Peter Vorobieff From ideal fluid flow to understanding lift and drag on wings

Transcription

1 Pete Voobieff Fom ideal fluid flow to undestanding lift and dag on wings These notes ae mostly based on the way the subjects of inteest ae pesented in Fundamental Mechanics of Fluids by I.G. Cuie, Pofesso Emeitus of Mechanical Engineeing at the Univesity of Toonto, to whom I owe a debt of gatitude. The eade would benefit fom some pio knowledge of diffeential equations and complex veiable theoy; no othe specialized skills ae equied. The notes ae not a eplacement fo the lectue mateials, but athe a supplement to them. It is also inevitable that some eades will likely find some topics hee explained in too much o too little detail. Thee is also some delibeate ovelap between these notes and those fom Pofesso Nitzche's class, so that the naative heein can be used both in combination with them o as a standalone document. A few emaks must be made on notation. In some of the equations hee, vecto ans scala quantities ae used togethe. Vecto quantities compised of two o thee components, depending on the dimensionality of the poblem ae epesented by bold lettes. 1. Intoduction of ideal-fluid flow What is efeed to as ideal fluid hee is to some extent the same substance that the scientists in the XVIIIth centuy consideed as just fluid - a woefully impecise, yet useful mathematical appoximation of the actual liquids and gases that flow in ives, pou out of bottles, fill ou lungs, and so on. The same appoximation also to an extent woks fo plasma. The main assumptions that diffeentiate the ideal fluid fom the eal thing ae that ideal fluid is inviscid and incompessible. Inviscid means that viscous dissipation in the medium is negligible. That assumption also leads to the possibility of solving a educed system of consevation equations, whee mass and momentum consevation laws uncouple fom the enegy equation. Incompessible means that the density ρ mass-tovolume atio emains constant any infinitesimal volume that moves with the flow. An impotant subcase of incompessible flow we will mostly deal with is ρ = const which means that density is not a vaiable we have to solve fo, although it may be possible to have an incompessible flow whee density is not constant fo example, a stably statified shea flow. In some cases, these assumptions ae petty easonable when dealing with eal-wold poblems. It is also notewothy that ideal fluid can be elegantly descibed and decently undestood mathematically, which, Leohnad Eule unfotunately, cannot be said fo the eal fluid. The Potait by Johann Geog Bucke 1756 govening equations fo ideal fluid ae quite appopiately To those who ask what the infinitely small known as Eule's equations, because Leonhad Eule quantity in mathematics is, we answe that it deived them and published the wok in 1757, when he is actually zeo. Hence thee ae not so woked at the Belin Academy, on a 25-yea beak fom many mysteies hidden in this concept as they ae usually believed to be.

2 the Russian Impeial Academy of Sciences. These equations in vecto fom ae: u=0 u u u= ρ p+ f t The fist one is efeed to as the continuity equation it follows fom the mass consevation law based on the assumption of incompessibility, the second is the momentum equation. Hee u is the velocity vecto, is the diffeential opeato known as del o nabla, epesents a scala poduct of two vectos, t is time, ρ is fluid density, p is pessue, and f epesents the foce pe mass. Both equations 1.1 can be deived fom geneal consevation consideations fo momentum and enegy fo a small fluid volume in Catesian coodinates. The physical meaning of the mass consevation equation is that the volume containing the same amount of fluid molecules emains the same. The momentum equation stictly speaking, it's one vecto equation o thee scala ones elates the change in momentum of the fluid volume as a combination of tempoal / t and convective u deivatives to the wok of pessue and mass foces ight side of equation. Hee we will not bothe with a fomal deivation. To go fom vecto notation in 1.1 to components, we must examine the components of individual vectos, taking into account that we have both space- and time-dependent vecto quantities fields and vecto diffeential opeatos. Conside, fo example, the quantity u, which epesents a diffeential opeato acting upon a field. If the opeato is applied to a function f defined on a one-dimensional domain x, it simply poduces its deivative as defined in calculus fo a function of a single vaiable: f = df/dx. Now conside the thee-dimensional case, whee the velocity u is a vecto compised of thee components: u x, y, z, t u=u, v, w= v x, y, z,t w x, y, z, t By analogy with the one-dimensional case x, the thee-dimensional x,y,z fom of the opeato becomes =,,, x y z whee the odinay deivative symbol d is eplaced with the patial deivative symbol, to denote dealing with multiple vaiables. Finally, to obtain the component fom fom the vecto fom, we must fomally apply the scala poduct to u, using the same ules we would to obtain the scala poduct of two odinay vectos: u v w u=,, u, v, w= + + x y z x y z The quantity u is also known as the divegence of the velocity field notation often used is fo divegence is div u. In incompessible flow, velocity divegence is zeo. Fo flow past obstacles, we must conside a bounday condition on a wall. If the diection nomal to the wall is n, and the wall is moving with velocity U, then fo the fluid nea the wall not to flow into the wall if it could, that wouldn't be much of a wall, would it?, we must equie that u n=u n Moeove, if the wall is stationay, the bounday condition becomes simply u n = 0. This is what is efeed to as the fee-slip bounday condition. In eal life, flow is slowed nea obstacles, [

