# 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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### Experimental study on free-surface deformation and forces on a finite submerged plate induced by a solitary wave

Physics of Fluids, Volume 32, Issue 8, August 2020.

Physical experiments are conducted to study the interaction between a solitary wave and a finite horizontal plate submerged at a depth equal to 1/4 of the water depth. The spatial and temporal variation of the three-dimensional (3D) surface deformation is measured using a multi-lens stereo reconstruction system. The hydrodynamic loads are measured by underwater load cells. The plate-induced shoaling causes 3D wave focusing, leading to an increased maximum elevation along the streamwise centerline of the plate. The detailed wave focusing process and the influence of wave amplitude on focusing are presented based on the results obtained through image processing. The characteristics of the horizontal forces, vertical forces, and pitching moments are discussed. A 6-stage loading process based on the maxima of vertical wave force and pitching moment is proposed. It is coupled with the synchronous surface deformation to reveal the loading mechanism. It proves that the vertical wave force on the plate reduces apparently compared with the results from 2D experiments. The surface elevation and wave-induced load data provide an excellent benchmark for further studies on the 3D nonlinear interaction between a solitary wave and a submerged plate.

Physical experiments are conducted to study the interaction between a solitary wave and a finite horizontal plate submerged at a depth equal to 1/4 of the water depth. The spatial and temporal variation of the three-dimensional (3D) surface deformation is measured using a multi-lens stereo reconstruction system. The hydrodynamic loads are measured by underwater load cells. The plate-induced shoaling causes 3D wave focusing, leading to an increased maximum elevation along the streamwise centerline of the plate. The detailed wave focusing process and the influence of wave amplitude on focusing are presented based on the results obtained through image processing. The characteristics of the horizontal forces, vertical forces, and pitching moments are discussed. A 6-stage loading process based on the maxima of vertical wave force and pitching moment is proposed. It is coupled with the synchronous surface deformation to reveal the loading mechanism. It proves that the vertical wave force on the plate reduces apparently compared with the results from 2D experiments. The surface elevation and wave-induced load data provide an excellent benchmark for further studies on the 3D nonlinear interaction between a solitary wave and a submerged plate.

Categories: Latest papers in fluid mechanics

### Uniform flow injection into a turbulent boundary layer for trailing edge noise reduction

Physics of Fluids, Volume 32, Issue 8, August 2020.

The hydrodynamic effects of inclined uniform continuous blowing on a turbulent boundary layer are investigated experimentally. A laminar flow is introduced into the boundary layer through a fence on a flat plate at a distance of 3.38δ0 upstream of the trailing edge. The effects of this open-loop technique of flow control are examined at different angles of injection and at different blowing rates. Surface pressure fluctuations acquired from flush-mounted microphones are used to estimate the trailing edge noise. Injection angles of 70° and 90° in combination with strong blowing rates enable a noise reduction of up to 15 dB at mid and high frequencies, f > 300 Hz. Similar aeroacoustic performances are obtained at a blowing angle of 50° but at lower blowing rates. At low frequencies, a penalty is expected, with the trailing edge noise increasing for all the injection angles and blowing rates under analysis. Mean velocity profiles from hot-wire anemometry reveal that high injection angles and strong blowing rates induce a flow separation, which is expected to deteriorate the aerodynamic performances. When applying a uniform blowing at 50°, however, no flow separation occurs. From an aeroacoustic and aerodynamic point of view, uniform blowing applied at 50° and at intermediate blowing rates is found to be the most promising setting.

The hydrodynamic effects of inclined uniform continuous blowing on a turbulent boundary layer are investigated experimentally. A laminar flow is introduced into the boundary layer through a fence on a flat plate at a distance of 3.38δ0 upstream of the trailing edge. The effects of this open-loop technique of flow control are examined at different angles of injection and at different blowing rates. Surface pressure fluctuations acquired from flush-mounted microphones are used to estimate the trailing edge noise. Injection angles of 70° and 90° in combination with strong blowing rates enable a noise reduction of up to 15 dB at mid and high frequencies, f > 300 Hz. Similar aeroacoustic performances are obtained at a blowing angle of 50° but at lower blowing rates. At low frequencies, a penalty is expected, with the trailing edge noise increasing for all the injection angles and blowing rates under analysis. Mean velocity profiles from hot-wire anemometry reveal that high injection angles and strong blowing rates induce a flow separation, which is expected to deteriorate the aerodynamic performances. When applying a uniform blowing at 50°, however, no flow separation occurs. From an aeroacoustic and aerodynamic point of view, uniform blowing applied at 50° and at intermediate blowing rates is found to be the most promising setting.

Categories: Latest papers in fluid mechanics

### Seiches in lateral cavities with simplified planform geometry: Oscillation modes and synchronization with the vortex shedding

Physics of Fluids, Volume 32, Issue 8, August 2020.

Lateral cavities adjacent to open-channel flows are dead zones located on one side of a main stream. With an approaching flow with a high (subcritical) Froude number, the free-surface of the dead-zone oscillates with high amplitudes and generates a so-called seiche. This configuration is reproduced in a rectangular cavity (with an interface length equal to the main stream channel width) in which the impact of the three dimensionless parameters (Froude number, dimensionless water depth, and geometrical aspect ratio) affecting the seiche is studied experimentally. For all configurations, a natural mode of the cavity is observed, this mode being either longitudinal or transverse, except in the case of a square cavity where bi-directional seiching occurs. Moreover, we show that while the approaching Froude number (0.55 < Fr < 0.7) and dimensionless water depth do not affect the oscillation mode, the selected natural mode is strongly dependent on the geometrical aspect ratio of the cavity. For narrow cavities (small [W + b]/b with W and b the cavity and channel widths, respectively), a longitudinal mode occurs while for wider cavities transverse modes occur, with an increasing number of nodes as the width of the cavity increases. Finally, measuring the time-resolved 2-dimensional field of free-surface deformation in the cavity and the adjacent main stream permits us to identify the vortices shed along the mixing layer at the cavity/main stream interface and thus to analyze the synchronization between the surface oscillation and vortex shedding (at the upstream edge) and impinging (at the downstream edge) processes.

