Drag reduction within radial turbine rotor passages using riblets

In this paper, reducing the friction losses in a radial inflow turbine rotor surface by adding engineered features (riblets) is explored. Initially, computational fluid dynamics analysis was used to study the operating mechanism of riblets and to test their ability to reduce drag within the rotor passage when running the turbine at the design point. Thereafter, riblets with different heights and spacing have been implemented at the rotor hub to study the effect of riblets geometry and arrangement on the drag reduction, which leads to determine the riblet geometry where the maximum benefit on turbine performance can be achieved. The effect of riblets on boundary layer development and on the secondary flow generation within the rotor passage has been examined. It was found that the introduction of riblets could reduce the wall shear stress at the hub surface, and on the other hand, they contribute to increasing the stream-wise vorticity within the rotor passage. The maximum wall shear reduction was achieved with riblet with relative height hrel = 2.5% equivalent to 19.3 wall units, while the maximum performance happens when using riblets with hrel = 1.5% equivalent to 11.8 wall units as the later contributes less in secondary flow generation within the passage. For riblets with height more than 19.3 wall units, the overall effect is negative, as they cause an increase in drag and give rise to secondary flow leading to lower turbine performance

Introducing an Efficient Modification of the Variational Iteration Method by Using Chebyshev Polynomials

In this article an efficient modification of the variational iteration method (VIM) is presented using Chebyshev polynomials. Special attention is given to study the convergence of the proposed method. The new modification is tested for some examples to demonstrate reliability and efficiency of the proposed method. A comparison of our numerical results those of the conventional numerical method, the fourth-order Runge-Kutta method (RK4) are given. The comparison shows that the solution using our modification is fast-convergent and is in excellent conformance with the exact solution. Finally, we conclude that the proposed method can be applied to a large class of linear and non-linear differential equations

Quasi-linearization method with rational Legendre collocation method for solving MHD flow over a stretching sheet with variable thickness and slip velocity which embedded in a porous medium

The quasi-linearization method (QLM) and the rational Legendre functions are introduced here to present the numerical solution for the Newtonian fluid flow past an impermeable stretching sheet which embedded in a porous medium with a power-law surface velocity, variable thickness and slip velocity. Firstly, due to the high nonlinearity which yielded from the ordinary differential equation which describes the proposed physical problem, we construct a sequence of linear ODEs by using the QLM, hence the resulted equations become a system of linear algebraic equations. The comparison with the available results in the literature review proves that the obtained results via QLM are accurate, and the method is reliable

Application of Taylor-Pade technique for obtaining approximate solution for system of linear Fredholm integro-dierential equations

In this article, we introduce a modification of the Taylor matrix method using Pad´e approximation to obtain an accurate solution of linear system of Fredholm integro-differential equations (FIDEs). This modification is based on, first, taking truncated Taylor series of the functions and then substituting their matrix forms into the given equations. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Finally, we use Pad´e approximation to obtain an accurate numerical solution of the proposed problem. To demonstrate the validity and the applicability of the proposed method, we present some numerical examples. A comparison with the standard Taylor matrix method is given

Analytical solution for determination the control parameter in the inverse parabolic equation using HAM

In this article, the homotopy analysis method (HAM) for obtaining the analytical solution of the inverse parabolic problem and computing the unknown time-dependent parameter is introduced. The series solution is developed and the recurrence relations are given explicitly. Special attention is given to satisfy the convergence of the proposed method. A comparison of HAM with the variational iteration method is made. In the HAM, we use the auxiliary parameter ~ to control with a simple way in the convergence region of the solution series. Applying this method with severa

Approximate Solutions for the Nonlinear Systems of Algebraic Equations Using the Power Series Method

In this paper, the approximate solutions for systems of nonlinear algebraic equations by the power series method (PSM) are presented. Illustrative examples have been presented to demonstrate the efficiency of the proposed method. In addition, the obtained results are compared with those obtained from the standard Adomian decomposition method. It turns out that the convergence of the proposed algorithm is rapid

Approximate Solutions for the Flow and Heat Transfer due to a Stretching Sheet Embedded in a Porous Medium with Variable Thickness, Variable Thermal Conductivity and Thermal Radiation using Laguerre Collocation Method

In this article, a numerical approach is given for studying the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing PDEs are transformed into a system of coupled non-linear ODEs which are using appropriate boundary conditions for various physical parameters. The proposed method is based on replacement of the unknown function by truncated series of well known Laguerre expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equations by means of collocation points to a system of algebraic equations with Laguerre coefficients. Thus, by solving this system of equations, the Laguerre coefficients are obtained. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. Comparison of obtained numerical results is made with previously published results in some special cases. The results attained in this paper confirm the idea that the proposed method is powerful mathematical tool and it can be applied to a large class of nonlinear problems arising in different fields of science and engineering

Implementation of the matrix differential transform method for obtaining an approximate solution of some nonlinear matrix evolution equations

This article introduces the matrix differential transform method (MDTM) to apply to matrix partial differential equations (MPDEs) and employs it for solving matrix Fisher equations, matrix Burgers equations and matrix KdV equations. We show how the MDTM applies to the linear part and nonlinear part of any MPDE and give various examples of MPDEs to illustrate the efficiency of the method. The results obtained are in excellent agreement with the exact solution and show that the proposed method is powerful, accurate, and easy

An Efficient Computational Method for Solving a System of FDEs via Fractional Finite Difference Method

This paper aims to provide a numerical method for solving systems of fractional (Caputo sense) differential equations (FDEs). This method is based on the fractional finite difference method (FDM), where we implemented the Grünwald-Letnikov’s approach. This method is computationally very efficient and gives very accurate solutions. In this study, the stability of the obtained numerical scheme is given. The numerical results show that the proposed approach is easy to be implemented and are accurate when applied to system of FDEs. The method introduces promising tool for solving many systems of FDEs. Two examples are given to demonstrate the applicability and the effectiveness of our method