# Latest papers in fluid mechanics

### Large-eddy simulation of turbulent natural-bed flow

The turbulent flow in natural rough beds is a complex subject, still poorly understood despite the longstanding effort of several researchers. In the present work, a turbulent open-channel flow experiment, with a pebble bed at Reynolds and Froude numbers, respectively, Re = 4.65 × 104 and Fr = 0.186, has been simulated using the Large-Eddy Simulation (LES) technique, in which the wall-adapting local eddy viscosity subgrid scale closure model is used and in the presence of an air-water interface to take into account the effects of the interface deformation in the flow turbulence statistics under a low relative submergence condition. The simulations have been compared with a companion experiment, where the channel bottom is constituted by four pebble layers. For the simulations, the pebble-bed surface has been captured with a high-resolution three-dimensional laser scanner and used to morphologically characterize the numerical channel bottom. Results are presented in terms of turbulence statistics and turbulent laws, showing a good agreement with those obtained in the experiment. Since a good convergence between simulation and experimental results was obtained, the LES dataset was used to compute the Turbulent Kinetic Energy (TKE) dissipation rate across the water depth. The mesh resolution allows showing a detailed TKE dissipation rate distribution across the water depth. Moreover, the equilibrium between TKE production and dissipation was checked to verify the overlap layer existence under low relative submergence condition. Finally, a new procedure for vortex-visualization is implemented, based on the relationship between the vorticity and the TKE dissipation rate.

### Dynamics of free-surface mutually perpendicular twin liquid sheets and their atomization characteristics

The radially expanding twin (circular and vertical) liquid sheets produced by impingement of a vertical cylindrical liquid jet onto a horizontally placed cone-disk deflector with a single slot were examined experimentally in the present work. Dynamics of these liquid sheets and the events leading to their breakup were studied by carrying out high-speed shadowgraphy simultaneously from side, front, and top views at a 5.4 kHz framing rate and for the jet Weber number (Wejet) range of 993 < Wejet < 3776. In the presence of the slot, the variation of the radial breakup distance of the circular sheet (Rb,CS) with Wejet changed from the monotonically decreasing trend ([math]) to a nonmonotonic increasing and decreasing one. Furthermore, Rb,CS was found to be lowered by about 42% compared to the breakup distance Rb,CS,no-slot of the circular sheet for the no-slot deflector. The vertical sheet breakup distance (Rb,Vs) was found to increase monotonically with the slot Weber number [math]. Three primary sources of droplet production, namely, the lower and front edges of the vertical sheet and the rim of the circular sheet, were identified. The smallest droplets were seen to originate from the front edge (D32,FE) and the largest droplets from the lower edge (D32,LE) of the vertical sheet. The measured droplet diameters followed [math] and [math], whereas the droplets originating at the rim of the circular sheet followed [math]. The droplets at all three edges were found to depend more strongly on the ligament thickness than the ligament length. Following conservation of mass, a linear relation between the droplet diameter, D32, and the ligament thickness, tlig, at each edge has been obtained.

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

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

### Papers Selected from the 10th National Congress on Fluid Mechanics of China, Hangzhou, 2018

### Surface tension and the origin of the circular hydraulic jump in a thin liquid film

Author(s): Alexis Duchesne, Anders Andersen, and Tomas Bohr

We discuss how to include surface tension in viscous thin film flows such as the circular hydraulic jump. We show that an energy term previously proposed can lead to a large overestimate of the influence of surface tension.

[Phys. Rev. Fluids 4, 084001] Published Fri Aug 02, 2019

### Conservation of the circulation for the Euler and Euler-Leray equations

Author(s): Jean Ginibre, Martine Le Berre, and Yves Pomeau

A simple proof of the Kelvin Theorem, namely conservation of circulation (CC) for solutions of the Euler equation, is given. The result is rewritten in terms of time rescaled variables leading to the Euler-Leray equations, and the implications of CC on the existence of self-similar solutions are discussed.

[Phys. Rev. Fluids 4, 084401] Published Fri Aug 02, 2019

### Flow induced by a rotating cone: Base flow and convective stability analysis

Author(s): Antonio Segalini and Simone Camarri

The boundary layer over a cone rotating in a still fluid is investigated. A self-similar correction to the classical von Kármán solution, taking into account the effect of the outer flow, is proposed and validated. Finally, the stability properties of the corrected base flow are assessed.