Lateral cavities adjacent to open-channel flows are dead zones located on one side of a main stream. With an approaching flow with a high (subcritical) Froude number, the free-surface of the dead-zone oscillates with high amplitudes and generates a so-called seiche. This configuration is reproduced in a rectangular cavity (with an interface length equal to the main stream channel width) in which the impact of the three dimensionless parameters (Froude number, dimensionless water depth, and geometrical aspect ratio) affecting the seiche is studied experimentally. For all configurations, a natural mode of the cavity is observed, this mode being either longitudinal or transverse, except in the case of a square cavity where bi-directional seiching occurs. Moreover, we show that while the approaching Froude number (0.55 < Fr < 0.7) and dimensionless water depth do not affect the oscillation mode, the selected natural mode is strongly dependent on the geometrical aspect ratio of the cavity. For narrow cavities (small [W + b]/b with W and b the cavity and channel widths, respectively), a longitudinal mode occurs while for wider cavities transverse modes occur, with an increasing number of nodes as the width of the cavity increases. Finally, measuring the time-resolved 2-dimensional field of free-surface deformation in the cavity and the adjacent main stream permits us to identify the vortices shed along the mixing layer at the cavity/main stream interface and thus to analyze the synchronization between the surface oscillation and vortex shedding (at the upstream edge) and impinging (at the downstream edge) processes.

Categories: Latest papers in fluid mechanics

### Effects of elasticity on unsteady forced convective heat transfer of viscoelastic fluid around a cylinder in the presence of viscous dissipation

Physics of Fluids, Volume 32, Issue 8, August 2020.

The effects of elasticity on flow structures and forced convection heat transfer of a viscoelastic fluid around a cylinder were studied numerically using a finite volume method for the first time. In addition, the effects of viscous dissipation on flow and heat transfer parameters were studied over a wide range of Brinkman number (Br). The accurate non-linear Phan-Thien–Tanner model was used to describe the viscoelastic behavior of a polymer solution with a high retardation ratio (β = 0.6) over a wide range of Weissenberg (Wi) and Reynolds numbers (Re). The fluid properties were considered to be temperature-dependent. The results showed that the role of elastic effects in vortex shedding is a depreciator for Wi < 0.5 and an amplifier for Wi > 0.5. The Nusselt number was monotonically increased up to 52.2% with increasing Wi, β, Prandtl number (Pr), and Br over a wide range. The elastic forces affected the physical parameters by contrasting with the viscous forces so that the effects of elastic forces were weaker for high Reynolds numbers. Both hydrodynamic and thermal boundary layers were thinner in a viscoelastic fluid, compared to a Newtonian fluid. In addition, a correlation for variations of the drag coefficient with Wi and two correlations for variations of drag coefficient and Strouhal number with Re for Newtonian and viscoelastic fluid were proposed. Finally, the effects of Wi, Pr, Br, and model parameters on the Nu, St, drag, and lift coefficient and the distribution of stress components, velocity, and temperature were examined.

The effects of elasticity on flow structures and forced convection heat transfer of a viscoelastic fluid around a cylinder were studied numerically using a finite volume method for the first time. In addition, the effects of viscous dissipation on flow and heat transfer parameters were studied over a wide range of Brinkman number (Br). The accurate non-linear Phan-Thien–Tanner model was used to describe the viscoelastic behavior of a polymer solution with a high retardation ratio (β = 0.6) over a wide range of Weissenberg (Wi) and Reynolds numbers (Re). The fluid properties were considered to be temperature-dependent. The results showed that the role of elastic effects in vortex shedding is a depreciator for Wi < 0.5 and an amplifier for Wi > 0.5. The Nusselt number was monotonically increased up to 52.2% with increasing Wi, β, Prandtl number (Pr), and Br over a wide range. The elastic forces affected the physical parameters by contrasting with the viscous forces so that the effects of elastic forces were weaker for high Reynolds numbers. Both hydrodynamic and thermal boundary layers were thinner in a viscoelastic fluid, compared to a Newtonian fluid. In addition, a correlation for variations of the drag coefficient with Wi and two correlations for variations of drag coefficient and Strouhal number with Re for Newtonian and viscoelastic fluid were proposed. Finally, the effects of Wi, Pr, Br, and model parameters on the Nu, St, drag, and lift coefficient and the distribution of stress components, velocity, and temperature were examined.

Categories: Latest papers in fluid mechanics

### Experimental and analytical investigation of meso-scale slug bubble dynamics in a square capillary channel

Physics of Fluids, Volume 32, Issue 8, August 2020.

The flow of dispersed gas bubbles in a viscous liquid can create a bubbly, slug bubble, or elongated bubble flow regime. A slug bubble flow, characterized by bubble sizes equal to the hydraulic diameter of the channel, is a transition regime with a complex local flow field that has received little attention in the past. In this study, dynamics of this flow regime in a square capillary with a cross-sectional area of 3 × 3 mm2 was studied analytically and experimentally. The main geometric parameters of the flow field, such as film and corner thicknesses and volume fraction, were calculated for different flow conditions based on a semi-empirical approach. Using velocity fields from particle image velocimetry (PIV), combined with the analytical equations derived, local mean variations of the film and corner flow thicknesses and velocity were analyzed in detail. Analysis of the results reveals a linear relation between the bubble speed and the liquid slug velocity that was obtained using sum-of-correlation PIV. Local backflow, where the liquid locally flows in the reverse direction, was demonstrated to occur in the slug bubble flow, and the theoretical analysis showed that it can be characterized based on the bubble cross-sectional area and ratio of the liquid slug and bubble speed. The backflow phenomenon is only contributed to the channel corners, where the speed of liquid can increase to the bubble speed. However, there is no evidence of reverse flow in the liquid film for the flow conditions analyzed in this study.

