A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

[EN] A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y′′=f(x,y,y′), it is a fourth order convergent method for the special second-order ordinary differential equation, y′′=f(x,y). Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

A numerical method for computing unsteady 2-D boundary layer flows

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results

Heat transfer analysis for falkner-skan boundary layer flow past a stationary wedge with slips boundary conditions considering temperature-dependent thermal conductivity

We studied the problem of heat transfer for Falkner-Skan boundary layer flow past a stationary wedge with momentum and thermal slip boundary conditions and the temperature dependent thermal conductivity. The governing partial differential equations for the physical situation are converted into a set of ordinary differential equations using scaling group of transformations. These are then numerically solved using the Runge-Kutta-Fehlberg fourth-fifth order numerical method. The momentum slip parameter δ leads to increase in the dimensionless velocity and the rate of heat transfer whilst it decreases the dimensionless temperature and the friction factor. The thermal slip parameter leads to the decrease rate of heat transfer as well as the dimensionless temperature. The dimensionless velocity, rate of heat transfer and the friction factor increase with the Falkner-Skan power law parameter m but the dimensionless fluid temperature decreases with m. The dimensionless fluid temperature and the heat transfer rate decrease as the thermal conductivity parameter A increases. Good agreements are found between the numerical results of the present paper with published results

Finite volume solution of the compressible boundary-layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data

Modeling magnetic nanopolymer flow with induction and nanoparticle solid volume fraction effects : solar magnetic nanopolymer fabrication simulation

A mathematical model is presented for the nonlinear steady, forced convection, hydromagnetic flow of electro-conductive magnetic nano-polymer with magnetic induction effects included. The transformed two-parameter, non-dimensional governing partial differential equations for mass, momentum, magnetic induction and heat conservation are solved with the local non-similarity method (LNM) subject to appropriate boundary conditions. Keller’s implicit finite difference “box” method (KBM) is used to validate solutions. Computations for four different nanoparticles and three different base fluids are included. Silver nanoparticles in combination with various base fluids enhance temperatures and induced magnetic field and accelerate the flow. An elevation in magnetic body force number decelerates the flow whereas an increase in magnetic Prandtl number elevates the magnetic induction. Furthermore, increasing nanoparticle solid volume fraction is found to substantially boost temperatures. Applications of the study arise in advanced magnetic solar nano-materials (fluids) processing technologies

The effects of passive wall porosity on the linear stability of boundary layers

A two-dimensional and quasi three-dimensional model has been developed which enables the stability characteristics of laminar boundary layer flows over porous surfaces to be evaluated. The theory assumes the porosity to be continu- ous over a fixed portion of the computational domain. The chosen configuration also assumes a continuous plenum chamber, of variable depth, beneath the thin porous boundary. It has been suggested that surfaces of this type may be of use in the delay of a laminar boundary-layer's transition to turbulence. Hence, a pro- gram of study, involving both simulation and experimentation, was initiated to catalogue the effects of porosity. The numerical model consists of two sets of coupled solutions; one describing the cavity dynamics and the other describing the flow above the porous surface. The two systems are coupled together using their common wall boundary condi- tion. This is defined in terms of a complex function which attempts to model the effects of the viscous and inertial stresses within the fluid as it periodically flows through the wall into the cavity. Essentially, the function defines the magnitude and phase relationship between the pressure across the porous boundary and the flow through it. The cavity dynamics are determined by an analytical solution to the Orr-Sommerfeld equation in the absence of a mean flow field. The upper boundary condition for this flow is provided by the admittance function of the porous surface. The Orr-Sommerfeld equation is then solved again for the boundary layer flow - with the mean flow provided by solutions to the Falkner-Skan (or Falkner-Skan-Cooke) group of similarity profiles. This solution uses the admittance function to define its lower boundary condition. A high accuracy spectral technique, using Cheby- shev polynomials, is used for the integration of the Orr-Sommerfeld equation. Numerical simulations suggest that the appropriate selection of cavity depth and porosity fraction can lead to a complete suppression of the Toflmien-Schlichting instability for the case of zero pressure gradient. The model also suggests that The cavity dynamics are determined by an analytical solution to the Orr-Sommerfeld equation in the absence of a mean flow field. The upper boundary condition for this flow is provided by the admittance function of the porous surface. The Orr-Sommerfeld equation is then solved again for the boundary layer flow - with the mean flow provided by solutions to the Falkner-Skan (or Falkner-Skan-Cooke) group of similarity profiles. This solution uses the admittance function to define its lower boundary condition. A high accuracy spectral technique, using Cheby- shev polynomials, is used for the integration of the Orr-Sommerfeld equation. Numerical simulations suggest that the appropriate selection of cavity depth and porosity fraction can lead to a complete suppression of the Toflmien-Schlichting instability for the case of zero pressure gradient. The model also suggests that surface produced a self-sustained cavity oscillation whose magnitude could be up to 40% of the free-stream flow speed. These oscillations were found to be caused by a shear layer instability, self-excited by the feedback of disturbances caused by the shear layer's own impingement on the downstream cavity edge. A theoretical model for the instability has been developed which gives qualitative agreement with the experimental results. However, various cavity baffle configurations ul- timately failed to remove the instability. Hence, linear experiments on the 10% porous surface were not possible. The final two chapters of the thesis concern the stability of a general three- dimensional boundary layer when the PPW boundary condition is applied. Flows with varying degrees of streamwise pressure gradient and sweep were considered. It was noted, for the zero sweep case, that flows which exhibited viscous instabil- ity with decelerating fluid (tending towards inviscid instability) performed rather poorly when influenced by the PPW boundary condition. Furthermore, a wholly inviscid mechanism, such as that exhibited by the crossflow instability, was seen to have massively increased growth rates under the action of passive wall poros- ity. These two observations where seen as independent evidence in support of the prediction that the PPW boudary-condition is only theoretically of use when instabilities are wholly viscous

