Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

VIT is one of the top ranked private universities in India according to NIRF, THE and QS Rankings.&nbsp;Govt. of India has recognized&nbsp;VIT, Vellore as an&nbsp;Institution of Eminence. This has allowed VIT to take independent quality initiatives and move up in world ranking.

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VIT University

The temperature distribution and distortion of fluid flow inside the closed cavities, square and triangle, are studied for different boundary conditions. Two different conditions of thermal boundary conditions are used for studying square cavities: (i) Left wall is hot, right wall is cold, top and bottom walls are adiabatic. (ii) Left and right walls are cold, top wall is adiabatic, bottom wall is hot. For triangular enclosure, the boundary conditions are (i) the vertical wall is insulated, bottom wall is hot. (ii) The vertical wall is hot, the bottom wall insulated and the inclined walls are kept cold in both conditions. The velocity of the flow is observed by means of stream function and the temperature distribution is displayed in the form of contours. The study is carried out in ANSYS software. The mathematical procedure for solving the nonlinear system of partial differential equations by penalty finite element method involving bi-quadratic elements is also discussed in detail. © M.H.M. JK and P. V, Published by EDP Sciences, 2021.

World University root node

Mathematics

School of Advanced Sciences

Chennai Campus

Fulltext

International Journal for Simulation and Multidisciplinary Design Optimization

Study of fluid flow inside closed cavities using computational numerical methods

Existing methods, such as Discrete Integration Algorithm (DIA) or Multiple DIA (MDIA) for evaluating Boltzmann integral to assess the nonlinear energy transfer within a given energy spectrum at a given location, do not account for all the contributing wave resonating quadruplets (QPs) for want of computational ease in the wave models such as WAM and WWIII. By virtue of employing the state-of-the-art Gauss–Legendre Quadrature Method (GLQM), the transfer integral becomes free of singularities, and fast estimation of all the contributing QPs is possible; and hence, this method provides both accuracy and efficiency. It also works for different frequency and angular resolutions of the input spectral grid. In this paper, GLQMis validated with EXACT-NL and WRT methods for a theoretical spectrum. It is then applied to a measured spectrum of a moored buoy of National Institute of Ocean Technology, off Machilipattinam (DS5) in deep waters for evaluating the nonlinear QP interactions based on one-month data during July 2005. A characteristic monthly averaged 1D frequency spectrum has been chosen which represented a double-peaked sea-dominated spectrum. It is then fitted with a theoretical JONSWAP spectrum with 99.5% confidence. The nonlinear energy transfer rate (Snl) between the higher and lower frequencies of this fitted spectrum has been evaluated using GLQM and are quantified. The nonlinear coupling between the sea and swell parts is found to be absent as the ratio of the swell and sea frequencies is less than 0.6 (Masson in J Phys Oceanogr 23:1249–1258, 1993 [1]). Few hypothetical cases have been studied to understand Snl behaviour further. © Springer Nature Singapore Pte Ltd. 2019.

Department of Software and Systems Engineering

School of Information Technology and Engineering

Vellore Campus

Lecture Notes in Civil Engineering Proceedings of the Fourth International Conference in Ocean Engineering (ICOE2018)

Assessment of Nonlinear Quadruplet Interactions for Measured Spectra in Deep Waters on the East Coast of India Through Gauss–Legendre Quadrature Method

A numerical method using Hybrid functions which are the orthogonal polynomials, formed from the combination of Block pulse function and Lagrange basis polynomial (HBL) are employed for the estimation of nonlinear energy transfers (NLT) occurring between set of four waves at finite water depths. The advantages and properties of HBL functions provide an easy way for estimating the quadruplets arising in NLT. The manuscript focuses on two aspects, namely sensitivity and efficiency of NLT at finite depths due to HBL. The sensitivity of NLT at finite depths to number of points on the locus curve, for various directional spectra, are studied in detail. With decreasing water depth (i) the locus curve grows and thus magnitude of NLT increases (ii) increase the number of points on the locus curve to achieve reliable (or) accurate results and (iii) the peak frequency starts shifting towards the left side of the spectrum. The efficiency of the HBL method to NLT is substantiated by comparison with the currently used Trapezoidal rule. Convergence results of NLT are established in both frequency and direction. The method is tested for its suitability in both circular and sector grids and the results confirm its adaptability to NLT. The method is validated against the methods implemented in the Wave models such as WRT and Discrete Interaction Approximation. © 2018, Springer International Publishing AG, part of Springer Nature.

GENERAL

Department of Ocean Engineering

Journal of Ocean Engineering and Marine Energy

Hybrid functions for nonlinear energy transfers at finite depths

This paper attempts to solve the definite integrals numerically using Gaussian quadrature rule with variable spacing. In this paper, the orthogonal polynomial is chosen to be the hybrid function formed from the block-pulse function of order N and Lagrange basis polynomial of order M. Here, the domain of the integral is partitioned into N sub-intervals, and the M roots of the Legendre polynomial of order M, are chosen as the nodes in each sub-interval. Having identified M nodes in each sub-interval, the Lagrange polynomial basis of order M is constructed. The total number of abscissas (or quadrature points) in the domain of the definite integral is NM. The weights can be easily obtained from the Lagrange polynomial. A significant advantage of this method is that the Hybrid matrix turns out to be an Identity matrix of order NM and the hybrid coefficients are simply the value of the integrand at the nodal points. The method is explained for single definite integrals, and this rule is extended to cover definite double and triple integrals with constant or variable limits. A comparative study of this method for both definite single and double integrals with Haar Wavelet and Hybrid function [2] reveals that better accuracy can be achieved with this quadrature rule with less number of points. It can be considered to cover a broad class of integrands. © 2016 Academic Publications, Ltd.

