7-Flow of Ideal Fluids

Transcription

1 7-Flow of Ideal Fluids 7.1 INTRODUCTION Flows at high Reynolds numbe eveal that the viscous effects ae confined within bounday layes. Fa away fom the solid suface, the flow is nealy in viscid d in many cases it is incompessible. We now aim at developing techniques f alyses of inviscid incompessible flows. Incompessible flow is a constant density flow, and we assume p to be nstant. We visualize a fluid element of defined mass moving along a steamline an incompessible flow. Because the density is constant, we can wite. V = (7.1) ove and above, if the fluid element does not otate as it moves along the eamline. to be pecise, if its motion is tanslational (and defmation with no ation) only. the flow is temed as iotational flow. It has aleady been shown Sec that the motion of a fluid element can in geneal have tanslation, fmation and otation. The ate of otation of the fluid element can be :asued by the aveage ate of otation of two pependicula line segments. The eage ate of otation w z about z-axis is expessed in tems of the gadients of locity components (efe to Chapte 3) as w z = 1 2 [ dv dx du dy Smilaly, the othe two components of otation ae w z = 1 2 [ dw dy dv dz and w y = 1 2 [ du dz dw dx As such, they ae components of o which is given by ω = 1 2 ( V ) In a two-dimensional flow, w z is the only non-tivial component of the ate of otation. Imagine a pathline of a fluid paticle shown in Fig Rate of spin of the paticle is w z. The flow in which this spin is zeo thoughout is known as iotational flow. A genealized statement is me appopiate: F iotational flows, V = in the flow field. Fig

2 Theefe f an iotational flow, the velocity V can be expessed as the gadient of a scala function called the velocity potential, denoted by φ V = φ (7.2) Combination of Eqs (7.1) and (7.2) yields 2 φ = (7.3) Fom Eq. (7.3) we see that an inviscid, incompessible, iotational flow is govened by Laplace s equation. Laplace s equation is linea, hence any numbe of paticula solutions of Eq. (7.3) added togethe will yield anothe solution. This concept fms the building-block of the solution of inviscid, incompessible, iotational flows. A complicated flow patten f an inviscid, incompessible, iotational flow can be synthesized by adding togethe a numbe of elementay flows which ae also in viscid, incompessible and iotational. The analysis of Laplace s Eq. (7.3) and finding out the potential functions ae known as potential flow they and the inviscid, incompessible, iotational flow is often called as potential flow. Howeve, the following elementay flows can constitute seveal complex potential-flow poblems 1. Unifm flow 2. Souce sink 3. Vtex 7.2 ELEMENTARY FLOWS IN A TWO DIMENSIONAL PLANE Unifm Flow In this flow. velocity is unifm along y-axis and thee exists only one component. velocity which is in the x diection. Magnitude of the velocity is Vo. om Eq. (7.2) we can wite îu + ĵ = ĵ dφ dx + ĵ dφ dy, dφ dx = u, dφ dy = Whence φ = u x + C 1 (7.4) Recall fom Sec that in a two dimensional flow field. flow can also 2

3 be escibed by steam function 1fI. In case of unifm flow dψ = u dy and dψ = dx so that ψ = u y + K 1 (7.5) The constants of integation C 1 and K 1 in Eqs (7.4) and (7.5) ae abitay. The alues of IfI and If f diffeent steamlines and velocity potential lines may change,ut flow patten is unalteed. The constants of integation may be omitted and it is,ossible to wite ψ = U y, φ = U x (7.6) These ae plotted in Fig. 7.2(a) and consist of a ectangula mesh of staight teamlines and thogonal staight potential-lines. It is conventional to put tows on the steamlines showing the diection of flow. n tems of pola ( - θ) codinate. Eq. (7.6)becomes ψ = U sin θ, ψ = U cos θ (7.7) If we conside a unifm steam at an angle a to the x-axis as shown in Fig. 7.2b. Ne equie that u= U cos α = dψ = dφ dy dx and v=u sin α = dψ dx = dφ dy (7.8) Integating. we obtain f a unifm velocity Vo at an angle a. the steam function and velocity potential espectively as ψ = U (y cos α xsinα), φ = U (x cos α + y sin α) (7.9) fig 7.2 b fig 7.2 a Souce Sink Conside a flow with staight steamlines emeging fom a point, whee velocity along each steamline vaies invesely with distance fom the poin shown in Fig Only the adial component of velocity is non-tivial 3

