Present Solution Methodology Research Articles

An Analytical Solution of Transient Wave Generation in the Wave Channel

Transient characteristics of wave generation in the wave channel can provide unique and important information in contrast to the steady and periodic motion of propagation waves. In this paper, a new analytical solution is proposed for a transient wavemaker problem in the wave channel. The mathematical model of the wavemaker problem is established based on the linear potential wave theory, and a new analytical solution for the corresponding initial and boundary-value problem is presented. The present solution methodology is motivated and developed from old methods shown in literature. The present solution can be mathematically reformulated and shown to be identical to the previous solution using different solution methodology. The present analytical solution is further compared with numerical results and experiments to validate the mathematical model. The present solution is used to calculate the steady state generated wave forms that compare very well with the steady wave theory both in wave length and wave period. The present solution is also used to study unsteady characteristics of wave heights and wave lengths of the leading waves. The present analytical solution methodology can provide an easier approach to obtain the analytical solution for transient wave generation problem in the wave channel.

Analysis of a magnetic field and Hall effects in nanoliquid flow under insertion of dust particles

AbstractIn this study, the two‐phase hydromagnetic flow of a viscous liquid through a suspension of dust and nanoparticles is considered. The influence of the Hall current is also taken into account. The similarity variables are utilized to transform the problem into one independent variable. The obtained expressions in one independent variable are solved through the Runge–Kutta–Fehlberg scheme connected with the shooting procedure. The computed results are sketched for employing multiple values of physical constraints on the temperature and velocity of the nanofluid and dust phase. The characterization of various nanoparticles like Cu, Al2O3, TiO2, and Ag on velocities and temperatures of both phases is made through plots. A comparative analysis in the limiting approach is presented to justify the present solution methodology. The range of emerging parameters is taken as 0 ≤ l ≤ 3, 0.1 ≤ βt ≤ 3, 0 ≤ m ≤ 2.5, 0 ≤ M2 ≤ 2, 0.1 ≤ βv ≤ 3, 0 ≤ ϕ ≤ 0.4, and −0.8 ≤ λ ≤ 0.8. From the study, it is revealed that βt has the opposite effect on the temperature of dust and nanofluid phases. The Hall parameter m raises the profiles of velocities in the nanoliquid and dust phases. Also, it is found that the transverse velocities h(η) and H((η) and temperatures θ(η) and θp(η) rise for larger ϕ.

Non-linear deflection and stress analysis of laminated composite sandwich plate with elliptical cutout under different transverse loadings in hygro-thermal environment

Abstract In this paper, non-linear transverse deflection, stress and stress concentration factors (SCF) of isotropic and laminated composite sandwich plate (LCSP) with and without elliptical cutouts subjected to various trans-verse loadings in hygrothermal environment are studied. The basic formulation is based on secant function-based shear deformation theory (SFSDT) with von-Karman nonlinearity. The governing equation of non-linear deflection is derived using C0 finite element method (FEM) through minimum potential energy approach. Normalized trans-verse maximum deflections (NTMD) along with stress concentration factor is determined by using Newton’s Raphson method through Gauss point stress extrapolation. Influence of fiber orientations, load parameters, fiber volume fractions, plate span to thickness ratios, aspect ratios, thickness of core and face, position of core, boundary conditions, environmental conditions and types of transverse loading in MATLAB R2015a environment are examined. The numerical results using present solution methodology are verified with the results available in the literatures.

Free convective MHD micropolar fluid flow with thermal radiation and radiation absorption: A numerical study

AbstractAn analysis is carried out for the free convective flow of an electrically conducting micropolar fluid through permeable stretching sheet in the presence of porous medium. Inclusion of thermal radiation to the energy equation enhances the thermal properties of polar fluid. In addition to that the radiation absorption parameter occurs due to the interaction of solutal concentration difference also considered in the heat transfer equation. Suitable similar transformation is used to convert the governing partial differential equations to ordinary differential equations. Furthermore, though analytical solutions of these complex nonlinear coupled equations are more complicated therefore, numerical solution such as Runge‐Kutta fourth order method associated with shooting technique is adopted. Behavior of characterizing parameters for the flow phenomena are presented via graphs and computed values of physical quantities of interest are obtained and shown in tabular form. Present result validates with that of earlier results in particular case which confirms the existence of present solution methodology. However, the main findings of contributing parameters are laid down as; angular velocity profile contributes a dual character from the point of contact at the middle of the channel. Fluid temperature is affected by the inclusion of absorption coefficient.

Simulation of three‐dimensional incompressible flows in generalized curvilinear coordinates using a high‐order compact finite‐difference lattice Boltzmann method

SummaryIn the present study, a high‐order compact finite‐difference lattice Boltzmann method is applied for accurately computing 3‐D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3‐D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth‐order compact finite‐difference scheme and a fourth‐order Runge‐Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3‐D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3‐D incompressible flow solver. A high‐order spectral‐type low‐pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3‐D generalized curvilinear coordinates. Numerical solutions of different 3‐D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2‐D cylindrical Couette flow, the decay of a 3‐D double shear wave, the cubic lid‐driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high‐order compact finite‐difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3‐D incompressible flows over practical and realistic geometries.

