# Latest papers in fluid mechanics

### Observation of von Kármán vortex street in a droplet breakup

We report the first observation of von Kármán vortex street in a droplet breakup induced by shock waves and high-speed fluid after the shock. To obtain these data, a novel experimental system is used to record the interaction between the droplet and shock wave and the following fluid. Details of flow fields and transients are also presented and discussed. Based on high-speed shadowgraphs, a Strouhal number of 0.28 ± 0.09 with a Reynolds number of 2817 is obtained, which is in good qualitative agreement with earlier experiments on the von Karman vortex street. The results suggest that the vortex-induced vibration may dominate the oscillation in the horizontal direction, which would result in resonance when the frequency of the oscillating flow matches the natural frequency of the droplet, thereby enhancing the deformation and breakup of the droplet. Our data may be useful to benchmark related multiphase flow models or nonlinear theories.

### A large-eddy simulation study on vortex-ring collisions upon round cylinders

A large-eddy simulation based numerical study was conducted on head-on collisions between vortex-rings and round cylinders. The vortex-ring Reynolds number was Re = 4000, while the ratio of the cylinder diameter to vortex-ring diameter (i.e., diameter ratio, D/d) was varied from 4 to 1. Vortical behavior predicted by the present simulations is observed to agree well with an earlier experimental study [New, T. H., and Zang, B., “Head-on collisions of vortex rings upon round cylinders,” J. Fluid Mech. 833, 648 (2017)]. The present simulations also reveal additional flow details on the vortex dynamics and vortex-core trajectories, which have not been observed previously. First, vortex-dipoles produced by D/d ≤ 2 cylinders are cross sections of elliptic vortex-ringlets formed via vortex disconnection/reconnection of secondary vortex-ring segments. Second, the aspect ratio of the elliptic vortex-ringlets increases when a smaller diameter-ratio cylinder is used, and finally, they undergo axis-switching behavior. Furthermore, up to three sets of tertiary vortex-ring cores are formed along the D/d = 2 and 1 cylinder straight-edges where they subsequently merge with the secondary vortex-ring cores within the confines of the primary vortex-ring cores. This merged vortex core moves toward the collision axis and forms an inner vortex-dipole with a wall separated vortex. Along the convex surface, up to two sets of tertiary vortex-ring cores are observed for D/d = 2 and 1 cylinders, and trajectories of the vortex-dipoles agree well with the past experimental results. These observations support the notion that higher vortex-stretching levels resulting from the use of small diameter-ratio cylinders with higher surface curvatures underpin the wide range of vortical behavior observed here.

### A large-eddy simulation study on vortex-ring collisions upon round cylinders

A large-eddy simulation based numerical study was conducted on head-on collisions between vortex-rings and round cylinders. The vortex-ring Reynolds number was Re = 4000, while the ratio of the cylinder diameter to vortex-ring diameter (i.e., diameter ratio, D/d) was varied from 4 to 1. Vortical behavior predicted by the present simulations is observed to agree well with an earlier experimental study [New, T. H., and Zang, B., “Head-on collisions of vortex rings upon round cylinders,” J. Fluid Mech. 833, 648 (2017)]. The present simulations also reveal additional flow details on the vortex dynamics and vortex-core trajectories, which have not been observed previously. First, vortex-dipoles produced by D/d ≤ 2 cylinders are cross sections of elliptic vortex-ringlets formed via vortex disconnection/reconnection of secondary vortex-ring segments. Second, the aspect ratio of the elliptic vortex-ringlets increases when a smaller diameter-ratio cylinder is used, and finally, they undergo axis-switching behavior. Furthermore, up to three sets of tertiary vortex-ring cores are formed along the D/d = 2 and 1 cylinder straight-edges where they subsequently merge with the secondary vortex-ring cores within the confines of the primary vortex-ring cores. This merged vortex core moves toward the collision axis and forms an inner vortex-dipole with a wall separated vortex. Along the convex surface, up to two sets of tertiary vortex-ring cores are observed for D/d = 2 and 1 cylinders, and trajectories of the vortex-dipoles agree well with the past experimental results. These observations support the notion that higher vortex-stretching levels resulting from the use of small diameter-ratio cylinders with higher surface curvatures underpin the wide range of vortical behavior observed here.