4 (v θ =, v z = ). fig 7.3 Such a flow is called souce flow. In a steady souce flow the amount of fluid cossing any given cylindical suface of adius and unit length is constant (m). m = 2πv ρ v = m 2πρ. 1 = 2π. 1 = K (7.1 a) whee, K is the souce stength K= m 2πρ = 2π (7.1 b) and is the volume flow ate Again ecall fom Sec that the definition of steam function in cylindical pola codinate states that v = 1 dψ and v dθ θ = dψ (7.11) d Now f the souce flow, it can be said that 1 dψ = K (7.12) dθ and dψ d = (7.13) Combining Eqs (7.12) and (7.13), we get ψ = Kθ + C 1 (7.14) Howeve, this flow is also iotational and we can wite îv + ĵv θ = î dφ + ĵ 1 dφ d dθ v = dψ and v d θ = = 1 dψ dθ dψ d = v = K and ψ = Kln + C 2 (7.15) Likewise in unifm flow, the integation constants C1 and C2 in Eqs (7.14) and (7.15) have no effect on the basic stuctue of velocity and pes- 4

5 sue in the flow. The equations f steamlines and velocity potential lines f souce flow become ψ = Kθ and φ = Kln (7.16) whee K is defined as the souce stength and is poptional to A which is the ate of volume flow fom the souce pe unit depth pependicula to the page as shown in Fig If A is negative, we have sink flow, whee the flow is in the opposite jiection of the souce flow. In Fig. 7.3, the point is the igin of the adial steamlines. We visualize that pointo is a point souce sink that induces adial flow in the neighbouhood. The point souce sink is a point of singulaity in the flow field (because v becomes infinite). It can also be visualized that point in Fig. 7.3 is simply a point fmed by the intesection of plane of the pape and a line pependicula to the pape. The line pependicula to the pape is a line souce, with volume flow ate ( ) pe unit length. Howeve, f sink, the steam function and velocity potential function ae ψ = Kθ and φ = Kln (7.17) Vtex Flow In this flow all the steamlines ae concentic cicles about a given point whee the velocity along each steamline is invesely poptional to the distance fom the cente, as shown in Fig Such a flow is called vtex (fee vtex)flow. This flow is necessaily iotational. Fig 7.4 In a puely ciculaty (fee vtex flow) motion, we can wite the tangential velocity as v θ = Ciculation constant Whee τ is ciculation v θ = τ/2π (7.18) Also, f puely ciculaty motion one can wite v = (7.19) 5

6 With the definition of steam function, it is evident that v θ = dψ and v d = 1 dψ dθ Combining Eqs (7.18) and (7.19) with the above said elations f steam function, it is possible to wite ψ = τ 2π ln + C 1 (7.2) Because of iotationality, it should satisfy îv + ĵv θ = î dφ + ĵ 1 dφ d dθ Eqs (7.18) and (7.19) and the above solution of Laplace s equation yields φ = τ 2π θ + C 2 (7.21) The integation constants C 1 and C z have no effect whatsoeve on the stuctue of velocities pessues in the flow. Theefe like othe elementay flows, we shall consistently igne such constants. It is clea that the steamlines f vtex flow ae cicles while the potential lines ae adial. These ae given by ψ = τ τ ln and φ = θ (7.22) 2π 2π In Fig. 7.4, point can be imagined as a point vtex that induces the ciculaty flow aound it. The point vtex is a singulaity in the flow field (ve becomes infinite). It is also discened that the point in Fig. 7.4 is simply a point fmed by the intesection of the plane of a pape and a line pependicula to the plane. This line is called vtex filament of stength, whee is the ciculation aound the vtex filament and the ciculation is defined as τ = V.d s (7.23) In Eq. (7.23), the line integal of the velocity component tangent to a cuve of elemental length ds, is taken aound a closed cuve. It may be stated that the ciculation f a closed path in an iotational flow field is zeo. Howeve, the ciculation f a given path in an iotational flow containing a finite numbe of singula points is constant. In geneal this ciculation constant denotes the algebaic stength of the vtex filament contained within the closed cuve. Fom Eq. (7.23) we can wite 6

