Two-layer Fluid Research Articles

Development of the wave motion induced by near-bottom periodic disturbances in a two-layer shear current

The behavior of wave motion arising in an ideal incompressible homogeneous fluid under switching-on periodic bottom disturbances is studied in the linear approximation for the two-dimensional non-stationary problem. In the undisturbed state, the velocities of two-layer fluid flow are linear functions of the vertical coordinate in each of the layers with different gradients and coincide on the boundary of the layers. The upper boundary of fluid can be either free or bounded by the rigid cover. The dispersion dependences and the group velocities of the appearing wave modes are determined. The vertical displacements of the free surface and the interface between the layers are calculated. A comparison with the solution for a single-layer fluid is carried out.

Investigating surface and internal waves and viscosity effects in two-layer fluids with a streamlined moving body using fully nonlinear simulation and experiment

Waves generated by a moving submerged moving body were analyzed in a two-layer density fluid. The experiments were performed in a two-dimensional towing tank and compared to the numerical results from a two-dimensional, fully nonlinear numerical towing tank (NTT) based on potential flow. The experimental conditions were determined by the relationship between the depth-dependent Froude number and the critical Froude number in baroclinic mode. The study was performed with various body velocities, changing the upper and lower fluid depth while maintaining the total water depth. An artificial damping coefficient was applied to the entire free surface and interface boundary condition to simulate the viscosity effect of the fluid interface. When the baroclinic mode was dominant, internal precursor soliton waves and depression waves were similar to the experimental results with strong nonlinearity. In contrast, the trailing waves were significantly different from the experimental results without applying an artificial damping scheme. A comparison with the experimental data showed that the effects of the fluid viscosity can be simulated with an appropriate damping coefficient. As the body velocity increases, the depressed or rising waves associated with the baroclinic mode are mixed with the trailing waves of the barotropic mode. As the depth of the upper fluid decreases, the characteristics of the baroclinic mode decrease, and the barotropic mode becomes dominant under the same velocity conditions.

Marginal stability analyses for thermochemical convection and its implications for the dynamics of continental lithosphere and core–mantle boundary regions

SUMMARY Significant compositional differences may exist in the lithospheric mantle and above the core–mantle boundary (CMB) relative to the ambient mantle. The intrinsic density differences may affect the development of thermal boundary layer (TBL) instabilities associated with lithospheric delamination and formation of thermochemical plumes. In this study, we explored the instability of two-layer thermochemical fluid using two different techniques: marginal stability analysis with a propagator-matrix method and finite element modelling. We investigated both the instabilities in lithospheric mantle (i.e. lithospheric instability) and the mantle above the CMB (i.e. plume-forming instability) using a background temperature Tbg(z) with the TBL. For lithospheric instability, we found that two-layer fluid with free-slip boundary conditions mainly undergoes the same three different convective modes (i.e. two oscillatory convection modes and one layered convection regime) as that with no-slip boundary conditions reported in Jaupart et al. However, with free-slip boundary conditions, the transitions between these convection modes occur at larger values of buoyancy number B. Free-slip boundary conditions lead to smaller critical Rayleigh number Rac, but larger convective wavelength and oscillation frequency ωc, compared with those with no-slip boundary conditions. Our numerical modelling results demonstrate that Rac and ωc predicted from the classical marginal stability analyses using Tbg(z) with TBL temperature may have significant errors when the oscillatory period is comparable with or larger than the timescale of lithospheric thermal diffusion that causes Tbg(z) to vary with time significantly. In this case, using a more gently sloped background temperature profile ignoring the TBL temperature, the stability analysis predicts more accurate stability conditions, thus presenting an effective remedy to the stability analysis. For plume-forming instability, because of the reduced viscosity in the hot and compositionally dense bottom layer, the transition to the layered convection occurs at significantly smaller B values, and in the oscillatory convection regime, Rac is larger but ωc is smaller, compared with those for lithospheric instability. Finally, our study provides a successful benchmark of numerical models of thermochemical convection by comparing Rac and ωc from numerical models with those from the marginal stability analysis.

A modified experimental approach for generating internal solitary waves using the dual paddle technique

A modification to the dual paddle wave generation method for internal solitary waves (ISWs) has been achieved through the implementation of the adjusted high-order unidirectional (aHOU) model and the Miyata–Choi–Camassa (MCC) model. This modified method utilizes layer-averaged horizontal velocities from ISWs in both the upper and lower fluid layers to control the operational velocities of the corresponding paddles. Experimental investigations conducted in a two-layer fluid with varied stratification conditions were employed to evaluate the applicability of the aHOU and MCC models. The validation of this approach involved examining the stability of generated ISWs by comparing experimental waveforms with theoretical predictions, demonstrating good agreement with prior research. Furthermore, evaluation of the modified wave generation method included an analysis of dimensionless phase velocity and characteristic frequency. The study also explored the quasi-linear relationship between the designed wave amplitude and the actual wave amplitude, establishing a polynomial fit between the actual wave amplitude and the layer thickness ratio. This approach offers a new method for stable and controllable laboratory generation of ISWs. Under finite water depth conditions, the method can produce ISWs that propagate stably over long distances and effectively control parameters such as wave amplitude and wave profile across a broad range of wave amplitudes and stratifications. These results confirm the effectiveness of the modified method and underscore its practical applicability in ISWs generation.

