# 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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### Numerical study on reflection of an oblique detonation wave on an outward turning wall

Physics of Fluids, Volume 32, Issue 4, April 2020.

Oblique detonation waves (ODWs) have been studied widely as the basis of detonation-based hypersonic engines, but there are few studies on ODWs in a confined space. This study simulates ODW reflection on a solid wall before an outward turning corner for a simplified combustor–nozzle flow based on a two-step kinetic model. Numerical results reveal three types of ODW structures: stable, critical, and unstable. When the reflection occurs at the turning point, the stable ODW structure remains almost the same as before reflection. When the wave reflects at the wall before the turning point, either the critical structure or the unstable structure arises, which has never been investigated before. Both structures have the same initial two-section detonation surface: but the critical one becomes stationary at a certain position, while the unstable one keeps traveling upstream. By adjusting the location of the expansion wave and degree of the turning angle, the difference of the two structures is attributed to the thermal choking appearing only in the unstable structure. The thermal choking is achieved by the merging of subsonic zones, whose dependence on the various parameters is discussed.

Oblique detonation waves (ODWs) have been studied widely as the basis of detonation-based hypersonic engines, but there are few studies on ODWs in a confined space. This study simulates ODW reflection on a solid wall before an outward turning corner for a simplified combustor–nozzle flow based on a two-step kinetic model. Numerical results reveal three types of ODW structures: stable, critical, and unstable. When the reflection occurs at the turning point, the stable ODW structure remains almost the same as before reflection. When the wave reflects at the wall before the turning point, either the critical structure or the unstable structure arises, which has never been investigated before. Both structures have the same initial two-section detonation surface: but the critical one becomes stationary at a certain position, while the unstable one keeps traveling upstream. By adjusting the location of the expansion wave and degree of the turning angle, the difference of the two structures is attributed to the thermal choking appearing only in the unstable structure. The thermal choking is achieved by the merging of subsonic zones, whose dependence on the various parameters is discussed.

Categories: Latest papers in fluid mechanics

### Agglomeration dynamics in liquid–solid particle-laden turbulent channel flows using an energy-based deterministic approach

Physics of Fluids, Volume 32, Issue 4, April 2020.

A deterministic particle–particle agglomeration technique is applied together with direct numerical simulation and four-way coupled Lagrangian particle tracking in order to accurately simulate and investigate fully coupled agglomerating particle-laden channel flows at a shear Reynolds number, Reτ = 180. The collision outcome determination (recoil or aggregate) is based on the balance between kinetic energy dispersed in the collision and the work required to overcome the van der Waals attractive potential. The influence of particle size (dP = 202 μm, 286 μm, and 405 μm), both at a fixed volume fraction (ϕP = 10−3) and a fixed primary injected particle number (NP = 109 313), on the resulting collision and agglomeration dynamics is investigated. Attention is also focused on how collision and agglomeration rates vary throughout the wall-normal regions of the channel flow. The results demonstrate that the normalized collision rates are similar for all particle sizes at the fixed volume fraction but increase with particle size at the fixed particle number, and a preference is observed for collisions to occur close to the walls. Despite this, in all cases considered here, agglomeration events are most frequent at the center of the channel, with agglomeration efficiencies also peaking in this region. In terms of particle diameter effects, the smallest particles exhibit the greatest preference to aggregate, given that a collision has already occurred. Furthermore, whereas normalized collision and agglomeration event counts show differing diameter-dependence based on whether the number of primary particles or the volume fraction is fixed, agglomeration rates show diameter-independence and as such are based solely on particle size and local dispersive properties. Analysis of the dynamic collision properties throughout the channel confirms that agglomeration is favored within the bulk flow region due to low relative particle velocities and small collision angles at this location. The temporal evolution of important interaction properties is investigated, all of which demonstrate stability over the course of the time simulated. Particle diameter is also shown to influence the long-term population of higher-order agglomerates, with (for a given volume fraction) smaller particles aggregating faster to form larger particles. The systems studied, which resemble those present in the processing of nuclear waste, all exhibit substantial agglomeration over the time considered. This reinforces the importance of accurately modeling agglomeration dynamics in flows where electrokinetic interactions are important in order to correctly predict multiphase flow properties over long timeframes.

