EGN 3353C Fluid Mechanics

Transcription

1 eture 5 Bukingham PI Theorem Reall dynami imilarity beteen a model and a rototye require that all dimenionle variable mut math. Ho do e determine the '? Ue the method of reeating variable 6 te Ste : Parameter are dimenional, nondimenional variable, and dimenional ontant. n inlude the deendent variable. Make ure that any lited indeendent arameter i indeed indeendent of the other, i.e., it annot be exreed in term of them. (e.g., don t inlude radiu r and area A = π r, ine r and A are not indeendent.) Ste 3: A a firt gue, et redution j equal to the number of rimary dimenion rereented in the roblem. The exeted number of ome from the Bukingham Pi theorem: k = n j. ' Ste 4: Sine the reeating arameter have the otential to aear in eah, be ure to hooe ommon arameter (ee Table 7 3). = = mt.... Can often be done by inetion. Ste 5: [ ] k Ste 6: = f,,,..., k 3 4 deendent indeendent arameter arameter

2 Examle Preure in a bubble.. it the arameter in the roblem: P f ( R σ ) Δ =. n = 3,. it the rimary dimenion of eah. [ P] [ R] [ σ ] Δ = m t, =, = mt 3. Gue redution j equal to the number of rimary dimenion = 3. So k = n j = 0! (not valid ine there mut be at leat ) Redue to j = o k =. 4. Chooe j = reeating arameter. In thi ae, hooe R, σ ine they are the indeendent arameter a a =, [ ] mt [ P][ R] [ σ ] m t [ ] a a 5. ΔPR σ or a + a a a mt = = = mt a a Δ = = m t equating exonent of mt,,... to 0 =, a = = Δ PR σ =Δ PR σ = f nothing = ont theory or ingle exeriment ho that 6. ( ) the ont = 4.

3 Examle ift on a -D (unet) ing of an b. Here, i the eed of ound. In thi ae, hooe for length ale, V for veloity ale (involve, t ) and ρ ine thi fluid roerty involve ma (better hoie than μ ). it the arameter in the roblem: (,,, α, ρμ,, ) F = f V b. n = 8. it the rimary dimenion of eah. [ F] [ V ] [ ] [ b] [ α],[ ρ] m,[ μ] m t,[ ] = mt, = t, =, =, = = = = t 3 3. Gue redution j equal to the number of rimary dimenion = 3. So k = n j = 5 4. Chooe j = 3 reeating arameter. a a a3 5. = FV ρ = b = aet ratio, 3 = α, 4 = ρv μ = Reynold number, ( ) 5 = V = Mah number 6. C f ( b, α,re, M) thee = F ρv C = F ρv lift oef f. =. Dynami imilarity mean all a model. ' mut be equal beteen a full-ale rototye and a. Often aumed that the model i -D o that ing aet ratio i not imortant ( F b. Eay to math αm = α. b).. Need to math Reynold number and Mah number. A

4 Reynold number and Mah number imilarity in an air ind tunnel ith an ideal ga ρ V m m m μ ρ V V V Vm Vm = and = = = here γ = ratio of eifi heat, R i the ideal ga μ γ R T γ R T m ontant and T i the ga tati temerature. If T Tm and P Pm m m m m, then fluid roertie are aroximately the ame: ρ = ρm, = m and μ = μm o Reynold number imilarity redue to V = V m m o Mah number imilarity redue to V = V o Can only have dynami imilarity if e tet a full-ale model ( m) In ratie, if flo i inomreible, e math Re (ine M ). m =!!!! If flo i omreible (tranoni airlane M 0.8), e math M and often try to tet at high Reynold number() and either find F i indeendent of Re (at high enough Re) OR extraolate (dangerou) OR erform numerial imulation (diffiult). Another otion i to hange the tet ga omoition or reure or temerature (e.g., ryogeni) to more loe math Re and M very exenive! o NASA angley National Tranoni Faility (NTF) htt://te.lar.naa.gov/failitie/aerodynami/national.fm

5 Examle Steady inomreible ie flo. In inomreible teady ie flo, the veloity rofile eventually beome fully-develoed, meaning that the veloity rofile doen t hange ith x (and, hene, neither doe τ ). A CV analyi ho: Ma onervation: m = ont; V = Vavg = ont x out in or x-mom: F = m ( V V ) = 0 ( ) ( ) ( π ) P x P x+δx D 4 τ πdδ x= 0 area In the limit a Δx 0, e obtain ( ) ( ) P x P x+δ x 4τ lim = or Δx 0 Δx D dp dx 4τ erimeter = = ont P mut dereae in a D linear fahion v. x.

6 Dimenional Analyi for τ. it the arameter in the roblem: τ f ( V, D, ε, ρ, μ). it the rimary dimenion of eah. =. n = 6 [ τ ] [ ] [ ] [ ε] 3 [ ρ] = m,[ μ] = m t = mt, V = t, D =, =, 3. Gue redution j equal to the number of rimary dimenion = 3. So k = n j = Chooe j = 3 reeating arameter. In thi ae, hooe D for length ale, V for veloity ale (involve, t ) and ρ ine thi fluid roerty involve ma (better hoie than μ ) 3 5. = τ D a V a ρ a = τ ρv = f = τ ρv = Dary frition fator 8 = ρvd μ = Reynold number, 3 = ε D = roughne ratio 6. 8 τ ρ ( ε,re) V = f D. Thi i valid for laminar and turbulent ier flo and i the ubjet of Chater 8.