# Latest papers in fluid mechanics

### Reynolds stresses transport in a turbulent channel flow subjected to streamwise traveling waves

Author(s): Mohammad Umair, Sedat Tardu, and Olivier Doche

Turbulent flows generate large skin friction over surfaces, dissipating significant energy and causing a significant waste of fuel. Various skin-friction control techniques have been proposed in the past, among which transverse wall oscillation (TWO) has been proven one of the most effective approaches. We examine the key modifications induced by the TWO technique in the turbulent flow field with the aim of elucidating the physical mechanism responsible for mitigating skin friction. We find that traveling-wave TWO significantly damps turbulence production and coherent structures, resulting in a decrease in turbulence intensity and hence skin friction.

[Phys. Rev. Fluids 7, 054601] Published Fri May 06, 2022

### The effects of nose bluntness on broadband disturbance receptivity in hypersonic flow

While nose bluntness is known to have a large impact on the stability of hypersonic vehicles, its influence on the freestream receptivity process has not been fully characterized for a wide range of conditions. This paper investigates the effects of nose bluntness on the second mode receptivity coefficients and the development of boundary layer disturbances over two 7° half-angle circular blunt cones at Mach 10 after perturbation with broadband freestream pulses of different types. The cones have nose radii of 9.525 mm (case B) and 5.08 mm (case I). Unsteady direct numerical simulation (DNS) and linear stability theory (LST) results compare well and predict stronger second mode growth for case I in all pulse cases. Unsteady DNS also shows variations in extramodal excitation between the cones depending on freestream disturbance type. Spectral receptivity coefficients are generated by decomposing the unsteady DNS data into discrete frequency Fourier modes, which are then corrected with LST N-factors. Fast acoustic disturbances demonstrate minimal variation in receptivity coefficients, while temperature and vorticity disturbances have much higher coefficients in case I. Planar slow acoustic pulses induce stronger disturbances outside of the second mode in case I, resulting in higher peak receptivity coefficients. Results show significant variation in receptivity response based on nose bluntness, pulse geometry, and the type of incident perturbation.

### Aerodynamic performance of a bio-inspired flapping wing with local sweep morphing

Birds and bats frequently reconfigure their wing planform through a combination of flapping and local sweep morphing, suggesting a possible approach for improving the performance of micro aerial vehicles. We explore the effects of combined flapping and local sweep morphing on aerodynamic performance by employing a bio-inspired two-jointed flapping wing with local sweep morphing. The bio-inspired wing consists of inner and outer sections, which flap around the root joint (shoulder) and the midspan joint (wrist), respectively. The aerodynamic forces and the unsteady vortex structures are evaluated by numerically solving the incompressible Navier–Stokes equations. The results show that combined flapping and local sweep morphing can significantly enhance the aerodynamic performance. In particular, the average lift coefficient is 1.50 times greater than that of simple gliding with single local sweep morphing. Combined flapping and local sweep morphing also have a relatively high pitch moment and shift the aerodynamic center position backward, producing advantages in terms of maneuverability/agility and stability. We find that the vortex structures associated with the combined motion feature midspan vortices, which arise from the leading-edge vortices of the inner wing and contribute to the enhanced aerodynamic performance. We show that the kinematics of combined flapping and local sweep morphing can be further optimized if the midspan vortices are captured by the outer wing.

### Variational principle of a shock stationed in a duct

This paper establishes a variational principle of a normal shock stationed in ducts of the various quasi-one-dimensional (quasi-1D) steady non-conservative flow systems. It is found that the locational stability of a shock inside these ducts, namely, the stationary property of the shock, can be judged by analyzing the second-order variation of the potential energy functional of flow impulse with respect to the locational function of the shock. It proves that, for a control volume containing a shock inside a duct, the real stationary location of the shock among all the possible locations satisfying the determined inlet and outlet boundary conditions of the duct is equivalent to that the potential energy of the cross-sectional flow impulse integrated through the entire duct is a minimal. First, the shock location in general duct flows is analyzed by a momentum relaxation method. Then, based on this method, this paper's variational principle is established referring to the principle of minimum potential energy and the principle of virtual displacement. Further, this principle is applied to the flows through a quasi-1D variable-area duct, a one-dimensional (1D) frictional constant-area duct, a 1D heat-exchange constant-area duct, and a 1D mass-additional constant-area duct, which verified the generality of the principle. At last, relevant examples are provided. This variational principle affords a unified and concise theoretical criterion to analyze the stationary property of a normal shock.