[Phys. Rev. Fluids 4, 084801] Published Fri Aug 02, 2019

### Diffuse interface immersed boundary method for low Mach number flows with heat transfer in enclosures

A novel diffuse interface immersed boundary (IB) approach in the finite volume framework is developed for non-Boussinesq flows with heat transfer. These flows are characterized by variable density, large temperature differences, nonzero velocity divergence, and low Mach numbers. The present IB methodology assumes that the solid body immersed in the domain is filled with a “virtual” fluid and constructs a unified momentum equation that is solved everywhere in the domain. The unified momentum equation is obtained as a convex combination of the Navier-Stokes equation and the no-slip boundary condition employing the solid volume fraction. The hydrodynamic pressure (p) that drives the flow is obtained by the solution of a variable density Poisson equation that is constructed by assuming that the velocity field inside the solid always remains solenoidal although the velocity divergence is nonzero in the fluid domain. The unified Poisson equation is also solved everywhere in the domain and has source terms that depend on the solid volume fraction, temperature gradients, and the spatially invariant thermodynamic pressure (P) that vanish in the Boussinesq limit. The thermodynamic pressure in closed domains follows from the principle of global mass conservation and is used to determine the density field everywhere in the domain except inside the solid where the density remains constant. Numerical simulations are carried out for natural and mixed convective flows in enclosures with stationary and moving heated bodies encompassing both Boussinesq and strongly non-Boussinesq flow regimes. The results of these investigations show that the local Nusselt number distribution over the body surface is oscillatory particularly when grid lines are not aligned with the surface of the body. However, the proposed approach can reasonably accurately compute the average heat transfer in both Boussinesq and non-Boussinesq flows. Investigations show that the heat transfer is significantly enhanced in the non-Boussinesq regime as compared to the Boussinesq regime. A comparison of results from the present approach with those obtained using a body-fitted finite volume solver for stationary bodies demonstrates that the proposed IB approach can compute the flow dynamics quite accurately even on Cartesian meshes that do not conform to the geometry. The IB approach presented herein is a generic approach for quasi-incompressible flows and may be applied to other low Mach number flows such as mixing and reacting flows.

### Wall pressure and secondary-flow origination in a square duct

The secondary flow of Prandtl’s second kind is a typical feature of turbulent flow in a duct with a noncircular cross section. In this paper, we proposed a physical mechanism on the formation of the secondary flow by analyzing the momentum balance along selected paths. The blocking effects of intersecting walls cause the pressure at the corner to be higher than that at the wall midpoint. Driven by the pressure gradient, fluids above the wall flow away from the corner, complemented by an inflow toward the corner along the diagonal. The proposed mechanism is confirmed using data from direct numerical simulation with the friction Reynolds number Reτ up to 900.

### Electrokinetic instability in microchannel viscoelastic fluid flows with conductivity gradients

Electrokinetic instability (EKI) is a flow instability that occurs in electric field-mediated microfluidic applications. It can be harnessed to enhance sample mixing or particle trapping but has to be avoided in particle separation. Current studies on EKI have been focused primarily on the flow of Newtonian fluids. However, many of the chemical and biological solutions exhibit non-Newtonian characteristics. This work presents the first experimental study of the EKI in viscoelastic fluid flows with conductivity gradients through a T-shaped microchannel. We find that the addition of polyethylene oxide (PEO) polymer into Newtonian buffer solutions alters the threshold electric field for the onset of EKI. Moreover, the speed and temporal frequency of the instability waves are significantly different from those in the pure buffer solutions. We develop a three-dimensional preliminary numerical model in COMSOL, which considers the increased viscosity and conductivity as well as the suppressed electroosmotic flow of the buffer-based PEO solutions. The numerically predicted threshold electric field and wave parameters compare favorably with the experimental data except at the highest PEO concentration. We attribute this deviation to the neglect of fluid elasticity effect in the current model that increases with the PEO concentration.

### Spatial characteristics of a zero-pressure-gradient turbulent boundary layer in the presence of free-stream turbulence

Author(s): Eda Dogan, R. Jason Hearst, Ronald E. Hanson, and Bharathram Ganapathisubramani

Particle image velocimetry measurements are performed to examine the spatial structure in boundary layers under the influence of free-stream turbulence (FST). A similarity of the structural organization inside the boundary layer is found between the present FST cases and the canonical flows.

