Application of the Method of Fundamental Solutions and the Radial Basis Functions for Laminar Flow and Heat Transfer in Internally Corrugated Tubes

Heat transfer characteristics for a laminar fully developed flow in an internally corrugated circular tube with axially uniform heat flux with peripherally uniform temperature are obtained using the method of fundamental solution and the radial basis functions. The internal shape of the tube is modelled by cosine function. The nonlinear governing equation of temperature field problem was converted into the hierarchy of non-homogenous problems using the Picard iteration method. The non-homogenous part of the equation was interpolated using the radial basis functions. On each iteration step the solution of the governing equation consists of general solution as linear combination of fundamental solutions and particular solution. Boundary conditions were satisfied using the boundary collocation method. The results of the numerical experiments, that are velocity profiles, temperature field, friction factor, Nusselt number, errors of method were presented. INTRODUCTION Heat transfer at fully developed steady laminar flow in circular pipes belongs to the classical problems in heat transfer analysis {[1], chapter 8}. Usually the authors consider heat transfer at constant surface heat flux or constant surface temperature. In literature one can find considerations related to heat transfer at fully developed steady laminar flow in noncircular conduits. In paper [2] square conduct was considered by means of point matching method. Flow and heat transfer characteristics in corrugated ducts confined by sinusoidal and arc curves were considered in paper [3] using finite difference technique. The numerical results for triangular, sine, rhombic and trapezoidal ducts are given in paper [4], where author use least-squares matching techniques. In paper [5] analyticalnumerical method based on Gram-Schmidt orthonormalization was used and as an example ducts of circular segment cross section were considered. The effects of duct shape on the heat transfer at fully developed laminar flow were considered in paper [6]. One of non-circular conduits important from practical standpoint of heat transfer is an internal finned tubes. The flow and heat transfer characteristics investigations have been carried out for several finned geometries under various physical and operating conditions. In this paper review other papers and our analysis are limited only to laminar and fully developed flow. In papers [8] and [10] only fully developed laminar flow was considered in finned tubes. In paper [8] authors using finite element method consider tubes with short triangularly shaped fins. Whereas in paper [10] flow region was divided in two subregions in which governing equation and some boundary conditions are fulfilled exactly and only matching velocity and its normal derivatives between sub-regions are fulfilled approximately. The theoretical analysis of laminar flow and heat transfer in internally finned tubes was considered in papers [7], [9], [1129]. Moreover, in paper [28] authors present the experimental verification of theoretical analysis. The shapes of the cross sections of the tube considered in these papers are given in Figure 1. To the best of our knowledge as of yet the internally corrugated tubes, presented in Figure 2, haven’t been considered in the literature. Fully developed laminar flow and heat transfer in such tubes is the subject of this paper. In the aforementioned papers for the numerical analysis of laminar flow and heat transfer in internally finned tubes traditional mesh method namely: finite element method ([9], [15], [16], [17], [18], [25], [26], [27], [29]), finite difference method ([11], [12], [13], [18], [28]), finite volume method ([14], [20], [21], [23], [24]) were used. In the last decades the