3 howeve, it is a manifestation of the same viscous dissipation that we chose to neglect when we decidied that the flow is inviscid. A special case of ideal flow is potential flow which is chaacteized by the existence of a function ϕ velocity potential such that the velocity field components u,v,w can be epesented as φ φ φ u=, v=, w=, x y z o, in vecto notation, u = ϕ. In this case, the continuity equation becomes 2 φ=0, and it with appopiate bounday conditions can be solved fo ϕ alone, without the momentum equation. Afte the solution fo ϕ is obtained, the velocity components must be plugged into the momentum equation to solve fo pessue p. 2. Theoy of two-dimensional potential flows. Fo the time being, let us conside two-dimensional flow. In eality, hydodynamic phenomena default to being thee-dimensional 3D any flow-elated function of inteest depends on spatial coodinates x,y,z, although physically inteesting flows whee the z-dependence is weak o negligible can be found. So two-dimensional 2D flow is a useful appoximation but one must always be awae that it is an appoximation, and question the limits of its applicability to any eal flow of inteest. With these caveats, let us assume that, z x y With the z-deivative neglected, the continuity equation fom 1.1 becomes u v + =0 x y Let the flow be steady / t = 0, the flow patten is time-independent. Futhe, let us assume that a velocity potential ϕ exists, so that φ φ u=, v=, x y Along with the potential ϕ, let us constuct a velocity steamfunction ψ such that ψ ψ u=, v= 2.1 y x Fo any ideal 2D flow, defining the velocity field also makes it possible to define the steamfunction. Conside two points in the flow sepaated by a small distance δ. Let q be the volume flow ate assuming that ou 2D plana flow has unit depth in the out-of-x-y-plane diection in the diection nomal to δ. Let δψ = qδ. Then let δ 0 and eaange the tems: dψ q= 2.2 d Now conside flow in a 2D plane with Catesian coodinates x,y. Let us assume that q is positive if, fo an obseve looking along an abitay axis in the diection of the flow, the flow appeas cossing the axis fom left to ight. Selecting the x- and y-axis as the diections of obsevation then would poduce 2.1. Also note the method of constuction elating the steamfunction to the flow ate we will use this featue late. One notable featue of the steamfunction is that the 2D flow defined via its steamfunction satisfies the continuity equation fom 1.1 by constuction:

4 2 ψ 2 ψ =0 x y y x Fo potential flow, this also means that the flow equations fo the velocity field defined via 2.1 ae satisfied automatically. Now let us define an impotant quantity based on the velocity field: voticity, which quantifies the about of swiling o sheaing motion in the flow. In the pevious section, we defined divegence div u = u using ules fo a scala poduct between the vectos and u. Similaly, one can fomally constuct a vecto poduct using the same quantities: i j k ω= u=det 2.3 x y z u v w Hee i,j,k is the tiad of unit vectos in the x,y,z diections coespondingly. Fo the time being, let us conside the 2D case, when w = 0 and / z = 0. In that case, only one component of the vecto poduct in the z-diection is non-zeo: v u ω=k =k ω x y Next let us conside iotational 2D flow: ω = 0. Using this condition of iotationality togethe with the definition of the steamfunction, we obtain: ψ v u ψ 0=ω= = = 2 ψ x y x x y y In othe wods, the steamfunction of the iotational flow also satisfies the Laplace equation. Isocontous of the steamfunction steamlines have aleady been intoduced in the peceding section of this couse, moeove, using the same easoning as we used to pove the steamfunction's existence, we can also pove that diffeence in the value of ψ between two steamlines equals the volume of fluid flowing between them. Additionally, the families of steamlines ψ = const and of potential lines ϕ = const ae othogonal at evey point in the flow. [ ] 3. Complex vaiable epesentation of 2D ideal flow. Let us ecall the Cauchy-Riemann condition fo a function F of complex vaiable z = x+iy to be holomophic1. If we wite this function as a sum of its eal and imaginay pats in Catesian coodinates Fz = ϕ x,y + i ψ x,y, 3.1 the Cauchy-Riemann condition necessay fo F to be holomophic is φ ψ φ ψ = ; = x y y x If we use potential ϕ as defined in Section 1 and steamfunction ψ as defined in Section 2 to constuct F in 3.1, the Cauchy-Riemann condition will be satisfied automatically and, fo diffeentiable ϕ and ψ, F will be a holomophic complex vaiable function that defines the flow by defining the potential, the steamfunction, satisfying the Laplace equation fo ϕ and ψ automatically, and being diectly elated to the flow velocity components u,v: 1 Holomophic - diffeentiable in a neighbohood of evey point within its domain. Fom hee on, we will assume that the eade is familia with the fundamentals of complex-vaiable theoy.

