Fluids, Volume 4, Issue 1 (March 2019) – 58 articles

Recently, the Whitham and capillary Whitham equations were shown to accurately model the evolution of surface waves on shallow water. In order to gain a deeper understanding of these equations, we compute periodic, traveling-wave solutions for both and study their stability. We present plots of a representative sampling of solutions for a range of wavelengths, wave speeds, wave heights, and surface tension values. Finally, we discuss the role these parameters play in the stability of these solutions. Full article

The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant to practical issues. Here, we deal only with the nonlinear character of this dynamics, which has been recognised as the main ingredient to rogue wave formation. In this perspective, the formation of rogue waves requires a non-vanishing and unstable background such as a nonlinear regular wave train with attractive self-interaction. The simplest, best known model of such dynamics is the universal nonlinear Schrödinger equation. This has proven to serve as a good approximation in various contexts and over a broad range of experimental settings. This model aims to give the slow evolution of the envelope of one monochromatic wave due to nonlinearity. Here, we naturally consider the same problem for the envelopes of two weakly resonant monochromatic waves. As for the nonlinear Schrödinger equation, which is integrable, we adopt an integrable model to describe the interaction of two waves. This is the system of two coupled nonlinear Schrödinger equations (Manakov model) with self- and cross-interactions that may be both defocussing and focussing. We first discuss the linear stability properties of the background by computing the spectrum for all values of the parameters such as coupling constants and amplitudes. In particular, we relate the instability bands to properties of the spectrum and compute the gain function (or growth rate). We also relate to the stability spectrum the value of the spectral variable, which corresponds to a rogue wave solution. In contrast with the nonlinear Schrödinger equation, different types of single rogue wave exist that correspond to different values of the spectral variable even in the same spectrum. For these critical values, which are completely classified, we give the corresponding explicit expression of the rogue wave solution that follows from the well known Darboux–Dressing transformation method. Although not all systems of two coupled nonlinear Schrödinger equations that have been derived in water wave dynamics are integrable, our investigation contributes to the understanding of new effects due to wave coupling, at least for model equations that, even if not integrable, are close enough to the model considered here. For instance, our findings lead to investigate rogue waves generated by instabilities due to self- and cross-interactions of defocusing type. An illustrative selection of two coupled rogue waves solutions is displayed. Full article

We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrödinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrödinger equation. When such an envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase in soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow non-stationary solitary wave. The dependences of the soliton parameters on the distance when it moves towards the shoaling are found from the conservation laws and analysed graphically. Estimates for the real ocean are presented. Full article

We study the horizontal dispersion of passive tracer particles on the free surface of gravity waves in deep water. For random linear waves with the JONSWAP spectrum, the Lagrangian particle trajectories are computed using an exact nonlinear model known as the John–Sclavounos equation. We show that the single-particle dispersion exhibits an unusual super-diffusive behavior. In particular, for large times t, the variance of the tracer $⟨|X(t){|}^2⟩$ increases as a quadratic function of time, i.e., ${⟨\left|X(t)\right|}^2⟩\sim t^2$ . This dispersion is markedly faster than Taylor’s single-particle dispersion theory which predicts that the variance of passive tracers grows linearly with time for large t. Our results imply that the wave motion significantly enhances the dispersion of fluid particles. We show that this super-diffusive behavior is a result of the long-term correlation of the Lagrangian velocities of fluid parcels on the free surface. Full article

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations $u_t+s{u}^p{u}_x+u_{xxx}$ (here $s=\pm 1$ and $p>0$ is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper. Full article

The Saffman–Taylor instability for yield stress fluids appears in various situations where two solid surfaces initially separated by such a material (paint, puree, concrete, yoghurt, glue, etc.) are moved away from each other. The theoretical treatment of this instability predicts fingering with a finite wavelength at vanishing velocity, and deposited materials behind the front advance, but the validity of this theory has been only partially tested so far. Here, after reviewing the basic results in that field, we propose a new series of experiments in traction to test the ability of this basic theory to predict data. We carried out tests with different initial volumes, distances and yield stresses of materials. It appears that the validity of the proposed instability criterion cannot really be tested under such experimental conditions, but at least we show that it effectively predicts the instability when it is observed. Furthermore, in agreement with the theoretical prediction for the finger size, a master curve is obtained when plotting the finger number as a function of the yield stress times the sample volume divided by the square initial thickness, in wide ranges of these parameters. This in particular shows that this traction test could be used for the estimation of the material yield stress. Full article