A deterministic particle–particle agglomeration technique is applied together with direct numerical simulation and four-way coupled Lagrangian particle tracking in order to accurately simulate and investigate fully coupled agglomerating particle-laden channel flows at a shear Reynolds number, Reτ = 180. The collision outcome determination (recoil or aggregate) is based on the balance between kinetic energy dispersed in the collision and the work required to overcome the van der Waals attractive potential. The influence of particle size (dP = 202 μm, 286 μm, and 405 μm), both at a fixed volume fraction (ϕP = 10−3) and a fixed primary injected particle number (NP = 109 313), on the resulting collision and agglomeration dynamics is investigated. Attention is also focused on how collision and agglomeration rates vary throughout the wall-normal regions of the channel flow. The results demonstrate that the normalized collision rates are similar for all particle sizes at the fixed volume fraction but increase with particle size at the fixed particle number, and a preference is observed for collisions to occur close to the walls. Despite this, in all cases considered here, agglomeration events are most frequent at the center of the channel, with agglomeration efficiencies also peaking in this region. In terms of particle diameter effects, the smallest particles exhibit the greatest preference to aggregate, given that a collision has already occurred. Furthermore, whereas normalized collision and agglomeration event counts show differing diameter-dependence based on whether the number of primary particles or the volume fraction is fixed, agglomeration rates show diameter-independence and as such are based solely on particle size and local dispersive properties. Analysis of the dynamic collision properties throughout the channel confirms that agglomeration is favored within the bulk flow region due to low relative particle velocities and small collision angles at this location. The temporal evolution of important interaction properties is investigated, all of which demonstrate stability over the course of the time simulated. Particle diameter is also shown to influence the long-term population of higher-order agglomerates, with (for a given volume fraction) smaller particles aggregating faster to form larger particles. The systems studied, which resemble those present in the processing of nuclear waste, all exhibit substantial agglomeration over the time considered. This reinforces the importance of accurately modeling agglomeration dynamics in flows where electrokinetic interactions are important in order to correctly predict multiphase flow properties over long timeframes.

Categories: Latest papers in fluid mechanics

### Development of a stagnation streamline model for thermochemical nonequilibrium flow

Physics of Fluids, Volume 32, Issue 4, April 2020.

A stagnation streamline model incorporating quantum-state-resolved chemistry is proposed to study hypersonic nonequilibrium flows along the stagnation streamline. This model is developed by reducing the full Navier–Stokes equations to the stagnation streamline with proper approximations for equation closure. The thermochemical nonequilibrium is described by either the state-to-state approach for detailed analysis or conventional two-temperature models for comparison purpose. The model is validated against various data, and nearly identical results are obtained as compared with those from full field computational fluid dynamics data. In addition, the calculated distributions agree well with the measurement data of a shock tube experiment for the dissociation and vibrational relaxation of O2, including the distributions of species mole fractions and vibrational temperature of the first excited state of O2 molecules. Furthermore, the results with the state-resolved chemistry show that the flow within a shock layer exhibits a strong thermochemical nonequilibrium behavior, which is beyond the capability of commonly used two-temperature models to correctly evaluate the dissociation rate and the associated reaction energy. The present model is also employed to calculate the nonequilibrium re-entry flow along the stagnation streamline for a five-species air mixture as an example to demonstrate the model capability. It is found that both species and internal energy are in a nonequilibrium state, especially the vibrational distributions are strongly deviated from the Boltzmann distribution right behind the bow shock and near the wall surface. The results demonstrate that the proposed stagnation streamline model is very useful to understand thermochemical nonequilibrium phenomena in hypersonic flows.

A stagnation streamline model incorporating quantum-state-resolved chemistry is proposed to study hypersonic nonequilibrium flows along the stagnation streamline. This model is developed by reducing the full Navier–Stokes equations to the stagnation streamline with proper approximations for equation closure. The thermochemical nonequilibrium is described by either the state-to-state approach for detailed analysis or conventional two-temperature models for comparison purpose. The model is validated against various data, and nearly identical results are obtained as compared with those from full field computational fluid dynamics data. In addition, the calculated distributions agree well with the measurement data of a shock tube experiment for the dissociation and vibrational relaxation of O2, including the distributions of species mole fractions and vibrational temperature of the first excited state of O2 molecules. Furthermore, the results with the state-resolved chemistry show that the flow within a shock layer exhibits a strong thermochemical nonequilibrium behavior, which is beyond the capability of commonly used two-temperature models to correctly evaluate the dissociation rate and the associated reaction energy. The present model is also employed to calculate the nonequilibrium re-entry flow along the stagnation streamline for a five-species air mixture as an example to demonstrate the model capability. It is found that both species and internal energy are in a nonequilibrium state, especially the vibrational distributions are strongly deviated from the Boltzmann distribution right behind the bow shock and near the wall surface. The results demonstrate that the proposed stagnation streamline model is very useful to understand thermochemical nonequilibrium phenomena in hypersonic flows.

Categories: Latest papers in fluid mechanics

### Numerical aero-thermal study of high-pressure turbine nozzle guide vane: Effects of inflow conditions

Physics of Fluids, Volume 32, Issue 3, March 2020.