5 Desciption and schematic of the flow Complex potential, velocity, etc. Unifom flow at angle α to x-axis F z =C e i α z y w z = df =C e i α =C cos α ic sin α dz u=c cos α, v=c sin α α x F z =C log z=c log e i θ=c log +iθ C eal, positive φ=c log, ψ=c θ df C C i θ w z = = = e dz z C u =, u θ=0 Souce stength dischage ate Point souce at oigin y x 2π 2π m= 0 u d θ= 0 C d θ=2 π C iθ Point sink at oigin F z =C log z=c log e =C log +iθ C eal, negative φ=c log, ψ=c θ df C C i θ w z = = = e dz z C u =, u θ=0 y x Figue 1. Some common examples of ideal 2D flow and thei epesentations with complex potential theoy.

6 Desciption and schematic of the flow Complex potential, velocity, etc. Point votex at oigin F z = ic log z= ic log e i θ = ic log +C θ C - eal, positive φ=c θ, ψ= C log df C C w z = = i = i e i θ dz z C u =0, uθ = y x z µ eal, positive doublet stength w z = 2 = 2 e i θ cos θ i sin θ z u = 2 cos θ u θ= 2 sin θ Doublet at oigin F z = y x Figue 1 continued. Some common examples of ideal 2D flow and thei epesentations with complex potential theoy. w = df/dz = u - iv Seveal useful flow fields can be constucted using some of the most common holomophic functions Fig. 1. While the constuction of the flow fields fo unifom flow, souce, and sink is faily obvious, the point votex and the doublet deseve some discussion. In the discussion that follows, we will use both the Catesian x,y and pola,θ coodinates in the plane of the complex vaiable z. Fist, in the case of the point votex, at the cente of otation oigin the complex potential eaches a singulaity, and so does the complex velocity with its adial component

7 undefined, and the tangential component going to infinity. Second, let us define ciculation of the votex along a closed contou L: Γ= u d l L Conside L as a cicle with adius and cente at the oigin to make integation simple. Then 2π Γ= u d l= 0 u θ d θ=2 π C L Note that fo all closed contous including the oigin ciculation Γ will emain the same. Moeove, fo any closed contou L' not including the oigin, Γ L ' = u d l 0 L' Now let us discuss the complex potential Fz = µ/z that poduces the doublet. Any steamline ψ = ψ0 can be poven to be a cicle of adius µ/2ψ0 and cente at x = 0, y = -µ/2ψ0. The fomula fo this potential can be deived by consideing a souce and a sink of equal stength at a small distance fom each othe, and letting this distance go to zeo in a cetain way. Thee ae othe useful flow pattens descibed by complex vaiables. Moeove, combinations of complex potentials e.g., a sum of two potentials by constuctions poduce moe complex potentials, some of which can also be useful. Similaly, tanslation of a complex potential likewise poduces a potential, but any singulaity the oiginal potential has will move accodingly. Take, fo example, a point votex with ciculation Γ at the oigin, whose complex potential is Fz = -iγ/2π log z. Based on this expession, the potential fo a point votex at z = z0 would be Fz = -iγ/2π log z. With the complex potentials at ou disposal, we can now exploe constuction some potentials of geate complexity, ultimately leading us to the theoy that allows to pedict pefomance of lifting sufaces. 4. Ideal 2D flow past a cylinde. Conside the sum of two potentials fom the pevious section Fig. 1 unifom flow α = 0 and flow past a doublet: F z =Uz+ z The potential and steamfunction of the combined flow can be obtained by finding the eal and imaginay pats of Fz: F z =Uei θ + i θ = U+ cos θ+i U sin θ e Thus the steamfunction is ψ= U sin θ Conside the steamline ψ = 0. It is easy to see that on this steamline, we must have U - µ/ = 0, thus the steamline ψ = 0 is simply a cicle of adius a =µ/u1/2. We can use this expession fo a to ewite the complex potential as a2 F z =U z + z We can also conside two limit cases hee. As z, the tem epesenting the unifom flow dominates, thus we have unifom flow at infinity. As z 0, the tem epesenting the doublet becomes dominant, thus we have a doublet at the oigin. Let us also emembe that ψ = 0 is a

