonic crystal behavior that can emerge from unique particle assembly in acoustic levitation.

Categories: Latest papers in fluid mechanics

### Quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation

Physics of Fluids, Volume 31, Issue 11, November 2019.

We theoretically and experimentally study the quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation in order to examine the role of the gel elasticity in the mass-diffusion-driven process. First, we model the diffusion-driven bubble growth with the classical Epstein-Plesset approach for quasistatic bubble growth, accounting for elasticity of the medium surrounding the bubbles. Next, we devise an experimental technique to visualize the bubble growth in an air-supersaturated gel of different gelatin concentrations and to obtain the growth rate of the bubble. We show, from comparisons between the theory and experiments, that the bubble growth is hindered by the gel elasticity.

We theoretically and experimentally study the quasistatic growth of bubbles in a gelatin gel under dissolved-gas supersaturation in order to examine the role of the gel elasticity in the mass-diffusion-driven process. First, we model the diffusion-driven bubble growth with the classical Epstein-Plesset approach for quasistatic bubble growth, accounting for elasticity of the medium surrounding the bubbles. Next, we devise an experimental technique to visualize the bubble growth in an air-supersaturated gel of different gelatin concentrations and to obtain the growth rate of the bubble. We show, from comparisons between the theory and experiments, that the bubble growth is hindered by the gel elasticity.

Categories: Latest papers in fluid mechanics

### Three-dimensional simulations of turbulent mixing in spherical implosions

Physics of Fluids, Volume 31, Issue 11, November 2019.

High-resolution large-eddy simulations of turbulent mixing at the inner surface of a dense shell which undergoes forced compression by a spherically imploding shock wave are presented. Perturbations on the inner surface grow as a result of Richtmyer-Meshkov and Rayleigh-Taylor instabilities and effects related to geometric convergence and compressibility. Three different cases with different initial surface perturbations, one broadband and two narrowband, are considered. The perturbation power spectrum is related to the mode number via Pℓ ∝ ℓn, where the case with broadband perturbations has n = −2, and modes in the range ℓ = 6–200. The narrowband perturbations have n = 0 and modes in the range ℓ = 50–100 and ℓ = 100–200. The simulations are carried out in spherical coordinates using the PLUTO hydrodynamics code. Results on the mix layer width, molecular mix, and turbulent kinetic energy distribution are presented, demonstrating clearly the impact of the amplitude and spectral form of the initial perturbation on the evolution of integral properties. A recently developed model predicting the growth of single mode perturbations in spherical implosions including shock waves is extended to predict mix layer amplitudes for broadband and narrowband cases, along with a model proposed by Mikaelian [“Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells,” Phys. Rev. A 42, 3400–3420 (1990)]. The resultant layer amplitude predictions from the new model are in good agreement with the numerical results while the longest wavelengths are not yet saturated, while Mikaelian’s model agrees well where the initial modes are saturated.

High-resolution large-eddy simulations of turbulent mixing at the inner surface of a dense shell which undergoes forced compression by a spherically imploding shock wave are presented. Perturbations on the inner surface grow as a result of Richtmyer-Meshkov and Rayleigh-Taylor instabilities and effects related to geometric convergence and compressibility. Three different cases with different initial surface perturbations, one broadband and two narrowband, are considered. The perturbation power spectrum is related to the mode number via Pℓ ∝ ℓn, where the case with broadband perturbations has n = −2, and modes in the range ℓ = 6–200. The narrowband perturbations have n = 0 and modes in the range ℓ = 50–100 and ℓ = 100–200. The simulations are carried out in spherical coordinates using the PLUTO hydrodynamics code. Results on the mix layer width, molecular mix, and turbulent kinetic energy distribution are presented, demonstrating clearly the impact of the amplitude and spectral form of the initial perturbation on the evolution of integral properties. A recently developed model predicting the growth of single mode perturbations in spherical implosions including shock waves is extended to predict mix layer amplitudes for broadband and narrowband cases, along with a model proposed by Mikaelian [“Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells,” Phys. Rev. A 42, 3400–3420 (1990)]. The resultant layer amplitude predictions from the new model are in good agreement with the numerical results while the longest wavelengths are not yet saturated, while Mikaelian’s model agrees well where the initial modes are saturated.

Categories: Latest papers in fluid mechanics

### Numerical and experimental investigation of the stability of a drop in a single-axis acoustic levitator

Physics of Fluids, Volume 31, Issue 11, November 2019.

