1. AE/ME 339 Computational Fluid Dynamics (CFD) K. M. Isaac
Professor of Aerospace
Engineering
2 January 2019
topic15_cylinder_flow

2. The transformation from (x,y) to polar
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 boundary-
fitted 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 = r1, where the
far field (free-stream) conditions can be applied.
2 January 2019
topic15_cylinder_flow

3. We can also use the transformation s = 1/r. With this tranformation,
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:
2 January 2019
topic15_cylinder_flow

4. These transformations can be checked by applying them to
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
2 January 2019
topic15_cylinder_flow

5. Substituting for the metrics give the following
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Substituting for the metrics give the following
2 January 2019
topic15_cylinder_flow

6. Substituting for the metrics give the following
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Substituting for the metrics give the following
2 January 2019
topic15_cylinder_flow

7. Similarly, get expressions for the other terms
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.
2 January 2019
topic15_cylinder_flow

8. For this problem the transformation s = 1/r can be used to transform
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.
2 January 2019
topic15_cylinder_flow

9. Computational Fluid Dynamics (AE/ME 339) K. M. Isaac MAEEM Dept., UMR
2 January 2019
topic15_cylinder_flow

10. The governing equation can now be written in terms of s as follows
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
2 January 2019
topic15_cylinder_flow

11. The above equation needs to be solved on the rectangular grid shown
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:
2 January 2019
topic15_cylinder_flow

12. Along the line of symmetry q-component of velocity .
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Along the line of symmetry q-component of velocity
2 January 2019
topic15_cylinder_flow

13. Flow over a cylinder is a classical problem
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
2 January 2019
topic15_cylinder_flow

14. Differentiation yields
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
2 January 2019
topic15_cylinder_flow

15. Bernoulli’s equation Recall
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Bernoulli’s equation
Recall
2 January 2019
topic15_cylinder_flow

16. Surface pressure distribution can now be obtained by substituting r=R
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.
2 January 2019
topic15_cylinder_flow