### Rarefied gas mixtures with large species mass ratio: Outflow into vacuum

The theoretical study of the processes of the outflow of a binary gas mixture from a source into a vacuum through an orifice in an infinitely thin wall is presented. Two mixtures with a large species mass ratio K are considered: Au–Ne (K = 9.76) and Au–He (K = 49.21). The work continues the study of the flow of Ag–He mixture (K = 26.95) started in Bykov and Zakharov [“Binary gas mixture outflow into vacuum through an orifice,” Phys. Fluids 32, 067109 (2020)]. The results of the direct simulation Monte Carlo made it possible to propose approximations of the mass flow rates of the species and the mixture depending on the species mass ratio, the flow rarefaction degree, and the mole fraction of light species in the source. It is shown that with an increase in the parameter K, an increase in the dimensionless mass flow rate of the mixture referred to the corresponding free molecular value is observed. The maximum dimensionless flow rate corresponds to the near-continuum regime and exceeds the value obtained using the hydrodynamic approximation and the equivalent single gas approach. A variation of K also leads to changes in the spatial distributions of the dimensionless density and velocity of the mixture and some axial focusing of the flow. An increase in the species mass ratio for the case of a small initial mole fraction of the heavy species in the source for a flow regime close to the hydrodynamic one leads to an increase in acceleration and axial focusing of the heavy species.

### Solidification of liquid metal droplet during impact in the presence of vertical magnetic field

We report a liquid metal droplet impacting onto a cold substrate under the influence of vertical magnetic field numerically. During the impacting dynamics, the spreading and the solidification of the droplet are seriously influenced by the magnetohydrodynamic effects. The numerical methodology is implemented by coupling the volume of fluid method and the implicit enthalpy approach, the former is used to track the liquid/solid–gas interface, while the latter is employed to simulate the solidification process. At first, the numerical method is validated against a series of benchmark problems. Then, by varying the impacting velocities, the thermal contact resistance and the magnetic strengths, the variations of the maximum spreading diameter against different dimensionless parameters are reported. An interpolation scheme between the impacting effect, the thermal effect, and the magnetohydrodynamic effect is proposed to predict the maximum spreading factor, and very good agreement is observed compared to our numerical results. After that, we identify different impacting behaviors in different parameter regimes. For non-isothermal cases, we find that the solidification makes the droplet transit from full rebound to adhesion on the cold substrate, and the participation of the magnetic field promotes the pinch off phenomena during the retraction of the liquid drop. Mechanisms for the transitions between different impacting regimes are discussed, and the comparisons with the available experimental results and analytical solutions are also delivered. At last, we identify that the thickness growth of the solidified splat can be predicted by solving the simple one-dimensional Stefan problem, implying that the thermal dynamics is dominating over the hydrodynamic or the magnetohydrodynamic effects during the melting process of the spreading droplet. Our work therefore provides a general framework to model and study more complex configurations, such as the droplet impacting problems in the metallurgical industry and Tokamak devices, in which environment the droplet dynamics significantly depend on the non-isothermal magnetohydrodynamic effects.

### The effect of trip wire on transition of boundary layer on a cylinder

The effect of height of a trip and its location on the transition of boundary layer on a cylinder is studied using large eddy simulations for [math]. The Reynolds number, Re, is based on the free stream speed and diameter of the cylinder (D). Two modes of transition are observed: (a) natural, for a relatively small trip of height [math], via formation of a laminar separation bubble (LSB) and (b) direct, for a large trip of height [math], wherein the formation of LSB is bypassed and the trip disturbs the flow enough to cause separation of the boundary layer and its subsequent turbulent reattachment. Transition delays the final separation leading to a very significant reduction in drag, often referred to as drag crisis. The delay is more for natural as compared to direct transition. Consequently, the drag at the end of crisis is lower for natural transition. The 1.0% trip at [math] leads to a more delayed flow separation than one at [math] from the front stagnation point. The drag crisis takes place in two stages for a cylinder with trip. During each of the two stages, asymmetric transition on the two sides results in generation of circulation and lift force. The effect of trip is felt even by the non-trip side. The cylinder experiences a relatively large “reverse lift” during the second stage of drag crisis. While natural transition is accompanied by intermittency of LSB, direct transition is associated with intermittency in laminar vs turbulent attachment of the flow following its separation at the trip.