Acoustic levitation can be employed to hold liquid drops in midair, enabling novel applications in X-ray scattering of proteins, amorphous crystallization of solutions, or contactless mixing. Multiple studies have characterized the physical behavior of a levitated drop inside an acoustic field. Here, we present a numerical and experimental study on the acoustic levitation of water drops in a single-axis acoustic levitator consisting of an ultrasonic transducer and an opposing reflector. Instead of modeling an abstract incident acoustic field, our model considers the shape of the drop as well as the real geometry of the levitator. We also use a high-speed camera to observe the disintegration and the undesired oscillations of the drops. Our results show that the insertion of a drop in the levitator provokes a shift in its resonant frequency that depends on the shape of the drop. Second, the levitation behavior depends on whether the levitator operates slightly below or above the resonance. Third, if the levitator is driven above the resonant frequency, it is possible to levitate with more strength and avoid disintegration of the drop. This research provides an insight on how to achieve more stable experiments that avoid the bursting and undesired oscillations of the levitated sample. We hope that it will facilitate numerous experiments involving acoustically levitated liquid drops.

Acoustic levitation can be employed to hold liquid drops in midair, enabling novel applications in X-ray scattering of proteins, amorphous crystallization of solutions, or contactless mixing. Multiple studies have characterized the physical behavior of a levitated drop inside an acoustic field. Here, we present a numerical and experimental study on the acoustic levitation of water drops in a single-axis acoustic levitator consisting of an ultrasonic transducer and an opposing reflector. Instead of modeling an abstract incident acoustic field, our model considers the shape of the drop as well as the real geometry of the levitator. We also use a high-speed camera to observe the disintegration and the undesired oscillations of the drops. Our results show that the insertion of a drop in the levitator provokes a shift in its resonant frequency that depends on the shape of the drop. Second, the levitation behavior depends on whether the levitator operates slightly below or above the resonance. Third, if the levitator is driven above the resonant frequency, it is possible to levitate with more strength and avoid disintegration of the drop. This research provides an insight on how to achieve more stable experiments that avoid the bursting and undesired oscillations of the levitated sample. We hope that it will facilitate numerous experiments involving acoustically levitated liquid drops.

Categories: Latest papers in fluid mechanics

### Numerical and experimental investigation of dough kneading in a three-dimensional spiral kneader

Physics of Fluids, Volume 31, Issue 11, November 2019.

This work reports on the first three-dimensional viscoelastic dough kneading simulation performed in a spiral kneader. Unstructured tetrahedral grids were generated using ICEM CFD 17.1. Viscoelastic volume-of-fluid simulations were performed using OpenFOAM v.4.0 in combination with the RheoTool package v.2.0. The White-Metzner model with a Bird-Carreau type of shear-rate dependency of the viscosity and relaxation time was utilized to describe the rheology of the dough matrix. We validated our numerical method by simulating the viscoelastic rod climbing benchmark problem in a cylindrical bowl. The temporal evolution of the dough surface was compared with screenshots obtained with a high-speed video camera during laboratory kneading. We found that the curvature of the free surface matches the experimental data well. With our numerical approach, we were able to predict the formation, extension, and breakup of dough pockets. The dough is convected around the inner stationary rod by the rotation of the outer cylindrical bowl, whereas the spiral arm located in between these two parts produces spiral flow patterns. Vertical mixing is not as good as radial mixing and may be enhanced by utilizing two spiral arms similar to hand kneading. Industrial kneading geometries and processes may be further optimized by performing such types of simulations.

This work reports on the first three-dimensional viscoelastic dough kneading simulation performed in a spiral kneader. Unstructured tetrahedral grids were generated using ICEM CFD 17.1. Viscoelastic volume-of-fluid simulations were performed using OpenFOAM v.4.0 in combination with the RheoTool package v.2.0. The White-Metzner model with a Bird-Carreau type of shear-rate dependency of the viscosity and relaxation time was utilized to describe the rheology of the dough matrix. We validated our numerical method by simulating the viscoelastic rod climbing benchmark problem in a cylindrical bowl. The temporal evolution of the dough surface was compared with screenshots obtained with a high-speed video camera during laboratory kneading. We found that the curvature of the free surface matches the experimental data well. With our numerical approach, we were able to predict the formation, extension, and breakup of dough pockets. The dough is convected around the inner stationary rod by the rotation of the outer cylindrical bowl, whereas the spiral arm located in between these two parts produces spiral flow patterns. Vertical mixing is not as good as radial mixing and may be enhanced by utilizing two spiral arms similar to hand kneading. Industrial kneading geometries and processes may be further optimized by performing such types of simulations.