8 steamline. The combined flow patten is shown in Fig. 2. One featue of ideal flow as defined in Section 1 that we can take to ou advantage is that the bounday condition on a solid suface is fee-slip velocity is paallel to the suface. Thus we can select any steamline as the solid-body bounday. Fo example, if we select the steamline ψ = 0 as the body bounday, then the flow field outside the cicle = a in Fig. 2 will epesent ideal 2D flow past a cylinde of adius a. Figue 2. Ideal flow past a cylinde. The flow field in Fig. 2 is chaacteized by a singulaity due to the doublet at the oigin. Let us also pay some attention to the patten of steamlines. Almost eveywhee in the field, thee is only one steamline going though a given point and a unique velocity. One exception to this ule is the singulaity itself but it would not be a pat of ou flow past the cylinde, as it would be inside of it. Two othe exceptions ae maked in the figue as fowad and ea stagnation points. At stagnation points in the flow, velocity dops to zeo, and steamline diection may be non-unique. Futhe, we can add ciculation to the flow by adding a point votex at the oigin to the complex potential: a2 iγ z 4.1 F z =U z + + log z 2π a The z/a scaling esults fom adding a constant to Fz to keep ψ = 0 at = a. The coesponding complex velocity is df a2 iγ 1 w= =U dz 2π z z In pola coodinates, w z =u iv= u iu θ e i θ Fo the velocity components in pola coodinates, we thus have

9 2 2 a a Γ cos θ, uθ= U 1+ 2 sin θ 2 2π Now conside the velocity on the cylinde suface, = a. Fo this case, Γ u =0, uθ = 2 U sin θ 2πa How does the addition of ciculation G affect the location of the stagnation points? Fo stagnation points on the cylinde to exist on the cylinde suface, the tangential velocity component at = a,θ must be zeo as well, so sin θ s= Γ 4πU a Depending on Γ, a and U, this equation can have two, one, o zeo solutions. In the latte case, while thee ae no stagnation points on the cylinde, they may exist elsewhee in the flow. Let us assume that a and U ae fixed, and Γ is inceased fom zeo. Fo small values of Γ, the flow patten will esemble one shown in Fig. 2. Subsequent changes with Γ ae illustated in Fig. 3. u =U 1 Γ <1 4 π Ua Two stagnation points on the cylinde Γ >1 4 π Ua No stagnation points on the cylinde, one stagnation point inside the cylinde not a pat of outside flow, one stagnation point at Γ 2 a 2 s= Γ + 4πU 4πU Figue 3. Stagnation points in flow past a cylinde with eciculation. 0< Γ =1 4 π Ua One stagnation point on the cylinde We will use this infomation about the effect of ciculation on the stagnation points in cylinde flow late. Now, howeve, let us use complex vaiable theoy to evaluate foces and moments on a body in fluid flow. 5. Foces and moments on a body in fluid flow, Blasius integal laws. To find the foce on a body in fluid flow in a geneal case, velocity components must be plugged into the momentum equation. The latte must then be solved to find pessue, and then the hydodynamic foce and moment can be found by appopiate integation ove the bodysuface in thee dimensions o contou in two dimensions. Fotunately, in tems of complex potential fo ideal 2D flows, the pocedue is much simple. Conside the complex foce in the fom

10 X - iy, whee X and Y ae the x- and y-components of the foce acting upon the body due to the fluid flow aound it. Then let C0 be an abitay contou fully enclosing the body. In , Blasius poved that both the fluid foce and the moment M about the body's cente of gavity can be evaluated as integals along C0: ρ X iy =i w 2 dz 5.1 2C ρ M = Rе z w 2 dz 2 C 0 0 Hee w = df/dz is the complex velocity. Both of these integals, when evaluated in accodance with complex vaiable theoy, can be easily esolved in tems of esidues associated with singulaities contained within the body contou. As an example and fo futue use, let us apply Blasius fist law 5.1 to the flow aound a cylinde with Paul Richad Heinich Blasius ciculation 4.1, 4.2: in df a2 iγ w= =U I tied to continue woking in tubulence, dz 2 πz z again without success, and finally told myself: Expession unde the integal in 5.1 is 'You ae not a scientist. 2U 2 a 2 U 2 a 4 i U Γ i U Γ a 2 Γ2 w 2=U πz z2 z4 π z3 4 π2 z 2 The sole singulaity of w2 is at z = 0. The integal can thus be tivially found with Cauchy's esidue theoem: ρ ρ iu Γ X iy =i w 2 dz=i 2 i π = iρu Γ 2C 2 π The esults fo the x-component of the foce on the cylinde efeed to as hydodynamic dag and the y-component hydodynamic lift ae thus 0 X=0 Y = ρuγ D'Alembet's paadox Zhukovsky-Kutta law Zeo dag esult actually applies to any body in a non-sepaated steady ideal fluid flow. It is efeed to as D'Alembet's paadox because in flows of eal fluids, dag is non-zeo. Howeve, unde the assumptions of ideal fluid theoy, dag cannot be popely evaluated because one of the main mechanisms esponsible fo it viscous dissipation is explicitly disegaded. Late we will discuss how to deal with this poblem, and how even in the context of viscous flow with dag, ideal-fluid theoy still can be put to good use. The esult fo the hydodynamic moment of ideal flow aound the cylinde can be similaly evaluated as M = 0. In the following sections, we will make good use of the esult fo lift we botained in this chapte, but fist we must develop mathematical appaatus that allows us to elate the flow aound a lifting suface to the flow aound the cylinde.