The flow of dispersed gas bubbles in a viscous liquid can create a bubbly, slug bubble, or elongated bubble flow regime. A slug bubble flow, characterized by bubble sizes equal to the hydraulic diameter of the channel, is a transition regime with a complex local flow field that has received little attention in the past. In this study, dynamics of this flow regime in a square capillary with a cross-sectional area of 3 × 3 mm2 was studied analytically and experimentally. The main geometric parameters of the flow field, such as film and corner thicknesses and volume fraction, were calculated for different flow conditions based on a semi-empirical approach. Using velocity fields from particle image velocimetry (PIV), combined with the analytical equations derived, local mean variations of the film and corner flow thicknesses and velocity were analyzed in detail. Analysis of the results reveals a linear relation between the bubble speed and the liquid slug velocity that was obtained using sum-of-correlation PIV. Local backflow, where the liquid locally flows in the reverse direction, was demonstrated to occur in the slug bubble flow, and the theoretical analysis showed that it can be characterized based on the bubble cross-sectional area and ratio of the liquid slug and bubble speed. The backflow phenomenon is only contributed to the channel corners, where the speed of liquid can increase to the bubble speed. However, there is no evidence of reverse flow in the liquid film for the flow conditions analyzed in this study.

Categories: Latest papers in fluid mechanics

### Electrokinetic membrane pumping flow model in a microchannel

Physics of Fluids, Volume 32, Issue 8, August 2020.

A microfluidic pumping flow model driven by electro-osmosis mechanisms is developed to analyze the flow characteristics of aqueous electrolytes. The pumping model is designed based on a single propagative rhythmic membrane contraction applied on the upper wall of a microchannel. The flow lubrication theory coupled with a nonlinear Poisson–Boltzmann equation is used to model the microchannel unsteady creeping flow and to describe the distribution of the electric potential across the electric double layer. A generic solution is obtained for the Poisson–Boltzmann equation without the Debye–Hückel linearization. The effects of zeta potential, Debye length, and electric field on the potential distribution, pressure distribution, velocity profiles, shear stress, and net flow rate are computed and interpreted in detail. The results have shown that this electrokinetic membrane pumping model can be used to understand microlevel transport phenomena in various physiological systems. The proposed model can also be integrated with other microfluidic devices for moving microvolume of liquids in artificial capillaries used in modern biomedical applications.

A microfluidic pumping flow model driven by electro-osmosis mechanisms is developed to analyze the flow characteristics of aqueous electrolytes. The pumping model is designed based on a single propagative rhythmic membrane contraction applied on the upper wall of a microchannel. The flow lubrication theory coupled with a nonlinear Poisson–Boltzmann equation is used to model the microchannel unsteady creeping flow and to describe the distribution of the electric potential across the electric double layer. A generic solution is obtained for the Poisson–Boltzmann equation without the Debye–Hückel linearization. The effects of zeta potential, Debye length, and electric field on the potential distribution, pressure distribution, velocity profiles, shear stress, and net flow rate are computed and interpreted in detail. The results have shown that this electrokinetic membrane pumping model can be used to understand microlevel transport phenomena in various physiological systems. The proposed model can also be integrated with other microfluidic devices for moving microvolume of liquids in artificial capillaries used in modern biomedical applications.

Categories: Latest papers in fluid mechanics

### Approach and breakup of Taylor bubble and Taylor drop in a Hele-Shaw cell

Physics of Fluids, Volume 32, Issue 8, August 2020.

The collision dynamics of a Taylor drop and a Taylor bubble is investigated in an immiscible surrounding liquid. The interaction of both the fluidic entities is studied using experiments and simulation in a vertically aligned Hele-Shaw cell. The steady rise of the bubble and fall of the drop are followed by a deceleration regime where their velocity has decreased due to the pressure imposed by the leading interfaces, indicated by the change in the curvature of their tip. Subsequently, the rapid outward expansion of the bubble has caused the swelling of the tip of the drop. The drop swell has then grown exponentially similar to Rayleigh–Taylor instability and resulted in a split of the bubble into two volumes.

The collision dynamics of a Taylor drop and a Taylor bubble is investigated in an immiscible surrounding liquid. The interaction of both the fluidic entities is studied using experiments and simulation in a vertically aligned Hele-Shaw cell. The steady rise of the bubble and fall of the drop are followed by a deceleration regime where their velocity has decreased due to the pressure imposed by the leading interfaces, indicated by the change in the curvature of their tip. Subsequently, the rapid outward expansion of the bubble has caused the swelling of the tip of the drop. The drop swell has then grown exponentially similar to Rayleigh–Taylor instability and resulted in a split of the bubble into two volumes.

Categories: Latest papers in fluid mechanics

### Liquid slippage on rough hydrophobic surfaces with and without entrapped bubbles

Physics of Fluids, Volume 32, Issue 8, August 2020.

The process of liquid slip on rough-walled hydrophobic surfaces with and without entrapped gas bubbles is modeled. Here, starting with the Navier–Stokes equations, a set of partial differential equations (PDE) and boundary conditions for the general effective slip tensor of a rough hydrophobic surface are constructed by an asymptotic analysis. The intrinsic slip and surface roughness are considered as the characteristics of the surface. The solution is based on a weak variation form that fully recovers the set of PDE and Navier slip boundary. For the surface with entrapped bubbles, a semi-analytical model based on eigenfunction expansion is developed. In addition to the surface characteristics, the size and contact angle of the bubbles are considered in the semi-analytical solution. Both models are validated with the published data as well as direct numerical simulation. Based on the model results, we present correlations of effective slip length with surface characteristics and entrapped bubbles. We found that surface roughness reduces liquid slippage on a surface. However, if the asperities on a surface are filled with gas bubbles, the effective slip length can significantly increase as long as the bubble contact angle is small. Interestingly, bubbles with a larger contact angle could act inversely and change a hydrophobic surface with a large intrinsic slip to a no-slip or even a sticky surface. These results shed light on the controversy over the order of magnitude of the actual slip length of water flow in carbon-based nanotubes and nanochannels. The model results also help understand the anomalies of high water production and high amounts of hydraulic fracturing fluid leak-off observed in tight oil and shale gas reservoirs.