Accurate predictability of high-pressure turbine nozzle guide vane aero-thermal performance is highly desired in the development campaign due to the exposure of the component to a frequent and high heat load. In this paper, the representative vane profile in modern aero-engines is numerically studied. Aerodynamics and aero-thermal validations of the blade profile have been performed in comparison with the available experimental data. It has been shown that a satisfactory agreement could be achieved with the use of the transitional turbulence model shear stress transport γ–θ due to its superiority in capturing the laminar–turbulent transition. Sensitivity studies on the increase in the inlet turbulence intensity, inlet endwall boundary layer thickness, and inlet total temperature profile have been performed to understand the impact of inflow conditions’ uncertainty on the aero-thermal predictability. Increasing the inlet turbulence intensity increases the pressure surface heat transfer coefficient and induces an earlier transition onset on the suction surface. Due to the rapid decay of turbulence intensity in the numerical model, the use of an artificially high inlet turbulence intensity has been shown to be effective in the prediction improvement. On the other hand, the change in the inlet boundary layer thickness influences the formation and strength of the secondary flow, namely, horseshoe vortex and passage vortex. These secondary flow phenomena affect the local blade surface heat transfer coefficient in the near-endwall region although the most significant rise in heat transfer is found on the endwall. The temperature distortion amplitude of a hot streak and its relative clocking position with the vane significantly affect the heat flux distribution. In contrast, the heat transfer coefficient is less sensitive to the change in hot streak conditions. However, it has been shown that increasing the temperature distortion amplitude could induce a larger difference among different clocking configurations. In addition, decreasing the difference between the fluid and wall temperature would delay the transition onset and stabilize the boundary layer. Further analysis of the unsteady effects has been carried out by comparing the steady and time-averaged flow solutions. It has been observed that the discrepancy between these solutions is attributed to the flow field nonlinearity. Thus, a significant discrepancy can be found in the laminar–turbulent transition as well as in the trailing edge region. However, since the contribution of these regions on the total area-averaged heat transfer is small, their influence on the total vane heat transfer is limited.

Accurate predictability of high-pressure turbine nozzle guide vane aero-thermal performance is highly desired in the development campaign due to the exposure of the component to a frequent and high heat load. In this paper, the representative vane profile in modern aero-engines is numerically studied. Aerodynamics and aero-thermal validations of the blade profile have been performed in comparison with the available experimental data. It has been shown that a satisfactory agreement could be achieved with the use of the transitional turbulence model shear stress transport γ–θ due to its superiority in capturing the laminar–turbulent transition. Sensitivity studies on the increase in the inlet turbulence intensity, inlet endwall boundary layer thickness, and inlet total temperature profile have been performed to understand the impact of inflow conditions’ uncertainty on the aero-thermal predictability. Increasing the inlet turbulence intensity increases the pressure surface heat transfer coefficient and induces an earlier transition onset on the suction surface. Due to the rapid decay of turbulence intensity in the numerical model, the use of an artificially high inlet turbulence intensity has been shown to be effective in the prediction improvement. On the other hand, the change in the inlet boundary layer thickness influences the formation and strength of the secondary flow, namely, horseshoe vortex and passage vortex. These secondary flow phenomena affect the local blade surface heat transfer coefficient in the near-endwall region although the most significant rise in heat transfer is found on the endwall. The temperature distortion amplitude of a hot streak and its relative clocking position with the vane significantly affect the heat flux distribution. In contrast, the heat transfer coefficient is less sensitive to the change in hot streak conditions. However, it has been shown that increasing the temperature distortion amplitude could induce a larger difference among different clocking configurations. In addition, decreasing the difference between the fluid and wall temperature would delay the transition onset and stabilize the boundary layer. Further analysis of the unsteady effects has been carried out by comparing the steady and time-averaged flow solutions. It has been observed that the discrepancy between these solutions is attributed to the flow field nonlinearity. Thus, a significant discrepancy can be found in the laminar–turbulent transition as well as in the trailing edge region. However, since the contribution of these regions on the total area-averaged heat transfer is small, their influence on the total vane heat transfer is limited.

Categories: Latest papers in fluid mechanics

### Vortex-induced vibrations of a confined circular cylinder for efficient flow power extraction

Physics of Fluids, Volume 32, Issue 3, March 2020.

The effects of confinement on vortex-induced vibration (VIV) of a circular cylinder (diameter D) and its flow power extraction capability are presented. A two-dimensional numerical study is performed on VIV of a circular cylinder inside a parallel plate channel of height H at Reynolds numbers 100 and 150. The blockage ratio (b = D/H) is varied from 1/4 to 1/2. The cylinder is elastically mounted with a spring such that it can only vibrate transversely to the flow. It has a fixed non-dimensional mass (m*) of 10. The energy extraction process is modeled as a damper (damping ratio ζ) attached to the cylinder. With increasing blockage, the vibration amplitude of the cylinder decreases, the lock-in happens at a larger non-dimensional natural frequency of the cylinder, and the initial branch of the vibration response of the cylinder shrinks. There is a minimum value of the channel height and Reynolds number for the existence of the initial branch. The extracted power is found to increase rapidly with the blockage. For the maximum blockage (b = 1/2), the flow power extracted by the cylinder is an order of magnitude larger as compared to what it would extract in an open domain with free stream velocity equal to the channel mean velocity. The optimal mass-damping (α = m*ζ) for extracting the maximum power is found to lie between 0.2 and 0.3. An expression is derived to predict the maximum extracted power from the undamped response of a confined/unconfined cylinder.