### Continuum simulation of non-local effects in a granular silo discharge flow using a regularized [math] rheology model

The effect of non-local momentum transport on a silo discharge process is numerically investigated using a continuum simulation with the [math] rheology model in which the gradient expansion model is adopted to account for the non-local effects due to the non-uniform field of inertial number I [Bouzid et al., Phys. Rev. Lett. 111, 238301 (2013)]. The singularity for I = 0 is handled with a regularization scheme [Lin and Yang, J. Comput. Phys. 420, 109708 (2020)]. Compared to the discharge dynamics predicted with the local [math] rheology model, the non-local effect enhances the velocity field to increase the volume discharge flow rate Q, especially when the silo orifice L is narrower. Both the local and non-local flow simulations conform to the Beverloo relation [math], where d is the intrinsic grain diameter but the non-local effects appear to lessen the orifice reduction effect coefficient k. The difference between the local and the non-local flow rates [math], made dimensionless by [math], grew monotonically with decreasing L/d with a slight enhancement if the silo height-to-width aspect ratio deviates from unity. Finally, we evaluated the ratio of the shear strain rate to the instantaneous maximum value to define a high-shear zone when the ratio is above a threshold and studied its evolution from the onset to the end of the discharge process. Interestingly, non-local momentum transport helped to reduce the size of the high-shear zone to give a more uniformly fluidized central zone above the orifice.

### Continuum simulation of non-local effects in a granular silo discharge flow using a regularized [math] rheology model

The effect of non-local momentum transport on a silo discharge process is numerically investigated using a continuum simulation with the [math] rheology model in which the gradient expansion model is adopted to account for the non-local effects due to the non-uniform field of inertial number I [Bouzid et al., Phys. Rev. Lett. 111, 238301 (2013)]. The singularity for I = 0 is handled with a regularization scheme [Lin and Yang, J. Comput. Phys. 420, 109708 (2020)]. Compared to the discharge dynamics predicted with the local [math] rheology model, the non-local effect enhances the velocity field to increase the volume discharge flow rate Q, especially when the silo orifice L is narrower. Both the local and non-local flow simulations conform to the Beverloo relation [math], where d is the intrinsic grain diameter but the non-local effects appear to lessen the orifice reduction effect coefficient k. The difference between the local and the non-local flow rates [math], made dimensionless by [math], grew monotonically with decreasing L/d with a slight enhancement if the silo height-to-width aspect ratio deviates from unity. Finally, we evaluated the ratio of the shear strain rate to the instantaneous maximum value to define a high-shear zone when the ratio is above a threshold and studied its evolution from the onset to the end of the discharge process. Interestingly, non-local momentum transport helped to reduce the size of the high-shear zone to give a more uniformly fluidized central zone above the orifice.

### Study of pressure-swirl atomizer with spiral path at design point and outside of design point

Studies on pressure-swirl atomizers have mainly focused on pressure-swirl atomizers with tangential input while there are limited studies on pressure-swirl atomizers with a spiral path. This study applies experimental and computational methods to provide a better understanding of flow development in this type of atomizer at the design point and outside the design point. Experimental results showed that as the pressure increases, the spray cone angle increases. This increase initially occurs with a higher slope and then the slope is toned down. While the drainage coefficient remains constant, the droplet diameter decreases as the pressure increases. It is observed that similar to the pressure-swirl atomizer with tangential input, the pressure-swirl atomizer with a spiral path has a conical hollow spray. At the constant mass flow rate, as the spiral path cross-section, the length of the swirl chamber and orifice diameter increase, the fluid film thickness and average diameter of droplets increase while the spray cone angle reduces. Further, increasing the number of spiral paths causes a wider spray cone angle, higher discharge coefficient, larger fluid film thickness, and larger droplet diameter. The results also showed that increasing the length of the orifice marginally affected the properties of the spray while significantly reducing the spray cone angle. It is important to note that the numerical results are in good agreement with the experimental data.