The process of liquid slip on rough-walled hydrophobic surfaces with and without entrapped gas bubbles is modeled. Here, starting with the Navier–Stokes equations, a set of partial differential equations (PDE) and boundary conditions for the general effective slip tensor of a rough hydrophobic surface are constructed by an asymptotic analysis. The intrinsic slip and surface roughness are considered as the characteristics of the surface. The solution is based on a weak variation form that fully recovers the set of PDE and Navier slip boundary. For the surface with entrapped bubbles, a semi-analytical model based on eigenfunction expansion is developed. In addition to the surface characteristics, the size and contact angle of the bubbles are considered in the semi-analytical solution. Both models are validated with the published data as well as direct numerical simulation. Based on the model results, we present correlations of effective slip length with surface characteristics and entrapped bubbles. We found that surface roughness reduces liquid slippage on a surface. However, if the asperities on a surface are filled with gas bubbles, the effective slip length can significantly increase as long as the bubble contact angle is small. Interestingly, bubbles with a larger contact angle could act inversely and change a hydrophobic surface with a large intrinsic slip to a no-slip or even a sticky surface. These results shed light on the controversy over the order of magnitude of the actual slip length of water flow in carbon-based nanotubes and nanochannels. The model results also help understand the anomalies of high water production and high amounts of hydraulic fracturing fluid leak-off observed in tight oil and shale gas reservoirs.

Categories: Latest papers in fluid mechanics

### Hydroplastic response of a square plate due to impact on calm water

Physics of Fluids, Volume 32, Issue 8, August 2020.

This paper investigates large, plastic deflections of a square plate due to impact on calm water. Most research in the area has examined linear elastic structural responses to such impact, but hydrodynamic responses during large, plastic deformations of engineering structures remain under-explored. A setup for an experimental drop test was designed for this purpose with equal emphasis on the hydrodynamical and structural mechanical aspects. Dual cameras were used to monitor the deforming plate from above during impact, and its deformation was tracked using a three-dimensional digital image correlation technique. The complex hydrodynamics of the impact were captured using a high-speed camera from below. The experimental results for flat impact showed a large air pocket under the deforming plate. The material properties of the plate were documented through separate tests. Hydroelastic theories were offered to account for large deformations and validated against the experimental results. Analytical hydroplastic theory shows that the maximum deflection is approximately equal to the velocity of impact times the square root of the ratio of the added mass to the plastic membrane capacity of the plate. An important source of error between the theory and the experiments was the effect of deceleration of the drop rig on deflection of the plate. This error was estimated using direct force integration and Wagner’s theory.

This paper investigates large, plastic deflections of a square plate due to impact on calm water. Most research in the area has examined linear elastic structural responses to such impact, but hydrodynamic responses during large, plastic deformations of engineering structures remain under-explored. A setup for an experimental drop test was designed for this purpose with equal emphasis on the hydrodynamical and structural mechanical aspects. Dual cameras were used to monitor the deforming plate from above during impact, and its deformation was tracked using a three-dimensional digital image correlation technique. The complex hydrodynamics of the impact were captured using a high-speed camera from below. The experimental results for flat impact showed a large air pocket under the deforming plate. The material properties of the plate were documented through separate tests. Hydroelastic theories were offered to account for large deformations and validated against the experimental results. Analytical hydroplastic theory shows that the maximum deflection is approximately equal to the velocity of impact times the square root of the ratio of the added mass to the plastic membrane capacity of the plate. An important source of error between the theory and the experiments was the effect of deceleration of the drop rig on deflection of the plate. This error was estimated using direct force integration and Wagner’s theory.

Categories: Latest papers in fluid mechanics

### On Helmholtz–Hodge decomposition of inertia on a discrete local frame of reference

Physics of Fluids, Volume 32, Issue 8, August 2020.

The notion of inertial reference frame is abandoned and replaced by a local reference frame on which the fundamental law of mechanics is expressed. The distant interactions of the cause and effect are modeled by the propagation of waves from one local reference frame to another. The derivation of the equation of motion on a straight segment serves to express the proper acceleration as the sum of the accelerations imposed on it, in the form of an orthogonal local Helmholtz–Hodge decomposition, in one divergence-free and another curl-free contribution. The inertia term is written in the form of a gradient of a scalar potential and a dual curl of a vector potential. The adopted formalism opens the way to a reformulation of the material derivative in terms of potentials and allows the removal of the fictitious forces from continuum mechanics. The discrete equation of motion, invariant by rotation at a constant angular velocity, is used to conserve the angular momentum per unit of mass, in addition to the conservation of energy per unit of mass and acceleration. All the variables in this equation are expressed only with two fundamental units, length and time.

The notion of inertial reference frame is abandoned and replaced by a local reference frame on which the fundamental law of mechanics is expressed. The distant interactions of the cause and effect are modeled by the propagation of waves from one local reference frame to another. The derivation of the equation of motion on a straight segment serves to express the proper acceleration as the sum of the accelerations imposed on it, in the form of an orthogonal local Helmholtz–Hodge decomposition, in one divergence-free and another curl-free contribution. The inertia term is written in the form of a gradient of a scalar potential and a dual curl of a vector potential. The adopted formalism opens the way to a reformulation of the material derivative in terms of potentials and allows the removal of the fictitious forces from continuum mechanics. The discrete equation of motion, invariant by rotation at a constant angular velocity, is used to conserve the angular momentum per unit of mass, in addition to the conservation of energy per unit of mass and acceleration. All the variables in this equation are expressed only with two fundamental units, length and time.

