1. topic13_grid_generation
AE/ME 339
Computational Fluid
Dynamics (CFD)
K. M. Isaac
February 16, 2019
topic13_grid_generation

2. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Grid Generation
February 16, 2019
topic13_grid_generation

3. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Algebraic Methods
Known functions are used to map irregular physical domain into
rectangular computational domains.
February 16, 2019
topic13_grid_generation

4. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Example: Grid stretching
may be necessary for some
problems such as flow with
boundary layers.
Let us consider the trans-
formation:
Inverse transformation
February 16, 2019
topic13_grid_generation

5. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
The following derivatives are used in the transformation
February 16, 2019
topic13_grid_generation

6. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
The following relation (Eq. 5.52) hold between increments Dy and Dh
February 16, 2019
topic13_grid_generation

7. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Therefore as h increases, Dy increases exponentially.
Thus we can choose Dh constant and still have an exponential stretching
of the grid in the y-direction.
February 16, 2019
topic13_grid_generation

8. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Eq. (5.57) is the continuity equation in the computational domain.
Thus we have transformed the continuity equation from the physical
space to the computational space.
The metrics carry the specifics of a particular transformation.
February 16, 2019
topic13_grid_generation

9. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Boundary Fitted Coordinate System (5.7)
Here we consider the flow through a divergent duct as given in Figure 5.6
(next slide). de is the curved upper wall and fg is the centerline.
Let ys = f(x) be the function that represents the upper wall.
The following transformation will give rise to a rectangular grid.
x = x (5.65)
h = y/ymax (5.66)
To test this choose ys = 1.5x and let x vary from 1 to 5.
At x = 1, x = 1, hmax = 1, and x = 5, x = 5, hmax = 1.
Thus the irregular domain is transformed into into a rectangular domain.
February 16, 2019
topic13_grid_generation

10. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Case where the
Nozzle wall is curved
February 16, 2019
topic13_grid_generation

11. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Example:
February 16, 2019
topic13_grid_generation

12. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
The above formulation is analytic
Could also be obtained using central differencing
if analytic expressions are not available
February 16, 2019
topic13_grid_generation

13. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Divide the domain into 4 segments each in the x and h directions
Therefore, Dx = 0.25 and Dh = 0.25
Consider a point in the domain where we have x = 1.75, h = 0.75
Let us calculate h_x analytically and numerically.
At this point:
February 16, 2019
topic13_grid_generation

14. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Numerical calculation using central differencing.
First calculate y at h = 0.5 and 1.0
February 16, 2019
topic13_grid_generation

15. topic13_grid_generation
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
MAEEM Dept., UMR
Numerically calculate the derivatives using CD
February 16, 2019
topic13_grid_generation

16. topic13_grid_generation
Program
Completed
University of Missouri-Rolla
Copyright 2002 Curators of University of Missouri
February 16, 2019
topic13_grid_generation