11 6. Confomal mapping and Zhukovsky tansfom. Conside two planes of complex vaiables: z = x+iy and ζ = ξ+iη. Let us define a confomal mapping ζ = fz, whee f is a holomophic function. Suppose we have a complicated contou in the z-plane that is educed by a much simple contou in the ζ-plane by this mapping Fig. 4. The hope would be that, instead of solving fo the flow in the z-plane, we could solve fo it with much simple bounday conditions in the ζ-plane, and then just emap the esults to the oiginal z-plane. Hee we ae helped by two featues of confomal mapping. Fist, if in the x,y plane we have 2ϕx,y = 0, then in the ξ,η plane it will hold that 2ϕξ,η = 0. In othe wods, confomal mapping peseves the Laplace equation. Second, we can show how complex velocities in the two planes ae elated: df df ζ d ζ d ζ w z = = = w ζ dz d ζ dz dz Likewise, wζ =dz/dζ wz. Moeove, confomal mapping can be poven to peseve the stength of souces and sinks, as well as the ciculation of votices. Thus it is possible to use confomal mapping to geatly simplify the bounday conditions fo the poblems of ideal 2D hydodynamics. A vaiety of useful confomal mappings exists, but hee we will use only one, named afte N.E. Zhukovsky, who pioneeed its use in lifting-suface studies in the late XIX and ealy XX centuy: c2 6.1 z =ζ+ ζ Let us note seveal popeties of the Zhukovsky tansfom 6.1. Fist, as ζ, ζ z. so the tansfom leaves the geomety of the fa field flow fa away fom the oigin the same in the z- and ζ-planes. Second, the velocity scaling facto fo the Zhukovsky tansfom is easiest to expess as dz c2 =1 2 dζ ζ Note that in confomal mapping, the points whee dz/dζ = 0 Nikolai Egoovich Zhukovsky ae called the citical points of the tansfom. Nea such points, confomal mapping does not peseve the angle: a Man will fly using the powe of his smooth cuve passing though such a point in the ζ-plane will intellect athe than the stength of his ams. be mapped to a cuve with a cusp in the z-plane. Fo Zhukovsky tansfom, the citical points ae ζ = ±c. Now let us begin the pocess of constuction of the potentials fo ideal steady flows past a vaiety of lifting sufaces in 2D. Ou method fo the constuction of such flows will ely on using flow past a cicula cylinde in the ζ plane, then applying Zhukovsky tansfom 6.1 to map the cylinde to the contou of inteest in the z-plane, then finding the appopiate flow patten and the lift foce on the body defined by that contou in the z-plane. 7. Ideal flow past a flat plate and the Zhukovsky-Chaplygin postulate. Ou pocedue of constucting flows past lifting sufaces begins with a flow past a cylinde in the ζ plane. Let us conside such a cylinde with adius c and a cente at the oigin. Thus both