Carbon sequestration in deep saline aquifers was recently developed at the industrial scale. CO₂ injection experiences in carbonates are quite limited, most of them coming from projects carried out in porous mediums in the USA and Canada. Hontomín (Spain) is the actual on-shore injection pilot in Europe, being a naturally fractured carbonate reservoir where innovative CO₂ injection strategies are being performed within the ENOS Project. CO₂ migration through the fracture network existing on site produces hydrodynamic, mechanical and geochemical effectsdifferent from those caused by the injection in mediums with a high matrix permeability. The interpretation of these effects is required to design safe and efficient injection strategies in these formations. For this, it is necessary to determine the evolution of pressure, temperature and flow rate during the injection, as well as the period of pressure recovery during the fall-off phase. The first results from the not-continuous injections (8–24 h) conducted at Hontomín reveal the injection of liquid CO₂ (density value of 0.828 t/m³) and the fluid transmissivity through the fractures. Taking into account the evolution of the pressure and flow rate showed variations of up to 23% and 30% respectively, which means that the relevant changes of injectivity took place. The results were modeled with a compositional dual media model which accounts for both temperature effects and multiphase flow hysteresis because alternative brine and CO₂ injections were conducted. Advanced modeling shows the lateral extension of CO₂ and the temperature disturbance away from the well. Full article

The present work investigates the performance of a mesoscopic Lagrangian approach for the prediction of gas-particle flows under the influence of different physical and numerical parameters. To this end, Geldart D particles with 1 mm diameter and density of 2500 kg/m ${}^3$ are simulated in a pseudo-2D fluidized bed using a Discrete Element Method (DEM)/Large-Eddy Simulation (LES) solver called YALES2. Time-averaged quantities are computed and compared with experimental results reported in the literature. A mesh sensitivity analysis showed that better predictions regarding the particulate phase are achieved when the mesh is finer. This is due to a better description of the local and instantaneous gas-particle interactions, leading to an accurate prediction of the particle dynamics. Slip and no slip wall conditions regarding the gas phase were tested and their effect was found negligible for the simulated regimes. Additional simulations showed that increasing either the particle-particle or the particle-wall friction coefficients tends to reduce bed expansion and to initiate bubble formation. A set of friction coefficients was retained for which the predictions were in good agreement with the experiments. Simulations for other Reynolds number and bed weight conditions are then carried out and satisfactory results were obtained. Full article

Turbulence in the edge plasma of a tokamak is a key actor in the determination of the confinement properties. The divertor configuration seems to be beneficial for confinement, suggesting an effect on turbulence of the particular magnetic geometry introduced by the X-point. Simulations with the 3D fluid turbulence code TOKAM3X are performed here to evaluate the impact of a diverted configuration on turbulence in the edge plasma, in an isothermal framework. The presence of the X-point is found, locally, to affect both the shape of turbulent structures and the amplitude of fluctuations, in qualitative agreement with recent experimental observations. In particular, a quiescent region is found in the divertor scrape-off layer (SOL), close to the separatrix. Globally, a mild transport barrier spontaneously forms in the closed flux surfaces region near the separatrix, differently from simulations in limiter configuration. The effect of turbulence-driven Reynolds stress on the formation of the barrier is found to be weak by dedicated simulations, while turbulence damping around the X-point seems to globally reduce turbulent transport on the whole flux surface. The magnetic shear is thus pointed out as a possible element that contributes to the formation of edge transport barriers. Full article