The effects of confinement on vortex-induced vibration (VIV) of a circular cylinder (diameter D) and its flow power extraction capability are presented. A two-dimensional numerical study is performed on VIV of a circular cylinder inside a parallel plate channel of height H at Reynolds numbers 100 and 150. The blockage ratio (b = D/H) is varied from 1/4 to 1/2. The cylinder is elastically mounted with a spring such that it can only vibrate transversely to the flow. It has a fixed non-dimensional mass (m*) of 10. The energy extraction process is modeled as a damper (damping ratio ζ) attached to the cylinder. With increasing blockage, the vibration amplitude of the cylinder decreases, the lock-in happens at a larger non-dimensional natural frequency of the cylinder, and the initial branch of the vibration response of the cylinder shrinks. There is a minimum value of the channel height and Reynolds number for the existence of the initial branch. The extracted power is found to increase rapidly with the blockage. For the maximum blockage (b = 1/2), the flow power extracted by the cylinder is an order of magnitude larger as compared to what it would extract in an open domain with free stream velocity equal to the channel mean velocity. The optimal mass-damping (α = m*ζ) for extracting the maximum power is found to lie between 0.2 and 0.3. An expression is derived to predict the maximum extracted power from the undamped response of a confined/unconfined cylinder.

Categories: Latest papers in fluid mechanics

### Energy dissipation analysis based on velocity gradient tensor decomposition

Physics of Fluids, Volume 32, Issue 3, March 2020.

A velocity gradient tensor decomposition method based on a normal frame is introduced in this paper. The velocity gradient tensor is decomposed into a compression–stretching tensor, pure rotation tensor, and pure shear tensor. The analysis shows that both the strain rate tensor and vorticity tensor in Helmholtz velocity decomposition contain shear tensor components, and the total pure shear tensor is the combination of shear components in the two tensors. Based on this decomposition and the physical meaning of each tensor term, the energy dissipation of the channel flow with or without a pressure gradient and a turbine passage flow are analyzed. The results show that the energy dissipation is caused by shear deformation and expansion and contraction deformation of the motion fluid, and pure rotation does not cause energy dissipation. In particular, the pure shear is the primary factor of energy dissipation. Shear accounts for 99.9% of energy dissipation in the fully developed turbulence of zero-pressure gradient channel flow, 99% of the energy dissipation in the separated boundary layer flow is caused by the pure shear, and 81% of the energy dissipation in the turbine stage flow is caused by pure shear.

A velocity gradient tensor decomposition method based on a normal frame is introduced in this paper. The velocity gradient tensor is decomposed into a compression–stretching tensor, pure rotation tensor, and pure shear tensor. The analysis shows that both the strain rate tensor and vorticity tensor in Helmholtz velocity decomposition contain shear tensor components, and the total pure shear tensor is the combination of shear components in the two tensors. Based on this decomposition and the physical meaning of each tensor term, the energy dissipation of the channel flow with or without a pressure gradient and a turbine passage flow are analyzed. The results show that the energy dissipation is caused by shear deformation and expansion and contraction deformation of the motion fluid, and pure rotation does not cause energy dissipation. In particular, the pure shear is the primary factor of energy dissipation. Shear accounts for 99.9% of energy dissipation in the fully developed turbulence of zero-pressure gradient channel flow, 99% of the energy dissipation in the separated boundary layer flow is caused by the pure shear, and 81% of the energy dissipation in the turbine stage flow is caused by pure shear.

Categories: Latest papers in fluid mechanics

### Investigation on the flow-induced noise propagation mechanism of centrifugal pump based on flow and sound fields synergy concept

Physics of Fluids, Volume 32, Issue 3, March 2020.

This paper explores the flow-induced noise propagation mechanism of centrifugal pump from the view of flow and sound field synergy concept. First, the unsteady synergetic relationship between flow and sound fields is deduced, and the synergy angle is defined to describe the synergy degree. It is shown that the domain-averaged synergy angle (θave) changes little with flow time, which implies that the synergy degree is basically unchanged with flow time. With increasing rotational speed or flow rate, the time-averaged θave (θtave) in the impeller and the volute moves far away from 90° gradually, i.e., the synergy degree increases. Meanwhile, the noise outside the pump increases, and the variation of both the noise outside the pump and θtave tends to be gradual. The results manifested that the flow-induced noise propagation mechanism of the centrifugal pump can be well described by the change in synergy degree and the increase in synergy degree can cause the noise tending to propagate outside. In addition, the impact of the blade outlet angle on the noise propagation characteristics is investigated. Considering the synergy degree in the impeller and the volute comprehensively, the deviation of θtave from 90° decreases from 6.48° to 4.74° as the angle increases from 15° to 35°, i.e., θtave tends to approach 90°, and the synergy degree decreases gradually, indicating that increasing the blade outlet angle can weaken the tendency of noise propagating outside by decreasing the synergy degree. These conclusions can guide noise control research and engineering design.