12 citical points of the tansfom will be on the cylinde suface. Fom 4.1 with no ciculation, the complex potential fo this flow with a feesteam velocity U would be c2 F ζ=u ζ+ ζ Let us next conside a moe geneal case, when the feesteam velocity is at an angle α with the hoizontal axis. Fo that, conside the oiginal flow in a plane ζ' otated with espect to the ζ plane, so that ζ' = exp-iαζ. Then, if in the ζ' plane c2 F ζ ' =U ζ '+ ζ ', then in the ζ plane we will have c2 i α i α F ζ=u ζ e + e ζ This cylinde flow will have stagnation points at ζ =±aeiα. Let us efe to the pola coodinates in the ζ plane as to ρ,ν to distinguish them fom the pola coodinates in the z-plane2. Now let us tansition to the z-plane. What happens to the cylinde ρ = c? Let us conside its paametic epesentation ρ = c exp iν and plug it into the Zhukovsky tansfom equation 6.1: c2 iν z =ce + i ν =c e i ν+e i ν =2c cos ν ce As -1 <cosν <1, this equation epesents a hoizontal line segment in the z-plane, stetching fom -2c to 2c Fig. 4. In effect, duing the tansfom hee, a ound hole is cut out of the ζ plane, and then stetched flat to close it in the z-plane, with the upwad-facing side of the segment epesenting the uppe half of the cicle 0 < ν π and the downwad-facing side epesenting its lowe half π < ν 2π. Figue 4. Zhukovsky tansfom poducing flow at an angle of attack past a flat plate. The flow patten on the left panel of Fig. 4 esembles a eal flow past a flat plate at an angle of attack, howeve, thee ae some impotant poblems with this steamline patten. Most notably, in ealizable flows past flat plates and slende lifting sufaces with a defined tailing edge, the ea 2 Note that the Geek lettes fo pola coodinates in the z-plane ae italicized to avoid confusion, because we aleady have non-italicized ρ denoting density.

13 stagnation point always is at the tailing edge Fig. 5. It has to deal with the viscous dissipation in eal fluid o gas binging the velocity on appoach to a stationay solid suface down to zeo. Zeo velocity condition on a solid bounday is known as the no-slip bounday condition. Sadly, in ideal fluid theoy thee is no way eithe to account fo viscous dissipation o to satisfy this bounday condition with non-tivial flow. Anothe poblem with shap edges is that in ideal-fluid theoy velocity of the flow aound a shap edge can become singula. This limitation could have poven fatal to any attempt to use ideal fluid theoy to estimate lift poduced by a wing. Howeve, a cunning kludge was contived by seveal eseaches in the beginning of the XX centuy. It is efeed to as the Kutta condition o moe accuately the Zhukovsky-Chaplygin postulate. The statement of the postulate is as follows: FOR BODIES WITH SHARP TRAILING EDGES AT MODEST ANGLES OF ATTACK TO THE FREESTREAM, THE REAR STAGNATION POINT WILL STAY AT THE TRAILING EDGE. This ensues that the velocity at the tailing edge is zeo, and the steamline patten is physically coect. Pactically, to satisfy the postulate, one needs to add ciculation to the flow in the ζplane to move the stagnation point in the z-plane to the tailing edge. In the case of the flat plate, this tailing edge is at z = 2c, which would coespond to ζ = c, while without ciculation, the ea stagnation point in the flow past a cylinde at angle α to the hoizontal axis esides at z = ceiα. Fom section 4, adding ciculation Γ moves the stagnation point on the cylinde of adius a by an angle θs, so that sin θ s= Γ 4πU a By letting a = c and sin θs = - sin α, we can select the ight Segey Chaplygin value of ciculation to move the stagnation point to ζ = c , Fig. 5: Heo of Socialist Labou Γ=4 π U c sin α 1 Febuay 1941 Now we can use Blasius integal law a.k.a. Zhukovsky-Kutta law, section 5 to elate lift Y on the plate to ciculation which, as section 6 mentions, should be peseved by the Zhukovsky tansfom. Accoding to the Zhukovsky-Kutta law, Y =ρ U Γ Thus fo the chosen value of Γ, Y =4 π ρu 2 c sin α This value epesents the lift of a flat plate with length 4c and unit span extent in diection nomal to the x-y plane, and is in supisingly good ageement with expeiment fo modeate velocities and angles of attack. Lift inceases quadatically with U and nealy linealy with α again, fo modest angles of attack. 8. Lift coefficient. Let us constuct a dimensionless paamete chaacteizing lift on a slende wing. To do that, we must non-dimensionalize the lift foce Y, using a physically elevant scaling facto constucted

14 fom the dimensional paametes of the poblem U, ρ, l, whee l is the chaacteistic length scale of the wing in the steamwise diection Fig. 6. Note that this non-dimensionalization will wok not just fo wings, but with appopiate modifications fo geomety fo bluff bodies as well. Fo example, in defining the lift coefficient fo a cylinde, the appopiate length scale would be its diamete. Figue 5. Top: smoke visualization of the flow past a lifting suface image cedit: Alexande Lippisch, Note the stagnation point on the tailing edge. Bottom: flat-plate flow with ciculation added to satisfy the Zhukovsky-Chaplygin postulate. Figue 6. Chod epesentative length scale of a lifting suface. In tems of the chod length l, the lift coefficient fo a lifting suface is then Y C L= 1 2 ρu l 2 Fo the flat-plate flow we consideed in the peceding section, l = 4c and C L=2 π sin α Thus the lift coefficient inceases with angle of attack. In eal life, this phenomenon is

