Free-surface Hydrodynamics Research Articles

A finite element method for partially erodible bed evolution coupled with multiphase flows

This paper introduces a continuous finite element model for two phase flows coupled with evolving bedforms. The method extends the positive definite non-oscillatory finite element algorithm (NFEM) capabilities to predict sediment transport, multiphase flow, and evolution of the resulting interfaces limiting fluids, erodible sediment layers and non-erodible strata. Free surface hydrodynamics is simulated by the integration of the governing equations for incompressible multiphase flows, including the advection and reinitialization of a phase function to update the interface location. Sign preservation of the algorithm is essential both for fluid interface tracking procedure and for the landform tracking. In the first case, the property avoids unphysical overshoots along the free surface. In the second case, a positive definite thickness of the erodible layer of sediment is mandatory to account for interaction between evolving cohesionless sediment layers and rigid beds. To compute sediment layer thickness, the method incorporates a conservation law to balance evolution of the bed position with sediment mass flux, including a general template for saturated flux calculation. The framework permits the simulation from bed load dominant transport to saltation dominant transport. Fluid/terrain interface is explicitly captured by a surface tracking methodology whereby the mesh is adapted to the bedform. Numerical experiments explore several saturated sediment flow formulae for interactive evolution of dune/atmosphere system, as well as stringent dam-break type problems on channels with dry beds and partially and totally erodible beds.

Modeling multiphase fluid flow, mass transfer, and chemical reactions in bioreactors using large-eddy simulation.

We present a transient large eddy simulation (LES) modeling approach for simulating the interlinked physics describing free surface hydrodynamics, multiphase mixing, reaction kinetics, and mass transport in bioreactor systems. Presented case-studies include non-reacting and reacting bioreactor systems, modeled through the inclusion of uniform reaction rates and more complex biochemical reactions described using Contois type kinetics. It is shown that the presence of reactions can result in a non-uniform spatially varying species concentration field, the magnitude and extent of which is directly related to the reaction rates and the underlying variations in the local volumetric mass transfer coefficient.

A review on advanced numerical methods for free-surface hydrodynamics

Environmental hydrodynamics is typically characterized by free-surface hydrostatic flows. Within such a framework a set of simplified models are derived from the Reynolds averaged Navier–Stokes equations under various simplifying assumptions. Numerically, a simple method for the 1D open channel flow is derived first. This method is then extended to the 2D_{xz} laterally averaged model and to the 2D_{xy} vertically averaged model. Next, a semi-implicit method for the 3D model is described and discussed.

Three-Dimensional Numerical Modeling of Flow Hydrodynamics and Cohesive Sediment Transport in Enid Lake, Mississippi

Enid Lake is one of the largest reservoirs located in Yazoo River Basin, the largest basin in the state of Mississippi. The lake was impounded by Enid Dam on the Yocona River in Yalobusha County and covers an area of 30 square kilometers. It provides significant natural and recreational resources. The soils in this region are highly erodible, resulting in a large amount of fine-grained cohesive sediment discharged into the lake. In this study, a 3D numerical model was developed to simulate the free surface hydrodynamics and transportation of cohesive sediment with a median diameter of 0.0025 to 0.003 mm in Enid Lake. Flow fields in the lake are generally induced by wind and upstream river inflow, and the sediment is also introduced from the inflow during storm events. The general processes of sediment flocculation and settling were considered in the model, and the erosion rate and deposition rate of cohesive sediment were calculated. In this model, the sediment simulation was coupled with flow simulation. In this research, remote sensing technology was applied to estimate the sediment concentration at the lake surface and provide validation data for numerical model simulation. The model results and remote sensing data help us to understand the transport, deposition and resuspension processes of cohesive sediment in large reservoirs due to wind-induced currents and upstream river flows.

Design and development of a 3D printed hydroblasting robot

This paper describes the design and development of a 3D printed hydroblasting robot called Hornbill. This robotic platform incorporates magnetic-based adhesion, differential wheel-based locomotion principle, and a hydroblasting unit consists of nozzles mounted on the rotating arm. One of the key indicators of the effectiveness of the hydroblasting process is how quick the area below the hydroblasting chamber can be covered by the water jets. Given the nozzle configuration in the chamber and the water jet flowrate, the water coverage depends on the water adhesion on the surface to be cleaned as well as the orientation of the surface. Hence, the water free surface hydrodynamics is modeled and studied using Computational Fluid Dynamics (CFD) simulations to evaluate how surface conditions and contact angle influence the water spreading on surface. The optimal contact angle is used in robot and through experiments, the hydroblasting capability on horizontal and vertical surfaces are validated.

