Free Surface Of Fluid Research Articles

A simple approximation for the drift rates of rotating polygons on a free fluid surface

We report on laboratory experiments investigating the dynamics of a free fluid surface in a cylindrical tank with a rotating bottom plate. The shear instability in the system creates polygonal structures, which propagate around the domain, along the fixed vertical sidewalls. We analyze the wavelengths and drift rates of these patterns, which are known from previous literature to be created by a complex interplay between centrifugal effects and gravity wave propagation in this unique geometry. We find that with an empirical correction factor, the drift rates of the polygonal vortices can be approximated fairly well by a surprisingly simple formula derived from the dispersion relation of linear gravity waves using easily observable parameters.

An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, https://doi.org/10.1002/cpa.3160180124 ). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $$\alpha \ge 0$$ and spatial period $$2\pi /\alpha $$ ; two-dimensional sloshing corresponds to the case $$\alpha = 0$$ . Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $$\alpha > 0$$ . In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $$\alpha \ge 0$$ and shallow radially symmetric containers with m azimuthal nodal lines, $$m = 0, 1$$ . Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $$\alpha = 0$$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.

Optimization of Performance Characteristics of Magnetofluidic Seals for Wind Power Plants

The use of magnetofluidic seals are a promising direction in sealing rotating shafts of wind power plants. Magnetofluidic seals are characterized by high tightness, simplicity of the design, low losses by friction. Magnetic fluid seal of the rotating shaft consists of a ring magnet and two concentrators of magnetic field, forming with the shaft a narrow ring gap in which the magnetic fluid retained by magnetic field is a hermetic seal. Magnetic forces provide balance of volume of magnetic fluid under the impact of pressure drop and centrifugal forces. With an increase in the speed of rotation of the shaft up to 10 m/s, deformation of the free surface of the magnetic fluid at the surface of the shaft in the form of a funnel is visually observed, which causes a decrease in the retained pressure drop. As the rotation speed increases, the funnel increases, part of the magnetic fluid is ejected from the working area, the retained pressure drop decreases, and at 50 m/s there is a complete release of the magnetic fluid and depressurization of the seal. In order to increase the stability of the free surface of the magnetic fluid in the field of centrifugal forces, multilayer carbon nanotubes were introduced into it. They are characterized by a high specific surface area and, accordingly, a strong Van der Waals attraction. In a magnetic fluid, multilayer carbon nanotubes form structures which are oriented along the magnetic field lines. The following increase in the static load retained by a layer of magnetic fluid has been experimentally determined with the introduction of nanocarbon structures: when coinciding with the axis of the structures the increase was by 100 %, for the normal direction of the load to the axis of the structure – by 50 %. In the seal, with an increase in the shaft rotation speed, deformation of the free surface of a magnetic fluid with nanocarbon structures was observed at 18 m/s at a distance of 3 mm from the shaft surface. The pressure drop retained by the seal increased in the speed range of 10–40 m/s, the maximum effect of 50 % was obtained at a speed of 40 m/s. Thus, the introduction of nanocarbon structures into the magnetic fluid made it possible to reduce the influence of centrifugal forces on the retained pressure drop and increase the efficiency of the magnetofluidic seal at high speeds of rotation of the shaft of wind power plants.

Prevention of edge fracture using a nontoxic liquid metal sealant

Edge fracture, the sudden indentation of a viscoelastic fluid's free surface, often occurs when a sufficiently strong shear is applied to the fluid, rendering rheological measurements at high shear rates difficult. We show that sealing the fluid sample's free surface with the nontoxic liquid metal Galinstan can delay fracture occurrence, extending the measurable shear rate range of a cone-and-plate fixture by a decade. The method's effectiveness is compared to the cone-partitioned plate fixture, an existing tool for mitigating the adverse effects of sample fracture. Our cheap yet effective way to prevent edge fracture will benefit the general rheological study of complex fluids.

Rod-climbing rheometry revisited.

