3.14 Lifting Surfaces

Transcription

1 .0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 3.4 Lifting Surfaces 3.4. D Symmetric Streamined Body No separation, even for arge Reynods numbers. stream ine Viscous effects ony in a thin boundary ayer. Sma Drag (ony skin friction). No Lift.

2 3.4. Asymmetric Body (a) Ange of attack α, chord ine α (b) or camber η(x), chord ine mean camber ine (c) or both chord ine amount of camber mean camber ine α ange of attack Lift to and Drag to

3 3.5 Potentia Fow and Kutta Condition From the P-Fow soution for fow past a body we obtain P-Fow soution, infinite veocity at traiing edge. Note that (a) the soution is not unique - we can aways superimpose a circuatory fow without vioating the boundary conditions, and (b) the veocity at the traiing edge. We must therefore, impose the Kutta condition, which states that the fow eaves tangentiay the traiing edge, i.e., the veocity at the traiing edge is finite. To satisfy the Kutta condition we need to add circuation. Circuatory fow ony. Superimposing the P-Fow soution pus circuatory fow, we obtain Figure : P-Fow soution pus circuatory fow. 3

4 3.5. Why Kutta condition? Consider a contro voume as iustrated beow. At t = 0, the foi is at rest (top contro voume). It starts moving impusivey with speed (midde contro voume). At t = 0 +, a starting vortex is created due to fow separation at the traiing edge. As the foi moves, viscous effects streamine the fow at the traiing edge (no separation for ater t), and the starting vortex is eft in the wake (bottom contro voume). t = 0 Γ = 0 + t = 0 Γ S Γ S starting vortex due to separation (a rea fuid effect, no infinite ve of potetia fow) for ater t Γ S Γ S no Γ starting vortex eft in wake Kevin s theorem: After a whie the Γ S dγ =0 Γ = 0 for t 0 ifγ(t = 0)=0 dt in the wake is far behind and we recover Figure. 4

5 3.5. How much Γ S? Just enough so that the Kutta condition is satisfied, so that no separation occurs. For exampe, consider a fat pate of chord and ange of attack α, as shown in the figure beow. chord ength Simpe P-Fow soution Γ= π sin α L = ργ = ρ π sin α L C L = =π sin α πα for sma α ρ }{{} ony for sma α However, notice that as α increases, separation occurs cose to the eading edge. Excessive ange of attack eads to separation at the eading edge. When the ange of attack exceeds a certain vaue (depends on the wing geometry) sta L occurs. The effects of staing on the ift coefficient (C L = ρ span ) are shown in the foowing figure. 5

6 C L This region independent of R, ν used ony to get Kutta condition sta ocation f(r) π sta O(5 o ) α In experiments, C L < πα for 3D foi - finite aspect ratio (finite span). With sharp eading edge, separation/sta to eary. sharp traiing edge round eading edge to forsta to deveop circuation staing 6

7 3.6 Thin Wing, Sma Ange of Attack Assumptions Fow: Steady, P-Fow. Wing: Let y (x), y L (x) denote the upper and ower vertica camber coordinates, respectivey. Aso, et x = /, x = / denote the horizonta coordinates of the eading and traiing edge, respectivey, as shown in the figure beow. y=y (x) For thin wing, at a sma ange of attack it is y y L, << dy dy L, dx dx << The probem is then inear and superposition appies. Let η(x) denote the camber ine t(x) η(x) = (y (x)+ y L (x)), and t(x) denote the haf-thickness t(x) = (y (x) y L (x)). Camber ine t(x) η(x) For inearized theory, i.e. thin wing at sma AoA, the ift on the wing depends ony on the camber ine but not on the wing thickness. Therefore, for the foowing anaysis we approximate the wing by the camber ine ony and ignore the wing thickness. 7

