An analytical solution for free liquid sloshing in a finite-length horizontal cylindrical container filled to an arbitrary depth

It is well established that vehicular traffic traveling over bridge-like structures can impart a dynamic load effect that is greater than vehicles’ static weight alone. In order to account for this increased load, bridge design codes use a factor known as the dynamic load allowance (IM) to amplify static vehicular live loads. In the current version of the American Association of State Highway and Transportation Officials (AASHTO) Manual for Bridge Evaluation (MBE), reductions in IM are allowed for bridges having span lengths greater than 12.2 m with road surfaces in good condition. In addition, the current AASHTO LRFD Bridge Design Specifications allow for a reduced IM for culverts with higher fill depth. However, many culverts have neither span lengths greater than 12.2 m nor higher fill depths and thus are not eligible for such IM reductions. This paper investigated whether similar IM reductions can be considered for culverts with smaller span lengths and fill depths. The field experiments conducted suggest that culverts having span lengths less than 12.2 m and fill depths less than 0.5 m could be considered for similar IM reductions allowed by the MBE.

Most of the commonly used biomass size reduction devices use rotary action. An alternative mode of size reduction is by linear action of cutting element directly on the biomass material. A linear action knife grid model device was developed to as a prototype to solve issues related to scale-up and determine cutting characteristics of selected biomass materials. Major components of the prototype were ram, feed block, knife grid, knife holder block, product block, and bottom tray. Tool steel hardened knifes were arranged in a square grid pattern and their spacing was adjustable. The whole assembly were mounted and tested on a universal testing machine. Dry corn stalks and switchgrass were selected as the test materials. Cutting experiments were conducted at various grid spacing, fill depths, and refill runs. The output variables measured were maximum force, peak stress, energy, material mat thickness, and mass of products after each refill runs. New surface generated by the cuts will be evaluated theoretically using the circle packing theory. Ultimate cutting stress and net energy requirements increased with increasing fill depth and decreasing knife grid spacing. A final product size of 50 mm required 1.69±0.81 kWh/t for corn stalks and 0.49±0.04 for switchgrass, while a product size of 101 mm required 0.48±0.08 kWh/t for corn stalks and 0.14±0.02 kWh/t for switchgrass. Mean values of specific energy based on new surface generated were estimated as 83.78±24.5 and 63.34±12.3 kN/m for corn stalks and 21.50±1.9 and 15.53±1.7 kN/m for switchgrass for product size of 50 and 101 mm, respectively. Corn stalks required 3.3 to 3.4, and 3.9 to 4.1 times higher cutting energy than switchgrass based on mass and new surface area generated energy basis, respectively. Scaling up is highly feasible for large product sizes because the determined specific energy values are small; and they are well below the reported values.

유체자유수면의 동적거동을 합리적으로 예측하기 위해서는 비선형 특성을 보이는 자유수면의 동역학적 경계조건을 고려해야할 뿐만 아니라 시간에 따라 변화하는 자유수면의 위치변화에 따른 운동학적 경계조건을 고려하여야 한다. 이러한 문제는 대상구조물이 3차원이 될 경우 더욱 복잡해지므로 3차원 비선형 유체자유수면의 해석은 이론해의 도출이 어려우며 수치해석 방법을 이용하는 것이 효과적이다. 본 연구에서는 수치해석 안정성이 높고 3차원 문제에서도 하나의 변수로 유체거동을 모사할 수 있는 arbitrary Lagrangian-Eulerian approach 를 경계요소에 적용하여 효율적이며 안정적인 유체 대변형 해석기법을 개발하였다. 개발된 기법은 향후 자유수면의 비선형 효과를 고려한 유체-구조물 상호작용 해석에 효과적으로 적용할 수 있을 것으로 판단된다. The solution to a liquid sloshing problem is challenge to the field of engineering. This is not only because the dynamic boundary condition at the free surface is nonlinear, but also because the position of the free surface varies with time in a manner not known a priori. Therefore, this nonlinear phenomenon, which is characterized by the oscillation of the unrestrained free surface of the fluid, is a difficult mathematical problem to solve numerically and analytically. In this study, three-dimensional boundary element method(BEM), which is based on the so-called an arbitrary Lagrangian-Eulerian(ALE) approach for the fluid flow problems with a free surface, was formulated to solve the behavior of the nonlinear free surface motion. An ALE-BEM has the advantage to track the free surface along any prescribed paths by using only one displacement variable, even for a three-dimensional problem. Also, some numerical examples were presented to demonstrate the validity and the applicability of the developed procedure.