### Moving immersed boundary method for fluid–solid interaction

A strongly coupled algorithm is presented for the incompressible fluid–rigid body interaction using the moving immersed boundary method. The pressure and the boundary force are treated as Lagrange multipliers to enforce the incompressibility and no-slip wall constraints. To compute the two unknowns from the velocity field, we adopt the fractional step algorithm and successively apply the two constraints. A Poisson equation and a moving force equation are derived for the pressure and the boundary force, respectively. As both coefficient matrices are formulated to be symmetric and positive-definite, the resulting linear systems are solved efficiently with the conjugate gradient solver. The strongly coupled nonlinear fluid–solid system is achieved by a fixed-point iteration. To improve the computational efficiency, we only iterate the moving force equation with the rigid body motion equation, and the time-consuming pressure Poisson equation is solved once the sub-iteration is finished. The proposed method is validated with various benchmark tests, and the results compare well with the literature.

### Drag force on a circular cylinder in stratified flow: A two-dimensional numerical study

Using two-dimensional numerical simulations, we investigate a circular cylinder in a stratified flow with a pycnocline of a smooth density profile. We are particularly concerned about the difference between stratified flows and their homogeneous, or unstratified, counterparts in drag coefficients. It is well known that the characteristics of a stratified flow depend on the Reynolds number Re and the internal Froude number Fr. First, we change the incoming flow velocity, with the Reynolds number and the internal Froude number varying simultaneously, in the ranges of [math] and [math], respectively. We find that the flow experiences a sequence of flow patterns, namely “multiple centerline structures,” “isolated mixed regions,” and “vortex shedding,” as we increase this combination, agreeing well with the previous experiments in the low Re and Fr ranges. Meanwhile, the mean drag coefficient decreases, and it drops below the value for a homogeneous state as an empirical stability parameter, k, drops below unity, indicating the transition from a lee-wave dominant wake to an unsteady, vortex shedding wake. This unique variation of drag coefficient is more clearly shown in another situation, when we fix the Reynolds number at Re = 234 while varying the Froude number within [math]. In such a situation, we observe a stable “double eddy” pattern at Fr = 1.87, around which a minimum of mean drag coefficient is reached. At this critical point, we understand that the stratification in the flow inhibits lee-side separation, while the internal waves have not yet played a significant role. Although we observe turbulence-like, strong mixing regions in the far downstream wake for some specific cases, it is argued that the current two-dimensional simulations might not be able to resolve the high Reynolds number or high Froude number cases.

### Synchronized states of hydrodynamically coupled filaments and their stability

Author(s): Smitha Maretvadakethope, Yongyun Hwang, and Eric E. Keaveny

Cilia and flagella are used throughout the natural world to facilitate microscale fluid motion. These active structures often appear in groups and their motion is coordinated. This paper explores the synchronized states of a pair of hydrodynamically coupled filaments and characterizes in detail the recently discovered bistability of two states. This study identifies the unstable edge state that exists between the two basins of attraction and shows how the bifurcations exhibited by the filament system can be recovered using an extension of Adler’s equation for coupled oscillators.

[Phys. Rev. Fluids 7, 053101] Published Thu May 05, 2022

### Flow structures and vertical transport in tilting salt fingers with a background shear

Author(s): Junyi Li and Yantao Yang

Fingering double diffusive convection plays an important role in the vertical mixing of the ocean, and is inevitably affected by background shear. Here we show that with a very weak shear salt fingers become horizontally well-organized, and the salinity flux can be enhanced. While for strong shear salt fingers are replaced by sheet-like structures and fluxes are suppressed compared to the cases without shear.

[Phys. Rev. Fluids 7, 053501] Published Thu May 05, 2022

### Effect of surface temperature strips on the evolution of supersonic and hypersonic Mack modes: Asymptotic theory and numerical results

Author(s): Lei Zhao and Ming Dong

The hypersonic boundary-layer transition is affected crucially by surface imperfections such as heating or cooling sources. In order to quantify this effect we develop an asymptotic theory, which not only reveals the interaction mechanism between oncoming instability modes and surface heating/cooling strips, but also provides quantitatively the change of the transition onset due to the scattering effect. The asymptotic predictions agree favorably with harmonic linearized Navier-Stokes calculations and direct numerical simulations.

