Fluids Lecture 3 Notes

Transcription

1 Fids Lectre 3 Notes Aerodynamic Forces and oments 2. Center of Pressre 3. Nondimensiona Coefficients Reading: Anderson Aerodynamics Forces and oments Srface force distribtion The fid fowing abot a body exerts a oca force/area (or stress) f on each oint of the body. Its norma and tangentia comonents are the ressre and the shear stress τ. r f r τ ds f force/area distribtion on airfoi R oca ressre and shear stress comonents ( magnitde greaty exaggerated) τ N L R V A restant force, and moment abot ref. oint aternative comonents of restant force The figre above greaty exaggerates the magnitde of the τ stress comonent jst to make it visibe. In tyica aerodynamic sitations, the ressre (or even the reative ressre ) is tyicay greater than τ by at east two orders of magnitde, and so f is very neary erendicar to the srface. Bt the sma τ often significanty contribtes to drag, so it cannot be negected entirey. The stress distribtion f integrated over the srface rodces a restant force R, and aso a moment abot some chosen moment-reference oint. In 2- cases, the sign convention for is ositive nose, as shown in the figre. Force comonents The restant force R has erendicar comonents aong any chosen axes. These axes are arbitrary, bt two articar choices are most sef in ractice. 1

2 Freestream Axes: The R comonents are the drag and the ift L, arae and erendicar to V. Body Axes: The R comonents are the axia force A and norma force N, arae and erendicar to the airfoi chord ine. If one set of comonents is comted, the other set can then be obtained by a sime axis transformation sing the ange of attack. Secificay, L and are obtained from N and A as foows. L = N cos A sin = N sin + A cos Force and moment cacation A cyindrica wing section of chord c and san b has force comonents A and N, and moment. In 2- it s more convenient to work with the nit-san qantities, with the san dimension divided ot. A A/b N N/b /b V y ) s ds τ ) s ds ) τ ) On the er srface, the nit-san force comonents acting on an eementa area of width ds are dn da And on the ower srface they are = ( cos τ sin ) ds = ( sin + τ cos ) ds dn = ( cos τ sin ) ds da = ( sin + τ cos ) ds Integration from the eading edge to the traiing edge oints rodces the tota nit-san forces. N A = = dn + dn da + da 2 x ds τ b c

3 The moment abot the origin (eading edge in this case) is the integra of these forces, weighted by their moment arms x and y, with aroriate signs. = x dn + x dn + y da + y da From the geometry, we have dy ds cos = dx ds sin = dy = dx dx which aows a the above integras to be erformed in x, sing the er and ower shaes of the airfoi y (x) and y (x). Anderson 1.5 has the comete exressions. Simifications In ractice, the shear stress τ has negigibe contribtions to the ift and moment, giving the foowing simified forms. ( ) c dy dy L = cos ( ) dx + sin dx dx dx [ ( ) ( )] c dy dy = x + y x + y dx dx dx A somewhat ess accrate bt sti common simification is to negect the sin term in L, and the dy/dx terms in. L ( ) dx ( ) x dx The shear stress τ cannot be negected when comting the drag on streamine bodies sch as airfois. This is becase for sch bodies the integrated contribtions of toward tend to mosty cance, eaving the sma contribtion of τ qite significant. Center of Pressre efinition The vae of the moment deends on the choice of reference oint. Using the simified form of the integra, the moment ref for an arbitrary reference oint x ref is ref = ( ) (x x ref ) dx = + L x ref This can be ositive, zero, or negative, deending on where x ref is chosen, as istrated in the figre. At one articar reference ocation x c, caed the center of ressre, the moment is defined to be zero. c = + L x c x c = /L 3

4 L L L x c or or < = < The center of ressre asymtotes to + or as the ift tends to zero. This awkward sitation can easiy occr in ractice, so the center of ressre is rarey sed in aerodynamics work. For reasons which wi become aarent when airfoi theory is stdied, it is advantageos to define the standard ocation for the moment reference oint of an airfoi to be at its qarter-chord ocation, or x ref = c/4. The corresonding standard moment is say written withot any sbscrits. c/4 = ( ) (x c/4) dx Aerodynamic Conventions As imied above, the aerodynamicist has the otion of icking any reference oint for the moment. The ift and the moment then reresent the integrated distribtion. Consider two ossibe reresentations: 1. A restant ift L acts at the center of ressre x = x c. The moment abot this oint is zero by definition: c =. The x c ocation moves with ange of attack in a comicated manner. 2. A restant ift L acts at the fixed qarter-chord oint x = c/4. The moment abot this oint is in genera nonzero: c/4. The figre shows how the L,, and x c change with ange of attack for a tyica cambered airfoi. Note that with reresentation 1, the x c ocation moves off the airfoi and tends to + as L aroaches zero. Fixing the moment reference oint, as in reresentation 2, is a simer and referabe aroach. Choosing the qarter-chord ocation for this is eseciay attractive, since then shows itte or no deendence on the ange of attack. This srrising fact wi come from a more detaied airfoi anaysis ater in the corse. 4

5 = 5 = = 5 1. L ( =) x c x c x c 2. L c/4 c/4 c/4 Nondimensiona Coefficients The forces and moment deend on a arge nmber of geometric and fow arameters. It is often advantageos to work with nondimensionaized forces and moment, for which most of these arameter deendencies are scaed ot. For this rose we define the foowing reference arameters: Reference area: S Reference ength: 1 ynamic ressre: q = ρv 2 2 The choices for S and are arbitrary, and deend on the tye of body invoved. For aircraft, traditiona choices are the wing area for S, and the wing chord or wing san for. The nondimensiona force and moment coefficients are then defined as foows: L Lift coefficient: C L q S rag coefficient: C q S oment coefficient: C q S For 2- bodies sch as airfois, the aroriate reference area/san is simy the chord c, and the reference ength is the chord as we. The oca coefficients are then defined as foows. Loca Lift coefficient: c q c Loca rag coefficient: c d q c Loca oment coefficient: c m q c 2 These oca coefficients are defined for each sanwise ocation on a wing, and may vary across the san. In contrast, the C L, C, C are singe nmbers which ay to the whoe wing. 5 L