This paper explores the flow-induced noise propagation mechanism of centrifugal pump from the view of flow and sound field synergy concept. First, the unsteady synergetic relationship between flow and sound fields is deduced, and the synergy angle is defined to describe the synergy degree. It is shown that the domain-averaged synergy angle (θave) changes little with flow time, which implies that the synergy degree is basically unchanged with flow time. With increasing rotational speed or flow rate, the time-averaged θave (θtave) in the impeller and the volute moves far away from 90° gradually, i.e., the synergy degree increases. Meanwhile, the noise outside the pump increases, and the variation of both the noise outside the pump and θtave tends to be gradual. The results manifested that the flow-induced noise propagation mechanism of the centrifugal pump can be well described by the change in synergy degree and the increase in synergy degree can cause the noise tending to propagate outside. In addition, the impact of the blade outlet angle on the noise propagation characteristics is investigated. Considering the synergy degree in the impeller and the volute comprehensively, the deviation of θtave from 90° decreases from 6.48° to 4.74° as the angle increases from 15° to 35°, i.e., θtave tends to approach 90°, and the synergy degree decreases gradually, indicating that increasing the blade outlet angle can weaken the tendency of noise propagating outside by decreasing the synergy degree. These conclusions can guide noise control research and engineering design.

Categories: Latest papers in fluid mechanics

### Transition effects on flow characteristics around a static two-dimensional airfoil

Physics of Fluids, Volume 32, Issue 3, March 2020.

Flows past a static NACA0015 airfoil are numerically investigated via Reynolds-averaged Navier–Stokes simulations at the Reynolds number 1.95 × 106, the Mach number 0.291, and the angle of attack (AoA) from 0° to 18°. Specifically, a one-equation local correlation-based transition model (γ model) coupled with Menter’s k–ω shear stress transport (SST) model (SST–γ model) is employed to approximate the unclosed Reynolds quantities in the governing equations. Distributions of mean velocity and Reynolds stresses as well as typical integral quantities, such as the drag coefficient, lift coefficient, and moment coefficient, are calculated and compared with previously reported experimental data and present numerical data based on Menter’s original k–ω SST model. It turns out that the SST–γ model enables the capture of a laminar separation bubble (LSB) near the leading edge of the airfoil and shows significant advantages over the traditional “fully turbulent” models for the prediction of static stall. As the AoA varies from 0° to 18°, the flow regime is affected by different processes, i.e., flow transition, flow separation, and interaction between the LSB and the trailing-edge separation bubble, which, respectively, correspond to the linear-lift stage, light-stall stage, and deep-stall stage.

Flows past a static NACA0015 airfoil are numerically investigated via Reynolds-averaged Navier–Stokes simulations at the Reynolds number 1.95 × 106, the Mach number 0.291, and the angle of attack (AoA) from 0° to 18°. Specifically, a one-equation local correlation-based transition model (γ model) coupled with Menter’s k–ω shear stress transport (SST) model (SST–γ model) is employed to approximate the unclosed Reynolds quantities in the governing equations. Distributions of mean velocity and Reynolds stresses as well as typical integral quantities, such as the drag coefficient, lift coefficient, and moment coefficient, are calculated and compared with previously reported experimental data and present numerical data based on Menter’s original k–ω SST model. It turns out that the SST–γ model enables the capture of a laminar separation bubble (LSB) near the leading edge of the airfoil and shows significant advantages over the traditional “fully turbulent” models for the prediction of static stall. As the AoA varies from 0° to 18°, the flow regime is affected by different processes, i.e., flow transition, flow separation, and interaction between the LSB and the trailing-edge separation bubble, which, respectively, correspond to the linear-lift stage, light-stall stage, and deep-stall stage.

Categories: Latest papers in fluid mechanics

### Heat transfer enhancement and reduction in low-Rayleigh number natural convection flow with polymer additives

Physics of Fluids, Volume 32, Issue 3, March 2020.

The effects of viscoelasticity, here caused by polymer additives, on Rayleigh Bénard convection flows are investigated via direct numerical simulations at a marginally turbulent Rayleigh number. Simulations with a range of polymer length and relaxation time scales show heat transfer enhancement (HTE) and reduction (HTR). The selection of HTE and HTR depends strongly on the maximum extensional viscosity of the solution, whereas the magnitude of heat transfer modification is a function of both the maximum extensional viscosity and relaxation time of the polymer solution. The underlying physics of HTE and HTR are explored, and a mechanism of the interaction between convection cells and polymers is proposed. The findings are extrapolated to high Ra to shed some new light onto experimental observations of HTR.

The effects of viscoelasticity, here caused by polymer additives, on Rayleigh Bénard convection flows are investigated via direct numerical simulations at a marginally turbulent Rayleigh number. Simulations with a range of polymer length and relaxation time scales show heat transfer enhancement (HTE) and reduction (HTR). The selection of HTE and HTR depends strongly on the maximum extensional viscosity of the solution, whereas the magnitude of heat transfer modification is a function of both the maximum extensional viscosity and relaxation time of the polymer solution. The underlying physics of HTE and HTR are explored, and a mechanism of the interaction between convection cells and polymers is proposed. The findings are extrapolated to high Ra to shed some new light onto experimental observations of HTR.

Categories: Latest papers in fluid mechanics

### Electroosmosis of a viscoelastic fluid over non-uniformly charged surfaces: Effect of fluid relaxation and retardation time

Physics of Fluids, Volume 32, Issue 3, March 2020.

