1. 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. 2 GOVERNING PDE

3. 3 R-K SHOOTING METHOD: GUESS #1 0 <  < 10 d 2 f/d  2 (0)=1 No more change with  f f’ f’’ Notice that f’ does not go to 1 Guess again

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

5. 5 R-K SHOOTING METHOD: GUESS #3 0 <  < 5 (just to make plots easier to see) d 2 f/d  2 (0)=0.33 Notice that f’ is getting very close to 1 What level of accuracy is required? Guess again f f’ f’’

6. 6 R-K SHOOTING METHOD: GUESS #4 0 <  < 5 (just to make plots easier to see) d 2 f/d  2 (0)=0.33206 Notice that f’ is getting very close to 1 What level of accuracy is required? Close enough

7. 7 R-K SHOOTING METHOD: GUESS #4 0 <  < 5 (just to make plots easier to see) d 2 f/d  2 (0)=0.33206

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

9. 9 SUMMARY

10. 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 1.Low velocity, where flow between each pair of plates is a plane Poiseuille flow 2.High velocity, where a boundary layer develops on each plate surface as though it were uninfluenced by the adjacent plates Calculate Re number Vh/ at which pressure drop in (1) and (2) are equal if L=10h