The thermomagnetic convection of paramagnetic gaseous oxygen in an enclosure under a magnetic field was numerically studied to simulate the thermomagnetic convection in a space environment with no gravity. The magnetic field in the enclosure was non-uniform and was generated by a permanent magnet which had a high magnetic energy product. The magnet was placed at different locations along one of the adiabatic walls with magnetic poles perpendicular to the hot and cold walls of the enclosure. The heat transfer performance, flow field, and temperature field were studied with each location of the magnet. The results show that the thermomagnetic convection in the enclosure was obviously affected by the location of the magnet. There was an optimum magnet location in terms of the best heat transfer performance in the enclosure. The optimum magnet location changed slightly and moved toward the hot wall as the magnetic flux density increased. The value of the Nusselt number, defined as the ratio of convection to conduction, reached up to 2.54 in the studied range of parameters. By optimizing the magnet location, the convection was enhanced by up to 77% at the optimum magnet location. Full article

We perform simulations to study the hydrodynamics of a conical-shaped swimming micro-robot that ejects catalytically produced bubbles from its inside. We underline the nontrivial dependency of the swimming velocity on the bubble deformability and on the geometry of the swimmer. We identify three distinct phases during the bubble evolution: immediately after nucleation the bubble is spherical and its inflation barely affects the swimming speed; then the bubble starts to deform due to the confinement gradient generating a force that propels the swimmer; while in the last phase, the bubble exits the cone, resulting in an increase in the swimmer velocity. Our results shed light on the fundamental hydrodynamics of the propulsion of catalytic conical swimmers and may help to improve the efficiency of these micro-machines. Full article

The waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four-wave interactions of water waves allows to describe the evolution of the two groups as a system of two equations. The fundamental consequence of this decomposition is the conservation of the “number of waves” in each particular group. The envelope approximation for the waves in each group of counter streaming waves is obtained. Full article

Incompressible magnetohydrodynamic (MHD) turbulence under influences of the Hall and the gyro-viscous terms was studied by means of direct numerical simulations of freely decaying, homogeneous and approximately isotropic turbulence. Numerical results were compared among MHD, Hall MHD, and extended MHD models focusing on differences of Hall and extended MHD turbulence from MHD turbulence at a fully relaxed state. Magnetic and kinetic energies, energy spectra, energy transfer, vorticity and current structures were studied. The Hall and gyro-viscous terms change the energy transfer in the equations of motions to be forward-transfer-dominant while the magnetic energy transfer remains backward-transfer-dominant. The gyro-viscosity works as a kind of hyper-diffusivity, attenuating the kinetic energy spectrum sharply at a high wave-number region. However, this term also induces high-vorticity events more frequently than MHD turbulence, making the turbulent field more intermittent. Vortices and currents were found to be transformed from sheet to tubular structures under the influences of the Hall and/or the gyro-viscous terms. These observations highlight features of fluid-dynamic aspect of turbulence in sub-ion-scales where turbulence is governed by the ion skin depth and ion Larmor radius. Full article

In this work, we examine the shear-banding flow in polymer-like micellar solutions with the generalized Bautista-Manero-Puig (BMP) model. The couplings between flow, structural parameters, and diffusion naturally arise in this model, derived from the extended irreversible thermodynamics (EIT) formalism. Full tensorial expressions derived from the constitutive equations of the model, in addition to the conservation equations, apply for the case of simple shear flow, in which gradients of the parameter representing the structure of the system and concentration vary in the velocity gradient direction. The model predicts shear-banding, concentration gradients, and jumps in the normal stresses across the interface in shear-banding flows. Full article