We investigate the electroosmotic flow of a quasi-linear viscoelastic fluid over a surface having charge modulation in narrow confinements. We obtain analytical solutions using a combination of regular and matched asymptotic expansions in order to describe the viscoelastic flow field and apparent slip velocity besides pinpointing variations of the flow rate and ionic currents due to the surface charge modulation. We demonstrate excellent agreement between the asymptotic analytical solution for the flow field and the full numerical solution in the limiting condition of a thin electrical double layer and weakly viscoelastic fluid. For a wide range of flow governing parameters, we analyze the flow velocity, vortex dynamics, flow rates, and streaming current. We demonstrate that the magnitude of the observed electroosmotic slip velocity is more sensitive to the thickness of the electrical double layer rather than the viscoelasticity of the fluid. We have observed that the contribution of fluid elasticity is prominent in breaking the axial symmetry in the electroosmotic flow with the presence of periodic charge distributions, which is in contrast to the symmetric electroosmotic flow field of a Newtonian fluid over the same charge modulated walls. The results hold the key toward understanding the flow of biological fluids in microfluidic flows by leveraging electrokinetic transport over charge modulated surfaces. We believe that the results of net throughput, streaming current, and vortex dynamics will aid our understanding of the complex fluid behavior and microfluidic mixers.

We investigate the electroosmotic flow of a quasi-linear viscoelastic fluid over a surface having charge modulation in narrow confinements. We obtain analytical solutions using a combination of regular and matched asymptotic expansions in order to describe the viscoelastic flow field and apparent slip velocity besides pinpointing variations of the flow rate and ionic currents due to the surface charge modulation. We demonstrate excellent agreement between the asymptotic analytical solution for the flow field and the full numerical solution in the limiting condition of a thin electrical double layer and weakly viscoelastic fluid. For a wide range of flow governing parameters, we analyze the flow velocity, vortex dynamics, flow rates, and streaming current. We demonstrate that the magnitude of the observed electroosmotic slip velocity is more sensitive to the thickness of the electrical double layer rather than the viscoelasticity of the fluid. We have observed that the contribution of fluid elasticity is prominent in breaking the axial symmetry in the electroosmotic flow with the presence of periodic charge distributions, which is in contrast to the symmetric electroosmotic flow field of a Newtonian fluid over the same charge modulated walls. The results hold the key toward understanding the flow of biological fluids in microfluidic flows by leveraging electrokinetic transport over charge modulated surfaces. We believe that the results of net throughput, streaming current, and vortex dynamics will aid our understanding of the complex fluid behavior and microfluidic mixers.

Categories: Latest papers in fluid mechanics

### Large eddy simulation of the separated flow transition on the suction surface of a high subsonic compressor airfoil

Physics of Fluids, Volume 32, Issue 3, March 2020.

A large eddy simulation (LES) was conducted to investigate the separated flow transition on the suction surface of a high subsonic compressor airfoil at two Reynolds number (Re) conditions (1.5 × 105 and 0.8 × 105). The detailed vortex evolution in the separated shear layer was revealed. The instability amplification in the transition process and the associated loss mechanism were clarified. At Re = 1.5 × 105, the two-dimensional spanwise vortices shed periodically and were further distorted with the interaction of the streamwise evolving vortices, and then, small vortices were generated in the streamwise pairing of the neighboring spanwise vortices. Finally, three-dimensional hairpin vortices broke down into small-scale turbulent structures near the reattachment, along with the “ejection-sweeping” process near the wall. When the Reynolds number decreased to 0.8 × 105, the initial vortex shedding was not periodic, but the subsequent vortex evolution process was very similar to the case of Re = 1.5 × 105. The results have demonstrated the importance of the Tollmien–Schlichting (T–S) mechanism for the initial growth of disturbances in the attached boundary layer, but the transition process that occurred in the separated shear layer was dominated by the inviscid Kelvin–Helmholtz (K–H) instability. Moreover, a secondary instability observed in the vortex pairing process was supposed to have a great impact on the onset of transition. With the decrease in Re, the shear layer instability declined to a lower level, leading to a delayed transition. In addition, the deformation works associated with the Reynolds shear stress was found to be mainly responsible for the loss generation in the transitional flow. Compared with the traditional Reynolds average Navier–Stokes method, the LES was more accurate in predicting the profile loss at a low Re.

A large eddy simulation (LES) was conducted to investigate the separated flow transition on the suction surface of a high subsonic compressor airfoil at two Reynolds number (Re) conditions (1.5 × 105 and 0.8 × 105). The detailed vortex evolution in the separated shear layer was revealed. The instability amplification in the transition process and the associated loss mechanism were clarified. At Re = 1.5 × 105, the two-dimensional spanwise vortices shed periodically and were further distorted with the interaction of the streamwise evolving vortices, and then, small vortices were generated in the streamwise pairing of the neighboring spanwise vortices. Finally, three-dimensional hairpin vortices broke down into small-scale turbulent structures near the reattachment, along with the “ejection-sweeping” process near the wall. When the Reynolds number decreased to 0.8 × 105, the initial vortex shedding was not periodic, but the subsequent vortex evolution process was very similar to the case of Re = 1.5 × 105. The results have demonstrated the importance of the Tollmien–Schlichting (T–S) mechanism for the initial growth of disturbances in the attached boundary layer, but the transition process that occurred in the separated shear layer was dominated by the inviscid Kelvin–Helmholtz (K–H) instability. Moreover, a secondary instability observed in the vortex pairing process was supposed to have a great impact on the onset of transition. With the decrease in Re, the shear layer instability declined to a lower level, leading to a delayed transition. In addition, the deformation works associated with the Reynolds shear stress was found to be mainly responsible for the loss generation in the transitional flow. Compared with the traditional Reynolds average Navier–Stokes method, the LES was more accurate in predicting the profile loss at a low Re.