Wave uplift forces on two dimensional platforms or docks in a variable depth medium has been studied theoretically and numerically. The incident waves are non-linear waves with maximum amplitude greater than the clearance between the still water surface and the underside of the platforms or docks. The flow is assumed to be inviscid, irrotational without ambient current. Thus, the theoretical model solves the Laplace equation along with full nonlinear free surface conditions. In order to conveniently handle the moving free surface in an irregular bottom topography with the presence of platform or dock an isoparametric mapping technique was used to transfer the fluid domain and its boundaries into a regular geometry. A Galerkin finite element model is developed to model the transformed fluid region. The resulting discrete equations are solved iteratively by using adaptive Line SOR (Successive-Over-Relaxation) technique. Artificial viscosity is included in both the dynamic and kinematic free surface equations to damp out the free surface oscillations in the front region of the platform or dock. The Runge-Kutta method is employed to integrate the time variation in the nonlinear free surface equations. Results obtained by the numerical method were compared to the available experimental data obtained by others in order to demonstrate the workability of the proposed algorithm.

Natural sloshing frequencies and modes in a rectangular tank with a slat-type screen

Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also called cylindrical functions, or cylindrical harmonics. This chapter is devoted to the construction of the generalized coherent state (GCS) and the theory of Bessel wavelets. The GCS is built by replacing the coefficient zn/n!,z∈C of the canonical CS by the cylindrical Bessel functions. Then, the Paley-Wiener space PW1 is discussed in the framework of a set of GCS related to the cylindrical Bessel functions and to the Legendre oscillator. We prove that the kernel of the finite Fourier transform (FFT) of L2-functions supported on −11 form a set of GCS. Otherwise, the wavelet transform is the special case of CS associated respectively with the Weyl-Heisenberg group (which gives the canonical CS) and with the affine group on the line. We recall the wavelet theory on R. As an application, we discuss the continuous Bessel wavelet. Thus, coherent state transformation (CST) and continuous Bessel wavelet transformation (CBWT) are defined. This chapter is mainly devoted to the application of the Bessel function.

<p>Liquid motion in partially filled tanks may cause large structural loads if the period of tank motion is close to the natural period of fluid inside the tank. This phenomenon is called sloshing. Sloshing means any motion of a free liquid surface inside a container. The effect of severe sloshing motion on global seagoing vessels is an important factor in safety design of such containers. In order to examine the sloshing effects, a shake table experiments were conducted for different water fill depth of aspect ratio 0.163, 0.325 and 0.488. The parametric studies were carried out to show the liquid sloshing effects in terms of slosh frequencies, maximum free surface elevation and hydrodynamic forces acting on the tank wall. Sloshing oscillation for the excitation frequency f<sub>1</sub>, f<sub>2</sub>, f<sub>3</sub>, f<sub>4 </sub>and f<sub>5</sub> are observed and analysed. The excitation frequencies is varied between 0.4566 Hz to 1.9757 Hz and constant amplitudes of 7.5mm was adopted. The movement of fluid in a rectangular tank has been studied using experimental approach and different baffle configurations were adopted for analysing the sloshing oscillation, natural frequencies and variation in wave deflection. The adopted porosities in the present study is 15% – 25 %. Porous screen is placed inside the tank at L/2 location and study is extended for single porous screen for better wave energy absorption. Capacitance wave probes have been placed at tank ends to record the free surface water elevation. Load cells are used to measure the sloshing force inside the tank. Linear variable displacement transducers is used to measure the displacement of shake table. In the present study single porous screen under the action of wave were analysed to understand the wave control performance due to porosity parameters. A boundary element model is developed to calculate problems of wave interaction with a porous screen structure. The numerical results from the present boundary element methods (BEM) are compared with series of experiments conducted in a rectangular tank with various baffle porosities and submerged depths.</p><p> </p>