7 τ = V.d s = (udx + vdy + wdz) F a two-dimensional flow τ = (udx + vdy) τ = V cos αds (7.24) Conside a fluid element as shown in Fig Ciculation is positive in the anticlockwise diection (not a mandaty but geneal convention). Fig 7.5 δτ = uδx + (v + dv du δx)δy (u + δy)δx vδy dx dy δτ = ( dv dx du dy )δxδy δτ = 2w z δa δτ = 2w δa z = Ω z s (7.25) Physically, ciculation pe unit aea is the vticity of the flow. as Now, f a fee vtex flow, the tangential velocity is given by Eq. (7.18) v θ = τ/2π = C F a cicula path (efe Fig. 7.5) α = V = v θ = C Thus, τ = 2π 1 C dθ = 2πC (7.26) It may be noted that although fee vtex is basically an iotational motion, the ciculation f a given path containing a singula point (including the igin) is constant (2πC) and independent of the adius of a cicula steamline. Howeve, if the ciculation is calculated in a fee vtex flow along any closed contou excluding the singula point (the igin), it should 7

8 be zeo. Let us look at Fig. 7.6 (a) and take a closed contou ABCD in de to find out ciculation about the point, P aound ABCD τ ABCD = v θab 1 dθ v BC ( 2 1 ) + v θcd 2 dθ + v DA ( 2 1 ) Thee is no adial flow v BC = v DA =, v θab = C 1 and v θcd = C 2 τ ABCD = C 1 1 dθ + C 2. 2 dθ = (7.27) If thee exists a solid body otation at constant W induced by some extenal mechanism, the flow should be called afced vtex motion (Fig. 7.6b) and we can wite v θ = w and τ = v θ ds = 2π ω.dθ = 2π 2 ω (7.28) Equation (7.28) pedicts that the ciculation is zeo at the igin and it inceases with inceasing adius. The vaiation is paabolic. It may be mentioned that the fee vtex (iotational) flow at the igin (Fig. 7.6a) is impossible because of mathematical singulaity. Howeve, physically thee should exist a otational (fced vtex) ce which is shown by the dotted line. Below ae given two statements which ae elated to Kelvin s ciculation theem (stated in 1869) and Cauchy s theem on iotational motion (stated in 1815) espectively (i) The ciculation aound any clo ed contou is invaiant with time in an inviscid fluid. (ii) A body of inviscid fluid in iotational motion continues to move iotationally. Fig SUPERPOSITION OF ELEMENTARY FLOWS We can now fm diffeent flow pattens by supeimposing the velocity potential and steam functions of the elementay flows stated above Doublet 8

9 In de to develop a doublet, imagine a souce and a sink of equal stength K at equal distance s fom the igin along x-axis as shown in Fig Fom any point P(x, y) in the field, t and z ae dawn to the souce and the sink. The pola codinates of this point (, θ) have been shown. Fig 7.7 The potential functions of the two flows may be supeimposed to descibe the potential f the combined flow at P as Similaly φ = Kln 1 Kln 2 (7.29) Ψ = K(θ 1 θ 2) = Kα (7.3) whee, α = (θ 2 θ 1 ) We can also wite tan θ 1 = y tan θ x+s 2 = y (7.31) x s 1 = 2 + s 2 + 2s cos θ and 2 = 2 + s 2 2s cos θ (7.32) Now using the above mentioned elations we find tan(θ 2 θ 1 ) = tan θ 2 tan θ 2 1+tan θ 2 tan θ 1 tan α = [ yx+ys yx+ys x 2 S 2 /(1 + y2 x 2 s 2 ) tan α = 2ys x 2 +y 2 s 2 (7.33) Hence the steam function and the velocity potential function ae fmed by combining Eqs (7.3) and (7.33), as well as Egs(7.29) and (7.32) espectively 2ys ψ = K tan 1 ( ) (7.34) x 2 +y 2 s 2 φ = K 2 ln( 2 +s 2 +2s cos θ 2 +s 2 2s cos θ ) (7.35) Doublet is a special case when a souce as well as a sink ae bought togethe in such a way that s and at the same time the stength K 9

