Estimate of Iteration Errors in Computational Fluid Dynamics

The solution of the conjugate problem of convective-conductive heat transfer in the channels of power plants is presented. The problem of convective heat transfer in the gas phase is solved by the integral theory of heat transfer. A one-dimensional problem of conductive heat transfer in the material of the wall is solved by the finite-difference method. Relative laws of heat and mass transfer and friction are obtained by numerical integration with respect to the boundary-layer thickness. The effect of the material and geometry of the wall on the level of problem “conjugation” is studied.

In this paper, a new technique of differential quadature method called the upwind difference - differential quadature method (UDDQM) for solving two-dimensional heat transfer (convection-diffusion) problems is proposed. Also, investigated the effects of physical quantities on behavior of flow problems, and combined effects of upwind difference mechanism together with differential quadrature method to modified the numerical solutions of heat transfer problems are presented. To validate our proposed UDDQM, two convection-diffusion problems ((i) Steady-state incompressible flow problem has exact solution and (ii) Natural convection motion of the incompressible fluid flow problem hasn't exact solution) are solving numerically. Graphical results on the effects of parameter variation on velocity, temperature, Peclet number, Grashof number, and Prandtl number are presented and discussed. Numerical experiments are conducted to test its accuracy and convergence and compare it with the standard DQM and other numerical methods that are available in literature. The numerical results show the efficiency of the proposed method to handle the problems, and it is more accurate and convergent than other methods.

An iterative method is introduced for solving noisy, ill-conditioned inverse problems. Analysis of the semi-convergence behavior identifies three error components iteration error, noise error, and initial guess error. A derived expression explains how the three errors are related to each other relative to the number of iterations. The Standard Tikhonov regularization method is just the first iteration of the iterative method and the derived noise damping filter is a generalization of the Standard Tikhonov filter. The derived filter is a function two parameters, a regularization parameter and the iteration number parameter. The new method is tested on image reconstruction from projections simulated data set.

A new efficient and accurate procedure for solving heat condution problems

Analytical methods exist to solve the problems of hydromechanics and heat transfer, but it is not possible to obtain the solution to some inhomogeneous and nonlinear problems of hydromechanics and heat transfer by analytical methods. The solution to such problems is carried out using numerical methods. Currently, there are many textbooks and monographs on numerical methods for solving problems of hydromechanics, thermal conductivity, heat and mass transfer, and others. The article presents the results of a numerical study of the flow structure in the flow around a flat plate. The calculations are based on the numerical solution of a system of nonstationary equations using a two-fluid turbulence model. For the numerical solution of these problems, schemes of the second and fourth order of accuracy were applied. The control volume method was used for the difference approximation of the initial equations, and the relationship between velocities and pressure was found using the SIMPLE procedure. To confirm the correctness of the numerical results, comparisons were made with each other and experimental data.

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

We study nonlinear inverse problems of the form F ( x ) = y , and their stable solution via iterative regularization methods, in particular by Newton-type methods, which are well-known for their fast convergence for well-posed problems. A basic step of Newton’s method consists of calculating the update Δ x k via solution of a linearized equation , which will in general be ill-posed if the nonlinear problem is. Thus, for ill-posed problems, the linearized equations have to be solved by some regularization method. In particular for large scale problems, e.g., inverse problems in partial differential equations, where F ( x ) is only defined implicitly via the solution of a PDE, iterative methods have to be used for this purpose. In order to keep the overall effort, i.e., the overall number of iterations, as small as possible, appropriate preconditioning has to be applied. We propose and analyse a general preconditioning strategy in Hilbert scales, and show that the overall number of iterations can be reduced to about the square root by preconditioning. Moreover, in many examples differential operators can be used as preconditioners, and thus preconditioning is almost for free. The theoretical results are illustrated in numerical examples. A comparison with preconditioned Landweber iteration shows that the iteration numbers can be further reduced, if fast iterative methods, e.g., the v -methods, are used for the solution of the linearized problems.

HOC Simulation of Double-diffusive Natural Convection in a Rectangular Cavity

Mixed boundary conditions of the adiabatic-isothermal type are often encountered in heat transfer problems. In some cases, singular behaviour may lead to erroneous results in determining the temperature and flow fields and evaluating the heat transfer. In this study, steady heat conduction in a square plate with mixed boundary condition on a straight boundary has been studied using finite difference and control volume methods. A method has been proposed to overcome the difficulties encountered in singular cases. Convergence criteria used in the numerical treatment of these problems have been discussed.

Numerical evaluation of the latent heat thermal energy storage performance enhancement by installing longitudinal fins

In the present study, a numerical investigation has been carried out into the fundamental problem of airflow past and heat transfer from a circular finned cylindrical tube, placed in a duct. The simulation is carried out using a finite volume method, based on laminar calculation of the transport quantities and employs an unsteady, 3-D, second order upwind scheme. As the work has importance in applications of air-cooled heat exchangers, practical values have been chosen for air velocity, air temperature, fin spacing and clearance between fin outer diameter and duct wall. In experimental determination of the performance of a finned-tube bundle, only overall average values such as drag coefficient and overall Nusselt number are possible. Local measurements are well nigh impossible, as any measurement instrument introduced into the narrow fin space will immediately change the flow field. This work gives an insight into variations of shear and heat transfer that will help the designer to optimize the fin spacing. The validity of the results for instantaneous velocity profiles and Nusselt number distribution comes from their physical plausibility. The agreement of the logical behavior of the studied variables when the fin space or fin clearance is modeled confirms the adequacy of the numerical simulation. The strong viscous effects caused by decreasing the fin space result in an increase in Cd and change in its frequency. Vortices generated on the rear section of the tube are damped, but the flow still shows small oscillations downstream of the tube, which indicates that vortex generation still exists, but has changed its location. Vortices augment the Nusselt number locally with increasing fin distance. Simultaneously, the opposite effect of converging boundary layers and therefore accelerated core flow in the fin space yields a maximum in the average Nusselt number as a function of fin space. The impact of decrease in the clearance between the tube and the duct wall is not as important as the effects of fin space.

Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

Preface to the classics edition Preface Acknowledgments Part I. The Method of Weighted Residuals: 1. Introduction 2. Boundary-value problems in heat and mass transfer 3. Eigenvalue and initial-value problems in heat and mass transfer 4. Applications to fluid mechanics 5. Chemical reaction systems 6. Convective instability problems Part II. Variational Principles: 7. Introduction to variational principles 8. Variational principles in fluid mechanics 9. Variational principles for heat and mass transfer problems 10. On the search for variational principles 11. Convergence and error bounds Author index Subject index.

A finite volume method (FVM) on distorted meshes for discretizing the energy equation's conduction term is presented. In this method, it is possible to compose the computational mesh of general quadrilateral elements (cells), namely, the cells are not required to be rectangular. The gradient of temperature on the cell's surface is computed to be second‐order accurate. Therefore, the error of numerical results by this method is smaller than using the traditional multilateral element method (MEM). The error does not depend on the degree of mesh distortion. The formulation based only on Taylor's theorem is straightforward. These are advantageous features to revise the fluid flow computation programs (based on FVM) that neglected the heat conduction term of the energy equation. The test calculations show that the convergence tendency of the numerical error using this method with the distorted mesh is the same as using an ordinary 2‐node central difference scheme on a constant‐interval rectangular mesh. By this method a conduction term was added to the energy equation of a SALE [1] program which had neglected that term originally, and z numerical calculation of a fluid flow with a heat transfer problem was performed. The numerical result of the present method with the distorted mesh well agrees with the analytical solution and the result of REM with a rectangular mesh. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20375