The rod-climbing or "Weissenberg" effect in which the free surface of a complex fluid climbs a thin rotating rod is a popular and convincing experiment demonstrating the existence of elasticity in polymeric fluids. The interface shape and steady-state climbing height depend on the rotation rate, fluid elasticity (through the presence of normal stresses), surface tension, and inertia. By solving the equations of motion in the low rotation rate limit for a second-order fluid, a mathematical relationship between the interface deflection and the fluid material functions, specifically the first and second normal stress differences, emerges. This relationship has been used in the past to measure the climbing constant, a combination of the first (Ψ1,0) and second (Ψ2,0) normal stress difference coefficients from experimental observations of rod-climbing in the low shear rate limit. However, a quantitative reconciliation of such observations with the capabilities of modern-day torsional rheometers is lacking. To this end, we combine rod-climbing experiments with both small amplitude oscillatory shear (SAOS) flow measurements and steady shear measurements of the first normal stress difference from commercial rheometers to quantify the values of both Ψ1,0 and Ψ2,0 for a series of polymer solutions. Furthermore, by retaining the oft-neglected inertial terms, we show that the "climbing constant"  = 0.5Ψ1,0 + 2Ψ2,0 can be measured even when the fluids, in fact, experience rod descending. A climbing condition derived by considering the competition between elasticity and inertial effects accurately predicts whether a fluid will undergo rod-climbing or rod-descending. Our results suggest a more general description, "rotating rod rheometry" instead of "rod-climbing rheometry", to be more apt and less restrictive. The analysis and observations presented in this study establish rotating rod rheometry combined with SAOS measurements as a prime candidate for measuring normal stress differences in complex fluids at low shear rates that are often below commercial rheometers' sensitivity limits.

Dynamic analysis of a partially-filled cylindrical–conical–cylindrical shell representing a pressure vessel

A base-supported partially-filled fluid–shell system representing the pressure vessel of a fast reactor is studied. The shell is made of two cylindrical parts joined through a conical part. The shell is characterized using Donnell’s theory, while the fluid is modeled using a velocity potential approach. Compatibility and interface conditions led to the energy functional, minimization of which yielded natural frequencies. The approach was validated against previously studied joined systems. Parametric studies revealed that vibration frequencies of the fluid-shell system (i.e., bulging frequencies) are comparable to shell-only vibration frequencies if only the lowermost part is filled. Vibration frequencies of the free fluid surface (i.e., sloshing frequencies) do not change materially with fluid height if most of the vessel is filled. Present study did not consider the coupling between bulging and sloshing modes, which may need to be considered for a completely-filled vessel with a large base radius (e.g., > 3 m) as the two sets of frequencies for such systems can be comparable.

Adaptive Mesh Refinement for Dam-Break Models using the Shallow Water Equations

The 2D shallow water equations are a common tool for the simulation of free surface fluid dynamics in civil engineering. However, the nonlinear structures of the equations' straightforward implementations lead to numerical problems, such as spurious oscillations and unphysical diffusion. Therefore, this research compared several strategies to overcome these problems, using various finite element formulations and combinations of stabilization methods and mesh options. The accuracy and performance of numerous approaches are examined on models of dam-break in one and two space dimensions. The analytical solution checks the numerical, derived shock wave heights and velocities for the 1D classical benchmark. The result showed that streamlined diffusion and shock capturing stabilization deal with the classical problems of spurious oscillations and numerical diffusion but still indicate similar problems locally in the vicinity of steep fronts and shock waves when used on fixed meshes. As adaptive meshing is the most promising method to deal with such situations, several concerned options are examined in detail. It is important to fine-tune the method to the model's needs, i.e. to adapt the maximum number of mesh refinements, the indicator functions, and the starting mesh. The use of adaptive meshing techniques leads to accurate solutions for the usual parameter range in 1D and 2D, requiring less computational resources than simulations on fixed meshes. Meanwhile, meshing reduces the model size of the 2D dam break model adaptive by almost one order of magnitude and the execution time by a factor of 20.

Numerical Simulation Research on the Anchor Last Deployment of Marine Submersible Buoy System Based on VOF Method

Marine submersible buoy systems are widely-used equipment for ocean resource development. The marine submersible buoy system studied in this paper consists of the submersible buoy, the battery compartment, the anchor block, the mooring line, and the power supply cable. To study the mechanical behavior and obtain the speed variation of each component during the anchor last deployment, this paper establishes the free surface computational fluid dynamics model of marine submersible buoy systems based on the VOF method. This model includes the incompressible Navier–Stokes equations, the Renormalization-Group turbulence model, and the fractional areas/volume obstacle representation method. The free fluid surface is tracked using the VOF method. The lumped mass method is used to simulate the mooring line and power supply cable. The results showed that the tension forces increase when the mooring lines were straightened. Subsequently, the tension forces gradually decrease with oscillations. After the anchor block sinks to the sea floor, the positive buoyancy of the battery compartment and the buoy will cause large tension on the mooring line and power supply cable.