Rossby waves with barotropic–baroclinic coherent structures and dynamics for the (2 + 1)-dimensional coupled cylindrical KP equations with variable coefficients

Starting from the classical quasi-geostrophic potential vorticity equation with equal depth two-layer fluid, the coupled cylindrical Kadomtsev–Petviashvili (KP) equations with variable coefficients for Rossby waves are studied. To be more general, the phase velocity is considered an indefinite integral about time and improves the analysis procedure. So the variable coefficients are obtained and some previous studies are reasonably explained. The cylindrical wave theory is therewith utilized to reduce the coupled cylindrical KP equations with variable coefficients, and based on the modified Hirota bilinear method, the lump solutions and interaction solutions are found. Through numerical simulations, the Rossby lump waves on both sides of the y axis move closer to the center, and their amplitude gradually decreases and tends to flatten with the generalized Rossby parameter growth. In the Rossby waves flow field, the dipole structures propagate to the east and lead to the appearance of the compress phenomenon during barotropic–baroclinic interaction. It is possibly useful for further theoretical research on atmospheric phenomena.

Sloshing analysis of a two-layered fluid in a tank with baffles using Level Set Method

Sloshing analysis of a two-layered fluid in a tank with baffles using Level Set Method

Boussinesq model for two-fluid system with surface- and interfacial tension

A Boussinesq model associated with surface and internal waves with surface tension and interfacial tension in a two-layer fluid is presented in the application range of weakly dispersive nonlinear. The nonlinear form of model definitions in terms of surface and interfacial tensions along with velocity potentials are provided. Based on the perturbation technique, the Boussinesq equations for both layers are derived. The present Boussinesq result is validated against existing published results and it has a good level of agreement. Further, the present model results of crests of surface and interface displacements are compared with existing published numerical OpenFOAM simulations. The dispersion relations for fast- and slow modes are derived in the quadratic form of wave frequencies in the presence of surface and interfacial tensions. The second-order surface and internal wave displacements along with velocity potentials for upper- and lower-layer fluids are presented. The effects of surface and interfacial tensions on dispersion characteristics, harmonic transfer functions, and shoaling gradients on different physical parameters associated with the present model are studied by analyzing several numerical results. In the end, a real physical problem is demonstrated in the application of surface tension with solvents in the gas-liquid interface.

Assessment of unidirectional internal solitary wave models in a two-layer fluid system through an improved experimental method

The suitability of unidirectional internal solitary wave (ISW) models in a two-layer fluid system is evaluated using an improved experimental method. A new method for generating internal solitary waves in the laboratory tank is proposed, utilizing a piston-type wave maker in a density stratified two-layer fluid. The prescribed velocities of the pistons are determined by the mean velocities derived from the adjusted High-order Unidirectional (aHOU) model for the upper and lower layers. Experimental conditions encompass a range of wave Reynolds numbers between 1.8×105 and 2.3×105. The repeatability and amplitude control of the proposed wave-making method are verified. The unidirectional models considered in this study encompass the Korteweg–de Vries (KdV), Gardner and aHOU model. Laboratory experiments and theoretical computations have been systematically compared in this study to examine the wave characteristics inherent in typical unidirectional internal solitary wave models. It is found that Reynolds number (Re) and nonlinear parameter (α=|a|/h) serve as suitable criteria for delineating the range of applicability of unidirectional models. According to these criteria, the KdV theory is suitable for ISWs with amplitudes α<0.02 across all Reynolds number cases, while the Gardner model is valid for cases where α<0.02 and Reω>2.2×105, with α<LimG, where LimG represents the non-dimensional limiting amplitude. The aHOU model exhibits the capability to capture fully nonlinear ISWs across all Reynolds numbers considered in this study.

