Howe_conformal_mapping

Howe_conformal_mapping - 3.8 THE JOUKOWSKI TRANSFORMATION...

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Unformatted text preview: 3.8 THE JOUKOWSKI TRANSFORMATION 167 To use these results we first combine (3.7.34) and (3.7.36) to obtain 1 ~_cos(6/2) _ _1"__ ,3 —sin(5/2)’ ’3 ” 4hU' Thus, when the vortex strength ,6 and offset hr are prescribed, Equations (3.7.38) and (3.7.39) can be solved for the deflection angle 6 and the vortex image angle 11/. The shape of the jet can then be plotted by use of (3.7.37), the upper and lower streamlines in the figure corresponding respectively to the intervals (—00, 0), (0, 00) of the real I axis. For the case shown in Figure 3.7.16, tan w = (3.7.39) 1“ E = ., = 3", =55.°. 4hU 05 6 51 w 7 hr=0, .3 3.8 The Joukowski transformation The Joukowski transformation §=§+ %—1, a>0, zl=x+iy, (3.8.1) .. z 1 1 . ' anditsmverse —=— {+— , §=§+lm a 2 C have important applications to the theory of airfoils. It is conformal everywhere except at z = 21:51, where dg/dz = 00 (i.e. at; = i1, where dz/d§ = O). In particular, this means that smooth contours in the 4“ plane in |§| >‘ 1 map into smooth contours in the z plane. Also, z ~ a; /2 when |z| and lgll are large, so that uniform flow at large distances from the origin in either plane maps into a uniform flow in the same direction in the other plane. _ V Consider steady flow in the g plane at speed V in a direction inclined at angle oz (0 < at < §) to the positive S axis, with complex potential Vge—i“. According to the circle theorem (§3.2), when a circular cylinder I; l = ,8 with centre at the origin and radius )6 is inserted into the flow the complex potential becomes- Zia w:V<;'e"“"+fi§e The streamlines of this flow are illustrated in Figure 3.8.1. We can plot them by first inverting (3.8.2) to obtain for g = E + in ‘ (3.8.2) wz (2V5 )2 §+in = fie”t + — 1), where w =' (p + iw; (3.8.3) the principal value of the square root is taken to ensure that to ~ V; e‘” as |§l —+ 00. This provides a parametric representation of the streamlines (on which 11/ = constant) in terms of<p (——oo < (0 < +00). 168 IDEAL FLOW IN TWO DIMENSIONS Figure 3.81 When the radius 5 > 1 the circular cylinder maps into a cylinder of elliptic cross section in the 2 plane [Figure 3.8.1(b)]. If g“ = fieie on the circular cylinder, the second of (3.8.1) gives I Z _ l i@ 1 —ie a _ 2 (fie + fie ) , so that the elliptic cylinder is given pararnetrically by x=§<fi+%>cos®, y=g<fl—%>sin®, x2 y2 V - a ,1 a 1 — ——= z- ._ —_—_ _# 3.4 13, AZ+B2 1, where A 2<fi+fi>, B 2<fi fl), (8) a=\/A2—B2, ,3: A+B>1.' A—B 3.8 THE JOUKOWSKI TRANSFORMATION 169 Figure 3.8.2 Thus in the 2 plane the mean flow in the at direction is around the elliptic cylinder. Now as lg], |z| —> 00, 22V w ~ §Ve"i°‘ ~ —e‘ a ict because g ~ 2z/a. Hence, if the speed in the z plane at large distances from the elliptic cylinder is U, we must set U = 2V/a, and the corresponding complex potential is then given by (3.8.2) and the first of (3.8.1) in the form U _ia '82 2 ia 11):? (Z‘l' zz—a2)e +'<z+;zf_azj ‘=%[<z+ aw We plot the streamline pattern around the elliptic cylinder [Figure 3.8.1(b)] by using (3.8.3) to calculate g = S + in in terms of w 2 go +i1/r and then substituting into the second of (3.8.1) to obtain the parametric representation of a point z = x + i y in terms of go (—00 < go < 00) on each streamline 1/r = constant. I It is evident from the symmetry of both of the ideal flows in Figure 3.8.1 that the fore—aft surface pressure forces are in equilibrium, and that there is no net force on either body (D’Alembert’s paradox). However, Bernoulli’s equation, 1 1 p = EpoUZ — EMWY; shows that the excess pressure p is a maximum on the surface at the stagnation points, labelled C, C’ in the figure. For the elliptic cylinder this means that there must be a net moment on the cylinder, tending to turn it in the clockwise direction, so that its major axis is facing the oncoming mean stream. To verify this the moment M3 (per unit span) about the origin is calculated by means of Blasius formula (3.3.3) (see Figure 3.8.2): loo dw 2 M3——R6[2 dZ], 170 IDEAL FLOW IN TWO DIMENSIONS where the integration can be taken around any contour enclosing the cylinder. In par- ticular, we can integrate around a circle of radius lzl —+ 00, where d_w_g —icv Z Zia _ Z alz—2{e [1+(Z2_a2)%]+fle [1 (Z2_42) NI— Then, by residues, . 