Enhanced resolution of the continuity equation in explicit weakly compressible SPH simulations of incompressible free‐surface fluid flows

This paper is dedicated to improved resolution of the continuity equation and thus more precise satisfaction of incompressibility conditions in explicit weakly compressible SPH (Smoothed Particle Hydrodynamics) simulations of incompressible free-surface fluid flows. In the explicit weakly compressible SPH methods, including δ-SPH with a widely adopted particle shifting scheme (or the so-called δ-plus-SPH), two incompressibility conditions corresponding to the invariant density condition and divergence-free velocity condition are not well resolved. To achieve enhanced resolution of the continuity equation as well as satisfaction of both incompressibility conditions, this paper presents two novel schemes, namely, Velocity-divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes. The VEM scheme corresponds to an additional term in the momentum equation that mitigates the velocity divergence error at every computational time step. The VEM term is derived by linking the instantaneous velocity divergence error to the undesired density time variations and accordingly to an error mitigating pressure pVEM obtained through an Equation of State (EOS). The corresponding pressure gradient term or acceleration leads to attenuation of numerical noise in the velocity divergence field. The VCS acts as a distinct particle shifting scheme to enforce local and thus global volume conservation. This distinct shifting is obtained through calculation of the gradient of a corresponding pressure, namely, pVCS, obtained through efficient solution of a Poisson Pressure Equation (PPE) derived from the concept of projection-based particle methods. By adopting both proposed schemes, the explicit SPH method is shown to provide enhanced satisfaction of both incompressibility conditions, i.e., divergence-free velocity and invariant density conditions, in simulation of incompressible free-surface fluid flows.

Planetary Core‐Style Rotating Convective Flows in Paraboloidal Laboratory Experiments

AbstractTurbulent convection in a planet's outer core is simulated here using a thermally‐driven, free surface paraboloidal laboratory annulus. We show that the rapidly rotating convection dynamics in free‐surface paraboloidal annuli are similar those in planetary spherical shell geometries. Three experimental cases are carried out, respectively, at 35 revolutions per minute (rpm), 50 and 60 rpm. Thermal Rossby waves are detected in full disk thermographic images of the fluid's free surface. Ultrasonic flow velocity measurements reveal the presence of multiple azimuthal (zonal) jets, with successively more jets forming in higher rotation rate cases. The jets' cylindrical radial extent is well approximated by the Rhines scale. Over time, the zonal jets migrate to larger radial position with migration rates in good agreement with prior theoretical estimates. Our results suggest that planetary core rotating convection will be comprised of flow structures found in other turbulent geophysical fluid dynamical systems: convective turbulence dominates the small‐scale flow field, and also act to flux energy into larger‐scale, slowly evolving zonal flow structures. How the ambient magnetic fields in planetary core settings affect such turbulent flows remains an open question.

New mathematical model to analysis fluid sloshing in 3D tanks with slotted middle baffle

This work studies the natural frequencies and modes of liquid sloshing in a rectangular 3D tank with the slotted baffles in middle. A new vibrational model incorporating the pressure difference between the two sides of the baffle, using a transmission function in Laplace space is presented. To this end, a part of the baffle that includes the wall and the slot is modeled in Fluent software, to express this pressure difference as a function of the fluid velocity passing through the baffle. Then, using the boundary element method, and boundary conditions, the equations of the fluid domain are developed. In this research, the rigid baffle is surrounded by a three-dimensional irrotational velocity field of an ideal liquid, and the potential flow conditions are considered. So, the equations from the boundary element method are coupled with the equations of potential flow and other boundary conditions, and then the fluid free surface mode shapes and frequencies are extracted. In the following, after validating the proposed model, the effects of different slot sizes on the liquid sloshing parameters are analyzed. Also, the effects of parameters of the mentioned new vibrational method on the frequencies and damping of the system are investigated.

Three-dimensional direct numerical simulation of free-surface magnetohydrodynamic wave turbulence.

We report on three-dimensional direct numerical simulation of wave turbulence on the free surface of a magnetic fluid subjected to an external horizontal magnetic field. A transition from capillary-wave turbulence to anisotropic magneto-capillary wave turbulence is observed for an increasing field. At high enough field, wave turbulence becomes highly anisotropic, cascading mainly perpendicularly to the field direction, in good agreement with the prediction of a phenomenological model, and with anisotropic Alfvén wave turbulence. Although surface waves on a magnetic fluid are different from Alfvén waves in plasma, a strong analogy is found with similar wave spectrum scalings and similar magnetic-field dependent dispersionless wave velocities.