15 complicated with viscous effects. As the esult, at some citical angle of attack, the flow on the uppe suface of the wing will sepaate, ceating a lage zone of eciculating flow, leading to decease of lift coefficient condition known as stall. At modest velocities, the stall angle is typically about 15. Flow sepaation can be contolled by vaiable wing geomety fo example, spoiles. 9. Wing with a Zhukovsky pofile. Section 7 povides us with an example of constucting a flow past a lifting suface using Zhukovsky tansfom applied to cylinde flow in the ζ plane. By vaying the size and position of the cylinde with espect to citical points of the tansfom ζ = ±c, we can poduce a vaiety of lifting-suface pofiles. To ceate a shap leading o tailing edge, one must position the cylinde to place the appopiate citical point on its suface. Conside, fo example, a pofile with a smooth leading edge and shap tailing edge a symmetical Zhukovsky aifoil, Fig. 7, top. To poduce such a pofile, the cente of the cylinde in the ζ-plane has to be shifted left fom the oigin by a distance m. Let m = εc, whee ε is small, and let us choose the adius of the cylinde so that the citical point ζ = c is on the suface of the cylinde: = a = c1+ε. The leading edge of the cylinde will then eside at ζ = c+2m. With lineaization neglecting quantities on the ode of ε2 and smalle, the chod length of the lifting suface in the z-plane coesponding to this cylinde is 4c, and the maximum thickness t of the pofile, occuing at z = c, is t=3 3 c ε The atio between t and chod length l is called the thickness atio of the wing. Fo the case of symmetical Zhukovsky aifoil, t 3 3 = ε l 4 Now conside the flow past this cylinde at an angle of attack α. The ea tailing edge of the wing in the z-plane is at z = 2c, coesponding to ζ = c. Thus we want to move ou stagnation point in the ζ-plane thee, which means similaly to the easoning of Section 7 that we have to add ciculation Γ = 4π a U sin α. Note that the cylinde adius fo this case is l 4 t l t a=c+m=c1+ε l 3 3 l To satisfy the Zhukovsky-Chaplygin postulate, we need to add ciculation 4 t Γ=4 πu a sin α=π U l 1+ sin α 3 3 l As t/l 0, this expession is educed to the expession fo the flat plate. Howeve, fo small but non-zeo thickness atio, the amount of ciculation is geate than that fo the flat plate. The lift coefficient fo the case of a symmetical Zhukovsky aifoil is t sin α l In othe wods, symmetical Zhukovsky aifoil has bette lift than the flat plate with the same chod length. Similaly, one can demonstate that an ac-shaped aifoil with cambe ac height h Fig. 7, second ow also has bette lift than the flat plate: h C L=2 π U c sin α+2 l C L 2 π

16 Then, to optimize lift, a suface with both thickness and cambe a geneal Zhukovsky pofile can be constucted, fo which the lift coefficient will be t 2h C L=2 π sin α+ l l Many pactical subsonic wing pofiles closely esemble the geneal Zhukovsky pofile. Symmetical Zhukovsky pofile Ac aifoil Zhukovsky pofile Figue 7. Constuction of lifting sufaces: top symmetical Zhukovsky pofile, middle ow ac aifoil, bottom geneal Zhukovsky pofile.