### Study of pressure-swirl atomizer with spiral path at design point and outside of design point

Studies on pressure-swirl atomizers have mainly focused on pressure-swirl atomizers with tangential input while there are limited studies on pressure-swirl atomizers with a spiral path. This study applies experimental and computational methods to provide a better understanding of flow development in this type of atomizer at the design point and outside the design point. Experimental results showed that as the pressure increases, the spray cone angle increases. This increase initially occurs with a higher slope and then the slope is toned down. While the drainage coefficient remains constant, the droplet diameter decreases as the pressure increases. It is observed that similar to the pressure-swirl atomizer with tangential input, the pressure-swirl atomizer with a spiral path has a conical hollow spray. At the constant mass flow rate, as the spiral path cross-section, the length of the swirl chamber and orifice diameter increase, the fluid film thickness and average diameter of droplets increase while the spray cone angle reduces. Further, increasing the number of spiral paths causes a wider spray cone angle, higher discharge coefficient, larger fluid film thickness, and larger droplet diameter. The results also showed that increasing the length of the orifice marginally affected the properties of the spray while significantly reducing the spray cone angle. It is important to note that the numerical results are in good agreement with the experimental data.

### Eliminating spurious currents in phase-field-theory-based lattice Boltzmann equation for two-phase flows

Spurious currents are frequently observed near an interface in the equilibrium multiphase flow system by lattice Boltzmann equation (LBE). These unphysical phenomena are the result of force imbalance of LBE at a discrete level. In this paper, we develop a well-balanced Cahn–Hilliard equation-based LBE for incompressible two-phase flows. The effects of small initial perturbation of order parameter or dynamic pressure and nonisotropic discretization of gradient in force term on eliminating the spurious currents are investigated systematically. Numerical simulations including flat interface and stationary droplet problems are carried out to show the capability of present LBE for eliminating the spurious currents and its accuracy. The results predicted by the present LBE are compared with those by mixed isotropic discretizations scheme (MIDS) frequently used in the LBE community. Numerical results show that the initial perturbation of order parameter or dynamic pressure and nonisotropic discretization of gradient term has no significant effect on eliminating the spurious currents by present LBE, while the MIDS is sensitive to them.

### Eliminating spurious currents in phase-field-theory-based lattice Boltzmann equation for two-phase flows

Spurious currents are frequently observed near an interface in the equilibrium multiphase flow system by lattice Boltzmann equation (LBE). These unphysical phenomena are the result of force imbalance of LBE at a discrete level. In this paper, we develop a well-balanced Cahn–Hilliard equation-based LBE for incompressible two-phase flows. The effects of small initial perturbation of order parameter or dynamic pressure and nonisotropic discretization of gradient in force term on eliminating the spurious currents are investigated systematically. Numerical simulations including flat interface and stationary droplet problems are carried out to show the capability of present LBE for eliminating the spurious currents and its accuracy. The results predicted by the present LBE are compared with those by mixed isotropic discretizations scheme (MIDS) frequently used in the LBE community. Numerical results show that the initial perturbation of order parameter or dynamic pressure and nonisotropic discretization of gradient term has no significant effect on eliminating the spurious currents by present LBE, while the MIDS is sensitive to them.

### Exact Beltramian solutions for hemispherically bounded cyclonic flowfields

This study focuses on the development of two analytical models that describe the wall-bounded cyclonic flowfield in a hemispherical domain. The closed-form solutions that we pursue are motivated by the need to characterize the swirling bidirectional motion engendered in an upper stage thrust chamber, namely, the VR35K-A VORTEX® engine, conceived and developed by Sierra Nevada Corporation. Our analysis proceeds from the Bragg–Hawthorne formulation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows. In this work, we show that two rotational solutions may be derived for particular specifications of the stagnation head and tangential angular momentum expressions that appear in the Bragg–Hawthorne equation. Then, with the parental streamfunctions in hand, other properties of interest are deduced and these include the main velocity and pressure variations, vorticities, crossflow velocities, extensional and shearing strain rates, virtual energy dissipation rates, and both axial and polar mantle distributions; the latter consist of pairs of rotating, non-translating interfacial layers, separating the so-called inner and outer bidirectional and bipolar regions, respectively. More specifically, two Beltramian solutions are identified with mantles that appear at 50% and 61.06% of the chamber radius, respectively. By matching the outlet radius of the chamber to the mantle location in the equatorial plane, the outflow is permitted to exit the chamber seamlessly. In both models, the axial and radial velocities vary linearly with the injection speed and a characteristic inflow parameter consisting of a geometric ratio of the inlet area and the chamber radius squared.