1 . moment = Re [_§p0fliU2a2 (6—21a7_ ’82)] 2 1 . = —%p,,Uzsin2oz = 774.42 — BZMOUZsinZoz. (3.8.6) Thus the mean flow exerts a clockwise couple on the cylinder, tending to turn it broadside to the stream, a conclusion that is applicable for any elongated body (such as a boat in a stream). The couple vanishes when at = 0, when the major axis of the cylinder is parallel to the stream. However, this configuration is clearly unstable. On the other hand, the other equilibrium position oz 2 g is stable. ' 3.8.1 The flat—plate airfoil When ,6 —> 1 elliptic image (3.8.4) in the z plane of the circular cylinder of Fig— ure 3.8.1(a) collapses onto the strip [xl < a, y = O, which can be regarded as a ‘flat—plate’ approximation to an airfoil (of infinite span). The points g” = $1 on the cylinder cor— respond respectively to the trailing edge 2 = —a and the leading edge z = +a of the airfoil. These are singular points at which dt/dz = 00, where the Joukowski transfor- mation (3.8.1) ceases to be conformal. The generation of lift by an airfoil requires it to be inclined at a small angle 0 f attack or to the mean flow direction. This is, of course, just the problem considered previously for the elliptic cylinder. We find that the velocity potential of the mean flow is given by setting V = aU/2 and fl = 1 in (3.8.2): .w = % (ge—za + . (3.8.7) In the 2 plane we have, setting ,8 = l in (3.8.5), warp—l—ir/r:U(zcosoz—isinon/z2—a2). (3.8.8) The streamline pattern (1/; = constant) is plotted in Figure 3.8.3. The flow exhibits fore—aft asymmetry, with leading and trailing stagnation points A and B where the excess pressure is a maximum. The complex velocity dw izsinoz —=U cosoz———- dz Z2_a2 3.8 THE JOUKOWSKI TRANSFORMATION Figure 38.3 is singular at the edges z = in, where streamlines are required to turn through 180°. There is no net force on the airfoil, but there is a couple of magnitude —%2m2 p0 U 2 sin 2a tending to rotate it in the clockwise direction. The flow past an airfoil in practice is subject to relatively strong viscous action in the neighbourhood of the trailing edge. When the motion starts from rest the flow everywhere is initially irrotational. The high—speed flow around the trailing edge must necessarily be rapidly slowed as it approaches the stagnation point B. In reality, however, the flow at the edge immediately separates resulting in the formation of a ‘Vstarting vortex’ of positive circulation F that is convected away by the mean flow. According to Kelvin’s circulation theorem the progressive shedding of circulation from the trailing edge must be compensated by the growth in an equal amount of negative circulation around the airfoil, the effect of which is to progressively shift the stagnation point B to the trailing edge where it ultimately cancels the irrotational flow edge singularityThus vorticity continues to be shed until the flow at the trailing edge becomes continuous, that is, until the edge flow is smooth and leaves the edge tangentially. The process is very rapid in practice, the shed vorticity coagulating into the starting vortex that is quickly carried away in the mean flow, so that it eventually ceases to affect the motion at the I airfoil. However, the influence of the circulation induced around the airfoil by shedding is to maintain the smooth flow from the trailing edge. The requirement that the flow be smooth at the trailing edge determines the magni- tude of P. In the r plane we obtain the velocity potential in the presence of the negative circulation about the airfoil by adding (iF/Zyr) ln 9“ to the right—hand side of (3.8.7): aU . em ir = — ‘1“ —— —— . 3.8.9 w 2<re +£>+2fllnr ( ) 172 IDEAL FLOW IN TWO DIMENSIONS o) I high speed, low pressure Figure 3.8.4 The complex velocity in the z plane then becomes dw izsinoz i1“ - — = U 0080: — —— + ———. 3.8.10 dz ( 22 — a2) Zyrx/ z2 — a2 ( ) The value of F is found by application of the Kumz—Joukowski hypothesis that the velocity should remain finite at z = a, which yields F = ZirUa sinoz, (3.8.11) in which case dw . . z—a — = Ucosoz—zUsmoz . 3.8.12 dz z + a ( ) The flow therefore leaves the trailing edge tangentially with velocity U cos at on both sides of the airfoil. The situation is illustrated in Figure 3.8.4. This compares the stream- line patterns (a) without circulation, and (b) with circulation (3.8.11) around the airfoil, when the airfoil and flow are rotated clockwise through the angle of attack at, so that the incident mean flow is horizontal. '“T 3.8 THE JOUKOWSKI TRANSFORMATION 173 pOFUsina ‘\ pol‘Ucosa Figure 3.8.5 3.8.2 Calculation of the lift There is no net force between the airfoil and fluid except in the presence of circulation. In this case the force is determined in two dimensions by use of the time-independent - form of Blasius formula (3.3.2). For steady flow dw /dz is given by (3.8.10) (with respect to coordinate axes orientated as in Figure 3.8.3), so that ‘ ’ . 