Problem Of Wave Interaction Research Articles

Modified Monotonicity Preserving Constraints for High-Resolution Optimized Compact Scheme

The monotonicity-preserving (MP) scheme is an accurate shock-capturing scheme. However, its performance is still inefficient for resolving high-frequency waves. In this paper, to improve the resolution characteristics, an upwind compact interpolation is proposed as a substitute to the original one in the MP scheme, and the coefficients of that were analytically optimized to minimize the dispersion and dissipation errors. Moreover, it was found that the limiting part of the original MP scheme degenerates the accuracy in a high-wavenumber region due to unnecessarily activation. This limitation is improved by applying a new indicator and criterion. The results of the nonlinear wave (N-wave) propagation demonstrate that the proposed scheme guarantees the robustness at the sharp discontinuity. At the same time, the solutions of linear wave propagation prove the excellent resolution of the proposed scheme. We intensively evaluated the performance for the standard and long-time situations of the shock-entropy wave interaction problems. The results prove that the usefulness of proposed scheme is more pronounced in the flow fields involving both of shock and waves.

Моделирование физической неустойчивости на контактных границах в течениях многокомпонентных сжимаемых газов гибридным методом крупных частиц

Статья посвящена развитию гибридного метода крупных частиц применительно к двумерным течениям с развитием физической неустойчивости на поверхностях раздела неоднородных газовых смесей. Высокая разрешающая способность метода продемонстрирована при решении задач взаимодействия ударной волны с цилиндрическим пузырем легкого или тяжелого газов в сравнении с экспериментом и расчетами по другим схемам повышенного порядка аппроксимации. This paper is devoted to the development of a large-particle hybrid method for two-dimensional flows with physical instability on the interface of inhomogeneous gas mixtures. The high resolving capacity of the method is shown for problems of shock wave interaction with a cylindrical bubble of a light or heavy gas in comparison with experiments and simulations using other schemes of higher-order approximation.

Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

Abstract This paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

Methods of simulating the front of the air shock wave for calculating the industrial structure

Introduction. The paper considers existing methods of simulating a wide front of an air shock wave for solving problems of shock wave interaction with an installation using gas-dynamic methods. When solving the problem of the air shock wave interaction with an installation in a dynamic setting, it was revealed that, when simulating a wide front of a distant explosion using point explosions, it is possible to obtain an underestimated time of the shock wave action. This results in a downward bias of loads to the installation. Thus, the loads obtained in this case do not correspond to the loads for which it is necessary to carry out the calculation of industrial installations protected from shock waves in accordance with domestic and international regulatory documents. To eliminate this drawback, another approach is proposed. It consists in setting the load on the computational region in the form of a pressure graph with specified parameters of overpressure and exposure time.
 Materials and methods. The interaction of the shock wave front with the installation is carried out using numerical simulation in a nonlinear dynamic setting using gas-dynamic methods in the LS-DYNA software package.
 Results. The following analyses were conducted in the scope of the study: an analysis of existing methods of forming the wide shock wave front of the distant explosion and an analysis of the parameters of the shock wave during the formation of the wide shock wave front of the distant explosion by setting the pressure graph with the specified parameters of the overpressure and the exposure time.
 Conclusions. The result of the analysis of methods for numerical simulation of the interaction of the air shock wave wide front with the installation showed that simulation of the explosion source in the form of volume elements and simulation of the shock wave using the CONWEP function of the LS-DYNA software package have disadvantages. These disadvantages do not allow obtaining the main parameters of the shock wave for the further use. A method for modeling the wide shock wave front is given by setting a pressure graph at the boundary of the computational region with the required overpressure parameters and exposure time.

Long Wave Interaction with a Partially Immersed Body. Part I: Mathematical Models

In the present article we consider the problem of wave interaction with a partially immersed, but floating body. We assume that the motion of the body is prescribed. The general mathematical formulation for this problem is presented in the framework of a hierarchy of mathematical models. Namely, in this first part we formulate the problem at every hierarchical level. The special attention is payed to fully nonlinear and weakly dispersive models since they are most likely to be used in practice. For this model we have to consider separately the inner (under the body) and outer domains. Various approached to the gluing of solutions at the boundary is discussed as well. We propose several strategies which ensure the global conservation or continuity of some important physical quantities.