### Exact Beltramian solutions for hemispherically bounded cyclonic flowfields

This study focuses on the development of two analytical models that describe the wall-bounded cyclonic flowfield in a hemispherical domain. The closed-form solutions that we pursue are motivated by the need to characterize the swirling bidirectional motion engendered in an upper stage thrust chamber, namely, the VR35K-A VORTEX® engine, conceived and developed by Sierra Nevada Corporation. Our analysis proceeds from the Bragg–Hawthorne formulation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows. In this work, we show that two rotational solutions may be derived for particular specifications of the stagnation head and tangential angular momentum expressions that appear in the Bragg–Hawthorne equation. Then, with the parental streamfunctions in hand, other properties of interest are deduced and these include the main velocity and pressure variations, vorticities, crossflow velocities, extensional and shearing strain rates, virtual energy dissipation rates, and both axial and polar mantle distributions; the latter consist of pairs of rotating, non-translating interfacial layers, separating the so-called inner and outer bidirectional and bipolar regions, respectively. More specifically, two Beltramian solutions are identified with mantles that appear at 50% and 61.06% of the chamber radius, respectively. By matching the outlet radius of the chamber to the mantle location in the equatorial plane, the outflow is permitted to exit the chamber seamlessly. In both models, the axial and radial velocities vary linearly with the injection speed and a characteristic inflow parameter consisting of a geometric ratio of the inlet area and the chamber radius squared.

### Exact Beltramian solutions for hemispherically bounded cyclonic flowfields

This study focuses on the development of two analytical models that describe the wall-bounded cyclonic flowfield in a hemispherical domain. The closed-form solutions that we pursue are motivated by the need to characterize the swirling bidirectional motion engendered in an upper stage thrust chamber, namely, the VR35K-A VORTEX® engine, conceived and developed by Sierra Nevada Corporation. Our analysis proceeds from the Bragg–Hawthorne formulation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows. In this work, we show that two rotational solutions may be derived for particular specifications of the stagnation head and tangential angular momentum expressions that appear in the Bragg–Hawthorne equation. Then, with the parental streamfunctions in hand, other properties of interest are deduced and these include the main velocity and pressure variations, vorticities, crossflow velocities, extensional and shearing strain rates, virtual energy dissipation rates, and both axial and polar mantle distributions; the latter consist of pairs of rotating, non-translating interfacial layers, separating the so-called inner and outer bidirectional and bipolar regions, respectively. More specifically, two Beltramian solutions are identified with mantles that appear at 50% and 61.06% of the chamber radius, respectively. By matching the outlet radius of the chamber to the mantle location in the equatorial plane, the outflow is permitted to exit the chamber seamlessly. In both models, the axial and radial velocities vary linearly with the injection speed and a characteristic inflow parameter consisting of a geometric ratio of the inlet area and the chamber radius squared.

### Propagation of two-dimensional vibroacoustic disturbances in a rarefied gas

Author(s): A. Manela and Y. Ben-Ami

The effect of gas rarefaction on the propagation of two-dimensional vibroacoustic disturbances generated by a nonuniformly oscillating plane is studied. Closed-form descriptions are obtained in the free-molecular (left panel of the figure) and continuum (right panel) limits and complemented by direct simulation Monte Carlo results. The impacts of gas rarefaction on signal decay rate and directivity pattern are highlighted and rationalized.

[Phys. Rev. Fluids 6, 093401] Published Wed Sep 01, 2021

### Instabilities of thermocapillary-buoyancy flow in a rotating annular pool for medium-Prandtl-number fluid

Author(s): Hao Liu, Jinchao He, Zhong Zeng, and Zhouhua Qiu

The instabilities of the steady axisymmetric thermocapillary-buoyancy flow in a rotating annular pool were investigated by linear stability analysis. The critical instability parameters for the thermocapillary-buoyancy flow (normal gravity) and the pure thermocapillary flow (microgravity) were compa...

