Two-dimensional Potential Flow Research Articles

On some properties of the pressure in nonlinear two-dimensional potential flow with a free surface

Two-dimensional nonlinear free surface flows are nowadays routinely simulated on computers. In the context of the potential flow theory, fine details of the free surface motion can be captured. However, few studies focus on the flow beneath the free surface. The novelty of the present work lies in identifying lines where the pressure gradient is colinear with one eigenvector of the pressure Hessian matrix. The spatial and temporal evolution of these lines offers new insights into the complexity of highly nonlinear flows. This pressure characteristic is used to revisit four more or less recent research studies on breaking waves.

Universality of finite-time ray focusing statistics in two-dimensional and one-dimensional potential flows.

In the investigation of extreme density or wave height statistics in a disordered medium, of special interest is the search for universal or fundamental properties shared by different types of disorder. In previous work [Chen and Kaplan, Entropy 25, 161 (2023)1099-430010.3390/e25010161] we have established a direct connection between the degree of stretching or focusing of ray trajectories and the density distribution. Here we demonstrate the universality of this connection for different physical contexts, and both analytically and numerically show a universal scaling relationship for the stretching exponent distribution in weak, small-angle scattering at finite times for different dispersion relations. We observe that the mean, skewness, and kurtosis of the stretching exponent all display universal nonmonotonic behavior on timescales comparable to the time of first caustic formation, corresponding to the first generation of hot spots in the density profile. In particular, the mean stretching exponent attains negative values before beginning its linear rise at large times. Using the correspondence between two-dimensional small-angle scattering and a one-dimensional kicked model, we show how higher moments of the distribution of the second derivative of the potential affect the statistics of the stretching exponents.

Homogenization of thin-structured surfaces for acoustics in the presence of a two-dimensional low Mach potential flow

A surface homogenization method for acoustic waves over thin microstructured surfaces in the presence of a fluid in a potential flow is presented. Sound hard surfaces are considered, the flow is considered two-dimensional and slow and a low Mach approximation is introduced. We consider acoustic waves with a typical wavelength 1 / k much larger than the array spacing h and thickness e . Owing to the small parameter ε = k h , with e / h = O ( 1 ) , a matched asymptotic expansion technique is applied to the low Mach potential wave equation in the frequency domain. A boundary condition is obtained on an equivalent flat wall, which links the acoustic velocity to its normal and tangential derivatives (of the Myers type). The accuracy of the effective model is tested numerically for various periodic shapes and the accuracy of the model in O ( ε 2 ) is validated.

Mathematical study on the potential flow past a vertical submerged flexible plate of non-uniform thickness

The present work emphasizes on the mathematical study of two-dimensional potential fluid flow past a vertical flexible plate of non-uniform thickness submerged in a water domain of finite depth. The boundary value problem associated with the described model is solved by two mathematical techniques while considering Kirchhoff’s thin plate theory. The first method is based on a perturbation analysis involving a small parameter ϵ, while the second method directly employs the classical method of integration and Green’s integral theorem to the governing equations to obtain the solution. Both the techniques result into two different systems of integral equations which are solved numerically using appropriate approximations. It has been established numerically that for smaller thickness variations depicted by particular values of ϵ (ϵ≤0.001), the two methods yield the same numerical results for all the hydrodynamic quantities. The correctness of the present results is verified by comparing them graphically with the ones already existing in the literature. Effects of the non-uniform plate thickness on various hydrodynamic parameters are presented graphically by choosing various thickness profiles. Also, comparisons are made with the results of constant thickness elastic plate to distinguish its hydrodynamic behaviour from the one having non-uniform thickness.

Free surface in two-dimensional potential flow: singularities, invariants and virtual fluid

We study a two-dimensional (2-D) potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution of the Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the ‘virtual’ fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time invariants associated with the topology of the conformal plane and Kelvin's theorem for a virtual fluid are explored. We called them ‘winding’ and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double exponential and supersedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with a free surface by unifying the problem of motion of vortex sheets, and the problem of 2-D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included.