The separation angle of the free surface of a viscous fluid at a straight edge

We rework some of the die-swell singularity analysis for Stokes flow, originally by Ramalingam (Ramalingam, 1994 Fiber spinning and rheology of liquid-crystalline polymers, PhD thesis, Massachusetts Institute of Technology) in appendix A of his PhD thesis, in an attempt to demonstrate that for capillary numbers in the range $(0,\infty )$ the curvature may enter into the normal stress balance on the free surface and lead to separation angles exceeding $180^{\circ }$ and infinite curvature at the separation point. The singular coefficients in the asymptotic solution and the free surface shape in a neighbourhood of the separation point cannot be determined by a local analysis of the Michael type (Michael, Mathematika, vol. 5, 1958, pp. 82–84) but must be found from matching with the solution valid away from the die edge. The numerical method that we use in the truncated die-swell domain is a boundary element method incorporating the singular solution near the separation point. Although there is some variation in the extrudate swell ratios at different capillary numbers reported in the numerical literature, our results for capillary numbers $Ca$ from $1$ to $1000$ are within the range of values published in earlier papers. The computed separation angles at different values of $Ca$ agree well with the range of separation angles to be found in experimental and numerical papers. The separation angle appears to converge to a value different from $180^{\circ }$ as $Ca$ increases, leading us to conclude that the case of zero surface tension ( $Ca=\infty$ , with corresponding separation angle of $180^{\circ }$ ), is a singular limit.

Damping of sloshing impact on bottom-layer fluid by adding a viscous top-layer fluid

This study examines the effects of the viscosity of stratified layers on liquid sloshing in a rectangular tank. It employs five different fluids to investigate viscosity in relation to free surface deformation and fluid impacts on sidewalls. A wide range of oscillation frequencies has been employed to provide resonant and non-resonant frequencies. Free surface deformations are tracked for one-layer and two-layer tests using various fluids and high-speed imaging. The experimental results have been comparatively analyzed, with a fair agreement being found between results. The viscous friction leads to energy dissipation associated with decaying pressure impacts and less free surface deformation. This phenomenon is utilized for reducing the deformation effects of one layer of fluid by adding an upper layer of fluids with a higher viscosity. The findings show the lower layer of fluid to experience a remarkable damping effect.

An immersed finite element material point (IFEMP) method for free surface fluid–structure interaction problems

The inherent nonlinearity of free surface fluid–structure interaction (FSI) problems challenges numerical methods in terms of efficiency and fidelity. In this article, we propose an immersed finite element material point method for the water entry fluid–structure interaction problems. In this method, the fluid domain is discretized by an improved incompressible material point method (iMPM) using both Eulerian and Lagrangian descriptions, while the solid domain is solved by finite element method (FEM). The interaction between the iMPM and FEM is handled by a sharp immersed interface approach. Moreover, weighted tracing points are designed to track the fluid–structure interface with a low time complexity; a particle rearranging method is developed to eliminate the numerical cavities, which are non-physical voids caused by the highly disordered particle distribution, from which the original iMPM for FSI problems suffers. Various free surface FSI problems are presented to demonstrate the accuracy and effectiveness of the proposed method. The computational results are compared with analytical, experimental, and simulation data from the literature, with good agreement in cases where such data is available. The proposed method is expected to be a powerful tool for free surface FSI problems.

Passive Suppression of Resonance Vibrations of a Plate and Parallel Plates Assembly, Interacting with a Fluid

We present the results of a series of experiments, which study the effect of the fluid layer height, viscosity and free surface waves on the vibration damping of a thin rectangular plates interacting with a fluid. The suppression technique is based on the shunt circuit connected to the piezoelectric elements. The possibility of using the previously proposed mathematical formulation and approach for calculating the optimal values of the parameters of a RL-circuit in systems with hydrodynamic dissipation is estimated. The experimental results presented confirm its reliability. It has been demonstrated that the efficiency of vibration damping of a single plate structure changes insignificantly with increasing height of the fluid layer except for some interval. At a small amount of the liquid, the waves on its free surface disturb the piezoelectric element reducing its efficiency. Using as an example an assembly of two parallel plates damping of “in-phase” and “out-of-phase” modes has been shown using one location of piezoelement.

Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes

The plane–cube intersection problem has been discussed in the literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding computing time, so a faster analytic solution to the plane–cube intersection would greatly reduce the computing time for such simulations. We derive an analytic solution for all intersection cases and compare it to the previous solution from Scardovelli and Zaleski (Scardovelli, R.; Zaleski, S. Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys.2000, 164, 228–237), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding computing time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton–Raphson is fastest with compiler optimization enabled, while analytic solutions perform better than iterative solutions without. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC—curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.

