MAE 5130: VISCOUS FLOWS

1. MAE 5130: VISCOUS FLOWS Similarity Solution for Steady 2D boundary layer flow November 4, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. GOVERNING PDE

3. R-K SHOOTING METHOD: GUESS #1 • 0 < h < 10 • d2f/dh2(0)=1 f No more change with h • Notice that f’ does not go to 1 • Guess again f’ f’’

4. R-K SHOOTING METHOD: GUESS #2 • 0 < h < 5 (just to make plots easier to see) • d2f/dh2(0)=0.5 f • Notice that f’ does not go to 1, but it is getting closer than previous guess • Guess again f’ f’’

5. R-K SHOOTING METHOD: GUESS #3 • 0 < h < 5 (just to make plots easier to see) • d2f/dh2(0)=0.33 f • Notice that f’ is getting very close to 1 • What level of accuracy is required? • Guess again f’ f’’

6. R-K SHOOTING METHOD: GUESS #4 • 0 < h < 5 (just to make plots easier to see) • d2f/dh2(0)=0.33206 • Notice that f’ is getting very close to 1 • What level of accuracy is required? • Close enough

7. R-K SHOOTING METHOD: GUESS #4 • 0 < h < 5 (just to make plots easier to see) • d2f/dh2(0)=0.33206

8. SUMMARY hff’f’’ hf f’f’’ u=99% of U∞ at h ~ 4.9 f’(h)=u/U∞=0.99 at h ~ 4.9

9. SUMMARY

10. EXAMPLE: FLAME ARRESTOR • A flame arrestor in the intake duct of a gasoline engine prevents the propagation of a flame should there be fuel flames in the intake air • Consider a flame arrestor made consisting of a series of thin parallel plates aligned with the intake flow, with spacing h and plate length L • Assuming the flow is incompressible, derive expressions for the pressure drop pin-pout between the inflow and outflow streams for the limiting cases of • Low velocity, where flow between each pair of plates is a plane Poiseuille flow • High velocity, where a boundary layer develops on each plate surface as though it were uninfluenced by the adjacent plates • Calculate Re number Vh/n at which pressure drop in (1) and (2) are equal if L=10h