10 ( /2π) is inceased to an infinite value. These ae assumed to be accomplished in a manne which makes the poduct of s and π afesaid cicumstances ψ =. 2ys 2π x 2 +y 2 s 2 [Since in the limiting case tan 1 α = α (in limiting case) a finite value X. Unde the ψ = x. y x 2 +y 2 = x sin θ (7.36) Fom Eg. (7.35), we get φ = 4π [1n(2 + s 2 + 2s cos θ) 1n( 2 + s 2 2s cos θ) φ = 4π [ln{(2 + s 2 2s cos θ )(1 + )} ln{( 2 + s 2 2s cos θ )(1 )} 2 +s 2 2 +s 2 2s cos φ = [{ 1 2s cos θ ( ) 2 } + 1 2s cos θ [ } 4π 2 +s s s 2 2s cos - { 1 2s cos θ ( ) 2 } 1 2s cos θ [ } 2 +s s s 2 φ = 4π [ 4s cos θ 2 +s s cos θ ( ) 3 } s 2 In the limiting condition the above expession can be witten as φ x cos θ 2 +s 2 φ x cos θ (7.37) We can see that the steamlines associated with the doublet ae x sin θ = C 1 If we eplacesin θby y/, and the minus sign be absbed in C 1, we get x y 2 = c 1 (7.38 a ) 1

11 In tems of catesian codinate, it is possible to wite x 2 + y 2 x C 1 y = (7.38b) Equation (7.38b) epesents a family of cicles. F x =, thee ae two values of y, one of which is zeo. The centes of the cicles fall on the y-axis. On the cicle, whee y =, x has to be zeo f all the values of the constant. It is obvious that the family of cicles fmed due to diffeent values of C 1 must be tangent to x-axis at the igin. These steamlines ae illustated in Fig Due to the initial positions of the souce and the sink in the development of the doublet, it is cetain that the flow will emege in the negative x diection fom the igin and it wi lconvege via the positive x diection of the igin. Fig 7.8 Howeve, the velocity potential lines ae x cos θ = K 1 In catesian codinate this equation becomes x 2 + y 2 x k 1 x = (7.39) Once again we shall obtain a family of cicles. The centes will fall on x-axis. F y = thee ae two values of x, one of which is zeo. When x =, y has to be zeo f all values of the constant. Theefe these cicles ae tangent to y-axis at the igin. The thogonality of constant,and constant ifj lines ae maintained as we ion out the pocedue of dawing constant value lines (Fig. 7.8). In addition to the detemination of the steam function and velocity potential, it is obseved fom Eq. (7.37) that f a doublet v = dφ d = x cos θ 2 (7.4) As the cente of the doublet is appoached; the adial velocity tends to be infinite. It shows that the doublet flow has a singulaity. Since the ciculation about a singula point of a souce a sink is zeo f any stength, it is obvious that the ciculation about the singula point in a doublet flow must be zeo. It follows that f all paths in a doublet flow τ = τ = V.d s = (7.41) 11

12 Applying Stokes Theem between the line integal and the aea-integal τ = ( V )d A = (7.42) Fom Eq. (7.42), the obvious conclusion is V = i.e., doublet flow is an iotational flow. At lage distances fom a doublet, the flow appoximates the distubances of a two dimensional aifoil. The influence of an aifoil as felt at distant walls may be appoximated mathematically by a combination of doublets with vaying stengths. Thus the cuise conditions of a two dimensional aifoil can be simulated by the supeposition of a unifm flow and a doublet sheet of vaying stengths Flow About a Cylinde Without Ciculation Inviscid-incompessible flow about a cylinde in unifm flow is equivalent to the supeposition of a unifm flow and a doublet. The doublet has its axis of development paallel to the diection of the unifm flow. The combined potential of this flow is given by φ = U x + x cos θ (7.43) and consequently the steam function becomes ψ = U y x sin θ (7.44) In ou analysis, we shall daw steamlines in the flow field. In twodimensional flow, a steamline may be intepeted as the edge of a suface on which the velocity vect should always be tangent and thee is no flow in the diection nmal to it. The latte is identically the chaacteistics of a solid impevious bounday. Hence, a steamline may also be consideed as the contou of an impevious two-dimensional body. Figue 7.9 shows a set of steamlines. The steamline C-D may be consideed as the edge of a two-dimensional body while the emaining steamlines fm the flow about the bounday. Fig. 7.9 Now we follow the essential steps involving the supeposition of elementay flows in de to fm a flow about the body of inteest. A steamline has to be detemined which encloses an aea whose shape is of pactical 12

