A Viscous Velocity Potential/Stream Function

The motion of fluids presents interesting phenomena including flow separation, wakes, turbulence etc. The physics of these are enshrined in the continuity equation and the NSE. Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, numerically, or theoretically. Theoretical studies using classical potential theory (CPT) have some gaps when compared to experiments. The present publication is part of a series introducing refined potential (RPT) that bridges these gaps. It leverages experimental observations, physical deductions and the match between CPT and experimentally observed flows in the theoretical development. It analytically imitates the numerical source/vortex panel method to describe how wall bounded eddies in a three-dimensional cylinder crossflow are linked to the detached wake eddies. Unlike discrete and arbitrary vortices/sources on the cylinder surface whose strengths are numerically determined in the panel method, the vortices/sources/sinks in RPT are mutually concentric and continuously distributed on the cylinder surface. Their strengths are analytically determined from CPT using physical deductions starting from Reynolds number dependence. This study results in the incompressible Kwasu function which is a Eulerian velocity potential/stream function that captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and turbulence. This Eulerian Kwasu function also theorizes streaklines. The Lagrangian form of the function is further exploited to obtain flow pathlines.

On the use of outgoing longwave radiation data in incorporating divergent part of the wind in analysis

In the present study, kinematic divergence computed using ECMWF grid point data at 850 hPa is enhanced by using the relationship between OLR and divergence. This new enhanced divergence is used to compute the velocity potential and then, the divergence part of the wind is obtained from velocity potetial. To obtain the rotational part of wind, we computed the vorticity from wind data, and subsequently stream function and obtained and the rotational part of the wind from the stream function. The total wind is the combination of divergent part obtained from modified velocity potential (using OLR data) and rotational part from unmodified stream function. This total wind field is used as initial guess for univariate objective analysis by optimum interpolation scheme so that Initial Guess field contained the more realistic divergent part of the wind. Consequently, the analysed field also will contain the divergent part of the wind.

Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

Three-dimensional interaction between uniform current and a submerged horizontal cylinder in an ice-covered channel

The problem of interaction of a uniform current with a submerged horizontal circular cylinder in an ice-covered channel is considered. The fluid flow is described by linearized velocity potential theory and the ice sheet is treated as a thin elastic plate. The potential due to a source or the Green function satisfying all boundary conditions apart from that on the body surface is first derived. This can be used to derive the boundary integral equation for a body of arbitrary shape. It can also be used to obtain the solution due to multipoles by differentiating the Green function with its position directly. For a transverse circular cylinder, through distributing multipoles along its centre line, the velocity potential can be written in an infinite series with unknown coefficients, which can be determined from the impermeable condition on a body surface. A major feature here is that different from the free surface problem, or a channel without the ice sheet cover, this problem is fully three-dimensional because of the constraints along the intersection of the ice sheet with the channel wall. It has been also confirmed that there is an infinite number of critical speeds. Whenever the current speed passes a critical value, the force on the body and wave pattern change rapidly, and two more wave components are generated at the far-field. Extensive results are provided for hydroelastic waves and hydrodynamic forces when the ice sheet is under different edge conditions, and the insight of their physical features is discussed.

Effect of Vertical Elastic Baffle on Liquid Sloshing in Rectangular Rigid Container

Sloshing may induce adverse loads to cause structural instability and damage. A vertical elastic baffle mounted at the inside bottom of a rectangular container is used as an anti-slosh device to attenuate the liquid oscillation. A semi-analytical model is presented to analyze the hydroelastic problem. The liquid is partitioned into four simple sub-domains with three hypothetical interfaces. The velocity potential of each sub-domain is analytically deduced by the separation of variables. The baffle deflection is expanded into the Fourier series by its dry modals. The eigenvalue equation is formulated by plugging the velocity potentials into the sloshing conditions, interface continuity conditions, as well as the dominant equation and compatibility conditions of the baffle. Then, the velocity potential is expressed by a complete basis of the coupled mode shapes for the system considered under lateral excitation. The system response equation is constituted by inserting the velocity potential into wave equations and baffle equation. The proposed method is verified by comparing the present results with the available data. In addition, numerical analyses are performed to examine the effects of baffle parameters on the natural frequencies, mode shapes and dynamic responses of the container. The sloshing frequency may be altered to a higher value due to the installation of the elastic baffle.