[Phys. Rev. E 104, 035101] Published Wed Sep 01, 2021

### Dynamics of wall jet flow under external pulsation

The present study aims to identify the dominant coherent structures in the wall jet flow subjected to external pulsation at Reynolds number 2600 (based on average jet exit velocity and nozzle diameter). The forcing frequency is varied between St = 0 and 0.99 (St is the Strouhal number). Quadrant analysis is employed to identify the relative contribution of different quadrant motions to the total Reynolds shear stress. Unlike boundary layer flows and channel flows, two distinct regions (inner shear region and outer shear region) are observed in the wall jet flows, and the characteristics of different quadrant motions change in these regions. About 70% of the total shear stress is contributed from the first and fourth quadrants in the outer shear region. We observe that ejection motion is more energetic than sweep motion in the downstream direction, although less frequent. The ejection motion is observed to be more violent for St = 0.44 than for the other frequencies. A proper orthogonal decomposition (POD) analysis reveals while the modal structures exist in different regions of the wall for different jet pulsation; there are no dominant modes (30 modes are required to recover about 75% of the total energy), and the energy is fairly distributed over a large number of modes. However, the POD analyses are capable of capturing the response of the wall jet to different jet pulsations. The most dominant and strongest modal structures are found nearer to the impingement region of the wall when St = 0.44 and the jet tends to laminarize for St > 0.9.

### Dynamics of wall jet flow under external pulsation

The present study aims to identify the dominant coherent structures in the wall jet flow subjected to external pulsation at Reynolds number 2600 (based on average jet exit velocity and nozzle diameter). The forcing frequency is varied between St = 0 and 0.99 (St is the Strouhal number). Quadrant analysis is employed to identify the relative contribution of different quadrant motions to the total Reynolds shear stress. Unlike boundary layer flows and channel flows, two distinct regions (inner shear region and outer shear region) are observed in the wall jet flows, and the characteristics of different quadrant motions change in these regions. About 70% of the total shear stress is contributed from the first and fourth quadrants in the outer shear region. We observe that ejection motion is more energetic than sweep motion in the downstream direction, although less frequent. The ejection motion is observed to be more violent for St = 0.44 than for the other frequencies. A proper orthogonal decomposition (POD) analysis reveals while the modal structures exist in different regions of the wall for different jet pulsation; there are no dominant modes (30 modes are required to recover about 75% of the total energy), and the energy is fairly distributed over a large number of modes. However, the POD analyses are capable of capturing the response of the wall jet to different jet pulsations. The most dominant and strongest modal structures are found nearer to the impingement region of the wall when St = 0.44 and the jet tends to laminarize for St > 0.9.

### Artificial neural network approach for turbulence models: A local framework

Author(s): Chenyue Xie, Xiangming Xiong, and Jianchun Wang

The Reynolds-averaged Navier-Stokes (RANS) unclosed terms can be reconstructed by the local artificial neural network (LANN) based on the local coordinate system which is orthogonal to the curved wall. The LANN model performs better than the Global artifical neural network (GANN), Spalart-Allmaras (SA), and Shear Stress Transport (SST) k−ω models in the predictions of the average velocity, wall-shear stress, and average pressure in the flows over periodic hills. The LANN framework has a great potential to be applied to complex wall-bounded turbulent flows over curved walls.

[Phys. Rev. Fluids 6, 084612] Published Tue Aug 31, 2021

### Shock-induced combustion of aluminum particle clusters investigated with resolved sharp-interface two-dimensional simulations

Author(s): Pratik Das and H. S. Udaykumar

The combustion of aluminum particle clusters in shocked flows is studied with two-dimensional numerical simulations. These simulations examine, for the first time, aspects of the vaporization and burning of molten aluminum particle clusters that are markedly different from an isolated burning aluminum particle. The flame-structure around a particle within a cluster is found to vary along the flow direction: particles at the cluster front end undergo kinetically limited combustion with the formation of a wake flame, while particles located downstream in the cluster burn with an envelope flame indicating combustion limited by transport and mixing.

[Phys. Rev. Fluids 6, 083201] Published Mon Aug 30, 2021

### Modeling transport of soft particles in porous media

Author(s): Shuaijun Li, Hong-hui Yu, and Jing Fan

Flow-driven transport of soft particles in porous media is ubiquitous in many natural and engineering processes, such as the gel treatment for enhanced oil recovery. In many of these processes, injected deformable particles block the pores and thus increase the overall pressure drop and reduce the p...

[Phys. Rev. E 104, 025112] Published Mon Aug 30, 2021

### Comment on “Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields”

Author(s): Juha Honkonen

Recently, asymptotic scaling behavior of the compressible randomly forced Navier-Stokes equation has been analyzed with the use of field-theoretic renormalization group near four dimensions [Phys. Rev. E **95**, 033120 (2017)]. Two infrared stable nontrivial asymptotic scaling patterns have been found a...

[Phys. Rev. E 104, 027101] Published Mon Aug 30, 2021