13 imptance in fluid flow. This steamline will descibe the bounday of a two-dimensional solid body. The emaining steamlines outside this solid egion will constitute the flow about this body. Let us look f the steamline whose value is zeo. Thus we obtain U y x sin θ (7.45) eplacing y by sin e, we have sin θ(u x ) = (7.46) If θ = θ = π the equation is satisfied. This indicates that the x-axis is a pat of the steamline ψ= O. When the quantity in the paentheses is zeo, the equation is identically satisfied. Hence it follows that = ( x U ) 1/2 (7.47) It can be said that thee is a cicle of adius ( x U ) 1/2 which is an intinsic pat of the steamline ψ =. This is shown in Fig Let us look at the points of intesection of the cicle and x- axis, i.e. the points A and B. The pola codinate od these points ae = ( x U ) 1/2, θ = π f point A = ( x U ) 1/2, θ =,f point B Fig 7.1 velocity at these points ae found out by taking patial deivatives of the,velocity potential in two thogonal diections and then substituting the pope values of the codinates. Thus v = dφ d = U cos θ x cos θ 2 (7.48 a) v θ = 1 At a point A [θ = π, = ( x U ) 1/2 dφ d = U cos θ x cos θ 2 (7.48 b) v =, v θ = 13

14 At a point B [θ =, = ( x U ) 1/2 v =, v θ = the points A and B ae clealy the stagnation points though which the flow livides and subsequently eunites fming a zone of cicula bluff body. The cicula egion, enclosed by pat of the steamline ψ= could be imagined IS a solid cylinde in an inviscid flow. At a lage distance fom the cylinde the low is moving unifmly in a coss-flow configuation. Figue 7.11 shows the steamlines of the flow. The steamlines outside the ;icle descibe the flow patten of the inviscid iotational flow acoss a cylinde. -Ioweve, the steamlines inside the cicle may be disegaded since this egion is ;onsideed as a solid obstacle. Fig Lift and Dag f Flow Past a Cylinde Without Ciculation Lift and dag ae the fces pe unit length on the cylinde in the diections nmal and paallel espectively, to the diection of unifm flow. Pessue f the combined doublet and unifm flow becomes unifm at lage distances fom the cylinde whee the influence of doublet is indeed small. Let us imagine the pessue P o is known as well as unifm velocity U. Now we can apply Benoulli s equation between infinity and the points on the bounday of the cylinde. Neglecting the vaiation of potential enegy between the afesaid point at infinity and any point on the suface of the cylinde, we can wite p ρ g + U 2 2 g = p b ρ g + U 2 b 2 g (7.49) whee, the subscipt b indicates the suface on the cylinde. As we know, since fluid cannot penetate the solid bounday, the velocity U b should be only in the tansvese diection, in othe wds, only v θ component of velocity is pesent on the steamline ψ =. Thus at = ( x U o ) 1/2 14

15 U b = v at=(x/u ) 1/2 = 1 dφ dθ (7.5) at=(x/u ) 1/2 = -2 U sin θ Fom Eqs (7.49) and (7.5) we obtain [ U p b = ρ 2 g 2 g + p ρ g (2U sin θ) 2 s 2 g (7.51) The dag is calculated by integating the fce components aising out of pessue, in the x diection on the bounday. Refeing to Fig. 7.12, the dag fce can be witten as Fif 7.12 D = 2π D = 2π D = - 2π [ Similaly, the lift fce p b cos θ( x U o ) 1/2 dθ ρg( x U ) 1/2 [ U 2 2 g + p ρg (2U sin θ) 2 L = - 2π p + ρu 2 2 (1 4 sin2 θ) 2 g cos θdθ ( x U ) 1/2 cos θdθ (7.52) p b sin θ( x U o ) 1/2 dθ (7.53) The Eqs (7.52) and (7.53) poduce D = and L= afte the integation is caied out. Howeve, in eality, the cylinde will always expeience some dag fce. This contadiction between the inviscid flow esult and the expeiment is usually known as D Almbet paadox. The eason f the discepancy lies in completely igning the viscous effects thoughout the flow field. Effect of the thin egion adjacent to the solid bounday is of paamount imptance in detemining dag fce. Howeve, the lift may often be pedicted by the pesent technique. We shall appeciate this fact in a subsequent section Flow About a Rotating Cylinde In addition to supeimposed unifm flow and a doublet, a vtex is thown at the doublet cente. This will simulate a otating cylinde in unifm steam. We shall see that the pessue distibution will esult in a fce, 15