Numerical simulation of wavepackets in a transitional boundary layer

Results from two-dimensional direct numerical simulations of the governing equations that model incompressible fluid flow over a flat plate are presented. The Navier-Stokes equations are cast in a novel velocity-vorticity formulation (see Davies and Carpenter (2001)) and discretized with a mixed pseudospectral and compact finite-difference scheme in space, and a three-level backward-difference scheme in time. A method to determine the envelope of a wavepacket (from numerical data) was developed. Based on the usual Hilbert Transform, new stages were incorporated to ensure a smooth envelope was found when the wavepacket was asymmetric. The early transitional stages of the Blasius flow (flow over a flat plate with zero streamwise pressure gradient) are investigated with particular regard to a weakly nonlinear effect called wave-envelope steepening. Blasius flow is linearly unstable and so-called Tollmien-Schlichting modes develop. As nonlinearities become significant, the envelope of the wavepacket starts to develop differently at its leading and trailing edges. Numerical results presented here show that the envelope becomes steeper at the leading edge than it is at the trailing edge. The effect of a non-zero streamwise pressure gradient on wave-envelope steepening is investigated by using Falkner-Skan profiles in place of the Blasius profile. Natural transition is triggered by randomly-modulated waves. A disturbance with a randomly- modulated envelope was modelled and its effect on wave-envelope steepening was studied. The higher-order Ginzburg-Landau equation was used to model the evolution of an envelope of a wavepacket disturbance. These results gave good qualitative comparison with the direct numerical simulations. Finally, in preparation for developing a three-dimensional nonlinear version of the code, the discretization of one of the governing equations (the Poisson equation) was extended to three dimensions. Results from this new three-dimensional version of the Poisson solver show good agreement with those from an iterative solver, and also demonstrate the robustness of the nu merical scheme

Nonlinear Stability and Control of Three-Dimensional Boundary Layers

The linear and nonlinear evolution of steady and traveling disturbances in three-dimensional incompressible boundary layer flows is studied using Parabolized Stability Equations (PSE). Extensive primary stability analyses for the model problems of Swept Hiemenz flow and the DLR Transition experiment on a swept flat plate are performed first. Second, and building upon these results, detailed secondary instability studies based on both the classical Floquet Theory and a novel approach that uses the nonlinear PSE are conducted. The investigations reveal a connection of unstable secondary eigenvalues to both the linear eigenvalue spectrum of the undisturbed mean flow and the continuous spectrum, as well as the existence of an absolute instability in the region of nonlinear amplitude saturation. Third, a passive technique for boundary layer transition control using leading edge roughness is examined utilizing a newly developed implicit solution method for the nonlinear PSE. The results confirm experimental observations and indicate possible means of delaying transition on swept wings. In the present work, both the solution of the boundary layer equations for the mean flow and the explicit PSE solver are based on a fourth-order-accurate compact scheme formulation in body-oriented coordinates. In the secondary instability analysis, the Implicitly Restarted Arnoldi Method is applied

Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages

Enactment of implicit two-step Obrechkoff-type block method on unsteady sedimentation analysis of spherical particles in Newtonian fluid media

Purpose: The analysis of the characteristics of particles motion is considered in this article, where a model which studies a Newtonian fluid media with specific interest on the analysis of unsteady sedimentation of particles is considered. The numerical solution of this first order differential equation model using an Obrechkoff-type block method is presented. Methodology: The algorithm for the conventional Nyström -type multistep scheme is considered with specific parameter choices in order to obtain the main k-step Obrechkoff-type block method and the required additional method. The unknown coefficients of these methods are obtained by using the concept of Taylor series expansion to obtain the required schemes for the block method which were combined as simultaneous integrators for the solution of the differential equation model.Findings: The block method gave highly accurate results as compared with the exact solution of the model. Furthermore, at selected values of the physical properties of nanoparticles, the solutions using the two-step Obrechkoff-type block method was compared with past literatures and the results were seen to be in agreement. The influence of the physical parameters on terminal velocity is also discussed