Preconditioned WENO finite-difference lattice Boltzmann method for simulation of incompressible turbulent flows

In this work, a preconditioned high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (WENO-LBM) is applied to deal with the incompressible turbulent flows. Two different turbulence models namely, the Spalart–Allmaras (SA) and k−ωSST models are used and applied in the solution method for this aim. The spatial derivatives of the two-dimensional (2D) preconditioned LB equation in the generalized curvilinear coordinates are discretized by using the fifth-order WENO finite-difference scheme and an implicit–explicit Runge–Kutta scheme is adopted for the time discretization. For the convective and diffusive terms of the turbulence transport equations, the third-order WENO and second-order central finite-difference schemes are used, respectively. The preconditioning technique along with the local time-stepping method are applied to the WENO-LBM to further accelerate the solution to the converged steady-state condition, and therefore, an accurate and efficient incompressible LB solver is provided for simulating turbulent flows. The accuracy and robustness of the proposed solution method are assessed by computing two test cases: the 2D turbulent flow over a flat plate at Re=1.0×107 and the 2D turbulent flow over a NACA0012 airfoil at Re=6.0×106 and different angles of attack. The present results are compared with the available numerical and experimental results which show excellent agreement. It is demonstrated that the present solution methodology based on the WENO-LBM provides more accurate results of the incompressible turbulent flows and it requires lower number of grid points compared to the traditional (low-order accurate) LB and Navier–Stokes solvers.

Item placement for questionnaire design for optimal reliability

This paper presents a methodology for placement of the items of surveys to obtain optimal reliability. A mathematical model is developed based on some specific structured assumptions for reliability and consistency. The problem is transformed into a mathematical optimization problem. We present solution methodology for the problem and also properties of the algorithm. An example is presented to illustrate our methodology.

Dynamic Behavior of Curved Panels of Rectangular Planar Form: An Analytical Approach

The dynamic behavior of singly and doubly curved panels of the rectangular planar form subjected to different types of loadings is presented. The mathematical formulation is based on the higher order shear deformation theory, and the principle of virtual work is used to derive the equations of motion. The fast converging finite double Chebyshev series and Houbolt time-marching scheme are used for evaluating the dynamic response of the panel. The effect of the magnitude and duration of pulse loadings on the transverse central displacement and bending moment responses is evaluated for different parameters. The accuracy of the present solution methodology is established by the convergence study of non-dimensional central deflection and central moments and comparison of the present results with those available in the literature. Some new results are presented for the hyperboloid panels.

Chebyshev collocation spectral lattice Boltzmann method in generalized curvilinear coordinates

Abstract In this work, the Chebyshev collocation spectral lattice Boltzmann method is implemented in the generalized curvilinear coordinates to provide an accurate and efficient low-speed LB-based flow solver to be capable of handling curved geometries with non-uniform grids. The low-speed form of the D2Q9 and D3Q19 lattice Boltzmann equations is transformed into the generalized curvilinear coordinates and then the spatial derivatives in the resulting equations are discretized by using the Chebyshev collocation spectral method and the temporal term is discretized with the fourth-order Runge–Kutta scheme to provide an accurate and efficient low-speed flow solver. All boundary conditions are implemented based on the solution of the governing equations in the generalized curvilinear coordinates. The accuracy and robustness of the solution methodology presented are demonstrated by computing different benchmark and practical low-speed flow problems that are 2D Couette flow between concentric moving cylinders, 2D flow in a gradual expansion duct, 2D regularized trapezoidal cavity flow, and 3D flow in curved ducts of rectangular cross-sections. Results obtained for these test cases are in good agreement with the existing analytical and numerical results. The computational efficiency of the proposed solution methodology based on the Chebyshev collocation spectral lattice Boltzmann method implemented in the generalized curvilinear coordinates is also examined by comparison with the developed second-order finite-difference lattice Boltzmann method that indicates the proposed method provides more accurate and efficient solutions in terms of the CPU time and memory usage. The study shows the present solution methodology is robust and accurate for solving 2D and 3D low-speed flows over practical geometries. Indications are that the solution algorithm based on the CCSLBM in the generalized curvilinear coordinates does not need any filtering or numerical dissipation for stability considerations and thus high accuracy solutions obtained by applying the CCSLBM can be used as benchmark solutions for the evaluation of other LBM-based flow solvers.

A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids

In this work, a spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional inviscid compressible flows on structured grids. The compressible form of the discrete Boltzmann–BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is applied. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function. For time accurate solutions, the fourth-order Runge–Kutta scheme is used to discretize the temporal term in the LB equation. Note that the procedure of implementing the SD method to solve the LB equation is nearly the same as that procedure developed in the literature for the solution of the Euler equations. The accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems. A sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size/distribution on the accuracy and performance of the solution. At first, three inviscid compressible flow problems, namely, the stationary isentropic vortex, the shock tube and the shock–vortex interaction are solved by using the SDLBM to demonstrate the accuracy and robustness of the present solution methodology. Results computed for these test cases are in good agreement with the analytical and the available numerical solutions. To more assess the accuracy and robustness of the SDLBM, a third-order finite volume LBM (FVLBM) is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other. It is demonstrated that the SDLBM is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible LB-based flow solvers. As the final problem, the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the SDLBM in reaching to a steady-state solution.