Validated analytical model for a cyclorotor wave energy device

We present a new analytical model for a horizontal axis cyclorotor-based wave energy converter (WEC). A number of cyclorotor-based WEC concepts and models, with different numbers of hydrofoils, have previously been studied. Our model is derived for a horizontal cyclorotor with 2 hydrofoils. The governing equations are optimised and converted to the polar coordinate system. The mechanical model is based on Newton's second law for rotation. Rotation is considered in two-dimensional potential flow, for both monochromatic and panchromatic waves, including waves generated by the rotating rotor, and viscous losses. The developed model is very convenient for modelling, analysis and control design for a cyclorotor based WEC.
 The authors of this work have derived new, exact, analytic functions for the free surface perturbation and induced fluid velocity field caused by hydrofoil rotation. These new formulae significantly decrease the model calculation time, compared to previous models, and increase the accuracy of the results. We present the results of free rotation simulations for the rotor in monochromatic and panchromatic waves obtained with the use of the newly derived equations.

A new approach to narrow waterways traffic routing with potential flow theory and CFD

Navigational planning is crucial for safety of navigation in narrow waterways, both for the ships and the environment. The movement of the ships is very complex in narrow waterways, which are in curved form with surface current and counter-deep current. A TSS (traffic separation scheme), a maritime traffic management route system, has been established worldwide in most narrow waterways and turning points close to shore. The ships plan their routes under the TSS in the region they navigate. In this research, the aim of this study is to determine the optimum route by taking into account the depth, width, water flow rate, movement of the water in curved regions, and upper and lower currents in order to create proposed TSS model. For this purpose, a region with turning points in the Bosphorus and where accidents are frequent were selected. The two-dimensional potential flow theory was used to determine the velocity and pressure distribution at surface in the region where the accidents were intense, and results of the velocity and pressure distributions obtained with CFD were verified against the analytical data obtained from the potential flow theory. As a result of the study, it has been revealed that the data obtained using the two-dimensional potential flow theory and CFD are compatible with each other, and the flow dynamics obtained using the potential flow theory and/or CFD can be used to determine the dynamic TSS to navigable waterways, channels and rivers in worldwide.

THE MODEL OF FREE SPREADING A FLOW RAPID BEHIND A RECTANGULAR PIPE

The article considers the free spreading of a turbulent stationary two-dimensional open potential water flow into a wide diverting riverbed behind a non-pressure pipe of a rectangular section. A system of nonlinear partial differential equations of motion has been adopted as the mathematical model of the flow in the physical plane. When moving to the plane of the velocity hodograph, the nonlinear system of equations is transformed into a linear system with respect to partial derivatives. Using the obtained system of equations, various problems along the flow of two-dimensional water streams have been solved analytically. The paper determines the flow kinetics parameter t and the angle q characterizing the direction of the local flow velocity vector at the intersection points of an arbitrary equipotential and an arbitrary current line. The X, Y coordinates of these points are found. The peculiarities of changing the angle θ during the transition of the vertical front of the XD are taken into account.
 Article proposes a module for the transition from a two-dimensional water flow model to a one-dimensional one. This module is necessary for using the laws of flow resistance and taking into account the resistance forces.
 The model proposed in this paper is a development of analytical methods for calculating potential flows with previously unknown boundaries and before the flow expands. It allows determining the entire range of geometric and kinematic parameters of the flow with an error not exceeding 10%.
 The adequacy of the model for all flow parameters improves the accuracy of previously existing methods. This allows the designers of road culverts to increase its reliability.

Stability of two-dimensional potential flows using bicomplex numbers.

The use of the complex velocity potential and the complex velocity is widely disseminated in the study of two-dimensional incompressible potential flows. The advantages of working with complex analytical functions made this representation of the flow ubiquitous in the field of theoretical aerodynamics. However, this representation is not usually employed in linear stability studies, where the representation of the velocity as real vectors is preferred by most authors, in order to allow the representation of the perturbation as the complex exponential function. Some of the classical attempts to use the complex velocity potential in stability studies suffer from formal errors. In this work, we present a framework that reconciles these two complex representations using bicomplex numbers. This framework is applied to the stability of the von Kármán vortex street and a generalized formula is found. It is shown that the classical results of the symmetric and staggered von Kármán vortex streets are just particular cases of the generalized dynamical system in bicomplex formulation.