Categories: Latest papers in fluid mechanics

### Analytical solutions to shock and expansion waves for non-ideal equations of state

Physics of Fluids, Volume 32, Issue 8, August 2020.

We present analytical solutions to the stationary normal shock and centered rarefaction waves, which are valid for arbitrary non-ideal equations of state (EOS). Generalized shock functions are defined, which are shown to be well-behaved and locally convex, facilitating rapid and exact computation of shock ratios. For rarefactions, a novel domain mapping is used to derive flow variables as closed-form analytical functions in space and time, independent of the EOS. Results are discussed for transcritical and supercritical CO2. The solutions enable researchers to test shock-capturing codes designed for non-ideal flows, and the derivation strategy opens possibilities to revisit nonlinear hyperbolic conservation problems that traditionally lack analytical solutions.

We present analytical solutions to the stationary normal shock and centered rarefaction waves, which are valid for arbitrary non-ideal equations of state (EOS). Generalized shock functions are defined, which are shown to be well-behaved and locally convex, facilitating rapid and exact computation of shock ratios. For rarefactions, a novel domain mapping is used to derive flow variables as closed-form analytical functions in space and time, independent of the EOS. Results are discussed for transcritical and supercritical CO2. The solutions enable researchers to test shock-capturing codes designed for non-ideal flows, and the derivation strategy opens possibilities to revisit nonlinear hyperbolic conservation problems that traditionally lack analytical solutions.

Categories: Latest papers in fluid mechanics

### Steady-state multiple near resonances of periodic interfacial waves with rigid boundary

Physics of Fluids, Volume 32, Issue 8, August 2020.

Steady-state resonant interfacial waves in a two-layer fluid within a frictionless duct are investigated theoretically. A combination of the homotopy analysis method (HAM) and Galerkin’s method is used to search for accurate steady-state resonant solutions with multiple near resonances. In the HAM, a piecewise parameter in the auxiliary linear operators is introduced to remove the small divisors caused by nearly resonant components. Convergent series solutions are then provided to the Galerkin iterations to accelerate the convergence rate. It is found that weakly nonlinear steady-state resonant waves form a continuum in the parameter space. As nonlinearity (wave steepness) increases, energy appears to be progressively shifted to sideband frequency components, effectively broadening the spectrum. The corresponding interfacial wave profile exhibits an almost fixed spatial pattern of repeated relatively high frequency, high-amplitude bursts followed by low-amplitude, longer waves. On examining the influence of density ratio, though changing slightly, the upper layer enlarges the amplitude of components near primary ones, which reduces the amplitude of higher frequency components, enlarges the wave steepness, and reduces the horizontal velocity in the wave field. Our results indicate that steady-state systems with resonant interactions among periodic interfacial wave components could occur naturally in the ocean. All these should enhance our understanding of periodic resonant interfacial waves.

Steady-state resonant interfacial waves in a two-layer fluid within a frictionless duct are investigated theoretically. A combination of the homotopy analysis method (HAM) and Galerkin’s method is used to search for accurate steady-state resonant solutions with multiple near resonances. In the HAM, a piecewise parameter in the auxiliary linear operators is introduced to remove the small divisors caused by nearly resonant components. Convergent series solutions are then provided to the Galerkin iterations to accelerate the convergence rate. It is found that weakly nonlinear steady-state resonant waves form a continuum in the parameter space. As nonlinearity (wave steepness) increases, energy appears to be progressively shifted to sideband frequency components, effectively broadening the spectrum. The corresponding interfacial wave profile exhibits an almost fixed spatial pattern of repeated relatively high frequency, high-amplitude bursts followed by low-amplitude, longer waves. On examining the influence of density ratio, though changing slightly, the upper layer enlarges the amplitude of components near primary ones, which reduces the amplitude of higher frequency components, enlarges the wave steepness, and reduces the horizontal velocity in the wave field. Our results indicate that steady-state systems with resonant interactions among periodic interfacial wave components could occur naturally in the ocean. All these should enhance our understanding of periodic resonant interfacial waves.

Categories: Latest papers in fluid mechanics

### Hierarchies of new invariants and conserved integrals in inviscid fluid flow

Physics of Fluids, Volume 32, Issue 8, August 2020.

A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flows in three dimensions. This approach describes invariants by means of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid velocity, which has the physical meaning of characterizing tensorial quantities that are frozen into the flow. Several new main results are obtained. First, simple algebraic and differential operators that can be applied recursively to derive a complete set of invariants starting from the basic known local and nonlocal invariants are constructed. Second, these operators are used to derive infinite hierarchies of local and nonlocal invariants for both adiabatic fluids and homentropic fluids that are either incompressible or compressible with barotropic and non-barotropic equations of state. Each hierarchy is complete in the sense that no further invariants can be generated from the basic local and nonlocal invariants. All of the resulting new invariants are generalizations of Ertel’s invariant, the Ertel–Rossby invariant, and Hollmann’s invariant. In particular, for an incompressible fluid flow in which the density is non-constant across different fluid streamlines, a new variant of Ertel’s invariant and several new variants of Hollmann’s invariant are derived, where the entropy gradient is replaced by the density gradient. Third, the physical meaning of these new invariants and the resulting conserved integrals is discussed, and their relationship to conserved helicities and cross-helicities is described.