Multiphase flow of oil, gas, and water occurs in a reservoir’s underground formation and also within the associated downstream pipeline and structures. Computer simulations of such phenomena are essential in order to achieve the behavior of parameters including but not limited to evolution of phase fractions, temperature, velocity, pressure, and flow regimes. However, within the oil and gas industry, due to the highly complex nature of such phenomena seen in unconventional assets, an accurate and fast calculation of the aforementioned parameters has not been successful using numerical simulation techniques, i.e., computational fluid dynamic (CFD). In this study, a fast-track data-driven method based on artificial intelligence (AI) is designed, applied, and investigated in one of the most well-known multiphase flow problems. This problem is a two-dimensional dam-break that consists of a rectangular tank with the fluid column at the left side of the tank behind the gate. Initially, the gate is opened, which leads to the collapse of the column of fluid and generates a complex flow structure, including water and captured bubbles. The necessary data were obtained from the experience and partially used in our fast-track data-driven model. We built our models using Levenberg Marquardt algorithm in a feed-forward back propagation technique. We combined our model with stochastic optimization in a way that it decreased the absolute error accumulated in following time-steps compared to numerical computation. First, we observed that our models predicted the dynamic behavior of multiphase flow at each time-step with higher speed, and hence lowered the run time when compared to the CFD numerical simulation. To be exact, the computations of our models were more than one hundred times faster than the CFD model, an order of 8 h to minutes using our models. Second, the accuracy of our predictions was within the limit of 10% in cascading condition compared to the numerical simulation. This was acceptable considering its application in underground formations with highly complex fluid flow phenomena. Our models help all engineering aspects of the oil and gas industry from drilling and well design to the future prediction of an efficient production. Full article

In this study, closure relations for total and turbulent convection fluxes of flame surface density and scalar dissipation rate were developed (i) by placing the focus of consideration on the flow velocity conditioned to the instantaneous flame within the mean flame brush and (ii) by considering the limiting behavior of this velocity at the leading and trailing edges of the flame brush. The model was tested against direct numerical simulation (DNS) data obtained from three statistically stationary, one-dimensional, planar, premixed turbulent flames associated with the flamelet regime of turbulent burning. While turbulent fluxes of flame surface density and scalar dissipation rate, obtained in the DNSs, showed the countergradient behavior, the model predicted the total fluxes reasonably well without using any tuning parameter. The model predictions were also compared with results computed using an alternative closure relation for the flame-conditioned velocity. Full article

A simple steady-state model for a 3-species mixture (ions, electrons, and neutrals) in a screw-pinch plasma configuration is developed. The model is applied to the central plasma column of the PROTO-SPHERA experiment. Degree of ionization, azimuthal current density, and azimuthal ion velocity are calculated. Full ionization is found at plasma temperatures above 1.5 eV, with neutrals confined in an outer shell where radial plasma flow develops and drives both azimuthal current and azimuthal flow. Full article

A series of three-dimensional unsteady Reynolds-averaged Navier–Stokes (RANS) simulations are conducted to investigate the formation of stall cells over a pitching NACA 0012 aerofoil. Periodic boundary conditions are applied to the spanwise ends of the computational domain. Several different pitching ranges and frequencies are adopted. The influence of the pitching range and frequency on the lift coefficient (C_L) hysteresis loop and the development of leading-edge vortex (LEV) agrees with earlier studies in the literature. Depending on pitching range and frequency, the flow structures on the suction side of the aerofoil can be categorized into three types: (i) strong oscillatory stall cells resembling what are often observed on a static aerofoil; (ii) weak stall cells which are smaller in size and less oscillatory; and (iii) no stall cells at all (i.e., flow remains two-dimensional) or only very weak oval-shaped structures that have little impact on C_L. A clear difference in C_L during the flow reattachment stage is observed between the cases with strong stall cells and with weak stall cells. For the cases with strong stall cells, arch-shaped flow structures are observed above the aerofoil. They resemble the Π-shaped vortices often observed over a pitching finite aspect ratio wing. Full article

The linear stability theory of wind-wave generation is revisited with an emphasis on the generation of wave groups. The outcome is the fundamental requirement that the group move with a real-valued group velocity. This implies that both the wave frequency and the wavenumber should be complex-valued, and in turn this then leads to a growth rate in the reference frame moving with the group velocity which is in general different from the temporal growth rate. In the weakly nonlinear regime, the amplitude envelope of the wave group is governed by a forced nonlinear Schrödinger equation. The effect of the wind forcing term is to enhance modulation instability both in terms of the wave growth and in terms of the domain of instability in the modulation wavenumber space. Also, the soliton solution for the wave envelope grows in amplitude at twice the linear growth rate. Full article