Ideal planar fluid flow over a submerged obstacle: Review and extension

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour. doi:10.1017/S1446181121000341

IDEAL PLANAR FLUID FLOW OVER A SUBMERGED OBSTACLE: REVIEW AND EXTENSION

AbstractA classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour.

The influence of flexible fluid structure interactions on sway induced tank sloshing dynamics

The analysis of liquid sloshing remains a challenging computational mechanics topic due to its complex underlying physics. The rapid simulation of sloshing problems requires accurate modelling of two-phase fluid dynamics and sloshing impacts on solid tank boundaries by suitable Flexible Fluid Structure Interaction (FFSI) models. This paper presents a hydroelastic model for the prediction of sway induced sloshing loads on flexible trapezoidal and rectangular tanks. Tank walls and a vertical baffle in way of the mid span of the tank bottom are idealized by Timoshenko beam structural dynamics. Hydroelastic analysis is enabled by a Boundary Element Method (BEM) that couples tank wall and baffle structural dynamics with free surface hydrodynamics to evaluate excitation forces and peak hydrodynamic pressures in way of the tank perimeter. Results show that for the case study presented accounting for the influence of hydroelasticity in a rectangular tank may lead to decrease of free surface oscillations and peak pressure by 20%. This is because the dynamics of tank flexibility are coupled with the angular frequency of the sway motion. These benefits amplify further for the case of trapezoidal tank designs for which the free surface and pressure of the trapezoidal tank with lateral angle θ=80° are decreased relative to the rectangular one by about 80% and 65%, respectively.

Kuramoto-Like Synchronization Mediated through Faraday Surface Waves

A new class of problems in free surface hydrodynamics appeared after the groundbreaking discovery by Yves Couder and Emmanuel Fort. A bouncing droplet in association with Faraday surface waves gives rise to new nonlinear dynamics, in analogy with the pilot-wave proposed by de Broglie. The droplet and the underlying vibrating bath are of silicon oil. A weakly viscous potential theory model should be used. Numerical simulations are presented with one and two bouncing droplets oscillating while confined to their cavities. These oscillators are implicitly coupled by the underlying surface wave field. In certain regimes, the oscillators can spontaneously synchronize, even when placed at a distance. Cavity parameters are varied in order to highlight the sensitive wave-mediated coupling. The present nonlinear wave-mediated oscillator synchronization is more general than that displayed by the celebrated Kuramoto model and therefore of general interest.

Application of SPH to Single and Multiphase Geophysical, Biophysical and Industrial Fluid Flows

A series of challenging fluid flow applications are used to demonstrate the powerful capabilities of the SPH method. The applications are classified according to whether they are industrial, geophysical or biophysical in nature. The versatility and flexibility of SPH allows it to be used to predict wide ranges of flow types with diverse coupled secondary physics and chemistry. The demonstration examples span free surface hydrodynamics, fluid-structure interactions, multiphase flows (bubbles and/or solids immersed in a fluid and multiple fluids with large density differences), and flows involving reactions and phase change. For the studies presented SPH demonstrates at least one and often several key advantages that make the method well suited to these applications. These include the natural handling of free surfaces (especially when splashing), strong advection (arising from the method being Lagrangian), very high deformation levels (due to being meshfree) and intrinsic history tracking (which provides specific benefits for flows with multiple materials, reactions and phase change).

Digital Design of Batch Cooling Crystallization Processes: Computational Fluid Dynamics Methodology for Modeling Free-Surface Hydrodynamics in Agitated Crystallizers