8 Definitions In genera, the ift on the wing is due to the tota circuation Γ around the wing. This tota circuation can be given in terms due to a distribution of circuation γ(x) (nits: [LT ]) inside the wing, i.e., γ (x) / Γ= γ(x)dx / Γ Noting that superposition appies, et the tota potentia Φ for this fow be expressed as the sum of two potentias The fow veocity can by expressed as Φ= x }{{} + φ }{{} Free stream potentia Disturbunce potentia v = Φ =( + u, v) where (u, v) are given by φ = (u, v) and denote the veocity disturbance, due to the presence of the wing. For inearized wing we can assume u v u, v <<, << Consider a fow property q, such as veocity, pressure etc. Then et q = q(x, 0 + ) and q L = q(x, 0 ) denote the vaues of q at the upper and ower wing surfaces, respectivey. 8

9 Lift due to circuation Appying Bernoui equation for steady, inviscid, rotationa fow, aong a streamine from to a point on the wing, we obtain ( ) p p = ρ v {( ) } p p = ρ (u ) + v = ρ(u + v u) u v v p p = ρu( + ) }{{} }{{} }{{} u << << Dropping terms of order u, v << we obtained the inearized Bernoui equation for thin wing at sma AoA p p = ρu Integrating the pressure aong the wing surface, we obtain an expression for the tota ift L on the wing / [( ) ( )] L = (p p )n y ds = p(x, 0 ) p p(x, 0 + ) p dx / / / ( ) ( ) L = p(x, 0 ) p(x, 0 + ) dx = ρ u(x, 0 ) u(x, 0 + ) dx () / / 9

10 To obtain the tota ift on the wing we wi seek an expression for u(x, 0 ± ). Consider a cosed contour on the wing, of negigibe thickness, as shown in the figure beow. γ (x) In this case we have t 0 u ( x,0+ ) u ( x,0 ) From Equations (), and () the tota ift can be expressed as / L = ρ γ(x)dx = ργ / }{{} δ x γ(x)δx = u(x, 0 + ) δx + u(x, 0 )δx γ(x) = u(x, 0 + ) + u(x, 0 ) For sma u/ we can argue that u(x, 0 + ) = u(x, 0 ), and obtain The same resut can be obtained from the Kutta-Joukowski aw (for noninear foi) / δl = ρδγ = ργ(x)δx L = ργ(x)δx = ργ / x γ(x) u(x, 0 ± )= () =Γ δ L = ρ δγ = ργ (x)δ x t 0 δ Γ = γ (x)δ x x δ x 0

11 Moment, with respect to mid-chord, due to circuation y L M x cp x δl(x) = ργ(x)δx δm = xδl(x) = ρxγ(x)δx / M = ρxγ(x)dx / C M = M ρ The center of pressure x cp, can be obtained by M = Lx cp / M xγ(x)dx / x cp = = L / γ(x)dx /

12 3.7 Simpe Cosed-Form Soutions for Theory. Fat pate at ange of attack α, i.e., η = αx. / / γ(x)dx from Linear Linear ifting theory gives γ(x), which can be integrated to give the ift coefficient C L, / L/span = ρ γ(x)dx = = ρ πα / C L = L/span ρ C L = πα ( exact noninear hydrofoi C L =π sin α) the moment coefficient C M, / M/span = ρ xγ(x)dx = = ρ πα 4 / M/span C M = ρ C M = πα and the center of pressure x cp x cp = 4 i.e., at quarter chord

13 . Paraboic camber η = η 0 { ( x ) }, at zero AoA α = 0. Linear ifting theory gives γ(x), which can be integrated to give the ift coefficient C L, L/span = ρ the moment coefficient C M, C L = 4π η 0 / γ(x)dx = =ρ πη 0 /, where η 0 camber ratio M/span = 0 (from symmetry) C M = 0 and the center of pressure x cp x cp =0 3

14 ( ( ) ) x 3. Linear superposition: Both AoA and camber η = αx + η 0. C L = C Lα + C Lη =πα +4π We can aso write the previous reation in a more genera form C L (α) =πα + C } L (α = 0) {{} 4π η 0 η 0 Lift coefficient C L as a function of the ange of attack α and η 0. In practice even if the camber is not paraboic, we sti make use of the previous reations, i.e., C L (α = 0) = 4πη 0 /. Aso note that the ange of attack for any camber is defined as η(/) η( /) α = and η 0 is determined from η, where y y L η = η αx. 4