Construction of global Riemann solutions with delta-type initial data for a thin film model with a perfectly soluble anti-surfactant solution

The global solutions of an extended Riemann problem for a thin film model with a perfectly soluble anti-surfactant solution belonging to the Temple class with delta-type initial data are constructed in fully explicit forms for all possible situations. The method used is to perturb the delta-type initial data into initial data consisting of three piecewise constant states, such that the wave interaction problems have been extensively studied, including the interaction between a delta shock wave with a rarefaction (or shock) wave. The global solutions of the extended Riemann problem are obtained by letting the perturbation parameter tend to zero. We find that the initial Dirac delta function propagates along either the line of delta shock wave or the line of contact discontinuity. In addition, we find that some interesting nonlinear phenomena occur during the process of constructing the global solutions, such as a delta shock wave that separates into a shock wave and a delta contact discontinuity.

Scattering of surface waves by a pair of asymmetric thin elliptic arc shaped plates with variable permeability

Abstract We consider the interaction between incident surface water waves and two asymmetric curved porous plates which assume the shape of ellipse. The permeability of the plates is not uniform. This is assumed to vary along the circumference of the elliptical arc. Two coupled integral equations are derived for the problem and are solved by employing an appropriate algorithm. Using these solutions, the numerical estimates for the reflection and the transmission coefficients, amplitudes of the hydrodynamic forces and energy dissipation coefficient are computed. The accuracy of the numerical results is established through an energy identity valid for permeable plates, and by comparing them with the published ones. A selection of new results is also presented for illustrative purposes. Importantly, in the present paper a mathematical method is developed to solve the problem of wave interaction with non-vertical (elliptic) plates with variable permeability. The derived numerical results indicate that the system of dual elliptic plates serves as a suitable model for a wave lens.

Hydrodynamic performance of a vertical slotted breakwater

The wave interaction problem with a vertical slotted breakwater, consisting of impermeable upper, lower parts and a permeable middle part, has been studied theoretically. An analytical model was presented for the estimation of reflection and transmission of monochromatic waves by a slotted breakwater. The far-field solution of the wave scattering involving nonlinear porous boundary condition was obtained using eigenfunction expansion method. The empirical formula for drag coefficient in the near-field, representing energy dissipation across the slotted barrier, was determined by curve fitting of the numerical solutions of 2-D channel flow using CFD code StarCCM+. The theoretical model was validated with laboratory experiments for various configurations of a slotted barrier. It showed that the developed analytical model can correctly predict the energy dissipation caused by turbulent eddies due to sudden contraction and expansion of a slotted barrier. The present paper provides a synergetic approach of the analytical and numerical modelling with minimum CPU time, for better estimation of the hydrodynamic performance of slotted breakwater.

Wave energy dissipation of a submerged horizontal flexible porous membrane under oblique wave interaction

An analytical model associated with oblique wave interaction with a submerged horizontal flexible porous membrane is presented under the assumption of linearized small amplitude and structural response in finite water depth. The expansion formula associated with Green's functions using fundamental source potentials based on reduced wave equation is derived. Further, a physical problem of oblique wave interaction with a moored finite horizontal submerged flexible porous membrane on wave energy dissipation is illustrated to analyse the different design parameter based on matched eigenfunction expansion method. The roots of the complex dispersion relation are obtained by applying a perturbation scheme and the accuracy of the numerical computation is demonstrated by analysing the convergence of reflection, transmission, and dissipation coefficients for different spring stiffness and porous-effect parameters. Further, the correctness of the present solution is checked by validating against analytical, numerical boundary element method, and experimental data available in the literature. The effects of spring stiffness, porous-effect parameter, submergence depth, structural length, angle of incidence, and flow pattern on reflection, transmission, dissipation coefficients, membrane deflection, vertical force, and contour plots are analysed. The present analysis on different design parameters will be helpful for better understanding for engineers to design as an effective breakwater/wave absorber.