16 a component of which will culminate in lift fce. The phenomenon of geneation of lift by a otating object placed in a steam is known as Magnus effect. The velocity potential and steam functions f the combination of doublet, vtex and unifm flow ae φ = U x + x cos θ ψ = U y + x sin θ τ θ(clockwise otation) (7.54) 2π τ ln (clockwise otation) (7.55) 2π By making use of eithe the steam function velocity potential function, the velocity components ae v = 1 v θ = 1 dψ dθ = (U x 2 ) cos θ (7.56) dφ dθ = (U + x 2 ) sin θ τ 2π (7.57) Implicit in the above deivation ae x = cos θ and y = sin θ. At the stagnation points the velocity components must vanish. Fom Eq. (7.56), we get cos θ(u x 2 ) = (7.58) Fom Eq. (7.58) it is evident that a zeo adial velocity component may occu at θ = ± π 2 and along the cicle, =( x U ) 1/2. Eq. (7.57) depicts that a zeo tansvese velocity equies [ sin θ = τ/2π u θ = sin 1 +( x 2 ) τ/2π u + x 2 (7.59) Howeve, at the stagnation point, both adial and tansvese velocity components must be zeo. So, the location of stagnation point occus at = ( x u ) 1/2 and θ = sin 1 { τ/(2π( x ) u 1/2 )} [ u +x/( x u ) θ = sin 1 [ τ 2π( x u ) 1/2. 1 2u 16

17 θ = sin 1 [ τ 4π(xu ) 1/2 (7.6) Thee will be two stagnation points since thee ae two angles f a given sine except f sin 1 (±). The steamline passing though these points may be detemined by evaluating ψ at these points. Substitution of the stagnation codinate (, θ) into the steam function (Eq. 7.55) yields [ ψ = U ( x u ) 1/2 x ( x ) u 1/2 [ ψ = [(u x) 1/2 (u x) 1/2 sin sin 1 [ τ 4π(xu ) 1/2 τ 4π(xu ) 1/2 + τ 2π in( x u ) 1/2 + τ 2π ln( x u ) 1/2 ψ stag = τ 2π ln( x u ) 1/2 (7.61) Equating the geneal expession f steam function to the above constant, we get u sin θ x sin θ + τ 2π ln = τ 2π ln( x u ) By eaanging we can wite [ sin θ u x + [ln τ ln( xu ) 1/2 2π = (7.62) All points along the cicle = ( x u ) 1/2 satisfy Eq. (7.62), since f this value of, each quantity within paentheses in the equation is zeo. Consideing the intei of the cicle (on which ψ = ) to be a solid cylinde, the oute steamline patten is shown in Fig Fig 7.13 A futhe look into Eq. (7.6) explains that at the stagnation point θ = sin 1 [ θ = sin 1 [ τ/2π 2(xU ) 1/2 τ/2π 2(U ) (7.63) The limiting case aises f (τ/2π) U two = 2, whee θ = sin 1 ( 1) = 9 and 17

18 stagnation points meet at the bottom as shown in Fig Fig 7.14 Howeve, in all these cases the effects of the vtex and doublet become negligibly small as one moves a lage distance fom the cylinde. The flow is assumed to be unifm at infinity. We have aleady seen that the change in stength of the vtex changes the flow patten, paticulaly the position of the stagnation points but the adius of the cylinde emains unchanged Lift and Dag f Flow About a Rotating Cylinde The pessue at lage distances fom the cylinde is unifm and given by Po. Deploying Benoulli s equation between the points at infinity and on the bounday of the cylinde, [ U p b = ρg 2 2 g + p ρ g The velocity U b is as such v θ Hence, U b = 1 U b 2 2 g =( x ) U 1/2 dφ dθ = 2U sin θ τ 2π (7.64) [ U x 1/2 (7.65) Fom Eqs (7.64) and (7.65) we can wite [ [ U p b = ρg 2 2 g + p 2U sin θ τ 2π ( U x ) 1/2 ρ g 2 g (7.66) The lift may calculated as (efe Fig. 7.12) L = 2π L = 2π L= - 2π { ρu 2 [ ρu 2 p b sin θ[ x U 1/2 dθ 2 + p [ 2U sin θ τ 2π ( U x ) 1/2 2 2 }[ x U 1/2 (sin θ)dθ ( x 2 U ) 1/2 sin θ + p ( x U ) 1/2 sin θ ρ {4U 2 2 sin 2 θ+ } ( x U ) 1/2 sin θ dθ 4U τ sin θ 2π ( U x ) 1/2 + τ 2 4π 2 [ U x L = 2π [ ρu 2 ( x 2 U ) 1/2 sin θ + p ( x U ) 1/2 sin θ 2ρU 2 sin 3 θ 18