Categories: Latest papers in fluid mechanics

### Variable viscosity effects for the steady flow past a sphere

Physics of Fluids, Volume 31, Issue 11, November 2019.

The effect of a spatially dependent viscosity in the unbounded flow around a rigid spherical particle that translates with constant velocity is investigated theoretically. The variable viscosity emulates the effect of a variable concentration of an additive material in a simple solvent. The analysis is performed utilizing a smooth function for the viscosity of the additive material, which can describe qualitatively both depletion and accumulation phenomena around the particle. Assuming steady state, creeping and isothermal conditions, and no external forces and torques, the momentum and mass balances are a generalization of the classical Stokes equations for which the linearity is preserved. Manipulating suitably the governing partial differential equations, a single ordinary integrodifferential equation for the radial part of the radial velocity component is derived. This equation is solved either numerically using a Chebyshev pseudospectral method or analytically using an asymptotic technique. A decrease in the total drag on the particle as the additive material increases is predicted in the depletion case. In the accumulation case, the total drag may increase or decrease in comparison to the simple Newtonian fluid. Analysis of the total drag to its individual contributions reveals that the friction drag (due to viscous forces) is affected substantially by the change of the viscosity, while the form drag (due to pressure) varies much smoother and milder. Finally, we investigate under which conditions the variable viscosity fluid can be approximated as a constant viscosity fluid with Navier type slip at the wall.

The effect of a spatially dependent viscosity in the unbounded flow around a rigid spherical particle that translates with constant velocity is investigated theoretically. The variable viscosity emulates the effect of a variable concentration of an additive material in a simple solvent. The analysis is performed utilizing a smooth function for the viscosity of the additive material, which can describe qualitatively both depletion and accumulation phenomena around the particle. Assuming steady state, creeping and isothermal conditions, and no external forces and torques, the momentum and mass balances are a generalization of the classical Stokes equations for which the linearity is preserved. Manipulating suitably the governing partial differential equations, a single ordinary integrodifferential equation for the radial part of the radial velocity component is derived. This equation is solved either numerically using a Chebyshev pseudospectral method or analytically using an asymptotic technique. A decrease in the total drag on the particle as the additive material increases is predicted in the depletion case. In the accumulation case, the total drag may increase or decrease in comparison to the simple Newtonian fluid. Analysis of the total drag to its individual contributions reveals that the friction drag (due to viscous forces) is affected substantially by the change of the viscosity, while the form drag (due to pressure) varies much smoother and milder. Finally, we investigate under which conditions the variable viscosity fluid can be approximated as a constant viscosity fluid with Navier type slip at the wall.

Categories: Latest papers in fluid mechanics

### Effect of incoming shear on unsteady wake in flow past surface mounted polygonal prism

Physics of Fluids, Volume 31, Issue 11, November 2019.

Three-dimensional numerical investigations for flow past surface mounted finite height prisms of different cross sections (circular, square, and triangle with apex and base facing) have been carried out using Open Source Field Operation and Manipulation. A general code has been written to consider the effect of impinging shear flow using GroovyBC and is integrated with the existing solver. The effect of impinging shear at the inlet (shear intensity, K) on three-dimensional vortex structures has been explored using iso-Q surfaces for the varying Reynolds number (Re) ranging from 60 to 200 and fixed aspect ratio equal to 5. Three different vortex shedding regimes have been investigated based on values of Re and K, viz., steady flow, symmetric, and asymmetric modes of vortex shedding. Interwave and intrawave frequency modulations and their effects on wake oscillation have been illustrated using Hilbert spectra of transverse velocity signals in the wake. Effects of K and Re on wake oscillation frequency have been presented in terms of marginal spectra of the velocity signals. The extent of nonlinear fluctuations in the wake has been quantified in terms of “degree of stationarity.” Moreover, different modes with their associated frequencies and growth rates that are responsible for transition from the asymmetric to the symmetric mode have been discussed using the computational technique named “Dynamic Mode Decomposition.” Variation in the mean drag coefficient with the change in values of K and Re has also been reported.