Potential flow around polygonal shaped cylinders using hypotrochoidal mapping function

In this paper, some investigations on two-dimensional steady potential flow over polygonal shaped cylinders are presented. The polygonal shapes with finite corner radius are obtained using Hypotrochoidal mapping function. The complex potential function derived using Milne-Thompson circle theorem, along with Hypotrochoidal mapping is employed to get various flow parameters over different shaped cylinders. The uniform and non-uniform flow conditions are arrived at, by taking infinite and finite distance of vortex from the cylinder, respectively. The velocity and pressure distribution over various shaped cylinders are presented and some of them are compared with the results from commercial software package (ANSYS) and available literature. Formulas are developed to obtain velocity and pressure distribution around various shaped geometries. Also, the effect of various geometrical parameters on velocity and pressure distribution is studied.

Inverse distance interpolation for used in unstructured mesh finite volume solver

This article discusses adjusting inverse distance interpolation for use in unstructured mesh finite volume solutions. The adjustment was made on the weight function of the inverse distance interpolation using the Laplacian of the flow variable inside a Voronoi-dual of finite volume cells. We tested the accuracy of the adjusted inverse distance interpolation on two-dimensional potential flows. It was found that the adjusted and standard inverse distance interpolations have a similar degree of accuracy when used in unstructured, Delaunay based, finite volume mesh. However, the L1 norm error of the adjusted version of the inverse distance interpolation was much smaller than the L1 norm error of the standard version.

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains {Ωμ:μ∈P}⊂RD where μ∈P⊂RP denotes a vector of parameters, (ii) a parameterized mapping Φμ between a reference domain Ω and the parameter-dependent domain Ωμ, and (iii) a finite element triangulation of Ω, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to “move” the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard “map-then-discretize” approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.

ОСНОВНЫЕ ПОЛОЖЕНИЯ В РАЗРАБОТКАХ МОДЕЛЕЙ ПО ТЕЧЕНИЮ ОТКРЫТЫХ ВОДНЫХ ПОТОКОВ

For the design of hydraulic structures, it is necessary to use special methods for calculating the water flow to determine the kinetic energy acting on the structures. A mathematical model of a two-dimensional in terms of stationary open water flow and boundary conditions in the problem of free spreading are formulated. The main solved and to be solved problems of determining the flow parameters are determined, its reduction to a dimensionless form by various transformations of coordinates and flow parameters. The method proposed by I.A. Sherenkov. The solution of the problem is described, which depends on the dimensionless parameter - the Froude criterion at the outlet of the flow from the pipe. With Froude numbers exceeding one or close to it, it is required to build a series of graphs or develop a unified theory, an algorithm for solving the problem. The general conclusions on the work are as follows: - the need for further research has been proven theoretically and experimentally. - tasks are formulated that must be performed, solved in order to obtain a result adequate to the real process of solving the problem of free spreading of a turbulent flow to the entire spectrum of parameters. - substantiated the need to continue research to determine the entire spectrum of parameters of a stationary open two-dimensional potential flow in terms of its outflow from a free-flow pipe into a wide horizontal smooth channel. - the requirements for the model taking into account the conjugation of a uniform flow with a radial flow in the form of a simple wave are determined The work was written with a critical assessment of existing methods for solving the problem and to substantiate the relevance of further scientific research.

Two-dimensional plan source, vortex and vortex source

Modern hydraulic structures require highly reliable water-supply channels, free-flow pipes and open spillways. Therefore, they must be built with considering dynamic properties of the affecting flow. The theory of one-dimensional open flows cannot answer many questions raised by the hydraulic engineering practice. Hence, this paper considers two-dimensional graphical open flows, with separate particular models following from the general theory, important for designing couplings, curves in more complex constructions (two different channels) required in hydraulic engineering (Hydraulics technical structures) (HTS). And the more such models are obtained, the more opportunities will appear for scientists and designers to synthesize more complex structures required in HTS.This study identifies three elements, and three simplest models of two-dimensional graphic open water flows similar to flat models: a source, a vortex and a vortex source. The number of such individual models will continue to increase and expand the range of possibilities for designing more complex water technical objects (WTO).The models are obtained analytically from solving the system of two-dimensional graphical open potential water flows in the plane of the velocity hodograph. The elementary construction of each model includes an analytical solution for the potential function and the stream function.The paper considers the problem of determining the parameters of a two-dimensional graphical open water flow at any point in the flow: a source, a vortex, a vortex source. The practical significance of the models lies in the possibility to use the results by the designers of hydraulic structures both at the first stage of solving problems and at the subsequent ones, with flow resistance forces taken into account.The study also represents the transition method from the plane of the flow velocity hodograph to the flow diagram by integrating the models in the plane of the velocity hodograph from the condition of the connection between the considered planes.

Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle

<p style='text-indent:20px;'>This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Hölder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.</p>

On the physical mechanism of front–back asymmetry of non-breaking gravity–capillary waves

Abstract

Two-dimensional numerical study of gap resonance coupling with motions of floating body moored close to a bottom-mounted wall

The piston mode fluid resonance in the narrow gap between a moored floating body and a bottom-mounted vertical wall is numerically investigated based on a two-dimensional potential flow model and viscous numerical simulations. This study focuses on understanding the effect of mooring stiffness on the coupling dynamics of the gap resonance and the sway or heave motion of the floating body in regular waves. Numerical studies show that the resonant wave amplitude in the gap is reduced by the sway and heave motions. The reduction is highly dependent on the mooring stiffness. Two resonant frequencies are confirmed, and both increase with the mooring stiffness. Different modes of motions are identified in terms of the phase difference between the oscillatory motions of the gap flow and the floating body. Higher harmonic components of responses are found for the specific mooring stiffness. The performance of potential flow models in predicting resonant responses is revisited based on the understanding that the overall damping effect consists of two parts: (1) radiation damping and (2) viscous dissipation. It is confirmed that a potential model is also able to produce reasonable predictions as radiation damping plays a dominant role, for example, at the second resonant frequencies of coupling the gap resonance with the sway motion. Otherwise, as viscous dissipation dominates radiation damping, noticeable over-predictions by a potential model occur as recognized before, for example, the present results at the second peak response of gap resonance with the heave motion. The relative viscous dissipation is quantified with the reflection coefficient of viscous numerical results, while the radiation damping is quantified based on a specially designed radiation potential model with inputs of viscous numerical solutions.

МЕТОД АНАЛОГИЙ МЕЖДУ ГИДРАВЛИКОЙ ДВУХМЕРНЫХ В ПЛАНЕ ВОДНЫХ ПОТОКОВ И ГАЗОВОЙ ДИНАМИКОЙ

Real potential two-dimensional flows of practical hydraulics are considered, which can be considered with a certain accuracy as two-dimensional in terms, which confirms the relevance of the work. It is proposed to use the method of analogies when applying gas dynamics equations in hydraulics. An important result was obtained for the development of the analytical theory of two-dimensional potential water flows.

Wave response of segmented floating plate and validation of its homogenized solution

Homogenization of a segmented plate floating on water is investigated. Firstly, the mathematical approach to solve wave responses of the segmented floating plate is developed. The approach is based on the eigen-function matching method, and resultant boundary-value problems are solved numerically. Secondly, the homogenization of the segmented floating plate, of which flexural rigidity alternates periodically, is derived by assuming the unit length is significantly smaller than the representative length. All problems are formulated by assuming two-dimensional linear potential flow, and the elastic plate is thin enough to be modeled as the surface condition. Wave responses of the floating plate with solo segment and four segments with variable thickness are firstly calculated for validation in comparison with referenced results. After that, homogenized solutions are compared with exact solutions in terms of segments. The average error between them decreases as the ratio of unit length and the effective elastic wavelength. Comparisons of reflection coefficient, wave elevation, bending moment, shear force, and surface strain are shown.

Impulsive Motion Inside a Cylindrical Cavity

Experimental studies of supercavitating models moving at speeds in the range from 400 m/s to 1000 m/s revealed a regime of bouncing motion, in which the rear part of an axisymmetric body periodically bounces against the free boundaries of the supercavity. The impulsive force generated by the impacts is the main concern in this paper. The analysis is performed in the approximation of two-dimensional potential flow of an ideal and incompressible liquid with negligible surface tension effects. The primary interest of the study is to determine the added mass taking into account the shape of the cavity. The theoretical study is based on the integral hodograph method, which makes it possible to obtain analytic expressions for the flow potential and for the complex velocity in an auxiliary parameter plane and obtain a parametric solution to the problem. The problem is reduced to a system of two integro-differential equations in two unknowns: the velocity magnitude on the cavity boundary and the slope of the velocity angle to the body. The equations are solved numerically using the method of successive approximations. The obtained results show that the added mass of an arc impacting a cylindrical cavity depends heavily on the arc angle. As the angle tends to zero or the radius of the cavity tends to infinity, the obtained solution predicts the added mass corresponding to a plate impacting a flat free surface.