Analytical modeling of water wave interaction with a bottom-mounted surface-piercing porous cylinder in front of a vertical wall

The interaction of water wave with a bottom-mounted surface-piercing porous cylinder near a rigid vertical wall is investigated by an analytical model newly developed in the present work within the context of linear potential flow theory. The image principle is used to transfer the original problem in bounded water into the equivalent problem of wave interaction with two symmetrical porous cylinders in open seas in the presence of bi-directional incident waves. The velocity potential is analytically derived by means of the eigenfunction expansion along with the matching technique. Furthermore, a new alternative method for the evaluation of wave force is developed via the application of the Haskind–Hanaoka relation to a porous structure. In this method, an auxiliary radiation potential is introduced to replace the diffraction potential for the calculation of wave force. The auxiliary radiation potential used here is due to the oscillation of a porous cylinder in front of a wall. The image principle is used again to search the solution of the wave radiation problem in bounded water and the original radiation problem is then transferred to that due to two porous cylinders undergoing in-phase or out-of-phase motions in open seas. After the validation of the developed model, detailed parametric study is carried out. The porosity of the cylinder, incident wave heading and spacing between the cylinder and the wall are systematically adjusted to investigate their effects on the wave force as well as the wave elevation. The extension of our model to the case of a cylinder array in front of a wall has also been performed, and the associated phenomenon has been explored.

Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient

We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.

Interaction of ocean waves with floating and submerged horizontal flexible structures in three-dimensions

A three-dimensional general mathematical hydroelastic model dealing with the problem of wave interaction with a floating and a submerged flexible structure is developed based on small amplitude wave theory and linear structural response. The horizontal floating and submerged flexible structures are modelled with a thin plate theory. The linearized long wave equations based on shallow water approximations are derived and results are compared. Three-dimensional Green’s functions are derived using fundamental source potentials in water of finite and infinite depths. The expansion formulae associated with orthogonal mode-coupling relations are derived based on the application of Fourier transform in finite and infinite depths in case of finite width in three-dimensions. The usefulness of the expansion formula is demonstrated by analysing a physical problem of surface gravity wave interaction with a moored finite floating elastic plate in the presence of a finite submerged flexible membrane in three-dimensions. The numerical accuracy of the method is demonstrated by computing the complex values of reflected wave amplitudes for different modes of oscillation and mooring stiffness. Further, the effect of compressive force and modes of oscillations on a free oscillation hydroelastic waves in a closed channel of finite width and length for floating and submerged elastic plate system is analysed.

Global Solutions of the Perturbed Riemann Problem for the Chromatography Equations

The Riemann problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.

Motion of a floating body in a harbour by domain decomposition method

A three-dimensional domain decomposition method is used to solve the problem of wave interaction with a ship floating inside a harbour with arbitrary shape. The linearized velocity potential theory is adopted. The total fluid domain is divided into two sub-ones: one for the harbour and the other for the external open sea. Boundary integral equations together with the free surface Green function are used in the both domains. Matching conditions are imposed on the interface of the two sub-domains to ensure the velocity and pressure continuity. The advantage of the domain decomposition method over the single domain method is that it removes the coastal surface from the boundary integral equation. This subsequently removes the need for elements on the coastal wall when the equation is discretized. The accuracy of the method is demonstrated through convergence study and through the comparison with the published data. Extensive results through the hydrodynamic coefficients, wave exciting forces and ship motions are provided. Highly oscillatory behaviour is observed and its mechanism is discussed. Finally, the effects of incident wave direction, ship location as well as the harbour topography are investigated in detail.

Interaction of wave with a body submerged below an ice sheet with multiple arbitrarily spaced cracks

The problem of wave interaction with a body submerged below an ice sheet with multiple arbitrarily spaced cracks is considered, based on the linearized velocity potential theory together with the boundary element method. The ice sheet is modeled as a thin elastic plate with uniform properties, and zero bending moment and shear force conditions are enforced at the cracks. The Green function satisfying all the boundary conditions including those at cracks, apart from that on the body surface, is derived and is expressed in an explicit integral form. The boundary integral equation for the velocity potential is constructed with an unknown source distribution over the body surface only. The wave/crack interaction problem without the body is first solved directly without the need for source. The convergence and comparison studies are undertaken to show the accuracy and reliability of the solution procedure. Detailed numerical results through the hydrodynamic coefficients and wave exciting forces are provided for a body submerged below double cracks and an array of cracks. Some unique features are observed, and their mechanisms are analyzed.

