Fluids Lecture 2 Notes

Transcription

1 Fluids Leture Notes. Airfoil orte Sheet Models. Thi-Airfoil Aalysis Problem Readig: Aderso.,.7 Airfoil orte Sheet Models Surfae orte Sheet Model A aurate meas of represetig the flow about a airfoil i a uiform flow is to plae a vorte sheet o the airfoil surfae. The total veloity (, z), whih is the vetor sum of the freestream veloity ad the vorte-sheet veloity, a be fored parallel to the airfoil surfae by suitably settig the sheet stregth distributio (s). (s) A pael method is ormally used to umerially ompute (s). By usig a suffiiet umber of paels, this result a be made as aurate as eeded. The mai drawbak of suh umerial alulatios is that they give limited isight ito how the flow is iflueed by hages i the agle of attak or the airfoil geometry. Suh isight, whih is importat for effetive aerodyami desig ad egieerig, is muh better provided by simple approimate aalyti solutios. The pael method a still be used for auray whe it s eeded. Sigle orte Sheet Model I order to simplify the problem suffiietly to allow aalyti solutio, we make the followig assumptios ad approimatios: ) The airfoil is assumed to be thi, with small maimum amber ad thikess relative to the hord, ad is assumed to operate at a small agle of attak, α. ) The upper ad lower vorte sheets are superimposed together ito a sigle vorte sheet = u + l, whih is plaed o the ais rather tha o the urved mea amber lie Z = (Z u + Z l )/. z z z () u Z u () Z() Z() () l Z () l () () 3) The flow-tagey oditio ˆ = is applied o the -ais at z =, rather tha o the amber lie at z = Z. But the ormal vetor ˆ is ormal to the atual amber lie shape, as show i the figure. ) Small-agle approimatios are assumed. The freestream veloity is the writte as follows. [ ] ( ) = (os α)î + (si α)ˆ k î + αˆ k

2 O the -ais where the vorte sheet lies, the sheet s veloity w(), whih is stritly i the z-diretio, is give by itegratig all the otributios alog the sheet. (ξ) w() = π( ξ) Addig this to the freestream veloity the gives the total veloity. ( ) ˆ (ξ) (, ) = + wk î + α k ˆ () π( ξ) The ormal uit vetor is obtaied from the slope of the amberlie shape Z(). dz () = î + k d ˆ, ˆ = z () dz/d α w ξ To fore the total veloity to be parallel to the amberlie, we ow apply the flow tagey oditio ˆ =. Performig this dot produt betwee () ad (), ad removig the ueessary fator / gives the fudametal equatio of thi airfoil theory. ( ) dz (ξ) α = (for < < ) (3) d π( ξ) Thi-Airfoil Aalysis Problem Flow tagey impositio For a give amberlie shape Z() ad agle of attak α, we ow seek to determie the vorte stregth distributio () suh that the fudametal equatio (3) is satisfied at every loatio. As show i the figure, this will result i the total veloity at every -loatio to be approimately parallel to the loal amberlie, produig a physially-orret flow about this amberlie. The thier the airfoil, the loser the amberlie is to the -ais where the flow tagey is atually imposed, ad the more aurate the approimatio beomes. Compared to typial airfoils, the height of the amberlie i the figure is eaggerated severalfold for the sake of illustratio.. =

3 Coordiate trasformatio To eable solutio of equatio (3), it is eessary to first perform a trigoometri substitutio for the oordiate, ad the dummy variable of itegratio ξ. = ( os θ o ) ξ = ( os θ) = si θ dθ θ = As show i the figure, θ rus from at the leadig edge, to π at the trailig edge. Sie ad θ are iterhagable, futios of a ow be treated as futios of θ. Equatio (3) the beomes ( ) π (θ) si θ dθ dz = α (for < θ o < π) () π os θ os θ o d where the kow amberlie slope dz/d is ow osidered a futio of θ o. This is a itegral equatio whih must be solved for the ukow (θ) distributio, with the additioal requiremet that it satisfy the Kutta oditio at the trailig edge poit, (π) = Symmetri airfoil ase I pratie, the amberlie slope dz/d a have ay arbitrary distributio alog the hord. For simpliity, we will first osider a symmetri airfoil. This has a flat amberlie, with Z = ad dz/d =. Equatio () the simplifies to θ ξ d θ / θ ο θ = π π (θ) si θ dθ π os θ os θ o = α (5) Solutio of this equatio is still formidable, ad is beyod sope here. Let us simply state that the solutio is + os θ (θ) = α or () = α si θ The shape of these distributios is show i the figure below. α 3 π/ θ π..5 /. 3

4 Note that at the trailig edge, = as required by the Kutta oditio, ad that at the leadig edge. The latter is of ourse ot physial, although the sigularity is weak (itegrable), ad the itegrated results for l ad m are i fat valid. The load distributio p l p u is obtaied usig the Beroulli equatio, together with the tagetial veloity jump properties aross the vorte sheet. ( ) ( ) p l p u = p o ρ l p o ρ u = ρ ( u + l ) ( u l ) = ρ The lift/spa o a elemet of the sheet is dl = (p l p u ) = ρ ad the total lift/spa is the obtaied by itegratig this load distributio. L = ρ (ξ) The itegral is also see to be the overall irulatio, makig this lift result osistet with the Kutta-Joukowsky Theorem. Γ = (ξ), L = ρ Γ z dl dm ξ l u The atual itegratio of the loadig or sheet stregth is most easily performed i the trigoometri oordiate θ. By diret substitutio, we have The lift/spa is the π Γ = (θ) si θ dθ = α ( + os θ) dθ = πα ad the orrespodig lift oeffiiet is π L = ρ Γ = πα ρ L l = ρ = This is a very importat result, showig that the lift is proportioal to the agle of attak, with a lift slope of d l = π dα These results very losely math the results of more omple pael method alulatios, as well as eperimetal data. π α

5 The pithig momet/spa o the sheet elemet, take about the leadig edge, is obtaied by weightig the lift by the momet arm ξ. dm The overall momet/spa is the LE = ξ dl = ρ ξ M = ρ ξ LE We a agai most easily itegrate this i the trigoometri oordiate. π π M = α ρ ( + os θ)( os θ) dθ = α ρ ( os θ) dθ = πα ρ LE The momet oeffiiet about the leadig edge poit is M LE πα l m,le = ρ = = ad the equivalet momet oeffiiet about the stadard quarter-hord poit at / = / is m,/ = m,le + l = This very importat result shows that a symmetri airfoil has zero momet about the quarterhord poit, for ay agle of attak. 5