Fluid Flow Around and Heat Transfer From an Infinite Circular Cylinder W. A. Khan J. R. Culham M. M. Yovanovich Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 In this study, an integral approach of the boundary layer analysis is employed to investigate fluid flow around and heat transfer from an infinite circular cylinder. The Von Karman–Pohlhausen method is used to solve momentum integral equation and the energy inte- gral equation is solved for both isothermal and isoflux boundary conditions. A fourth-order velocity profile in the hydrodynamic boundary layer and a third-order temperature profile in the ther- mal boundary layer are used to solve both integral equations. Closed form expressions are obtained for the drag and the aver- age heat transfer coefficients which can be used for a wide range of Reynolds and Prandtl numbers. The results for both drag and heat transfer coefficients are in good agreement with experimental/numerical data for a circular cylinder. DOI: 10.1115/1.1924629 Introduction The equations describing fluid flow and heat transfer in forced convection are complicated by being nonlinear. These nonlineari- ties arise from the inertial and convective terms in the momentum and energy equations, respectively. From a mathematical point of view, the presence of the pressure gradient term in the momentum equation for forced convection further complicates the problem. The energy equation depends on the velocity through the convec- tive terms and, as a result, is coupled with the momentum equa- tion. Because of these mathematical difficulties, the theoretical in- vestigations about fluid flow around and heat transfer from circu- lar cylinders have mainly centered upon asymptotic solutions. These solutions are well documented in the open literature and are valid for very large 2  10 5  and small 1 Reynolds num- bers. However, no theoretical investigation could be found that can be used to determine drag coefficients and average heat trans- fer from cylinders for low to moderate Reynolds numbers 1–2  10 5  as well as for large Prandtl numbers 0.71. For this range of Reynolds numbers and for selected fluids, there has been heavy reliance on both experiments and numerical methods. These approaches are not only expensive and time consuming but their results are applicable over a fixed range of conditions. Unfortunately, many situations arise where solutions are re- quired for low to moderate Reynolds numbers and for fluids hav- ing Pr  0.71. Such solutions are of particular interest to thermal engineers involved with cylinders and fluids other than air or wa- ter. In this study a circular cylinder is considered in cross flow to investigate the fluid flow and heat transfer from a cylinder for a wide range of Reynolds and Prandtl numbers. A review of existing literature reveals that most of the studies related to a single isolated cylinder are experimental or numerical. They are applicable over a fixed range of conditions. Furthermore, no analytical study gives a closed form solution for the fluid flow and heat transfer from a circular cylinder for a wide range of Reynolds and Prandtl numbers. At most, they provide a solution at the front stagnation point or a solution of boundary layer equa- tions for very low Reynolds numbers. In this study, a closed form solution is obtained for the drag coefficients and Nusselt number, which can be used for a wide range of parameters. For this pur- pose, the Von Karman–Pohlhausen method is used, which was first introduced by Pohlhausen 1 at the suggestion of Von Kar- man 2 and then modified by Walz 3 and Holstein and Bohlen 4. Schlichting 5 has explained and applied this method to the general problem of a two-dimensional boundary layer with pres- sure gradient. He obtained general solutions for the velocity pro- files and the thermal boundary layers and compared them with the exact solution of a flat plate at zero incidence. Analysis Consider a uniform flow of a Newtonian fluid past a fixed cir- cular cylinder of diameter D, with vanishing circulation around it, as shown in Fig. 1. The approaching velocity of the fluid is U app and the ambient temperature is assumed to be T a . The surface temperature of the wall is T w T a  in the case of the isothermal cylinder and the heat flux is q for the isoflux boundary condition. The flow is assumed to be laminar, steady, and two-dimensional. The potential flow velocity just outside the boundary layer is de- noted by Us. Using order-of-magnitude analysis, the reduced equations of continuity, momentum and energy in the curvilinear system of coordinates Fig. 1 for an incompressible fluid can be written as: Continuity: u s + v  =0 1 s-Momentum: u u s + v u  =- 1  dP ds +   2 u  2 2 -Momentum: dP d =0 3 Bernoulli equation: - 1  dP ds = Us dUs ds 4 Energy: u T s + v T  =   2 T  2 5 Hydrodynamic Boundary Conditions. At the cylinder sur- face, i.e., at  =0 u = 0 and  2 u  2 = 1  P s 6 At the edge of the boundary layer, i.e., at  = s u = Us, u  = 0 and  2 u  2 =0 7 Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 25, 2004. Final manuscript received October 25, 2004. Review conducted by: N. K. Anand. Journal of Heat Transfer JULY 2005, Vol. 127 / 785 Copyright © 2005 by ASME