Three-dimensional numerical investigations for flow past surface mounted finite height prisms of different cross sections (circular, square, and triangle with apex and base facing) have been carried out using Open Source Field Operation and Manipulation. A general code has been written to consider the effect of impinging shear flow using GroovyBC and is integrated with the existing solver. The effect of impinging shear at the inlet (shear intensity, K) on three-dimensional vortex structures has been explored using iso-Q surfaces for the varying Reynolds number (Re) ranging from 60 to 200 and fixed aspect ratio equal to 5. Three different vortex shedding regimes have been investigated based on values of Re and K, viz., steady flow, symmetric, and asymmetric modes of vortex shedding. Interwave and intrawave frequency modulations and their effects on wake oscillation have been illustrated using Hilbert spectra of transverse velocity signals in the wake. Effects of K and Re on wake oscillation frequency have been presented in terms of marginal spectra of the velocity signals. The extent of nonlinear fluctuations in the wake has been quantified in terms of “degree of stationarity.” Moreover, different modes with their associated frequencies and growth rates that are responsible for transition from the asymmetric to the symmetric mode have been discussed using the computational technique named “Dynamic Mode Decomposition.” Variation in the mean drag coefficient with the change in values of K and Re has also been reported.

Categories: Latest papers in fluid mechanics

### Transition to chaos in a cross-corrugated channel at low Reynolds numbers

Physics of Fluids, Volume 31, Issue 11, November 2019.

The pressure-gradient driven flow in a three-dimensional cross-corrugated channel is investigated based on large eddy simulations. The channel geometry is highly tortuous so that the flow unsteadiness can be triggered at a moderate Reynolds number. The objective of this paper is to provide a better understanding of the laminar-to-chaotic transition in this type of channel. The transition from steady to chaos is found to occur at a low Reynolds number range between 77.4 and 113.1. The nonlinear dynamics is analyzed based on the power spectra of the velocity and reconstructed phase space. The route to chaos is identified, which favors the Ruelle-Takens-Newhouse scenario. Moreover, the transition from quasiperiodic mode to chaotic mode is accompanied with temporal intermittencies. The fluid dynamics is analyzed. It is found that the cross-corrugated geometry prompts a recirculating-and-rotating wake behind each contact corner. Each wake is enfolded by a pair of curved free shear layers. They are destabilized by the Kelvin-Helmholtz instability, leading to the periodic flow oscillation. Subsequently, the centrifugal instability sets in and promotes a type of primary vortex structures, forming streamwise “zig-zag” vortex streets. The competition between the adjacent vortex streets leads to a quasiperiodic flow. Temporal intermittencies emerge as the Reynolds number is increased. Finally, the periodicities in both the streamwise and spanwise directions are broken, and the flow becomes chaotic. When further increasing the Reynolds number to 343.1, Taylor-Görtler-like vortexes and necklace-like vortexes are formed in the channel.

The pressure-gradient driven flow in a three-dimensional cross-corrugated channel is investigated based on large eddy simulations. The channel geometry is highly tortuous so that the flow unsteadiness can be triggered at a moderate Reynolds number. The objective of this paper is to provide a better understanding of the laminar-to-chaotic transition in this type of channel. The transition from steady to chaos is found to occur at a low Reynolds number range between 77.4 and 113.1. The nonlinear dynamics is analyzed based on the power spectra of the velocity and reconstructed phase space. The route to chaos is identified, which favors the Ruelle-Takens-Newhouse scenario. Moreover, the transition from quasiperiodic mode to chaotic mode is accompanied with temporal intermittencies. The fluid dynamics is analyzed. It is found that the cross-corrugated geometry prompts a recirculating-and-rotating wake behind each contact corner. Each wake is enfolded by a pair of curved free shear layers. They are destabilized by the Kelvin-Helmholtz instability, leading to the periodic flow oscillation. Subsequently, the centrifugal instability sets in and promotes a type of primary vortex structures, forming streamwise “zig-zag” vortex streets. The competition between the adjacent vortex streets leads to a quasiperiodic flow. Temporal intermittencies emerge as the Reynolds number is increased. Finally, the periodicities in both the streamwise and spanwise directions are broken, and the flow becomes chaotic. When further increasing the Reynolds number to 343.1, Taylor-Görtler-like vortexes and necklace-like vortexes are formed in the channel.