A two-phase bubbly flow is often found in the process industry. For the efficient operation of such devices, it is important to know the details of the flow. The paper presents a numerical simulation of the rising bubble in a stagnant liquid column. The interFOAM solver from the open source Computational Fluid Dynamics (CFD) toolbox OpenFOAM was used to obtain the necessary data. The constant and dynamic computational grids were used in the numerical simulation. The results of the calculation were compared with the measured values. As expected, by using the dynamic mesh, the bubble trajectory was closer to the experimental results due to the more detailed description of the gas–liquid interface. Full article

In this study, the ability of standard one- or two-equation turbulence models to predict mean and turbulence profiles, the Reynolds stress, and the turbulent heat flux in hypersonic cold-wall boundary-layer applications is investigated. The turbulence models under investigation include the one-equation model of Spalart–Allmaras, the baseline $k\text{-}\omega$ model by Menter, as well as the shear-stress transport $k\text{-}\omega$ model by Menter. Reynolds-Averaged Navier-Stokes (RANS) simulations with the different turbulence models are conducted for a flat-plate, zero-pressure-gradient turbulent boundary layer with a nominal free-stream Mach number of 8 and wall-to-recovery temperature ratio of $0.48$ , and the RANS results are compared with those of direct numerical simulations (DNS) under similar conditions. The study shows that the selected eddy-viscosity turbulence models, in combination with a constant Prandtl number model for turbulent heat flux, give good predictions of the skin friction, wall heat flux, and boundary-layer mean profiles. The Boussinesq assumption leads to essentially correct predictions of the Reynolds shear stress, but gives wrong predictions of the Reynolds normal stresses. The constant Prandtl number model gives an adequate prediction of the normal turbulent heat flux, while it fails to predict transverse turbulent heat fluxes. The discrepancy in model predictions among the three eddy-viscosity models under investigation is small. Full article

Thermomagnetic convection is still a phenomenon which generates interest among researchers. The authors decided to focus their attention on the magnetic field influence on forced convection and analyze the extended Graetz–Brinkman problem. A numerical model based on a commonly available solver implemented with user-defined functions was used. The results exhibited the variety of possible flow structures depending on the dimensionless parameters, namely Prandtl and Reynolds numbers. Three flow structure classes were distinguished, and they provide a platform for further research. Full article

This paper discusses a turbulent intermittency model introduced in 1990, the B-model. It was found that the original manuscript which introduced the B-model contained a couple arithmetic errors in the equations. This work goes over corrections to the original equations, and explains where problems arose in the derivations. These corrections cause the results to differ from those in the original manuscript, and these differences are discussed. A generalization of this B-model is then introduced to explore the range of behaviors this style of model provides. To distinguish between the different intermittency models discussed in this paper requires structure function power exponents of order greater than 12. As a source of comparison, data from a flume experiment is introduced, and, with the corrections introduced, this data seems to imply that an intermittency coefficient between $0.17$ and $0.2$ gives good agreement. Higher quality future measurements of high order moments could help with distinguishing the different intermittency models. Full article

Transient flows result in unbalanced forces and high pressure in pipelines. Under these conditions, the combined effects of flow-induced forces along with sudden pipe displacements can create cracks in the pipeline, especially at the junctions. This situation consequently results in water leakage and reduced operational efficiency of the pipeline. In this study, displacements and stresses in a buried pressurized water transmission pipe installed by pipe jacking method are investigated using numerical modeling and considering interactions between fluid, pipe, and soil. The analyses were performed consecutively under no-flow, steady flow, and transient flow conditions, in order to investigate the effects of flow conditions on displacements and stresses in the system. Analyses of the results show that displacements and stresses in the jointed concrete pipes are significant under transient flow conditions. Moreover, because of pressure transient effects, maximum tensile stresses exceed the tensile strength of concrete at the junctions, leading to cracks and consequent water leakage. Full article