Categories: Latest papers in fluid mechanics

### Effect of temperature on gelation and cross-linking of gelatin methacryloyl for biomedical applications

Physics of Fluids, Volume 32, Issue 3, March 2020.

Hydrogels with or without chemical cross-linking have been studied and used for biomedical applications, such as tissue repair, surgical sealants, and three dimensional biofabrication. These materials often undergo a physical sol–gel or gel–sol transition between room and body temperatures and can also be chemically cross-linked at these temperatures to give dimensional stability. However, few studies have clearly shown the effect of heating/cooling rates on such transitions. Moreover, only a little is known about the effect of cross-linking temperature or the state on the modulus after cross-linking. We have established rheological methods to study these effects, an approach to determine transition temperatures, and a method to prevent sample drying during measurements. All the rheological measurements were performed minimizing the normal stress build-up to compensate for the shrinking and expansion due to temperature and phase changes. We chemically modified gelatin to give gelatin methacryloyl and determined the degree of methacryloylation by proton nuclear magnetic resonance. Using the gelatin methacryloyl as an example, we have found that the gel state or lower temperature can give more rigid gelatin-based polymers by cross-linking under visible light than the sol state or higher temperature. These methods and results can guide researchers to perform appropriate studies on material design and map applications, such as the optimal operating temperature of hydrogels for biomedical applications. We have also found that gelation temperatures strongly depend on the cooling rate, while solation temperatures are independent of the heating rate.

Hydrogels with or without chemical cross-linking have been studied and used for biomedical applications, such as tissue repair, surgical sealants, and three dimensional biofabrication. These materials often undergo a physical sol–gel or gel–sol transition between room and body temperatures and can also be chemically cross-linked at these temperatures to give dimensional stability. However, few studies have clearly shown the effect of heating/cooling rates on such transitions. Moreover, only a little is known about the effect of cross-linking temperature or the state on the modulus after cross-linking. We have established rheological methods to study these effects, an approach to determine transition temperatures, and a method to prevent sample drying during measurements. All the rheological measurements were performed minimizing the normal stress build-up to compensate for the shrinking and expansion due to temperature and phase changes. We chemically modified gelatin to give gelatin methacryloyl and determined the degree of methacryloylation by proton nuclear magnetic resonance. Using the gelatin methacryloyl as an example, we have found that the gel state or lower temperature can give more rigid gelatin-based polymers by cross-linking under visible light than the sol state or higher temperature. These methods and results can guide researchers to perform appropriate studies on material design and map applications, such as the optimal operating temperature of hydrogels for biomedical applications. We have also found that gelation temperatures strongly depend on the cooling rate, while solation temperatures are independent of the heating rate.

Categories: Latest papers in fluid mechanics

### Secondary instability of stationary Görtler vortices originating from first/second Mack mode

Physics of Fluids, Volume 32, Issue 3, March 2020.

This work investigates the origination of the secondary instability in Görtler vortices using the linear stability theory, BiGlobal analysis, three-dimensional linear parabolized stability equations (3DLPSEs), and direct numerical simulation (DNS). The flow over a concave wall suffering from the Görtler instability and first/second Mack mode instability is selected. Furthermore, this work simulates the evolution of infinitesimal Mack mode disturbance in a flow perturbed by finite-amplitude Görtler vortices by using DNS and 3DLPSE methods. The 3DLPSE approach accurately predicts the process of Mack mode disturbance evolving into the secondary instability of Görtler vortices, and a perfect agreement with results by DNS is obtained. The results indicate that the secondary instability of stationary Görtler vortices can originate from the first/second Mack mode. The evolutions of first/second Mack mode with different spanwise wavenumbers are performed based on 3DLPSE and compared against the BiGlobal method. The results show that the shape functions and growth rates of disturbances always tend to the results of dominant modes obtained by the BiGlobal method. Because the dominant mode might shift from one to another, the overall evolution cannot be predicted only by the BiGlobal method based on a fixed mode. According to our computations, it is deduced that the Mack modes with the same frequency and symmetric characteristics would finally develop into the secondary instability with similar shapes.