A comparative study of the cumulant lattice Boltzmann method in a single-phase free-surface model of violent flows

Many coastal and ocean engineering flows, such as tsunamis inundating urban areas, tend to be violent. The characteristics of these damaging flow fields are three-dimensional, highly non-linear and non-hydrostatic. A fully three-dimensional free-surface fluid model is required to simulate such a flow field. Fluid simulations in the field of coastal engineering are often large-scale since large areas are the subject of the simulations. The numerical model must be not only accurate but also efficient. In recent years, the lattice Boltzmann method (LBM) has attracted much attention as a novel simulation method and has been successfully applied to various engineering fields. The LBM calculates complex phenomena in a simple framework. The density field determines the pressure field with the equation of state. This means that the LBM does not have to solve the pressure Poisson equation. However, the existing gas–liquid multi-phase flow models that employ the LBM have a critical problem. Pressure calculations with large density ratios easily become unstable, and the application of these models to violent flow fields is limited. The single-phase free-surface fluid model provides good approximations for flow problems in which the gas dynamics can be neglected. Moreover, the cumulant LBM has attracted attention because it has excellent numerical stability even for high Reynolds number flows. The single-phase free-surface flow model using the cumulant LBM is a suitable approach for simulating violent flow fields in coastal engineering. In this study, we propose a single-phase free-surface flow model based on the cumulant LBM using the volume-of-fluid (VOF) model to capture the interface. We demonstrate that the cumulant LBM is stable under violent flows and reproduces the density field well compared with the traditional single relaxation time model. We find that a larger bulk viscosity can reduce the numerical oscillation of the impact pressure acting on a structure, although a bulk viscosity that is too large reduces the accuracy and stability. The results of the proposed model are in good agreement with previous experimental results.

ЗАДАЧИ ШОУОЛТЕРА–СИДОРОВА И КОШИ ДЛЯ ЛИНЕЙНОГО УРАВНЕНИЯ ДЗЕКЦЕРА С КРАЕВЫМИ УСЛОВИЯМИ ВЕНТЦЕЛЯ И РОБЕНА В ОГРАНИЧЕННОЙ ОБЛАСТИ

Deterministic and stochastic initial boundary value problems for the Dzekzer equation describing the evolution of the free surface of a filtering fluid in a bounded region with a smooth boundary are considered. Wentzell and Robin conditions are set on the boundary of the domain, and either the Showalter-Sidorov condition or the Cauchy condition is taken as the initial condition. Note that for the filtration model under study, the Wentzell condition is considered, which is not a classical condition. In recent years, the boundary condition has been considered in the mathematical literature from two points of view (classical and neoclassical). Since Cauchy and Showalter–Sidorov initial conditions have been studied earlier in various situations, in this work, in the particular case of classical Wentzell and Robin conditions, by methods of the theory of degenerate holomorphic semigroups, exact solutions have been constructed, which allow to determine quantitative predictions of changes in geochemical regime of groundwater under unpressurized filtration. Nelson–Glicklich derivative theory was used in stochastic case. In particular, the investigation of the set problems in the context of Wentzell boundary conditions allowed to determine the processes occurring at the boundary of two media (in the region and at its boundary).

Numerical investigation of non-Newtonian power law flows using B-spline material point method

• A weakly compressible BSMPM method for simulating non-Newtonian flow is proposed. • Complex free surface flow and fluid structure interaction problems for power law fluid are studied. • Shear thickening/thinning, jet buckling/splashing, and vortex whirling are well represented. • Results demonstrate the efficacy of the proposed method for non-Newtonian power law flow problems. Simulating the complex flow with free surface and large deformation of non-Newtonian fluid is a challenging problem and plays a significant role in many industrial areas. In this paper, a weakly compressible B-spline material point method (WC-BSMPM) is presented and extended to solve the complex free surface flow of non-Newtonian fluid with a power law rheology model. Numerical examples including two-dimensional problems of plane Poiseuille, dam break, and container filling flow as well as three-dimensional sphere water entry problems are adopted to demonstrate the effectiveness and capabilities of the proposed method. The simulated results show good agreements with analytical solutions or published experimental data. Moreover, significant flow features for non-Newtonian power law fluid, including shear thickening, shear thinning, jet buckling, jet splashing, vortex whirling, and fluid-structure interactions can be well captured. Meanwhile, the numerical noise of the traditional MPM can be eliminated by the proposed method, which confirms that the WC-BSMPM method is a promising alternative to simulate the complex free surface flow and fluid-structure interactions for the non-Newtonian power law fluid.