A framework for the digital design of batch cooling crystallization processes is presented comprising three stages, which are based on different levels of process complexity, integrating crystallizer hydrodynamics with crystallization kinetics and consequently with expected crystal size distribution. In the first stage of the framework, a computational fluid dynamics methodology is developed to accurately assess hydrodynamics in a typical batch crystallizer configuration, comprising a 20 L scale dish-bottom vessel with a single beavertail baffle agitated by a retreat curve impeller, used in the pharmaceutical as well as in the fine chemical industries. The hydrodynamics of crystallizers with such configurations is characterized by vortex formation on the free liquid surface. It is therefore important to model the free surface using the Volume-of-Fluid (VoF) method. Comparison of the predicted mean velocity components with experimental measurements using laser Doppler anemometry reveals that improved predictions are obtained using a differential Reynolds-stress transport model for turbulence coupled with the VoF for modeling the gas-liquid interface compared with those using the Shear-stress transport model and with a flat liquid surface. This study demonstrates that an accurate treatment of the liquid free surface for capturing vortex formation is essential for reliable predictions of the crystallizer’s flow field. While the vortex depth is predicted to increase with increasing impeller Reynolds number, the dependence of hydrodynamic macroparameters, including power number, impeller flow number, and secondary circulation flow number, on Reynolds number reveals that they are essentially constant within the turbulent regime but fluctuate when the flow is in the transitional and laminar regimes as fluid viscosity increases.

Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations

In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.

Role of impinging powder particles on melt pool hydrodynamics, thermal behaviour and microstructure in laser-assisted DED process: A particle-scale DEM – CFD – CA approach

High speed imaging of molten pool free-surface hydrodynamics in laser-assisted directed energy deposition process clearly revealed a highly oscillatory and dynamic melt flow due to impinging powder particles. Surprisingly, most of the reported computational work exclude the injection of powder particles and rather adopt a homogeneous mass and energy addition approach, and therefore provides less accurate predictions. In this work, we develop a coupled multi-physics particle-scale approach utilizing the discrete element method for particle trajectory prediction, the computational fluid dynamics for free-surface thermo-fluidic modelling and the cellular automata method for grain growth evolution. In the model, the governing physical phenomena, such as laser-powder interaction, in-flight particle heating, phase change (melting, vaporization and solidification), free-surface evolution, molten pool hydrodynamics and impinging particles-melt interaction have been considered. Experiments for the deposition of Inconel-625 on an Inconel-625 substrate are carried out, and the model predictions are validated with the experimental measurements. For the first time, the predicted thermo-fluidic simulation results reveal highly oscillatory, chaotic and random melt flow attributed to the impinging powder particles. During the deposition, it is found that the role of the Marangoni convection is less significant as compared to the momentum imparted by the impinging powder particles in the melt pool. Using the simulated thermal undercooling data, cellular automata-based grain growth simulation predicts elongated columnar dendrites in the melt pool that grows epitaxially from the melt pool interface and stretches towards the centre. Using the Kurz-Fisher model, the effect of local thermodynamic solidification conditions on the size of dendritic microstructure is also described. The predicted melt pool geometry, temperature field and grain structure compare well with the experimental measurements.

Poles and Branch Cuts in Free Surface Hydrodynamics

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics though the time-dependent conformal mapping $z=x+iy=z(w,t)$ of the lower complex half plane of the conformal variable $w$ into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for $1/z_w$ and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second and the first order moving pole.

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$. We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.

Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models

This paper presents the development of a new boundary treatment for free-surface hydrodynamics using the smoothed particle hydrodynamics (SPH) method accelerated with a graphics processing unit (GPU). The new solid boundary formulation uses a local uniform stencil (LUST) of fictitious particles that surround and move with each fluid particle and are only activated when they are located inside a boundary. This addresses the issues currently affecting boundary conditions in SPH, namely the accuracy, robustness and applicability while being amenable to easy parallelization such as on a GPU. In 3-D, the methodology uses triangles to represent the geometry with a ray tracing procedure to identify when the LUST particles are activated. A new correction is proposed to the popular density diffusion term treatment to correct for pressure errors at the boundary. The methodology is applicable to complex arbitrary geometries without the need of special treatments for corners and curvature is presented. The paper presents the results from 2-D and 3-D Poiseuille flows showing convergence rates typical for weakly compressible SPH. Still water in a complex 3-D geometry with a pyramid demonstrates the robustness of the technique with excellent agreement for the pressure distributions. The method is finally applied to the SPHERIC benchmark of a dry-bed dam-break impacting an obstacle showing satisfactory agreement and convergence for a violent flow.