MATHEMATICAL MODELS APPLICATION TO PREDICT HEAT AND CHEMICAL AIR POLLUTION IN WORKING AREARS

Problem statement. The problem of prediction the level of air pollution in working areas is considered on the basis of mathematical models of aerodynamics and heat and mass transfer. The task is to calculate the concentration field of chemically hazardous substances and the temperature field in the working zones. The purpose of the article. Construction of numerical models that allow determine the distribution of temperature and concentration of chemically hazardous substances in work areas with a complex geometric shape. Methodology. For numerical modeling of the process of air pollution in working areas during the spread of chemically hazardous substances, G. Marchuk's equation is used, which takes into account the transfer of a chemically hazardous substance due to convection, as well as due to turbulent diffusion. The energy equation is used to model the thermal contamination of work areas. To simulate the wind speed field in the presence of various kinds of obstacles, the Laplace equation for the speed potential is used. The integration of the modeling equations is carried out on a rectangular grid. For the numerical integration of the equation describing the propagation of a chemically hazardous substance in the air of working areas, a finite-difference splitting scheme is used. For the numerical integration of the Laplace equation for the velocity potential, two splitting schemes are used. The unknown value of the velocity potential at each splitting step is calculated using an explicit formula. Numerical integration of the energy equation is carried out using an explicit difference scheme. Scientific novelty. The constructed numerical models that allow to calculate the zones of chemical and thermal pollution, taking into account a set of important physical factors. A feature of numerical models is the speed of calculation, which is important when serial calculations are carrying out in practice. Practical significance. A complex of applied programs was created on the basis of the developed numerical models. This complex of programs allows to analyze and predict the intensity and size of zones of thermal or chemical pollution. This set of programs can be useful in determining the affected areas in case of extreme situations at chemically hazardous facilities. Conclusions. Numerical models have been developed. On the basis of these models a complex of applied programs has been created that allow to study multiparameter processes of chemical and thermal air pollution of working areas using the method of computer modeling. The complex of programs can be implemented on computers of low and medium power. The results of a computational experiment are presented.

3D MODELING OF BIOLOGICAL WASTEWATER TREATMENT IN AERATION TANK

Purpose. The main purpose of the article is to develop a 3D CFD model for modeling the process of biological wastewater treatment in an aeration tank. Methodology. For mathematical modeling of the process of biological wastewater treatment in the reactor, taking into account the flow hydrodynamics, geometric shape of the aeration tank, convective-diffusion transfer of the substrate and activated sludge, a 3D CFD model was built. The model is based on the three-dimensional equation of motion of an ideal liquid and the equation of mass conservation for the substrate, activated sludge. The field of sewage flow rate in the aeration tank is calculated based on the velocity potential equation. The process of biological transformation of the substrate is calculated on the basis of the Monod model. The splitting scheme was used for numerical integration of the equations of convective-diffusion transfer of activated sludge and substrate. The splitting is carried out in such a way to take into account the transfer of substrate (activated sludge) in only one direction at each step of splitting. The calculation of the unknown value of the substrate (activated sludge) concentration is carried out according to an explicit scheme. The Richardson method is used to numerically integrate the three-dimensional equation for the velocity potential, and the unknown value of the velocity potential is calculated by an explicit formula. Euler's method is used for numerical integration of equations describing the process of substrate transformation and change in activated sludge concentration (Monod model). Findings. The software implementation of the constructed 3D CFD model is carried out. A description of the structure of the developed software package is provided. The results of a computer experiment to study the process of wastewater treatment in an aeration tank with additional elements are presented. Originality. A new multifactor 3D CFD model has been developed, which allows quick assessing the efficiency of biological treatment in an aeration tank. Practical value. The constructed 3D CFD model can be used to analyze the efficiency of the aeration tank under different operating conditions at the stage of sketch design of wastewater treatment systems.

Local Dynamic Path Planning for an Ambulance Based on Driving Risk and Attraction Field

Path planning is one of the most important aspects for ambulance driving. A local dynamic path planning method based on the potential field theory is presented in this paper. The potential field model includes two components—repulsive potential and attractive potential. Repulsive potential includes road potential, lane potential and obstacle potential. Considering the driving distinction between an ambulance and a regular vehicle, especially in congested traffic, an adaptive potential function for a lane line is constructed in association with traffic conditions. The attractive potential is constructed with target potential, lane-velocity potential and tailgating potential. The design of lane-velocity potential is to characterize the influence of velocity on other lanes so as to prevent unnecessary lane-changing behavior for the sake of time-efficiency. The results obtained from simulation demonstrate that the proposed method yields a good performance for ambulance driving in an urban area, which can provide support for designing an ambulance support system for the ambulance personnel and dispatcher.

Flow and residence time in a laboratory aquifer recharged by rainfall

<p>During rainfall, water infiltrates the soil, and percolates through the unsaturated zone until it reaches the water table. Groundwater then flows through the aquifer, and eventually emerges into streams to feed surface runoff. We reproduce this process in a  two-dimensional laboratory aquifer recharged by artificial rainfall. As rainwater infiltrates, it forms a body of groundwater which can exit the aquifer only through one of its sides. The outlet is located high above the base of the aquifer, and drives the flow upwards. The resulting vertical flow component violates the Dupuit-Boussinesq approximation. In this configuration, the velocity potential that drives the flow obeys the Laplace equation, the solution of which crucially depends on the boundary conditions. Noting that the water table barely deviates from the horizontal, we linearize the boundary condition at the free surface, and solve the flow equations in steady state. We derive an expression for the velocity potential, which accounts for the shape of the experimental streamlines and for the propagation rate of tracers through the aquifer. This theory allows us to calculate the travel times of tracers through the experimental aquifer, which are in agreement with the observations. The travel time distribution has an exponential tail, with a characteristic time that depends on the aspect ratio of the aquifer. This distribution depends essentially on the geometry of the groundwater flow, and is weakly sensitive to the hydrodynamic dispersion that occurs at the pore scale.</p>