Flow analysis of a two-layered micropolar fluid in a catheterized oesophageal tube under the influence of a dilating amplitude: Application to pre-diagnosis of oesophageal motility disorder

This study employs the Homotopy perturbation method to analyze the behavior of immiscible, incompressible fluids within a cylindrical coaxial tube, focusing on scenarios relevant to physiological fluid dynamics, particularly in the catheterized oesophagus and similar biological systems. Adopting long-wavelength and low Reynolds number approximations, a two-layered model is proposed with a micropolar fluid in the core and a Newtonian fluid in periphery regions. Parameters such as velocity, flux, friction, pressure, and impedance variations are formulated, particularly under the influence of dilating wave amplitude. Generally, when a catheter is introduced, pressure rises. It is further found that while pressure falls with increasing micropolar parameter, it rises with coupling number upon catheter insertion. Thus feeding patients with micropolar fluids during catheter-assisted pre-diagnosis is impractical due to associated pressure rise. Observations suggest a complex pressure profile during bolus passage through the oesophagus due to the broadening of the catheter size. Additionally, impedance exponentially increases with catheter size, influenced by the micropolar parameters and the coupling numbers, with micropolar fluids exhibiting higher impedance than that with Newtonian fluids. However, this study underscores the significant impact of catheterization on physiological fluid dynamics, notably increasing oesophageal impedance by two to threefold. This highlights the critical role of catheters in altering flow characteristics, emphasizing the need for a careful medical intervention during pre-diagnostic assessments.

Diffraction and Interaction of Interfacial Solitons in a Two-Layer Fluid of Great Depth

Diffraction and Interaction of Interfacial Solitons in a Two-Layer Fluid of Great Depth

Heat transfer in shear flow over a still nanofluid: A direct integration approach

Heat transfer in a two-layer fluid system, including shear-driven viscous fluid in the upper layer and still nanofluid in the lower layer, is investigated in this work. Following single phase model for nanofluid, three different types of nanofluids are considered. Different from existing literatures on two-layer fluid flow problem, this work introduces the heat transfer of shear-driven viscous fluid over a still nanofluid which makes this investigation new. The main equations which are partial in nature are reduced to a set of coupled nonlinear ordinary differential equation (ODE) by employing suitable similarity transformations. These ODEs together with the coupled interfacial boundary conditions are supplied to MATLAB software to get the solution numerically. Comparing the results obtained in this study with the previous authoritative research results, the accuracy and correctness of the present research methodology are ensured. The results show that the temperature of the upper fluid and the lower fluid are controlled by the thermophysical properties of different types of nanofluids. Also, opposite behavior of temperature distribution for the upper and the lower fluid are observed.

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi &amp; Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Hartmann Flow of Two-Layered Fluids in Horizontal and Inclined Channels

The effect of a transverse magnetic field on two-phase stratified flow in horizontal and inclined channels is studied. The lower heavier phase is assumed to be an electrical conductor (e.g., liquid metal), while the upper lighter phase is fully dielectric (e.g., gas). The flow is defined by prescribed flow rates in each phase, so the unknown frictional pressure gradient and location of the interface separating the phases (holdup) are found as part of the whole solution. It is shown that the solution of such a two-phase Hartmann flow is determined by four dimensionless parameters: the phases’ viscosity and flow-rate ratios, the inclination parameter, and the Hartmann number. The changes in velocity profiles, holdups, and pressure gradients with variations in the magnetic field and the phases’ flow-rate ratio are reported. The potential lubrication effect of the gas layer and pumping power reduction are found to be limited to low magnetic field strength. The effect of the magnetic field strength on the possibility of obtaining countercurrent flow and multiple flow states in concurrent upward and downward flows, and the associated flow characteristics, such as velocity profiles, back-flow phenomena, and pressure gradient, are explored. It is shown that increasing the magnetic field strength reduces the flow-rate range for which multiple solutions are obtained in concurrent flows and the flow-rate range where countercurrent flow is feasible.

Heat and concentration analysis of two-layered muco-ciliary third grade fluid flow in human airways

Understanding the mechanisms involved in the movement and clearance of mucus within the human airways is important from a physiological perspective. A mathematical model of muco-ciliary two-layered fluid flow in a finite two-dimensional channel is proposed. Due to the presence of both the airway ciliary layer (ACL) and the periciliary liquid layer (PCL) in airways epithelial cells, the two-layered model is adopted. The third grade fluid models is used to characterize the viscoelastic nature of mucus secreted in the airways. The mathematical modeling of two-layer flow problem is simplified by using long wavelength and small Reynolds number approximation. A formulated set of partial differential equations (PDEs) is solved for series-form solutions of velocity, temperature, concentration, and pressure gradient by using the Adomian decomposition method (ADM). The analysis delineated that with an increase in pressure gradient at airways entrance ξ the velocity and temperature increases while concentration decreases. Deborah numbers impact ACL and PCL velocities differently, with De(A) influencing flow direction and De(P) causing fluid to flow faster in the forward direction. It also emphasizes the significance of the rate of thermal diffusion, viscous dissipation, and relaxation time. The results showed that the suggested two-layered muco-ciliary model provides a better description of mucus transport in the human airways. The pressure gradient at the airways entrance has a significant impact on mucus transport through the airways.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