A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flows in three dimensions. This approach describes invariants by means of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid velocity, which has the physical meaning of characterizing tensorial quantities that are frozen into the flow. Several new main results are obtained. First, simple algebraic and differential operators that can be applied recursively to derive a complete set of invariants starting from the basic known local and nonlocal invariants are constructed. Second, these operators are used to derive infinite hierarchies of local and nonlocal invariants for both adiabatic fluids and homentropic fluids that are either incompressible or compressible with barotropic and non-barotropic equations of state. Each hierarchy is complete in the sense that no further invariants can be generated from the basic local and nonlocal invariants. All of the resulting new invariants are generalizations of Ertel’s invariant, the Ertel–Rossby invariant, and Hollmann’s invariant. In particular, for an incompressible fluid flow in which the density is non-constant across different fluid streamlines, a new variant of Ertel’s invariant and several new variants of Hollmann’s invariant are derived, where the entropy gradient is replaced by the density gradient. Third, the physical meaning of these new invariants and the resulting conserved integrals is discussed, and their relationship to conserved helicities and cross-helicities is described.

Categories: Latest papers in fluid mechanics

### Variance-reduction kinetic simulation of low-speed rarefied gas flow through long microchannels of annular cross sections

Physics of Fluids, Volume 32, Issue 8, August 2020.

In micro/nano-devices, the low-speed transport of mass, momentum, and energy through long-ducts is frequently encountered, thereby necessitating scientific investigations. Here, long-ducts of various annular cross sections conducting low-speed gas flows under the influence of a small pressure gradient are considered, in order to understand how the mass flow rate is affected by rarefaction, variations in the radius ratio, and eccentricity of annular geometries. The Boltzmann model equation is treated by a low-variance formulation and simulated by a stochastic kinetic particle-based approach, which addresses the deviation of the molecular distribution function from equilibrium to reduce computational cost significantly. An efficient parallel solver has also been developed and utilized in this research, which is validated against the reported results in the literature. The efficient kinetic particle treatment provides a powerful simulation tool to reveal multi-scale flow physics, which is essential to develop and optimize micro/nano-fluidic devices.

In micro/nano-devices, the low-speed transport of mass, momentum, and energy through long-ducts is frequently encountered, thereby necessitating scientific investigations. Here, long-ducts of various annular cross sections conducting low-speed gas flows under the influence of a small pressure gradient are considered, in order to understand how the mass flow rate is affected by rarefaction, variations in the radius ratio, and eccentricity of annular geometries. The Boltzmann model equation is treated by a low-variance formulation and simulated by a stochastic kinetic particle-based approach, which addresses the deviation of the molecular distribution function from equilibrium to reduce computational cost significantly. An efficient parallel solver has also been developed and utilized in this research, which is validated against the reported results in the literature. The efficient kinetic particle treatment provides a powerful simulation tool to reveal multi-scale flow physics, which is essential to develop and optimize micro/nano-fluidic devices.

Categories: Latest papers in fluid mechanics

### Investigation of the drag reduction performance of bionic flexible coating

Physics of Fluids, Volume 32, Issue 8, August 2020.

The drag is a crucial factor in reducing the speed of movement and increasing unnecessary energy loss. In this work, inspired by dolphins, five bionic flexible coatings with drag reduction performance were designed and manufactured. First and foremost, the mixed solution, composed of the polydimethylsiloxane and ethyl acetate, was sprayed on aluminum disks with a spray gun, and the bionic flexible coatings were obtained by heating the aluminum disks sprayed with the mixed solution. Afterward, the mechanical properties and surface characteristics of the flexible coatings were characterized. The experimental results for the flexible coatings of drag reduction performance were obtained by using the drag force device. Above all, the parametric study focusing on the flexible coating of the mechanical properties affects the station of flow, which is performed to analyze the impact on drag reduction. Selecting the aluminum disk without any coating as a reference, numerical simulation methods were introduced to explore the drag reduction mechanism of the bionic flexible coating. The results evidence that the drag reduction ratio is 21.6% at the rotation velocity 50 rpm. Under the action of frictional resistance, the coating of elastic deformation caused by the viscoelasticity of the coating like the dolphin skin results in a decrease in frictional resistance of the wall.

The drag is a crucial factor in reducing the speed of movement and increasing unnecessary energy loss. In this work, inspired by dolphins, five bionic flexible coatings with drag reduction performance were designed and manufactured. First and foremost, the mixed solution, composed of the polydimethylsiloxane and ethyl acetate, was sprayed on aluminum disks with a spray gun, and the bionic flexible coatings were obtained by heating the aluminum disks sprayed with the mixed solution. Afterward, the mechanical properties and surface characteristics of the flexible coatings were characterized. The experimental results for the flexible coatings of drag reduction performance were obtained by using the drag force device. Above all, the parametric study focusing on the flexible coating of the mechanical properties affects the station of flow, which is performed to analyze the impact on drag reduction. Selecting the aluminum disk without any coating as a reference, numerical simulation methods were introduced to explore the drag reduction mechanism of the bionic flexible coating. The results evidence that the drag reduction ratio is 21.6% at the rotation velocity 50 rpm. Under the action of frictional resistance, the coating of elastic deformation caused by the viscoelasticity of the coating like the dolphin skin results in a decrease in frictional resistance of the wall.

Categories: Latest papers in fluid mechanics

### Harnessing flow-induced vibration of a D-section cylinder for convective heat transfer augmentation in laminar channel flow

Physics of Fluids, Volume 32, Issue 8, August 2020.

Flow-induced vibration (FIV) of a D-section cylinder is computationally studied and utilized to augment convective heat transfer in a heated laminar channel flow. An in-house fluid–structure interaction (FSI) solver, based on a sharp-interface immersed boundary method, is employed to solve the flow and thermal fields. In conjunction, a spring–mass system is utilized to solve for the rigid structural dynamics. Using numerical simulations, we highlight that the oscillations of a D-section cylinder are driven by vortex-induced vibration and galloping. It is observed that as the cylinder vibrates, vortices are shed from the apex of the cylinder due to the separating shear layers. These vortices, categorized using shedding patterns, interact with the heated channel walls. This interaction results in disruption of the thermal boundary layer (TBL), thus leading to heat transfer augmentation. The enhancement in thermal performance has been quantified using time and space-averaged Nusselt numbers at the channel walls. It is observed that the oscillation amplitude of the D-section cylinder and the extent of vortex–TBL interaction are crucial for determining heat transfer augmentation. Both symmetric and asymmetric thermal augmentation at the top and bottom channel walls have been reported. Finally, to assess the effectiveness of heat transfer augmentation, the D-section cylinder FIV is compared to other FSI systems operating under similar conditions.

