Nonsimilar second order viscoelastic boundary layer flow at a stagnation point over a moving wall

An analysis is presented to investigate the fluid dynamic characteristics of a steady, laminar second order viscoelastic boundary layer flow at a two-dimensional stagnation point over a moving wall. The governing boundary layer equations have been solved by means of a series solution approach. Numerical solutions for the series functions have been given in tabular form. The development of the velocity distribution has been illustrated for several positive and negative values of the wall velocity. The values of the Weissenberg numbers ranged from 0 to 0.3.     

Characteristic based split (CBS) meshfree method modeling for viscoelastic flow

A semi-implicit characteristic based split meshfree algorithm is proposed for the numerical solution of incompressible viscoelastic flow in the paper, in which the governing equations are discretized by element free Galerkin method in spatial space and the characteristic based split method is adopted in temporal space. In order to obtain stable and convergent solution at high Weissenberg number, streamline upwind method is chosen to tackle the convective terms in constitutive equations of viscoelastic flow. Meanwhile, mass lumped technique is adopted to accelerate computational efficiency. The method allows the equal order basis to approximate pressure, velocity and extra stress, especially the linear basis that is easy to implement. The planar Poiseuile flow and 4:1 planar contraction flow for an Oldroyd B fluid are investigated. Through these numerical experiments, we find that the method has good stability and the numerical results are in agreement well with those reported by other papers.     

Time-dependent fundamental solutions for homogeneous diffusion problems

This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.     

Green's function method for axisymmetric flows: analysis of the Taylor-Couette flow

A numerical method for the solution of the Navier-Stokes equations in rotationally symmetric flow problems is presented. The numerical procedure is based on a boundary integral equation formulation with the fundamental solution of the Stokes' equation accounting for the rotational symmetry. The proposed methodology has been applied to the study of the Taylor-Couette flow between two concentric rotating cylinders of infinite axial length. A comparison with the available theoretical, experimental or numerical findings is performed to evaluate the accuracy of the present results. As predicted by the analytical theory and confirmed by the experiments, multiple solutions that are found for Reynolds numbers higher than the critical value, indicate the proposed methodology as a useful tool to get physical insight on the instabilities occurring in the solution of the Navier-Stokes equations.     

Heat transfer in a second order viscoelastic boundary layer flow at a stagnation point

The steady and transient heat transfer characteristics of a second order viscoelastic boundary layer flow at a stagnation point have been studied in this paper. The implicit cubic spline numerical procedure is used to solve the governing boundary layer equations. The details of the temperature profiles and wall heat flux rates have been graphically illustrated. The range of values of the Prandtl number was from 5 to 1000 while the Weissenberg number was varied from 0.1 to 0.3.     

CFD shape optimization using an incomplete‐gradient adjoint formulation

A design methodology based on the adjoint approach for flow problems governed by the incompressible Euler equations is presented. The main feature of the algorithm is that it avoids solving the adjoint equations, which saves an important amount of CPU time. Furthermore, the methodology is general in the sense it does not depend on the geometry representation. All the grid points on the surface to be optimized can be chosen as design parameters. In addition, the methodology can be applied to any type of mesh. The partial derivatives of the flow equations with respect to the design parameters are computed by finite differences. In this way, this computation is independent of the numerical scheme employed to obtain the flow solution. Once the design parameters have been updated, the new solid surface is obtained with a pseudo‐shell approach in such a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Some 2D and 3D numerical examples are shown to demonstrate the proposed methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

An implicit potential method for incompressible flows

In this paper, we consider a novel numerical scheme for solving incompressible flows on collocated grids. The implicit potential method utilizes an implicit potential velocity obtained from a Helmholtz decomposition for the mass conservation and employs a modified form of Bernoulli's law for the coupling of the velocity–pressure corrections. It requires the solution only of the momentum equations, does not involve the solution of additional partial differential equations for the pressure, and is applied on a collocated grid. The accuracy of the method is tested through comparison with analytical, experimental, and numerical data from the literature, and its efficiency and robustness are evaluated by solving several benchmark problems such as flow around a circular cylinder and in curved square and circular ducts.Copyright © 2013 John Wiley & Sons, Ltd.     

Heat transfer in a viscoelastic boundary layer flow through a porous medium

This paper examines the viscoelastic fluid flow and heat transfer characteristics in a saturated porous medium over an impermeable stretching surface with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for two different heating processes, namely (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). The governing equations for the boundary layer flow problem result similar solutions. For the specified five boundary conditions, it is not possible to solve directly the resulting sixth-order nonlinear ordinary differential equation. For the present incompressible boundary layer flow problem with constant physical parameters, the momentum equation is decoupled from the energy equation. Two closed–form solutions for the momentum equation are obtained and identified the realistic solution of the physical problem. Exact solution for the velocity field and the skin-friction are obtained. Also, the solution for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Asymptotic results for the temperature function for large Prandtl numbers are presented. The work due to deformation in the energy equation, which is essential and escaped from the attention of researchers while formulating the visco-elastic boundary layer flow problems, is considered. Drastic variation in the values of heat transfer coefficient is observed when the work due to deformation is ignored.The authors would like to thank the reviewers for their valuable comments/ suggestions to improve the clarity of the paper.     

Determination of multiple solutions of load flow equations

This paper is concerned with the problem of finding all the real solutions (all components of the solution vector must be real values) of load flow equations. Solutions in which some of the components are complex values are of no interest as they have no physical significance as a load flow solution. This problem is significant not only because of its theoretical challenge but also, its relationship with several system behavior related issues. Approaches suggested so far for solving this problem are rather ad hoc, computationally demanding and have been demonstrated only on very small systems. Further, it has been subsequently shown by others that many of these methods are not capable of finding all solutions. In this work a new approach is proposed which is more systematic and seems to have the potential to handle even large problems. We show that for any system it is possible to find the multiple load flow solutions (MLFS) corresponding to a given operating point extremely easily, starting from a set of points that are referred to as zero load solutions (ZLS) in this paper. It is shown that the complete set of ZLS is unique for a system and MLFS for any other operating point can be obtained starting from these ZLS using only the Newton’s load flow method. The set of procedures for implementing the proposed scheme are illustrated and their features are highlighted by considering several sample systems.     