Categories: Latest papers in fluid mechanics

### Forced convection in a fluid saturated anisotropic porous channel with isoflux boundaries

Physics of Fluids, Volume 31, Issue 11, November 2019.

Fully developed forced convective flow inside a channel filled with a porous material bounded by two impermeable walls subject to a constant heat flux is considered. We consider the Brinkman-Forchheimer equation to govern the flow inside the porous medium, which accounts for the presence of the inertial term. We assume that the porous medium is anisotropic in nature and the permeability is varying along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We have obtained velocity, temperature, and Nusselt number numerically due to the presence of the nonlinear quadratic term in the momentum equation. Asymptotic solutions for small Darcy number (∼10−3) and high Darcy number (∼10) are obtained. The asymptotic behavior of the Nusselt number is discussed. The key purpose of this paper is to study the effect of anisotropic permeability ratio, anisotropic angle, and inertial parameter on the hydrodynamic quantities and heat transfer for the configuration considered. In particular, we observe that for the moderate range of Darcy number, 10−2 to 102, inertia plays a significant role in the Nusselt number. We observe that inclusion of anisotropic permeability enhances the relative heat transfer rate by almost 20% compared to the corresponding isotropic situation. We present a detailed analysis about the inclusion of the permeability matrix in the Brinkman-Forchheimer extended Darcy momentum equation.

Fully developed forced convective flow inside a channel filled with a porous material bounded by two impermeable walls subject to a constant heat flux is considered. We consider the Brinkman-Forchheimer equation to govern the flow inside the porous medium, which accounts for the presence of the inertial term. We assume that the porous medium is anisotropic in nature and the permeability is varying along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We have obtained velocity, temperature, and Nusselt number numerically due to the presence of the nonlinear quadratic term in the momentum equation. Asymptotic solutions for small Darcy number (∼10−3) and high Darcy number (∼10) are obtained. The asymptotic behavior of the Nusselt number is discussed. The key purpose of this paper is to study the effect of anisotropic permeability ratio, anisotropic angle, and inertial parameter on the hydrodynamic quantities and heat transfer for the configuration considered. In particular, we observe that for the moderate range of Darcy number, 10−2 to 102, inertia plays a significant role in the Nusselt number. We observe that inclusion of anisotropic permeability enhances the relative heat transfer rate by almost 20% compared to the corresponding isotropic situation. We present a detailed analysis about the inclusion of the permeability matrix in the Brinkman-Forchheimer extended Darcy momentum equation.

Categories: Latest papers in fluid mechanics

### Effects of the flapping frequency on the thrust performance for three-dimensional bionic multi-wings in a schooling

Physics of Fluids, Volume 31, Issue 11, November 2019.

The excellent performance of many creatures using flapping wings has attracted a lot of research on the performance of a single flapping wing. However, many species generally choose highly organized movements rather than alone in the animal world; there is a very popular and interesting biological clustering phenomenon known as schooling. Understanding the flow mechanisms and thrust performance of flapping multiwings in a schooling could be applied to novel bionic flapping wing aircraft formation design. We perform numerical simulations employing the immersed boundary-lattice Boltzmann method for flow over a single flapping wing and the flapping multiwings in a diamond schooling at different St numbers. Meanwhile, the effects of the difference in individual flapping frequency on the overall propulsive performance of the schooling were investigated. We present the spectra of aerodynamic forces for a single flapping wing and each wing in a diamond schooling at different individual flapping frequencies. Numerical results indicate that the flapping frequency has great effects on the thrust performance of a single wing and the multiwings in a schooling. The average thrust coefficient of a single flapping wing grows with the increase in the St. However, there is an optimal St number to obtain the maximum propulsive efficiency. For a schooling that maintains the same flapping frequency, the overall schooling or each wing in a schooling shows the same trend as a single wing. For a schooling with different individual flapping frequencies, the aerodynamic characteristics of the last downstream wing are more affected by the frequency difference.

