AEME 339 Computational Fluid Dynamics CFD K M

AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac Professor of Aerospace Engineering 11/1/2021 topic 15_cylinder_flow 1

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Flow Over a Cylinder The transformation from (x, y) to polar coordinates (r, q) will give boundaryfitted coordinates. (Recall r = R (constant) is the cylinder surface. Here the domain is between r = R and r = infinity. Computationally, the outer boundary needs to be finite, say r = r 1, where the far field (free-stream) conditions can be applied. 11/1/2021 topic 15_cylinder_flow 2

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR We can also use the transformation s = 1/r. With this tranformation, s = 1/R represents the cylinder surface and s = 0 represents conditions at infinity. In the new (s, q) system, the computational space lies between s = 0 (far field) and 1/R (surface). First consider the polar coordinates. The relationship between the two systems is as follows: x = r cos(q) y = r sin(q) Transformation metrics: 11/1/2021 topic 15_cylinder_flow 3

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR These transformations can be checked by applying them to Laplace’s equation, which in polar coordinates is given by 11/1/2021 topic 15_cylinder_flow 4

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Substituting for the metrics give the following 11/1/2021 topic 15_cylinder_flow 5

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Substituting for the metrics give the following 11/1/2021 topic 15_cylinder_flow 6

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Similarly, get expressions for the other terms And substitute in Laplace’s equation in polar coordinates to transoform The equations to cartesian coordinates. 11/1/2021 topic 15_cylinder_flow 7

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR For this problem the transformation s = 1/r can be used to transform The original equation into in terms of s. Also since the flow is symmetric about q = 0 line, we need to solve Only one half of the flow domain. 11/1/2021 topic 15_cylinder_flow 8

Computational Fluid Dynamics (AE/ME 339) 11/1/2021 topic 15_cylinder_flow K. M. Isaac MAEEM Dept. , UMR 9

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The governing equation can now be written in terms of s as follows Using the above relations in Laplaces equation gives 11/1/2021 topic 15_cylinder_flow 10

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR The above equation needs to be solved on the rectangular grid shown In the figure. Before starting the solution, the boundary conditions also must be Transformed. At r = R: 11/1/2021 topic 15_cylinder_flow 11

Computational Fluid Dynamics (AE/ME 339) Along the line of symmetry q-component of velocity 11/1/2021 topic 15_cylinder_flow K. M. Isaac MAEEM Dept. , UMR . 12

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Flow over a cylinder is a classical problem Potential flow over a cylinder is equivalent to a doublet in uniform flow See potential flow theory for details. The analytical solution for f is as follows Velocity component at any point is given by 11/1/2021 topic 15_cylinder_flow 13

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Differentiation yields Pressure distribution can be represented in terms of the pressure Coefficient Cp 11/1/2021 topic 15_cylinder_flow 14

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Bernoulli’s equation Recall 11/1/2021 topic 15_cylinder_flow 15

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept. , UMR Surface pressure distribution can now be obtained by substituting r=R Note that the surface pressure distribution is independent of the cylinder radius. 11/1/2021 topic 15_cylinder_flow 16