With the aim of providing a first step in the quest for a reduction of the aerodynamic drag on the rear-end of a car, we study the phenomena of separation and reattachment of an incompressible flow by focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25 ${}^{\circ }$ of inclination. The ensuing recirculation bubble provides the basis for an analytical and numerical investigation of streamwise-streak generation, lift-up effect, and turbulent-wake and Kelvin–Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of a variational formulation, adjoint methods, penalization schemes, and an orthogonalization algorithm. Dealing with the transient growth of spanwise-periodic perturbations, and inspired by the need of physically-realizable disturbances, we finally provide a procedure attaining a kinetic-energy maximal gain on the order of ${10}^6$ , with respect to the power introduced by the external forcing. Full article

The study is motivated by monitoring the space orientation of a hydrolic fracture used in oil production. Streaming potential arises due to the leakage of ionic fracking fluid under the rock elastic forces which make the fracture disclosure disappear after pumping stops. The vector of electric field correlates with the fracture space orientation since the fluid leakage is directed normally to the fracture surfaces. We develop a mathematical model for the numerical evaluation of the streaming potential magnitude. To this end, we perform an asymptotic analysis taking advantage of scale separation between the fracture disclosure and its length. The contrast between the virgin rock fluid and the fluid invading from the fracture is proved to be crucial in a build up of a net charge at the invasion front. Calculations reveal that an increase of the viscosity and resistivity contrast parameters results in an increase of the streaming potential magnitude. Such a conclusion agrees with laboratory experiments. Full article

Propagation of either an infinitely thin interface or a reaction wave of a nonzero thickness in forced, constant-density, statistically stationary, homogeneous, isotropic turbulence is simulated by solving unsteady 3D Navier–Stokes equations and either a level set (G) or a reaction-diffusion equation, respectively, with all other things being equal. In the case of the interface, the fully developed bulk consumption velocity normalized using the laminar-wave speed S_L depends linearly on the normalized rms velocity u′/S_L. In the case of the reaction wave of a nonzero thickness, dependencies of the normalized bulk consumption velocity on u′/S_L show bending, with the effect being increased by a ratio of the laminar-wave thickness to the turbulence length scale. The obtained bending effect is controlled by a decrease in the rate of an increase δA_F in the reaction-zone-surface area with increasing u′/S_L. In its turn, the bending of the δA_F(u′/S_L)-curves stems from inefficiency of small-scale turbulent eddies in wrinkling the reaction-zone surface, because such small-scale wrinkles characterized by a high local curvature are smoothed out by molecular transport within the reaction wave. Full article

In this paper, we study the pressure-driven thin film flow of an inhomogeneous incompressible fluid in which its viscosity depends on the density. The constitutive response of this class of fluids can be derived using a thermodynamical framework put into place to describe the dissipative response of materials where the materials’ stored energy depends on the gradient of the density (Mechanics of Materials, 2006, 38, pp. 233–242). Assuming a small aspect ratio for the channel, we use the lubrication approximation and focus on the leading order problem. We show the mathematical problem reduce to a nonlinear first order partial differential equation (PDE) for the density in which the coefficients are integral operators. The problem is solved numerically and plots that describe the evolution of the density in the fluid domain are displayed. We also show that it was possible to determine an analytical solution of the problem when the boundary data are small perturbations of the homogeneous case. Finally, we use such an analytical solution to validate the numerical scheme. Full article

Different continuous phases formulated with ecofriendly ingredients such as AMIDET® N, an ecological surfactant, as well as welan and rhamsan gums were developed. An experimental design strategy was been in order to study the influence of the ratio of these two polysaccharides and the homogenization pressure applied in a microfluidizer on the critical shear stress for the continuous phases developed. A pure rhamsan gum solution was selected as the starting point for further study based on the production of thyme oil-in-water emulsions. The effect of the homogenization pressure on the physical stability, critical shear stress and droplet size distribution was analyzed for emulsions with optimized values of the rhamsan–welan ratio. These bioactive thyme oil-in-water emulgels could be considered as delivery systems with potential applications in the food industry. Full article