A study on the interaction of waves with a submerged vertical elastic plate present in either layer of a two-layer fluid

The problems of interactions between an elastic plate and surface as well as interface waves are considered and analyzed using the theory of linearized water waves. Considering the ocean’s multilayered nature, a two-layered fluid system that serves as the foundation for a more complex multilayered fluid model is examined. It is taken into consideration when an elastic plate is present in either layer of the two-layered water domain. Applying Green’s function technique on the boundary value problem related to the flexible plate and its end conditions, an expression of the normal derivative of the potential function is derived. Employing Green’s identity on the fundamental potential function and the scattered potential, an alternative expression for the normal derivative of the potential function is obtained. Comparison of alternative expressions of the normal derivative of the potential function allows the deduction of a Hypersingular Integral Equation (HSIE) in terms of the unknown potential difference across the plate. Then, using the Expansion-Collocation approach, the HSIE is resolved. The resulting solution provides insights into the reflection coefficient and plate deflection as they vary with different parameters.

Investigations into Motion Responses of Suspended Submersible in Internal Solitary Wave Field

When the underwater submersible encounters an internal solitary wave (ISW), its loadings and motions are significantly disturbed. To investigate the interaction mechanism between the suspended submersible and the ISW, a three-dimensional ISW–submersible-interaction numerical model was established, based on the computational fluid dynamics (CFD) method. The generation and propagation of the ISW was simulated in a two-layer fluid numerical wave tank, according to the eKdV theory. The standard operation equation of the submersible was introduced to simulate the six degree of freedom (6DoF) motions of the submersible combined with the overset dynamic mesh method. The motion simulation method was effectively validated by comparing it with published experimental results on the motion responses of a slender body under the ISW. Based on the constructed numerical model, the dynamic mechanisms between the suspended submersible and the ISW were studied, and the effects of the initial submerged depths and the ISW amplitudes on the dynamic responses of the submersible were revealed. According to the numerical results, the motions of the submersible have been significantly determined by its initial submerged depths. The submersible located above the ISW interface has a significant motion along the propagation direction of the ISW and its motion trajectory resembles a counterclockwise semi ellipse. The motion of the submersible located below the ISW interface follows the trace of the lower layer of fluid, which presents as an unclosed clockwise ellipse. The corresponding motions of the submersible would be increased with the increase in the ISW amplitudes.

Hydrodynamic of surge radiation by a floating cylinder in the presence of bottom-mounted cylindrical obstacle in a two-layer fluid system

Hydrodynamic of surge radiation by a floating cylinder in the presence of bottom-mounted cylindrical obstacle in a two-layer fluid system

Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids

Rogue waves and breathers emerge as significant solitons, result from ubiquitous nonlinear effects of dynamics in the natural world, science and engineering. In this paper, we study the AB system, a model derived from a two-layer fluid that is widely used to investigate nonlinear fluid dynamics. We use the Darboux transformation to construct analytical solutions for higher-order hybrid rogue waves and breathers, which are expressed by symmetric algebraic matrices. We use these solutions and the spectrum parameters to explore the complex interaction dynamics of the hybrid rogue waves and breathers. We find that the interactions cause significant changes in the phases and shapes. Moreover, we discover an interesting phenomenon: the collisions between the rogue waves and breathers are semi-elastic, meaning that only the phases of the rogue waves change, while the speeds, shapes and phases of the breathers remain the same, before and after the collisions. This study provides new insights into the dynamics of the AB system, and helps us understand nonlinear phenomena in various two-layer fluid systems.

Dynamics of a two-layer immiscible fluid system exposed to ultrasound.

The relocation dynamics of a two-layer immiscible fluid system exposed to bulk acoustic waves using simulations and experiments are reported. A theoretical formulation of the acoustic radiation pressure (ARP) acting on the interface reveals that ARP is a nonlinear function of the impedance contrast. It has been shown that the force acting on the interface is the simple sum of the ARP and the interfacial tension, which is dependent on the angle of the interface. It was discovered that although the acoustic radiation force is directed from high-impedance fluid (HIF) to low-impedance fluid (LIF), the final steady-state configuration depends on the wall-fluid contact angle (CA). Our study reveals that the HIF and LIF would relocate to the channel center for CA>110°, and CA<70°, respectively, while complete flipping of the fluids is observed for intermediate angles. The forces relocate the fluids in the channel, generally, by a clockwise or anticlockwise rotation. Here, it is demonstrated that the direction of this twist can be determined by the relative densities and wettabilities of the two fluids.