[Phys. Rev. Fluids 4, 084601] Published Thu Aug 01, 2019

### Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids in cylindrical enclosures and cylindrical annuli

An analytical study of linear and nonlinear Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids confined in a cylindrical porous enclosure is made. The effect of concentric insertion of a solid cylinder into the hollow circular cylinder on onset and heat transport is also investigated. An axisymmetric mode is considered, and the Bessel functions are the eigenfunctions for the problem. The two-phase model is used in the case of nanoliquids. Weakly nonlinear stability analysis is performed by considering the double Fourier-Bessel series expansion for velocity, temperature, and nanoparticle concentration fields. Water well-dispersed with copper nanoparticles of very high thermal conductivity, and one of the five different shapes is chosen as the working medium. The thermophysical properties of nanoliquids are calculated using the phenomenological laws and the mixture theory. It is found that the effect of concentric insertion of a solid cylinder into the hollow cylinder is to enhance the heat transport. The results of rectangular enclosures are obtained as limiting cases of the present study. In general, curvature enhances the heat transport and hence the heat transport is maximum in the case of a cylindrical annulus followed by that in cylindrical and rectangular enclosures. Among the five different shapes of nanoparticles, blade-shaped nanoparticles help transport maximum heat. An analytical expression is obtained for the Hopf bifurcation point in the cases of the fifth-order and the third-order Lorenz models. Regular, chaotic, mildly chaotic, and periodic behaviors of the Lorenz system are discussed using plots of the maximum Lyapunov exponent and the bifurcation diagram.

### Vibration-induced floatation of a heavy liquid drop on a lighter liquid film

We carry out a theoretical study of vibration-induced saturation of the Rayleigh-Taylor instability for an isolated liquid drop on the surface of a less dense finite-thickness carrier film. Without vibration, a heavy drop falls through the carrier film by forming a stretching liquid column until the bottom tip of the column reaches the solid substrate and the carrier film ruptures. We show that an externally applied vertical vibration prevents the rupture of the film and enables stable flotation of the drop. A hydrodynamic model is used to study the effect of inertia on the long-time dynamics of the drop. It is shown that rupture can only be prevented when the Reynolds number is nonzero.

### Direct simulation Monte Carlo on petaflop supercomputers and beyond

The gold-standard definition of the Direct Simulation Monte Carlo (DSMC) method is given in the 1994 book by Bird [Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon Press, Oxford, UK, 1994)], which refined his pioneering earlier papers in which he first formulated the method. In the intervening 25 years, DSMC has become the method of choice for modeling rarefied gas dynamics in a variety of scenarios. The chief barrier to applying DSMC to more dense or even continuum flows is its computational expense compared to continuum computational fluid dynamics methods. The dramatic (nearly billion-fold) increase in speed of the largest supercomputers over the last 30 years has thus been a key enabling factor in using DSMC to model a richer variety of flows, due to the method’s inherent parallelism. We have developed the open-source SPARTA DSMC code with the goal of running DSMC efficiently on the largest machines, both current and future. It is largely an implementation of Bird’s 1994 formulation. Here, we describe algorithms used in SPARTA to enable DSMC to operate in parallel at the scale of many billions of particles or grid cells, or with billions of surface elements. We give a few examples of the kinds of fundamental physics questions and engineering applications that DSMC can address at these scales.

### Improved upper bounds on the energy dissipation rate for shear flow with injection and suction

Improved upper bounds on viscous energy dissipation rates of wall-driven shear flow subject to uniform injection and suction rates are computationally determined. The so-called “background” variational formulation is implemented via a time-stepping numerical scheme to determine optimal estimates. Shear flow Reynolds numbers range from 50 to 40 000 with injection angles up to 2°. The computed upper bounds for preselected angles of injection at high Reynolds numbers significantly improve the rigorously estimated ones. Our results suggest that the steady laminar flow is nonlinearly stable for angles of injection greater than 2°.