Simulation of two-phase liquid-vapor flows using a high-order compact finite-difference lattice Boltzmann method

A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios.

On application of high-order compact finite-difference schemes to compressible vorticity confinement method

The main goal of this study is to assess the application of high-order compact finite-difference schemes for the solution of the Euler equations in conjunction with the compressible vorticity confinement method on both uniform Cartesian and curvilinear grids. Here, the spatial discretization of the governing equations is performed by the fourth-order compact finite-difference scheme and the temporal term is discretized by the fourth-order Runge–Kutta method. To stabilize the numerical solution, appropriate dissipation terms are applied and a detail assessment is performed to study the effects of the values of confinement and dissipation coefficients on the solution to reasonably preserve the structure of vortical flows. The accuracy and performance of the proposed solution procedure by applying the fourth-order compact finite-difference scheme are also examined by comparison with the solution obtained by the second-order central finite-difference scheme. Low-pass high-order filters are also applied to the fourth-order compact finite-difference scheme to investigate the performance of the vorticity confinement with the filtering scheme. The advection of a 2D isentropic vortex and a shock–vortex interaction problem are two test cases simulated for the assessment of the present solution methodology. The study shows that the high-order compact finite-difference schemes in conjunction with the vorticity confinement method can reasonably preserve the structure of vortical flows in coarse uniform Cartesian and curvilinear grids. Indications are that the present solution methodology is accurate and effective for solving compressible flow problems with vortical structures when coarse grids are used.

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

SUMMARYA high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.

Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates

In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the implementation of the high-order compact finite-difference Lattice Boltzmann method (CFDLBM) in the generalized curvilinear coordinates is robust, efficient and accurate for solving steady and unsteady incompressible flows over practical geometries.

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.

Free-Vibration Analysis of Ring-Stiffened Branched Composite Shells of Revolution

Application of the multisegment numerical integration technique is extended to the free-vibration analysis of macroscopically anisotropic filament-wound branched shells of revolution with ring stiffeners, considering the variation of the thickness and winding angle. The solution procedure is based on a modified-frequency trial method, which processes on the numerically integrated transformed fundamental shell equations that are obtained in terms of finite exponential Fourier transform of the fundamental shell variables. The full macroscopically anisotropic form of the constitutive relations, including first-order transverse shear deformation and all components of translatory and rotary inertia, are included in the analysis. To handle branched shells of revolution, modifications that are necessary to incorporate junctions are added to the solution procedure. Inclusion of asymmetric circumferential stiffeners, with respect to the middle surface of the shell, into the semi-analytical solution method is demonstrated by presenting two alternative methods of analysis. The present solution methodology also incorporates the variation of the thickness and winding angle along the meridian of filament-wound shells of revolution, with general meridional curvature, by assuming placement of filaments along the geodesic fiber path on the surface of the shell of revolution.

Higher-order theory based boundary-discontinuous Fourier analysis of simply supported thick cross-ply doubly curved panels

A boundary-discontinuous double Fourier series based solution methodology is employed to solve the problem of a HSDT-based thick cross-ply doubly curved panel, characterized by a system of five highly coupled linear partial differential equations, with the SS1-type simply supported boundary condition prescribed at all four edges. For derivation of the complementary solution, the complementary boundary constraints are introduced through boundary discontinuities of some of the particular solution functions and their partial derivatives. Such discontinuities form a set of measure zero. The present solution methodology is based on the fact that the complementary boundary constraints, which are inequalities, play as important a role as the prescribed (admissible) boundary conditions, which are equalities. The effects of curvature, lamination, material property, thickness as well as their interactions are investigated in detail. Numerical results presented herein illustrate the severity of the effects of those parameters.

Multiobjective optimal placement of convectively cooled electronic components on printed wiring boards

This paper presents a solution methodology for multiobjective optimization problems in the context of models for the placement of components on printed wiring boards (PWB's). The methodology combines the use of a flow and heat transfer solver, a genetic algorithm for the adaptive search of optimal or near-optimal solutions, and a multiobjective optimization strategy [Pareto optimization or multiattribute utility analysis (MUA)]. Using as the optimization criterion, the minimization of an estimate of the failure rate of the system of components due to thermal overheating (via an Arrhenius relation), the effectiveness of the present solution methodology is demonstrated by reference to a case with known optimal solutions. The results obtained using the same solution methodology for a multiobjective optimization problem (a variation of the case study) involving the minimization of the aforementioned total failure rate of the system as well as the minimization of the total wiring length (given some interconnectivity requirements) are presented and discussed for both Pareto optimization and MUA.