НОВЫЕ ИНТЕГРАЛЫ ДВИЖЕНИЯ И НЕКАНОНИЧЕСКАЯ ГАМИЛЬТОНОВА СТРУКТУРА ДЛЯ ДВУМЕРНОЙ ГИДРОДИНАМИКИ СО СВОБОДНОЙ ПОВЕРХНОСТЬЮ

We consider a potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensions with gravity forces and surface tension. A time-dependent conformal mapping z(w, t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. We study the dynamics of singularities of both z(w, t) and the complex fluid potential Π(w, t) in the upper complex half-plane of w. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables (Dyachenko et al., submitted), the imaginary part of z(w, t) and the real part at Π(w, t) (both evaluated of fluid’s free surface). The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of (Zakharov, 1968) (valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface). In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations as in (Lushnikov and Zubarev, 2018) with the powerful reductions which allowed to find general classes of particular solutions. In Eulerian case we show the existence of solutions with an arbitrary finite number N of complex poles in zw(w, t) and Πw(w, t) which are the derivatives of z(w, t) and Π(w, t) over w (Dyachenko et al., submitted). These solutions are not purely rational because they generally have branch points at other positions of the upper complex halfplane with generally the infinite number of sheets of the Riemann surface for z(w, t) and Π(w, t) (Lushnikov, 2016). The order of poles is arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of zw(w, t) at these N points are new, previously unknown constants of motion. These constants of motion commute with each other with respect to the Poisson bracket. There are more integrals of motion beyond these residues. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For higher order poles the number of the integrals is increasing. These nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics. Work of A. Dyachenko, P. Lushnikov and V. Zakharov was supported by state assignment «Dynamics of the complex materials».

Design, analysis and verification of a volume-of-fluid model with interface-capturing scheme

The Volume-of-Fluid (VoF) model with interface-capturing scheme, available in many commercial and open-source CFD packages, may be widely used for simulations of free-surface hydrodynamics but is certainly not used without difficulties. While practitioners have come to expect conservation of mass and momentum, unconditional stability, second-order of accuracy and good iterative convergence, none of these qualities are evident when a free surface gets involved. In this paper, we investigate how these qualities are affected by the discontinuous nature of the volume fraction, which requires specialized numerical techniques for advection, for the interpolation of material properties, for the hydrostatic balance and for the pressure-weighted interpolation used in finite-volume methods with co-located variables. We find that conservation properties and good iterative convergence can still be attained, but not second-order of accuracy and unconditional stability. Code verification is presented to substantiate these findings, with cases ranging from one to three dimensions, from uniform to locally refined grids and from the stand-alone volume fraction equation to the complete set of mass, momentum and volume fraction equations. For the latter, a novel manufactured solution based on a sinusoidal wave in deep water is introduced. These tests permit not only to quantitatively assess the results for comparison with theoretical predictions but also ensure that our findings are characteristic of the VoF method and not of a particular implementation.

Heat Transfer and Thermocapillary Convection during the Laser Deposition of Metal Powders Implemented in Additive Technologies

Heat-transfer- and thermocapillary-convection macroprocesses observed during direct laser metal deposition (DLMD) with coaxial powder injection are examined. The study is performed using the 3D mathematical model incorporating self-consistent equations for free surface evolution, heat transfer, and hydrodynamics, which allow for powder-particle embedding into the thermocapillary convection zone under DLMD. The processes under consideration refer to the main ones underlying additive laser technologies, which determine the microstructural properties and quality of synthesized parts. The convection-diffusion equations are numerically solved using the final volume method. Calculations are carried out for the thermocapillary convection of H13 steel powder. The influence of laser-radiation characteristics (power, scanning rate, intensity distribution in the beam) and the powder-mass flow velocity on temperature fields, the structure of convective melt flow (including a maximum melt velocity), and the geometric characteristics (height and width) of the object formed is investigated.

New asymptotic heat transfer model in thin liquid films

In this article, we present a model of heat transfer occurring through a li\-quid film flowing down a vertical wall. This new model is formally derived using the method of asymptotic expansions by introducing appropriately chosen dimensionless variables. In our study the small parameter, known as the film parameter, is chosen as the ratio of the flow depth to the characteristic wavelength. A new Nusselt solution should be explained, taking into account the hydrodynamic free surface variations and the contributions of the higher order terms coming from temperature variation effects. Comparisons are made with numerical solutions of the full Fourier equations in a steady state frame. The flow and heat transfer are coupled through Marangoni and temperature dependent viscosity effects. Even if these effects have been considered separately before, here a fully coupled model is proposed. Another novelty consists in the asymptotic approach in contrast to the weighted residual approach which have been formerly applied to these problems.