19 ( x U ) 1/2 ρu τ sin 2 θ ρτ 2 ( x π 8π 2 U ) 1/2 sin θ dθ L = ρu τ The dag fce, which includes the multiplication by cos θ (and integation ove 2π) is zeo. Thus the inviscid flow also demonstates lift. It can be seen that the lift becomes a simple fmula involving only the density of the medium, fee steam velocity and ciculation. In addition, it can also be shown that in two dimensional incompessible steady flow about a bounday of any shape, the lift is always a poduct of these thee quantities. The validity of Eq. (7.67) f any two-dimensional incompessible steady potential flow aound a body of any shape, not necessaily a cicula cylinde, is known as the Kutta-Joukowski theem named afte the Geman fluid dynamist Wilhelm Kutta ( ) and Russian mathematician Nikolai J. Joukowski ( ). A vey popula example of the lift fce acting on a otating body is obseved in the game of socce. If a playe impats otation on the ball while shooting it, instead of following the usual tajecty, the ball will sweve in the ai and puzzle the goalkeepe. The sweve in the ai can be contolled by vaying the stength of ciculation, i.e., the amount of otation. In 1924, a man named Flettne had a ship built in Gemany which possessed two otating cylindes to geneate thust nmal to wind blowing past the ship. The Flettne design did not gain any populaity but it is of consideable scientific inteest (shown in Fig. 7.15). Fig AEROFOIL THEORY Aeofoils ae steamline shaped wings which ae used in aiplanes and tubo machiney. These shapes ae such that the dag fce is a vey small faction of the lift. The following nomenclatues ae used f defining an aeofoil (efe to Fig. 7.16). The chd (c) is the distance between the leading edge and tailing edge. The length of an aeofoil, nmal to the coss-section (i.e., nmal to the plane of a pape) is called the span of aeofoil. The cambe line epesents the mean pofile of the aeofoil. Some imptant geometical paametes f an aeofoil ae the atio of maximum thickness to chd (tic) and the atio 19

20 of maximum cambe to chd (hie). When these atios ae small, an aeofoil can be consideed to be thin. F the analysis of flow, a thin aeofoil is epesented by its cambe. Fig The they of thick cambeed aeofoils is an advanced topic. Basically it uses a complex-vaiable mapping which tansfms the in viscid flow acoss a otating cylinde into the flow about an aeofoil shape with ciculation Flow Aound a Thin Aeofoil Thin aeofoil they is based upon the supeposition of unifm flow at infinity and a continuous distibution of clockwise fee vtex on the cambe line having ciculation density γ(s) pe unit length. The ciculation densityγ(s) should be such that the esultant flow is tangent to the cambe line at evey point. Since the slope of the cambe line is assumed to be small, γ(s)ds = γ(n)dn (efe Fig. 7.17). The total ciculation aound the pofile is given by τ = c γ(η)dη (7.68) vtical motion of stength γdη at x = η develops a velocity at the point P which may be expessed as dv = γ(η)dη 2π(η x) acting upwads The total induced velocity in the upwad diection at P due to the entie vtex distibution along the cambe line is Fig v(x)= 1 c γ(η)dη (7.69) 2π (η xs) F a small cambe(having small α), this expession is identically valid f the induced velocity at p due to the vtex sheet of vaiable stength γ(s) on the cambe line. The esultant velocity due to U and v(x) must be tangential to the cambe line so that the slope of a cambe line may be expessed as 2