The excellent performance of many creatures using flapping wings has attracted a lot of research on the performance of a single flapping wing. However, many species generally choose highly organized movements rather than alone in the animal world; there is a very popular and interesting biological clustering phenomenon known as schooling. Understanding the flow mechanisms and thrust performance of flapping multiwings in a schooling could be applied to novel bionic flapping wing aircraft formation design. We perform numerical simulations employing the immersed boundary-lattice Boltzmann method for flow over a single flapping wing and the flapping multiwings in a diamond schooling at different St numbers. Meanwhile, the effects of the difference in individual flapping frequency on the overall propulsive performance of the schooling were investigated. We present the spectra of aerodynamic forces for a single flapping wing and each wing in a diamond schooling at different individual flapping frequencies. Numerical results indicate that the flapping frequency has great effects on the thrust performance of a single wing and the multiwings in a schooling. The average thrust coefficient of a single flapping wing grows with the increase in the St. However, there is an optimal St number to obtain the maximum propulsive efficiency. For a schooling that maintains the same flapping frequency, the overall schooling or each wing in a schooling shows the same trend as a single wing. For a schooling with different individual flapping frequencies, the aerodynamic characteristics of the last downstream wing are more affected by the frequency difference.

Categories: Latest papers in fluid mechanics

### Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contexts

Physics of Fluids, Volume DSMC2019, Issue 1, November 2019.

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.

Categories: Latest papers in fluid mechanics

### Temperature dependence of shock wave structure in helium and neon

Physics of Fluids, Volume DSMC2019, Issue 1, November 2019.

Temperature dependence of a planar shock wave in helium and neon is studied by the direct simulation Monte Carlo method based on ab initio potentials. A quantum approach to interatomic interactions used here allows us to carry out calculations over a wide temperature range beginning from 1 K up to 5000 K. Moreover, for high temperatures, the quantum approach requires less computational effort than the classical one. Three gaseous species are considered: helium-3, helium-4, and neon. The problem is solved for three values of the Mach number Ma = 2, 5, 10. No influence of the quantum effects has been detected within the numerical error for the temperature of 300 K and higher. For temperatures lower than 300 K, the influence of the quantum effects in helium exceeds the numerical error and reaches 230%. In the case of neon, the quantum effect does not exceed 2% in the whole temperature range considered in the present work. A comparative analysis of flow-fields in shock waves at various temperatures points out a strong influence of the temperature ahead of a shock wave on its structure. The numerical data provided in the supplementary material can be used to model any flow of helium and neon in a wide range of temperatures.

Temperature dependence of a planar shock wave in helium and neon is studied by the direct simulation Monte Carlo method based on ab initio potentials. A quantum approach to interatomic interactions used here allows us to carry out calculations over a wide temperature range beginning from 1 K up to 5000 K. Moreover, for high temperatures, the quantum approach requires less computational effort than the classical one. Three gaseous species are considered: helium-3, helium-4, and neon. The problem is solved for three values of the Mach number Ma = 2, 5, 10. No influence of the quantum effects has been detected within the numerical error for the temperature of 300 K and higher. For temperatures lower than 300 K, the influence of the quantum effects in helium exceeds the numerical error and reaches 230%. In the case of neon, the quantum effect does not exceed 2% in the whole temperature range considered in the present work. A comparative analysis of flow-fields in shock waves at various temperatures points out a strong influence of the temperature ahead of a shock wave on its structure. The numerical data provided in the supplementary material can be used to model any flow of helium and neon in a wide range of temperatures.

Categories: Latest papers in fluid mechanics

### Characterizing deviation from equilibrium in direct simulation Monte Carlo simulations

Physics of Fluids, Volume DSMC2019, Issue 1, November 2019.

A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated using two benchmark problems—Couette flow (over a range of Knudsen numbers) and structure of a normal shock (for upstream Mach numbers of 1.5, 3, and 5)—solved using the DSMC method. While the current work only presents results for benchmark one-dimensional DSMC simulations, the approach can be extended easily to rarefied flows in higher dimensions. Therefore, the proposed parameter has the potential to be used for understanding the nature of VDF and its deviation from near-equilibrium conditions at all locations in a flow field without the need for explicitly sampling the VDF.

A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated using two benchmark problems—Couette flow (over a range of Knudsen numbers) and structure of a normal shock (for upstream Mach numbers of 1.5, 3, and 5)—solved using the DSMC method. While the current work only presents results for benchmark one-dimensional DSMC simulations, the approach can be extended easily to rarefied flows in higher dimensions. Therefore, the proposed parameter has the potential to be used for understanding the nature of VDF and its deviation from near-equilibrium conditions at all locations in a flow field without the need for explicitly sampling the VDF.

Categories: Latest papers in fluid mechanics