This work investigates the origination of the secondary instability in Görtler vortices using the linear stability theory, BiGlobal analysis, three-dimensional linear parabolized stability equations (3DLPSEs), and direct numerical simulation (DNS). The flow over a concave wall suffering from the Görtler instability and first/second Mack mode instability is selected. Furthermore, this work simulates the evolution of infinitesimal Mack mode disturbance in a flow perturbed by finite-amplitude Görtler vortices by using DNS and 3DLPSE methods. The 3DLPSE approach accurately predicts the process of Mack mode disturbance evolving into the secondary instability of Görtler vortices, and a perfect agreement with results by DNS is obtained. The results indicate that the secondary instability of stationary Görtler vortices can originate from the first/second Mack mode. The evolutions of first/second Mack mode with different spanwise wavenumbers are performed based on 3DLPSE and compared against the BiGlobal method. The results show that the shape functions and growth rates of disturbances always tend to the results of dominant modes obtained by the BiGlobal method. Because the dominant mode might shift from one to another, the overall evolution cannot be predicted only by the BiGlobal method based on a fixed mode. According to our computations, it is deduced that the Mack modes with the same frequency and symmetric characteristics would finally develop into the secondary instability with similar shapes.

Categories: Latest papers in fluid mechanics

### Pattern method for higher harmonics of first normal stress difference from molecular orientation in oscillatory shear flow

Physics of Fluids, Volume 32, Issue 3, March 2020.

This study examines the simplest relevant molecular model of a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow: rigid dumbbells suspended in a Newtonian solvent. For such suspensions, the viscoelastic response of the polymeric liquid depends exclusively on the dynamics of dumbbell orientation. Previously, the explicit analytical expressions of the zeroth, second, and fourth harmonics of the alternating first normal stress difference response in LAOS have been derived. In this paper, we correct and extend these expressions by seeking an understanding of the next higher harmonic. Specifically, this paper continues a series of studies that shed light on molecular theory as a useful approach in investigating the response of polymeric liquids to oscillatory shear. Following the general method of Bird and Armstrong [“Time-dependent flows of dilute solutions of rodlike macromolecules,” J. Chem. Phys. 56, 3680 (1972)], we derive the expression of the first normal stress coefficient up to and including the sixth harmonic. Our analysis relies on the extension of the orientation distribution function to the sixth power of the shear rate. Our expression is the only one to have been derived from a molecular theory for a sixth harmonic and thus provides the first glimpse of the molecular origins of a first normal stress difference higher than the fourth.

This study examines the simplest relevant molecular model of a polymeric liquid in large-amplitude oscillatory shear (LAOS) flow: rigid dumbbells suspended in a Newtonian solvent. For such suspensions, the viscoelastic response of the polymeric liquid depends exclusively on the dynamics of dumbbell orientation. Previously, the explicit analytical expressions of the zeroth, second, and fourth harmonics of the alternating first normal stress difference response in LAOS have been derived. In this paper, we correct and extend these expressions by seeking an understanding of the next higher harmonic. Specifically, this paper continues a series of studies that shed light on molecular theory as a useful approach in investigating the response of polymeric liquids to oscillatory shear. Following the general method of Bird and Armstrong [“Time-dependent flows of dilute solutions of rodlike macromolecules,” J. Chem. Phys. 56, 3680 (1972)], we derive the expression of the first normal stress coefficient up to and including the sixth harmonic. Our analysis relies on the extension of the orientation distribution function to the sixth power of the shear rate. Our expression is the only one to have been derived from a molecular theory for a sixth harmonic and thus provides the first glimpse of the molecular origins of a first normal stress difference higher than the fourth.

Categories: Latest papers in fluid mechanics

### Universal scaling parameter for a counter jet drag reduction technique in supersonic flows

Physics of Fluids, Volume 32, Issue 3, March 2020.

The reduction in aerodynamic drag by injecting a gaseous jet from the nose of a blunt body into a supersonic stream is investigated numerically. The penetration of the jet into the supersonic flow modifies the shock structure around the body and creates a low pressure recirculation zone, thereby decreasing the wave drag significantly. Combining various theoretical estimates of different flow features and numerical simulations, we identify a universal parameter, called the jet to freestream momentum ratio (RmA), which uniquely governs the drag on the blunt body. The momentum ratio fundamentally decides the penetration of the jet as well as the extent of low pressure envelope around the body. In addition, various influencing parameters reported in the literature are reviewed for different steady jet flow conditions. Furthermore, their limitations in regulating the flowfield are explained by correlating the facts with the jet to freestream momentum ratio. We perform the simulations for various combinations of physical and flow parameters of the jet and the freestream to show a universal dependence of drag on the momentum ratio.

The reduction in aerodynamic drag by injecting a gaseous jet from the nose of a blunt body into a supersonic stream is investigated numerically. The penetration of the jet into the supersonic flow modifies the shock structure around the body and creates a low pressure recirculation zone, thereby decreasing the wave drag significantly. Combining various theoretical estimates of different flow features and numerical simulations, we identify a universal parameter, called the jet to freestream momentum ratio (RmA), which uniquely governs the drag on the blunt body. The momentum ratio fundamentally decides the penetration of the jet as well as the extent of low pressure envelope around the body. In addition, various influencing parameters reported in the literature are reviewed for different steady jet flow conditions. Furthermore, their limitations in regulating the flowfield are explained by correlating the facts with the jet to freestream momentum ratio. We perform the simulations for various combinations of physical and flow parameters of the jet and the freestream to show a universal dependence of drag on the momentum ratio.

Categories: Latest papers in fluid mechanics