Flow-induced vibration (FIV) of a D-section cylinder is computationally studied and utilized to augment convective heat transfer in a heated laminar channel flow. An in-house fluid–structure interaction (FSI) solver, based on a sharp-interface immersed boundary method, is employed to solve the flow and thermal fields. In conjunction, a spring–mass system is utilized to solve for the rigid structural dynamics. Using numerical simulations, we highlight that the oscillations of a D-section cylinder are driven by vortex-induced vibration and galloping. It is observed that as the cylinder vibrates, vortices are shed from the apex of the cylinder due to the separating shear layers. These vortices, categorized using shedding patterns, interact with the heated channel walls. This interaction results in disruption of the thermal boundary layer (TBL), thus leading to heat transfer augmentation. The enhancement in thermal performance has been quantified using time and space-averaged Nusselt numbers at the channel walls. It is observed that the oscillation amplitude of the D-section cylinder and the extent of vortex–TBL interaction are crucial for determining heat transfer augmentation. Both symmetric and asymmetric thermal augmentation at the top and bottom channel walls have been reported. Finally, to assess the effectiveness of heat transfer augmentation, the D-section cylinder FIV is compared to other FSI systems operating under similar conditions.

Categories: Latest papers in fluid mechanics

### Crystallization and jamming in narrow fluidized beds

Physics of Fluids, Volume 32, Issue 8, August 2020.

A fluidized bed is basically a suspension of granular material by an ascending fluid in a tube, and it has a rich dynamics that includes clustering and pattern formation. When the ratio between the tube and grain diameters is small, different behaviors can be induced by high confinement effects. Some unexpected and curious behaviors that we investigate in this paper are the crystallization and jamming of grains in liquids with velocities higher than those for incipient fluidization, supposed to maintain the grains fluidized. In our experiments, performed in a vertical tube of transparent material, different grains, water velocities, resting times, and velocity decelerations were used. An analysis of the bed evolution based on image processing shows that, after a decreasing flow that reaches a velocity still higher than that for incipient fluidization, grains become organized in lattice structures of high compactness, where they are trapped though with small fluctuations. These structures are initially localized and grow along time, in a similar manner as happens in phase transitions and glass formation. After a certain time, if the liquid velocity is slightly increased, jamming occurs, with grains being completely blocked and their fluctuation disappearing. We show that different lattice structures appear depending on the grain type. Our results provide new insights into fluidization conditions, glass-like formation, and jamming.

A fluidized bed is basically a suspension of granular material by an ascending fluid in a tube, and it has a rich dynamics that includes clustering and pattern formation. When the ratio between the tube and grain diameters is small, different behaviors can be induced by high confinement effects. Some unexpected and curious behaviors that we investigate in this paper are the crystallization and jamming of grains in liquids with velocities higher than those for incipient fluidization, supposed to maintain the grains fluidized. In our experiments, performed in a vertical tube of transparent material, different grains, water velocities, resting times, and velocity decelerations were used. An analysis of the bed evolution based on image processing shows that, after a decreasing flow that reaches a velocity still higher than that for incipient fluidization, grains become organized in lattice structures of high compactness, where they are trapped though with small fluctuations. These structures are initially localized and grow along time, in a similar manner as happens in phase transitions and glass formation. After a certain time, if the liquid velocity is slightly increased, jamming occurs, with grains being completely blocked and their fluctuation disappearing. We show that different lattice structures appear depending on the grain type. Our results provide new insights into fluidization conditions, glass-like formation, and jamming.

Categories: Latest papers in fluid mechanics

### Numerical investigation of the bevelled effects on shock structure and screech noise in planar supersonic jets

Physics of Fluids, Volume 32, Issue 8, August 2020.

Rectangular supersonic jets exist widely in propulsion systems of aircrafts. When they are imperfectly expanded under certain conditions, the upstream traveling waves referred to as screech tones will be produced, which may cause structural fatigue failure. In this work, high fidelity simulations are employed to investigate the bevelled effects due to the asymmetric lips of nozzles on shock structures and screech noise in planar supersonic jets. The present results are in agreement with previous experimental and numerical data for the symmetric case. For asymmetric cases, it is found that the bevelled effects will affect the shear layer transition, noise radiation, and shock cell oscillations. The level of screech noise generally decreases with increasing the length difference of two lips. The maximum 7.9 dB drop is identified, and the deflection angle of the mainstream of 9.35° is achieved when this length difference reaches the height of the nozzle. Moreover, dynamic mode decomposition (DMD) is specifically utilized to analyze shock cell oscillations. The results show that the bevelled effects suppress the most energetic DMD mode, corresponding to the dominant frequency of shock screech. The phenomenon of shock leakage is detected in the symmetric case, which is assumed to be an important screech noise source, while it seems to be weakened when the nozzle is bevelled. The longitudinal flapping motion of shock cells is substantially weakened due to the bevelled effects, which might be responsible for the suppression of shock leakage and the screech noise reduction.