[Phys. Rev. Fluids 7, 053901] Published Thu May 05, 2022

### Spectra and structure functions of the temperature and velocity fields in supergravitational thermal turbulence

We analyze the power spectra and structure functions (SFs) of the temperature and radial velocity fields, calculated in the radial and azimuthal directions, in annular centrifugal Rayleigh–Bénard convection (ACRBC) for Rayleigh number Ra [math], Prandtl number Pr = 10.7, and inverse Rossby number [math] using the spatial data obtained by quasi-two-dimensional direct numerical simulation. Bolgiano and Obukhov-like (BO59-like) scalings for the energy spectrum in both the azimuthal and radial directions and thermal spectrum in the azimuthal direction are observed. The range of BO59-like scaling becomes wider as Ra increases. At [math], it is found that BO59-like scaling [math] spans nearly two decades for the energy spectrum calculated in the radial direction. Power-law fittings in the range larger than the Bolgiano scales, the scaling exponents of transverse and longitudinal velocity SFs vs the order coincide with the theoretical prediction of BO59 scaling [math] basically. The second-order temperature SFs exhibit a gradual transition from the Obukhov–Corrsin behavior at scales smaller than the Bolgiano scales to the BO59 behavior at scales larger than the Bolgiano scales. The slopes from the third to sixth-order temperature SFs are similar, which is similar to classical Rayleigh–Bénard convection and Rayleigh–Taylor turbulence. The probability density functions (p.d.f.) of temperature fluctuations [math] reveal the cold plumes are strong and the p.d.f. in different regions at high Ra are similar. The stronger turbulent-mixing and larger centrifugal buoyancy in ACRBC may result in the BO59-like scaling.

### On the estimation of bulk viscosity of dilute nitrogen gas using equilibrium molecular dynamics approach

In this work, we present a study for the estimation of bulk viscosity using the equilibrium molecular dynamics-based Green–Kubo method. We have performed a parametric study to find optimal hyper-parameters to estimate bulk viscosity using the Green–Kubo method. Although similar studies exist for shear viscosity, none has been reported so far specifically for bulk viscosity. The expected uncertainty in bulk viscosity for a given length and number of molecular dynamics trajectories used in statistical averaging is determined. The effect of system size, temperature, and pressure on bulk viscosity has also been studied. The study reveals that the decay of autocorrelation function for bulk viscosity is slower than that for shear viscosity and hence requires a longer correlation length. A novel observation has been made that the autocorrelation length required for convergence in the Green–Kubo method for both shear and bulk viscosity of dilute nitrogen gas is of the same mean collision time length units irrespective of simulation pressure. However, when the temperature is varied, the required autocorrelation length remains unaffected for shear viscosity but increases slightly with temperature for bulk viscosity. The results obtained from the Green–Kubo method are compared with experimental and numerical results from the literature with special emphasis on their comparison with the results from the nonequilibrium molecular dynamics-based continuous expansion/compression method. Although the primary focus and novelty of this work are the discussion on bulk viscosity, a similar discussion on shear viscosity has also been added.

### Analytical simulation of normal shock waves in turbulent flow

The focus of the work is on analytical modeling of normal shock wave propagation in a turbulent adiabatic gas flow. For this, a modified Rankine–Hugoniot model was developed. A solution is obtained for the Rankine–Hugoniot conditions in a turbulent gas flow with different turbulence intensity. Variation of the velocity of an adiabatic turbulent gas flow during its passage through a normal shock wave is elucidated depending on the turbulence intensity. The equation of the modified Hugoniot adiabat is also obtained.