International Journal of Pure and Applied Mathematics

Numerical integration using hybrid of block-pulse functions and lagrange polynomial

A new attempt is made to compute the nonlinear four wave interactions in the discrete spectral wave model represented by the 6-D Boltzmann integral using fully Gauss-Legendre quadrature method. An advantage of this method is that the singularities of the integral are naturally avoided. The method can be used to obtain accurate results for the nonlinear energy transfer rate. In this paper, the evaluation of the transfer integral based on the locus parameters arc length (s) and direction (θ) are considered separately using Gauss-Legendre quadrature method. Results obtained using this method for the deep water nonlinear energy transfer rate corresponding to the input standard JONSWAP wave spectra with angular distributions in the cases 2-6 of Hasselmann and Hasselmann [2] [hereafter referred as HH]are presented. A comparison study of the present 1-D nonlinear results indicate good agreement with the corresponding results of the exact WRT method [9]. © 2016 Academic Publications, Ltd.

A fully quadrature method for the nonlinear four-wave interactions

A wavelet-based attempt is made to estimate the nonlinear interactions for wind wave spectra using Haar wavelets. The nonlinear interactions have been synthesized using the orthogonal basis of the Haar wavelets. The analysis of the nonlinear interactions using wavelets provides an easy way of computing the transfer integral in the Webb–Resio–Tracy’s (WRT) method. The one-dimensional and two-dimensional results confirm the applicability of the wavelets to the nonlinear wave–wave interactions and the approach of multi-resolution analysis ensures the convergence and the accuracy. © 2016, Springer International Publishing Switzerland.

VIT Unitversity

Department of Communication Engineering

School of Electronics Engineering

A wavelet approach for computing nonlinear wave–wave interactions in discrete spectral wave models

A Polar Method based on parametric Cubic Spline technique (PM-CS) is presented for obtaining wave resonating quadruplets {K1,K2,K3,K4} in the calculation of the nonlinear source term of the wave model, with results for both deep and finite depths. This polar method is applied in the WRT method for the nonlinear term, and may be considered in addition to the Polar Method with Adaptive Stepping (PM-AS) by Van Vledder (2000). The PM-CS is used to obtain an uniform distribution of points on the locus of K2, for input pair (K1,K3), similar to PM-AS. However, it differs from PM-AS in two ways: (i) no assumption has been made regarding the shape of the locus, and (ii) iterative procedure is avoided, except for locus points on the symmetry in finite depths. For a smaller set of points on the locus, the present PM-CS with variable spacing can be employed. This is illustrated through the convergence of the nonlinear source term by varying points on the locus and the transfer integral evaluated using the Gauss-Legendre quadrature method. Results presented for 1-D nonlinear source term using WRT method are shown comparable with existing WRT results of Resio and Perrie (1991) and Vledder and Hashimoto (2013). © 2015 Elsevier Ltd. All rights reserved.

Ocean Engineering

A Polar Method using cubic spline approach for obtaining wave resonating quadruplets

An attempt is made to obtain the Finite Element Solutions to steady state and transient state heat conduction problems using the hybrid of Block-Pulse functions and Lagrange interpolation polynomial. This method can be used to obtain accurate results. The finite element results are compared with the corresponding results obtained using quadrature methods and found to be in good agreement. © Research India Publications.

Global Journal of Pure and Applied Mathematics

Finite element solutions using hybrid of block-pulse functions and lagrange polynomial

A polar method for obtaining wave resonating quadruplets {K1, K2, K3, K4} in the computation of nonlinear wave-wave interaction source term of the wave model is presented with results for both deep and finite water depths. The method first determines the end radial points of the locus equation for K2, for each set of input wave vectors (K1, K3) on the symmetry. The locus of K2 (and hence K4) is then traced in the anti-clockwise direction starting with the maximum radial point on the line of symmetry. It is shown that when k3&gt;k1, the number of points on the locus varies when the orientations of the input wave vectors are changed and reduces when the difference in the magnitude of the input wave vectors is increased. A significant advantage in this method is that the angular increment on the locus for K2 can be kept constant. © 2005 Elsevier Ltd. All rights reserved.

A polar method for obtaining wave resonating quadruplets in finite depths

An accurate method using composite Gauss-Legendre N-point quadrature formula is presented for solving the nonlinear wave-wave interaction source term in deep waters. This method employs a polar grid in the wavenumber space with a constant geometric ratio γ and uses the scaling relation for the transfer integral. The accuracy of the method can be tested for different γs by increasing N. This increase in γ will help in calculating the nonlinear source term at less number of frequency points, resulting in reduced computation time. We also included the procedure for obtaining the nonlinear results at more number of frequency points. A study of 1D nonlinear source term S nl(f) with results of exact methods, indicates that the present results are comparable and qualitatively in good agreement with Webb-Resio-Tracy (WRT) method in spite of slight differences at higher frequencies. This slight difference is due to different polar grids employed for the input vectors in the present method, whereas they are the same in the WRT method. The present method thus considers possibilities that are not explored in the WRT method.

Current Science

A quadrature method for computing nonlinear source term due to wave-wave interactions

The pressure variations exerted on a vertical wall in a constant water depth are computed from the calculation of a short-crested wave system using the Fourier series approximation method. The numerical results have been compared with experimental results from literature. The comparison with the linear solution of the present theory gives a good estimate for the variation of dynamic pressures. The double peak formation in the experimental results for pressure curves at still water level and at the bottom for the steeper waves was not observed for the linear case. In the case of surface elevation, deviations with the third order perturbation results are observed for higher waves. In addition, deviations are also observed within the elevation results for higher waves. © 2000 Elsevier Science Ltd. All rights reserved.