17 10. Dealing with dag. The methods descibed above help pedict lift, but by constuction cannot do anything about thedag foce. To deal with the dag foce, one needs to conside viscous flow, which has govening equations that ae much moe complicated than ideal flow: as viscous dissipation takes place, the enegy equation has to be consideed. That intoduces additional vaiables, such as intenal enegy and tempeatue, and equies additional equations fo example, equations of state and/o assumptions fo example, Newtonian viscosity to close the system. One of the most common models of viscous flow is known as the Navie-Stokes equations. Fo incompessible flow, A vey useful simplification of the Navie-Stokes equations is the bounday laye appoximation. Unde this appoximation, the flow outside the immediate vicinity of a solid suface is consideed as inviscid, while the consideation of the viscous flow in the naow zone nea the suface uses simplifying assumptions based on ode-of-magnitude analysis to disegad some tems, thus avoiding solving the full Navie-Stokes equations. Futhe, in the bounday laye, the exact functional foms of the solutions of the bounday laye equations can be eplaced with polynomial epesentations the Pohlhausen von Kaman method. Then the consevation equations fo mass, momentum, etc. can be solved togethe with the bounday conditions to find the extent of the bounday laye and the polynomial coefficients. At the edge of the bounday laye, the bounday laye solution has to be asymptotically matched with the inviscid oute flow solution. 11. Expeimental measuements of lift and dag. In laboatoy, lift and dag on a model of a lifting suface can be neasued with an aeodynamic scale. In addition, flow visualization helps undestand to what extent the ideal-flow pattens elate to the actual flow field. 12. Modeling lifting-suface flow with Xfoil. Xfoil is a package witten pimaily by Pofesso Mak Dela MIT fo analysis and design of subsonic isolated aifoils. The oiginal vesion was eleased in It can pefom both inviscid and viscous flow modeling. Fo the inviscid analysis, the aifoil is modeled with a linea-voticity steamfunction panel method. The aifoil suface is divided into piecewise staight line segments o panels o bounday elements, and votex sheets of stength γ ae placed on each panel. Think of a votex sheet as of a line with a votex at evey point. The ciculation a segment ds of the line thus imposes on the flow is dγ. By geneating ciculation and thus lift along the contou of the lifting suface, votex sheets mimic the effect of the bounday laye aound aifoils. In the latte, viscous effects poduce shea along the suface, and shea leads to voticity and thus ciculation deposition in the flow. Fo viscous modeling, the flow is sepaated into thee zones oute flow, bounday laye, wake. The oute flow is consideed as inviscid. Pofiles of vaiables in the bounday laye ae appoximated with an integal method. We will not use all the featues of Xfoil, but use it to pedict lift and dag on lifting sufaces unde conditions close to those we will ealize in laboatoy expeiment. Xfoil is command-line diven, and vesions fo all moden opeating systems exist. Let us stat hee with an assumption that you have eithe downloaded and installed an achive with appopiate Xfoil

18 binaies on a Windows-based machine, the default diectoy fo the installation would be c:\xfoil, o successfully compiled Xfoil fom the souce files on you system. The following naative will use Windows envionment fo an example, mostly because a typical Windows use is in the geatest need of guidance. Stat with opening a command-line window Win* R, then type cmd in the Run window that pops up. This should bing up a window. Stetch it out a little bit with the cuso, so that you should be able to see about fifty lines of text, change diectoy to the diectoy whee the xfoil.exe file esides, then stat xfoil: cd \xfoil\bin xfoil The Xfoil summay including the list of submenus should display, and the command pompt should change to something like XFOIL c> Now let us load a wing pofile by typing NACA 2412 Note that the NACA fou-digit designato defines the wing pofile by: 1. Fist digit descibing maximum cambe as pecentage of the chod. 2. Second digit descibing the distance of maximum cambe fom the aifoil leading edge in tens of pecent of the chod. 3. Last two digits descibing maximum thickness of the aifoil as pecent of the chod. The NACA 2412 aifoil that we loaded has a maximum cambe of 2% located 40% 0.4 chods fom the leading edge with a maximum thickness of 12% of the chod. Fou-digit seies aifoils by default have maximum thickness at 30% of the chod 0.3 chods fom the leading edge. Xfoil should espond by confiming some basic infomation about the wing thickness, cambe, etc.. Just in case, it's nice to foce cleanup of the geomety as loaded: pane Then let us move to the OPER submenu to pefom some opeations commands ae not casesensitive. OPER The pompt should change to.operi c> Note the i afte OPER - this means that you ae in inviscid mode. Now let us specify the angle of attack fo which we need to pefom the calculations: alfa 0 This should bing up a window showing velocity and pessue distibutions on the wing suface fo the inviscid case. The lift coefficient CL is also displayed. Please note that Xfoil assumes that the chod length of the pofile is unity thus the esults may need escaling to be compaed with theoy o expeiment. Now let's switch to viscous flow by typing visc In esponse, xfoil will pompt you fo the Reynolds numbe. Let's set it to 3e5 Let us edo the calculations fo the viscous case: * Windows key. In this naative, text blocks with gay highlights denote keyboad keys o shotcut combinations, and text in Couie font shows inteaction with the compute.

19 alfa 0 Along with the lift coefficient, the output window will now display the dag coefficient CD Fig. x. Also note that the inviscid esults fo velocity and pessue on the wing ae also displayed fo compaison, with dashed lines. In the bottom of the display window, the pogam will show the bounday laye and wake aeas. Despite the vey cude assumptions of the inviscid model, its pediction fo the lift coefficient tuns out to be vey close to the pediction of the model taking into consideation the viscous effects 0.235, and to the expeiment as well. Howeve, dag coefficient in this case cannot be pedicted with ideal theoy. Figue x. Xfoil display output with featues of inteest labeled.