In the present study, a robust modified Incompressible Smoothed Particle Hydrodynamics (ISPH) method is developed and applied to model a fluid mixer with a rotating stirrer and a tank body with free surface and horizontal oscillations. The mixer consists of a rectangular tank in which a stirrer rotates to mix the fluid. For the fluid in the tank, the free surface condition is considered. According to the Reynolds number, it is assumed that the mixing process is turbulent, and so a turbulent viscosity is defined. Although the smoothed particle hydrodynamics (SPH) method can result in some complexities, it is generally an easy and appropriate method for modeling mixing flow, free surface flow, and moving body problems, which will simultaneously be applied in the present study. The method is improved with kernel gradient corrective tensor, shifting particle algorithm, and turbulent viscosity, and it is validated against other well-known test cases and problems and is applied to model the mixing phenomenon. This study aims to increase the mixing rate and decrease the mixing time of the described mixer. To do so, the effect of the rotation of the stirrer on the mixing process is first investigated, after which the linear oscillation of the body is added and then the optimal shape of the tank is examined. The results show that the stirrer’s rotating speed, the linear oscillation of the tank body, and streamlining all have major effects on the mixing rate.

Oscillation dynamics of free vortex surface in uncovered unbaffled stirred vessels

When a cylindrical tank installed on the ground, such as an oil tank or a liquid storage tank, receives strong seismic waves, including a long-period component, motion of free liquid surface inside a tank called sloshing may occur. If high-amplitude sloshing occurs and waves of free liquid surface collide with tank roof, it may lead to accidents such as damage to tank roof or outflow of internal liquid. There is also a possibility that tank sidewall may be damaged due to fluid force generated by high liquid surface level of sloshing. When sloshing wave height is small, liquid surface level and velocity can be calculated based on the linear velocity potential theory. When sloshing wave height increases, sloshing becomes nonlinear, and the difference between actual value and calculated value based on the linear velocity potential theory increases. Sloshing loads acting on roof and sidewall should be evaluated in consideration of nonlinear sloshing. In the previous study, simplified equations of nonlinear liquid surface shape and nonlinear liquid surface velocity distribution were proposed, as input parameters of evaluating sloshing loads. In this paper, a new method is proposed for nonlinear sloshing condition of cylindrical tanks, which can evaluate the vertical load acting on roof and the horizontal load acting on sidewall. In the proposed method, new pressure models modified from existing models are combined with simplified equations of nonlinear liquid surface shape and nonlinear liquid surface velocity distribution proposed in the previous study. A long calculation time as CFD analysis is not needed, because the proposed method is consisted of simplified equations.

Numerical simulation of parametric liquid sloshing in a horizontally baffled rectangular container

Nonlinear hydrodynamic modeling of an offshore stationary multi-oscillating water column platform

Due to the high computational costs of the Eulerian multiphase model, which solves the conservation equations for each considered phase, a two-phase mixture model is proposed to reduce these costs in the current study. Only one single equation for each the momentum and enthalpy equations has to be solved for the mixture phase. The Navier–Stokes and energy equations were solved using the 3D finite volume method. The model was used to simulate the liquid–solid phase transformation of a Fe-0.82wt%C steel alloy under the effect of both thermocapillary and buoyancy convections. The alloy was cooled in a rectangular ingot (100 × 100 × 10 mm3) from the bottom cold surface to the top hot free surface by applying a heat transfer coefficient of h = 600 W/m2/K, which allows for heat exchange with the outer medium. The purpose of this work is to study the effect of the surface tension on the flow and segregation patterns. The results before solidification show that Marangoni flow was formed at the free surface of the molten alloy, extending into the liquid depth and creating polygonized hexagonal patterns. The size and the number of these hexagons were found to be dependent on the Marangoni number, where the number of convective cells increases with the increase in the Marangoni number. During solidification, the solid front grew in a concave morphology, as the centers of the cells were hotter; a macro-segregation pattern with hexagonal cells was formed, which was analogous to the hexagonal flow cells generated by the Marangoni effect. After full solidification, the segregation was found to be in perfect hexagonal shapes with a strong compositional variation at the free surface. This study illuminates the crucial role of surface-tension-driven Marangoni flow in producing hexagonal patterns before and during the solidification process and provides valuable insights into the complex interplay between the Marangoni flow, buoyancy convection, and solidification phenomena.

On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions

This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.