Rectangular supersonic jets exist widely in propulsion systems of aircrafts. When they are imperfectly expanded under certain conditions, the upstream traveling waves referred to as screech tones will be produced, which may cause structural fatigue failure. In this work, high fidelity simulations are employed to investigate the bevelled effects due to the asymmetric lips of nozzles on shock structures and screech noise in planar supersonic jets. The present results are in agreement with previous experimental and numerical data for the symmetric case. For asymmetric cases, it is found that the bevelled effects will affect the shear layer transition, noise radiation, and shock cell oscillations. The level of screech noise generally decreases with increasing the length difference of two lips. The maximum 7.9 dB drop is identified, and the deflection angle of the mainstream of 9.35° is achieved when this length difference reaches the height of the nozzle. Moreover, dynamic mode decomposition (DMD) is specifically utilized to analyze shock cell oscillations. The results show that the bevelled effects suppress the most energetic DMD mode, corresponding to the dominant frequency of shock screech. The phenomenon of shock leakage is detected in the symmetric case, which is assumed to be an important screech noise source, while it seems to be weakened when the nozzle is bevelled. The longitudinal flapping motion of shock cells is substantially weakened due to the bevelled effects, which might be responsible for the suppression of shock leakage and the screech noise reduction.

Categories: Latest papers in fluid mechanics

### Ferro-advection aided evaporation kinetics of ferrofluid droplets in magnetic field ambience

Physics of Fluids, Volume 32, Issue 8, August 2020.

The present article discusses the physics and mechanics of evaporation of pendant, aqueous ferrofluid droplets, and modulation of the same by an external magnetic field. We show experimentally and by mathematical analysis that the presence of a horizontal magnetic field augments the evaporation rates of pendant ferrofluid droplets. First, we tackle the question of improved evaporation of the colloidal droplets compared to water and propose physical mechanisms to explain the same. Experiments show that the changes in evaporation rates aided by the magnetic field cannot be explained on the basis of changes in surface tension or based on classical diffusion driven evaporation models. Probing via particle image velocimetry shows that the internal advection kinetics of such droplets plays a direct role toward the augmented evaporation rates by modulating the associated Stefan flow. Infrared thermography reveals changes in thermal gradients within the droplet and evaluating the dynamic surface tension reveals the presence of solutal gradients within the droplet, both brought about by the external field. Based on the premise, a scaling analysis of the internal magneto-thermal and magneto-solutal ferroadvection behavior is presented. The model incorporates the role of the governing Hartmann number, the magneto-thermal Prandtl number, and the magneto-solutal Schmidt number. The analysis and stability maps reveal that the magneto-solutal ferroadvection is the more dominant mechanism, and the model is able to predict the internal advection velocities with accuracy. Furthermore, another scaling model to predict the modified Stefan flow is proposed and is found to accurately predict the improved evaporation rates.

The present article discusses the physics and mechanics of evaporation of pendant, aqueous ferrofluid droplets, and modulation of the same by an external magnetic field. We show experimentally and by mathematical analysis that the presence of a horizontal magnetic field augments the evaporation rates of pendant ferrofluid droplets. First, we tackle the question of improved evaporation of the colloidal droplets compared to water and propose physical mechanisms to explain the same. Experiments show that the changes in evaporation rates aided by the magnetic field cannot be explained on the basis of changes in surface tension or based on classical diffusion driven evaporation models. Probing via particle image velocimetry shows that the internal advection kinetics of such droplets plays a direct role toward the augmented evaporation rates by modulating the associated Stefan flow. Infrared thermography reveals changes in thermal gradients within the droplet and evaluating the dynamic surface tension reveals the presence of solutal gradients within the droplet, both brought about by the external field. Based on the premise, a scaling analysis of the internal magneto-thermal and magneto-solutal ferroadvection behavior is presented. The model incorporates the role of the governing Hartmann number, the magneto-thermal Prandtl number, and the magneto-solutal Schmidt number. The analysis and stability maps reveal that the magneto-solutal ferroadvection is the more dominant mechanism, and the model is able to predict the internal advection velocities with accuracy. Furthermore, another scaling model to predict the modified Stefan flow is proposed and is found to accurately predict the improved evaporation rates.

Categories: Latest papers in fluid mechanics

### The dispersion of spherical droplets in source–sink flows and their relevance to the COVID-19 pandemic

Physics of Fluids, Volume 32, Issue 8, August 2020.

In this paper, we investigate the dynamics of spherical droplets in the presence of a source–sink pair flow field. The dynamics of the droplets is governed by the Maxey–Riley equation with the Basset–Boussinesq history term neglected. We find that, in the absence of gravity, there are two distinct behaviors for the droplets: small droplets cannot go further than a specific distance, which we determine analytically, from the source before getting pulled into the sink. Larger droplets can travel further from the source before getting pulled into the sink by virtue of their larger inertia, and their maximum traveled distance is determined analytically. We investigate the effects of gravity, and we find that there are three distinct droplet behaviors categorized by their relative sizes: small, intermediate-sized, and large. Counterintuitively, we find that the droplets with a minimum horizontal range are neither small nor large, but of intermediate size. Furthermore, we show that in conditions of regular human respiration, these intermediate-sized droplets range in size from a few μm to a few hundred μm. The result that such droplets have a very short range could have important implications for the interpretation of existing data on droplet dispersion.

In this paper, we investigate the dynamics of spherical droplets in the presence of a source–sink pair flow field. The dynamics of the droplets is governed by the Maxey–Riley equation with the Basset–Boussinesq history term neglected. We find that, in the absence of gravity, there are two distinct behaviors for the droplets: small droplets cannot go further than a specific distance, which we determine analytically, from the source before getting pulled into the sink. Larger droplets can travel further from the source before getting pulled into the sink by virtue of their larger inertia, and their maximum traveled distance is determined analytically. We investigate the effects of gravity, and we find that there are three distinct droplet behaviors categorized by their relative sizes: small, intermediate-sized, and large. Counterintuitively, we find that the droplets with a minimum horizontal range are neither small nor large, but of intermediate size. Furthermore, we show that in conditions of regular human respiration, these intermediate-sized droplets range in size from a few μm to a few hundred μm. The result that such droplets have a very short range could have important implications for the interpretation of existing data on droplet dispersion.

Categories: Latest papers in fluid mechanics