HWENO Schemes Based on Compact Differencefor Hyperbolic Conservation Laws

In this paper, in order to simplify the high order Hermite weighted essentially non-oscillatory (HWENO) finite difference schemes of Liu and Qiu (J Sci Comput 63:548–572, 2015), a new type of HWENO schemes based on compact difference schemes, termed CHWENO (compact HWENO) schemes, is proposed for solving both one and two dimensional hyperbolic conservation laws. The idea of reconstruction in CHWENO schemes is similar to HWENO schemes, however the first derivative values of solution are solved by the compact difference method and only one numerical flux is used in CHWENO schemes, while the derivative equation is needed to be solved and two numerical fluxes are used in HWENO schemes. Compared with the original finite difference weighted essentially non-oscillatory schemes, CHWENO schemes maintain the compactness of HWENO schemes, which means that only three points are needed for a fifth order CHWENO schemes. Compared with the HWENO schemes, CHWENO schemes avoid solving the complex derivative equations, which can considerably expedite calculating speed. Several numerical tests, including the 1D shock density wave interaction problem and 2D Riemann problem, are presented to demonstrate the efficiency of CHWENO schemes.

Numerical Modelling of Surface Water Wave Interaction with a Moving Wall

In the present manuscript, we consider the practical problem of wave interaction with a vertical wall. However, the novelty here consists in the fact that the wall can move horizontally due to a system of springs. The water wave evolution is described with the free surface potential flow model. Then, a semi-analytical numerical method is presented. It is based on a mapping technique and a finite difference scheme in the transformed domain. The idea is to pose the equations on a fixed domain. This method is thoroughly tested and validated in our study. By choosing specific values of spring parameters, this system can be used to damp (or in other words to extract the energy of) incident water waves.

Interactions of Composite Waves of the Two-Dimensional Full Euler Equations for Van Der Waals Gases

This paper studies two types of wave interactions arising from two-dimensional (2D) Riemann problems for the full Euler equations: interaction of jump-fan composite waves and interaction of fan-jump-fan composite waves, which are absent in polytropic gases, but appear in van der Waals gases. Because of the occurrence of shock waves, the flow considered here is unsteady, nonisentropic, and rotational. In order to construct the solutions to these wave interaction problems, we consider two types of discontinuous Goursat problems for a 2D self-similar Euler system. Global piecewise smooth solutions to these discontinuous Goursat problems are obtained by the method of characteristics. The 2D self-similar Euler system is a mixed-type system, and the type in each point is determined by the local pseudo-Mach number. So, one of the main difficulties for the global existence is that hyperbolicity is hard to control in the composite wave interaction zones. By constructing some invariant regions of wave characteristic angles, we prove that the system is uniformly hyperbolic in these zones. In order to obtain the a priori gradient estimates of solutions to these discontinuous Goursat problems, we derive some characteristic decompositions for the 2D self-similar full Euler system. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems.

Numerical Modelling of crack wave interaction with BEM

The novel BEM for dynamic crack wave interaction problem is presented here. The various concepts, key ideas and theories are discussed in brief. As the theory suggests, cracks behave as stress concentrators and/or stress distributors. Therefore, two parameters associated with these two phenomena are calculated, namely, the Stress Intensity Factor (SIF) that is near–field characteristic and the far–field displacement in the scattered or total wave field. The developed numerical method is validated against the existing analytical solution. Far field crack wave interactions are plotted for perpendicular and inclined wave field. The method captures the complex crack wave interaction with ease.

Fast Iterative Adaptive Multi-quadric Radial Basis Function Method for Edges Detection of Piecewise Functions—I: Uniform Mesh

In Jung et al. (Appl Numer Math 61:77–91, 2011), an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy $$O(n^3)$$ computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman–Morrison–Woodbury formula. The $$O(n^2)$$ Levinson–Durbin recursive algorithm that employed Yule–Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Several classical benchmark examples show that the IAMQ-RBF-Fast based edges detection method can be at least three times faster than the original IAMQ-RBF based one. And it can capture an edge with fewer grid points than the multi-resolution analysis (Harten in J Comput Phys 49:357–393, 1983) and just as good as if not better than the L1PA method (Denker and Gelb in SIAM J Sci Comput 39(2):A559–A592, 2017). Preliminary results in the density solution of the 1D Mach 3 extended shock–density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection method demonstrating an excellent performance in term of speed and accuracy, are also shown.