### Ultra-local model-based control of the square-back Ahmed body wake flow

This paper presents a new model-free control approach applied to a dynamical fluidic system. The main objective is to evaluate the ability of this closed-loop control technique to control the bistability of a turbulent wake flow past to a square-back Ahmed body. This bistable behavior occurs for some configurations depending mainly on the ground clearance. Due to the unsteady position of the wake vortex cores, the bistable phenomenon is responsible of a strong variation of the lateral force (drift force) and of a slight drag increase. Consequently, mitigating the wake symmetry-breaking modes can induce a substantial drag reduction. The feedback controller controls the drift using its ultralocal approximation and the estimation of its dynamics. The control signal is then applied to lateral blower actuators to suppress the spanwise bistability. The drift force is used as feedback to sense the wake flow, and concomitant velocity, forces, and pressure measurements are performed at a nominal Reynolds number of Reh = 2.86 × 105 to quantify and demonstrate the effectiveness of the present closed-loop control. Results show that for various actuation velocity ratios, the bistability suppression can lead to a drag reduction up to 2.5% with an energy consumption evaluated to be less than 0.6% of the aerodynamic power saving for the worst investigated case.

### A deep learning enabler for nonintrusive reduced order modeling of fluid flows

In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly nonintrusive since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate and can be used as a robust predictive tool for nonintrusive model order reduction of complex fluid flows.

### Diffusivity ratio effect on the onset of the buoyancy-driven instability of an A + B → C chemical reaction system in a Hele-Shaw cell: Numerical simulations and comparison with experiments

The effect of different diffusivities on the evolution of buoyancy-driven instability in a reactive-diffusion system is analyzed. For an instantaneous A + B → C chemical reaction in a Hele-Shaw cell, where a less dense phase of A is layered on top of a denser solution of B, the temporal evolution of the instability motion is traced numerically by using the Fourier spectral method. As expected, the evolution of instabilities can be controlled by chemical factors, such as the ratios of diffusivities, reactant concentrations, and densification coefficients. Double diffusive effects accelerate and hinder the evolution of instabilities and induce the onset of instabilities without an adverse density gradient. The present numerical simulation explains the previous experiments for the NH3 + CH3COOH → CH3COONH4 reaction in a Hele-Shaw cell, which was devised to explain the impact of the chemical reaction in geological CO2 sequestration.

### Numerical study of droplet formation in the ordinary and modified T-junctions

This work presents a flexible manipulation solution on droplet formation based on the modified T-junction with a rectangular rib to reduce the droplet size and improve monodispersity. The droplet formation in the ordinary and modified T-junctions is numerically investigated using the verified three-dimensional volume of fluid method. The results reveal that the modified T-junction can significantly enlarge the dripping regime and droplet-generable regimes while decreasing the jetting regime. In the modified T-junction, the droplet detachment is much easier as the detachment driving forces are strengthened, while the resistance forces are weakened. By investigating the droplet formation in the ordinary and different modified T-junctions with change in viscosity, surface tension, and wall wettability, it is found that the dominant geometric factor affecting the droplet formation is the rib height, not the rib width. Based on the rib height, two modified scaling laws are proposed to predict droplet size in squeezing and dripping regimes. The wall wettability can deteriorate the droplet formation in the ordinary T-junction, while the rib in the modified T-junction can weaken this adverse effect.

### Single diffusive magnetohydrodynamic pressure driven miscible displacement flows in a channel

We investigate the influence of a magnetic field on the single diffusive pressure driven miscible displacement of a low viscous fluid by a high viscous one in a channel using the streamline upwind Petrov-Galerkin based finite element method. We perform transient numerical simulations of the governing continuity and Navier–Stokes equations with magnetohydrodynamic effects coupled with the convection–diffusion solute concentration equation. We have assumed concentration-dependent viscosity and neglected the density contrast. Our computational results are found to match quite well with the other results from the literature. We report that the presence of a magnetic field can suppress the interface instabilities characterized by intense convective mixing and roll-up phenomena for the classical situation of a less viscous fluid displacing a more viscous one. We have found various new types of instability patterns with the combined influences of the Hartmann number, Reynolds number, and Schmidt number. We show that the mushroomlike structure at the tip of the leading finger grows in volume with enhancing magnetic field strength, whereas follows the reverse trend as the Reynolds number is increased. Finally, to examine the effect of magnetic field on the global stability characteristics, we have performed a dynamic mode decomposition analysis. Our analysis demonstrates that by effectively maneuvering the dimensionless parameters, the displacement rate can be enhanced, and this is attributed to the acceleration in fluid mixing. Apart from the fundamental importance, we trust that the results obtained from this study may help in improving the operating efficiency of the modern generation process industries.