### Euler–Euler modeling of reactive bubbly flow in a bubble column

To clarify the mass transfer mechanism and reaction behavior in the multiphase flow process, numerical simulations were implemented in the process of NaOH absorbing CO2 by means of the Euler–Euler two-fluid model coupled with the net production rate model of species. Results show that the chemisorption process can be divided into three stages, which are dominated by different chemical reaction equations, and the influence of the water ionization cannot be neglected at pH <10. The concentration change of each species is slightly earlier than those in the literature, while the time-dependent pH value is in good agreement with the experimental results and the consistency between the simulation and the experiment of the hydrodynamic parameters such as the axial component of bubble velocity and the oscillation frequency of bubble flow can be achieved. Therefore, it can be concluded that the mathematical model proposed in this paper can better reproduce the detailed characteristics of the reactive bubbly flow. Furthermore, the effect of bubble induced turbulence on the interfacial reaction behavior is also discussed, and the influence of bubble induced turbulence can be negligible.

### Eddy diffusivity operator in homogeneous isotropic turbulence

Author(s): Yasaman Shirian and Ali Mani

We present direct quantification of the scale-dependent eddy-diffusivity for homogeneous isotropic turbulence. This analysis has implications in closure modeling of transport equations involving turbulence. The computational methodology is based on the previously developed macroscopic forcing method which uses macroscopic forcing in transport equations and collects the field response. Eddy diffusivity is then obtained from statistics of the forced system after mapping the response field to the average space. In the limit of large scales the eddy diffusivity is consistent with the Boussinesq approximation; however, it deviates proportional to the scale in the limit of small scales.

[Phys. Rev. Fluids 7, L052601] Published Wed May 04, 2022

### Turbulent vertical convection under vertical vibration

Vertical convection (VC) under the action of vertical vibration in a square cavity has been investigated using direct numerical simulation. The simulations are conducted with Prandtl number Pr fixed at 4.38 and Rayleigh number Ra ranging from [math] to [math]. To examine the influence of vertical vibration, the dimensionless vibration frequency is varied in the range of [math] and a small dimensionless amplitude is fixed at [math]. First, for low vibration frequency, trivial results are obtained where flow structures and the scalings of Nu and Re resemble that of the standard VC cases. In contrast, when the vibration frequency ω increases beyond a critical value [math], a strong shearing effect from vibration leads to abundant eruptions of thermal plumes from sidewalls, and thus a laminar-turbulent transition of the bulk flow. As a result, heat-transport is greatly enhanced and the scaling exponent β of [math] substantially increases in such the vibration-dominated regime. In specific, the scaling relations obtained transit from [math] and [math] at ω = 0 in the laminar regime to [math] and [math] at [math] in the turbulent regime. Analysis of the mean flow field shows that the vibration thins the thermal boundary layer and enhances the thermal dissipation rate in the bulk region. Furthermore, we found that the trend of Nu and Re can be well described by the vibrational Rayleigh number [math]. In particular, Nu is insensitive to [math] for [math], whereas [math] for [math], where the critical vibrational Rayleigh number exhibits a scaling relation [math] obtained from numerical results.

### Modal analysis of the propeller wake under the heavy loading condition

Knowledge of the dynamics of propeller wakes is fundamental to design and optimize the next-generation propellers. This work aims at investigating the wake instability of a propeller operating under the heavy loading condition. Modal decomposition techniques are used to analyze the evolution characteristics of the propeller wake achieved by previous numerical simulations using different turbulence models [Wang et al., “Numerical simulation of the wake instabilities of a propeller,” Phys. Fluids 33, 125125 (2021)]. Modal analysis is performed on snapshots extracted from improved delay-detached eddy simulations and large eddy simulations of the propeller wake topologies under the high loading condition. In particular, proper orthogonal decomposition and dynamic mode decomposition are employed to identify the modes that play dominant roles in the destabilization physics of the propeller wake. The present study further extends knowledge of propeller wake instability inception mechanisms under heavy loading conditions.

### Spatiotemporal optimization on cross correlation for particle image velocimetry

We introduce an optimization method for the cross-correlation operation in particle image velocimetry by locating the correlation peaks assisted with constraint conditions. In this study, an objective function was constructed to include the residual of the normalized cross-correlation term, a component in charge of spatial smoothness (inspired by the optical flow method as used in a previous study) and a component for temporal smoothness (inspired by the concept of trajectory selection in particle tracking velocimetry). Minimizing the objective function gives optimized velocity fields for a series of tracer images for spatiotemporal smoothness. The proposed method was examined in synthetic images of turbulent flow and Batchelor vortex and in a laboratory experiment of vortex rings. The effect of image background noises and the initial guess for the optimization process were examined and discussed.