On the numerical solution of fractional differential equations under white noise processes

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.     

The symplectic approach for two-dimensional thermo-viscoelastic analysis

This paper presents an exact symplectic approach for two dimensional isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the state space method and the Laplace transform, all general solutions of the governing equations are obtained analytically. By applying the inverse integral transform, the time domain adjoint symplectic relationships between the general solutions are established. Therefore, the problems of the particular solution and the boundary conditions can be analysed either in the Laplace domain or directly in the time domain. As its applications, the boundary condition problems are discussed in the numerical calculations. The results show that, due to the displacement constraints and the temperature influence, local effects are distinct near the boundary, and the effects decay rapidly with the distance from the boundary.     

A note on plane wave propagation in a linear viscoelastic solid

A new formulation of the governing equations for one spatial dimension wave propagation in a linear viscoelastic solid with more than one discrete relaxation time is proposed. The resulting system of three equations is treated as a strictly hyperbolic system of first order hyperbolic partial differential equations and the method of characteristics is adapted to obtain numerical solutions. Results are presented for a solid with two discrete relaxation times and are compared with those obtained from a predictor-corrector scheme.     

An efficient semi-implicit finite element scheme for two-dimensional tidal flow computations

A semi-implicit finite element scheme is proposed for two-dimensional tidal flow computations. In the scheme, each term of the governing equations, rather than each dependent variable, is expanded in terms of the unknown nodal values and it helps to reduce computer execution time. The friction terms are represented semi-implicitly to improve stability, but this requires no additional computational effort. Test cases where analytic solutions have been obtained for the shallow water equations are employed to test the proposed scheme and the test results show that the scheme is efficient and stable. An numerical experiment is also included to compare the proposed scheme with another finite element scheme employing Serendipity-type Hermitian cubic basis functions. A numerical model of an actual bay is constructed based on the proposed scheme and computed tidal flows bear close resemblance to flows measured in field survey.     

Response of gradient-viscoelastic bar to static and dynamic axial load

Summary. The response of a bar to static or dynamic axial load is studied analytically on the basis of a simple linear theory of gradient viscoelasticity. The governing equations of axial equilibrium and motion are first obtained by combining the basic equations. They are also obtained by a variational statement, which provides in addition all possible boundary conditions. A correspondence principle between the gradient elastic and gradient viscoelastic formulation and solution is established. Thus, the Laplace transformed with respect to time viscoelastic solution is obtained from the corresponding elastic one by simply replacing the elastic modulus by its Laplace transform times the Laplace transform parameter. The time domain response is finally obtained by a numerical inversion of the transformed solution. Two boundary value problems, one quasi-static and one dynamic, are studied and the gradient viscoelasticity effect on the solutions is assessed.     

A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations

In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations.     

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.     

Accelerated and robust finite volume Navier-Stokes solver for all speeds

The paper describes the results obtained from a multigrid accelerated Navier-Stokes solver. The method is based on 2-D explicit cell-centred finite volume Reynolds averaged Navier-Stokes (N-S) flow solver for speeds from near-incompressible Mach numbers to high hypersonic Mach numbers including flows at high angles of attack. The time integration is done using a hybrid 5-stage Runge-Kutta local time stepping scheme. With the help of a simple technique, the capability of the Jameson-Schmidt-Turkel numerical dissipation scheme has been enhanced to include hypersonic flows. The iterative procedure is accelerated significantly by incorporating a multigrid technique which has been used in all computations up to about supersonic speeds. Systematic numerical experiments were conducted to evolve guidelines to generate airfoil grid that could offer reliable flow simulations. The computed results are in very good agreement with experimental data where available, especially from the point of view of predicting large suction peaks and shock locations where considerable departures are often seen in the literature. Further, the highly accelerated computations make this code a useful tool of practical interest in preliminary aerodynamic design.     

A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.     

Operator splitting and upwinding for the Navier-Stokes equations

We present an operator splitting scheme for the unsteady Navier-Stokes equations for incompressible viscous fluid flow. Like other operator splitting methods applied to these equations, the difficulties associated with the nonlinearity and the incompressibility condition are decoupled. At each time step we obtain two subproblems of Stokes type and a linear one of elliptic type. The linear problem gives us uncoupled scalar problems of transport type; then, we may take advantage of well known upwind techniques for such kind of problems in order to handle large Reynolds numbers flow with coarse meshes. To show the efficiency of the scheme we report numerical results up to Reynolds numbers Re=4000 obtained with very coarse meshes.     

The method of fundamental solutions for inverse 2D Stokes problems

A numerical scheme based on the method of fundamental solutions is proposed for the solution of two-dimensional boundary inverse Stokes problems, which involve over-specified or under-specified boundary conditions. The coefficients of the fundamental solutions for the inverse problems are determined by properly selecting the number of collocation points using all the known boundary values of the field variables. The boundary points of the inverse problems are collocated using the Stokeslet as the source points. Validation results obtained for two test cases of inverse Stokes flow in a circular cavity, without involving any iterative procedure, indicate the proposed method is able to predict results close to the analytical solutions. The effects of the number and the radius of the source points on the accuracy of numerical predictions have also been investigated. The capability of the method is demonstrated by solving different types of inverse problems obtained by assuming mixed combinations of field variables on varying number of under- and over-specified boundary segments.