21 dy dx = U sin α+v U cos α = tan α + v U cos α dy dx = α + v U [since α is vey small (7.7) Fom Eqs (7.69) and (7.7) we can wite dy dx == α + 1 2πU c γ(η)dη η x (7.71)* Let us conside an element ds on the cambe line. Conside a small ectangle (dawn with dotted line) aound ds. The uppe and lowe sides of the ectangle ae vey close to each othe and these ae paallel to the cambe line. The othe two sides ae nmal to the cambe line. The ciculation along the ectangle is measued in clockwise diection as V 1 V 2 ds = γds should be zeo [nmal components of velocity at the cambe line V 1 V 2 = γ (7.72) If the mean velocity in the tangential diection at the cambe line is given by V s = (V 1 + V 2 )/2, it can be ewitten as V 1 = V s + γ 2 and V 2 = V s γ 2 In the event, it can be said that if v is vey small [v << U, V s becomes equal to U. The diffeence in velocity acoss the cambe line bought about by the vtex sheet of vaiable stength y (s) causes pessue diffeence and geneates lift fce Geneation of Vtices Aound a Wing The lift aound an aeofoil is geneated following Kutta-Joukowski theem. Lift is a poduct of ρ, U and the ciculation Γ.Mechanism of induction of ciculation is to be undestood clealy. When the motion of a wing stats fom est, vtices ae fmed at the tailing edge (efe Fig. 7.18). At the stat, thee is a velocity discontinuity at the tailing edge. This is eventual because nea the tailing edge, the velocity at the bottom suface is 21

22 * F a given aeofoil, the left hand side tem of the integal Eq. (7.71) is a known function. Finding out γ(η) fom it is a fmidable task. This execise is not being discussed in this text. Inteested eades may efe to the books by Glauet [1 and Batchel [2. If γ(η) is detemined, the ciculation Γ and consequently the lift L = ρu Γ can easily be calculated. highe than that at the top suface. This discepancy in velocity culminates in the fmation of vtices at the tailing edge. Figue 7.18 (a) depicts the fmation of stating vtex by impulsively moving aeofoil. Howeve, the stating vtices induce a counte ciculation as shown in Figue 7.18 (b). The ciculation aound a path (ABCD) enclosing the wing and just shed (stating) vtex must be zeo. Hee we efe to Kelvin s theem once again. Fig 7.18 Initially the flow stats with the zeo ciculation aound the closed path. Theeafte, due to the change in angle of attack flow velocity, if a fesh stating vtex is shed, the ciculation aound the wing will adjust itself so that a net zeo vticity is set aound the closed path. The discussions in the pevious section wee f two-dimensional, infinite span wings. But eal wings have finite span finite aspect atio γ, defined as λ = b2 A s (7.73) whee b is the span length and A s is the plan fm aea as seen fom the top. F a wing of finite span, the end conditions affect both the lift and the dag. In the leading edge egion, pessue at the bottom suface of a wing is highe than that at the top suface. The longitudinal vtices ae geneated at the edges of finite wing owing to pessue diffeences between the bottom suface diectly facing the flow and the top suface (efe Fig. 7.19). This is vey pominent f small aspect atio delta wings which ae used in high-pefmance aicafts as shown in Fig.7.2. Fig 7.19 Fig. 7.2 Howeve, ciculation aound a wing gives ise to bound vtices that move along with the wing. In 1918, Pandtl successfully modelled such flows by 22

23 eplacing the wing with a lifting line. The bound vtices aound this lifting line, the stating vtices and the longitudinal vtices fmed at the edges, constitute a closed vtex ing as shown in Fig Summay This chapte has given a bief desciption of inviscid, incompessible, iotational flows. Iotationality leads to the condition V = which demands V = φ, whee φ is known as a potential function. F a potential flow 2 φ =. The steam function Ψ also obeys the Laplace s equation 2 ψ = f the potential flows. Laplace s equation is linea, hence any numbe of paticula solutions of Laplace s equation added togethe will yield anothe solution. So a complicated flow f an in viscid, incompessible, iotational condition can be synthesized by adding togethe a numbe of elementay flows which ae also inviscid, incompessible and iotational. This is called the method of supeposition. Some in viscid flow configuations of pactical imptance ae solved by using the method of supeposition. The ciculation in a flow field is defined as Γ = V.d s. subsequently, the velocity may be defined as ciculation pe unit aea. The ciculation f a closed path in an iotational flow field is zeo. Howeve, the ciculation f a given closed path is an iotational flow containinga finite numbe of singula points is a non -zeo constant. The lift aound an immesed body is geneated when the flow field pocesses ciculation. The lift aound a body of any shape is given by L = ρu τ, whee ρ is the density and U is the velocity in the steamwise diection. 23