7 . . . . 7 _ 1,00 dw ‘ zpo zzU smoz ll" “ 5—15: _ <_) dz: _— (mosa— _ +___ dz. 2 5 dz 2 S /z2_a2 .271 /Z2_a2 The integrand is regular throughout the region occupied by the fluid, and the value of the integral is therefore the same as that evaluated on a large circle lzl = R —> 00. On this circle only the term in the integrand that behaves like C / 2 (C = constant) can make a nontrivial contribution to the integral, by an amount equal precisely to 217i C. Hence 1171—in = *pol‘Usinoz — ipol‘Ucosa, i.e., (F1, F2) 2 pOPU(— sinoz, cos oz). The net force F on the airfoil (orientated as in Figure 3.8.3) accordingly consists of a component ,oOFUsinoc in the negative x direction and a component 'poFU cosoz in the y direction. The overall force is therefore a lift force of magnitude lift = poFU a 27rp0U2a sina per unit span, (3.8.13) directed as indicated in Figure 3.8.4(b) at right angles to the impinging mean flow. The / lift is the resultant of a component of magnitude poPU cos or normal to the plane of the airfoil and the component poFU sina parallel to the airfoil (Figure 3.8.5); the latter is produced by leading—edge suction. There is no suction at the trailing edge because the flow velocity remains finite there. 3.8.3 Lift calculated from the Kirchhoff vector force formula The lift can also be calculated in a very convenient fashion by use of formula (4.5.15) of Chapter 4..For steady motion in the x direction and when viscous forces are neglected? the lift (in the y direction) is given by lift: p; / VX2 . w /\ vreld3x, (3.8.14) V 174 IDEAL FLOW IN TWO DIMENSIONS (6!) y (b) Circulation/1‘ Lift/poI'U 0 5 10 15 20 25 Ut/a Figure 3.8.6 where 2600 is the y component of the Kirchhoff vector (the velocity potential of flow past the airfoil having unit speed in the y direction at large distances from the airfoil) and vrel is the convection velocity of the shed vorticity relative to the airfoil. When the motion is steady, all of the wake vorticity shed from the trailing edge has been swept away towards x = +00. At such points X2 ~ y, where it may be assumed that the convection velocity vrel = (U, 0): lift per unit span = pO/ w(x — Ur, y)dedy E pOFU, V where a) is the shed vorticity distribution of total circulation 1". 3.8.4 Lift developed by a starting airfoil When the angle of attack or is small it is possible to obtain an analytical representation of the growth of the lift for an airfoil that starts impulsively from rest at z‘ = O and proceeds to translate at constant speed U (Wagner 1925). Let the motion be in the negative x direction in an otherwise stationary fluid [Figure 3.8.6(a)], and let the coordinate origin translate with the airfoil at the midchord position. In a linearised approximation 3.9 THE JOUKOWSKI AIRFOIL 175 (to first order in a) it may be assumed that the vorticity a) shed from the trailing edge is confined to a vortex sheet lying along the x axis between x = a and x = a + Ur [that is, that vrel = (U, 0)], so that a)(x, t) = y0(x, t)6(y), ' (3.8.15) where y0(x, t) is the circulation per unit length of the sheet. It may then be shown (by the method discussed below in §3.12) that 2a U 00 e—ik(Ut-x)dk VOL“: I) = — ——.—<1>_“Tr ‘1 < x < a + Ur, (3.8.16) ” -°° (k+10)[H0 (ka) +iH1 {an} where H81) and H51) are Hankel functions, and the integration path passes just above any singularities on the real k axis. This integral is easily evaluated numerically and can be used with formula (3.8.14) to calculate the lift force as a function of time. The circulation in the wake at time t is a+Ut / yo(x, t)dx, ll which is effectively equal to the total shed circulation l" N ZyromU when U r /a exceeds about 10; it is plotted as the dashed curve in Figure 3.8.6(b). p Because to ~ 0(a), the Kirchhoff vector X’g in (3.8.14) can be approximated by its value for an airfoilat zero angle of attack (Table 2.19.1), namely X2 = Re (4m). in terms of which y0(x, t)xdx When U r /a >> 1 the shed vorticity is far downstream of the airfoil, where 3X2 / 3 y —-> 1, and the lift tends to the Kutta—Joukowski value 271mm,, U2 E p01“ U. The approach to this limit is plotted as the solid curve in Figure 3.8.6(b), which also indicates that the lift immediately jumps to half its final value as soon as the airfoil begins to move. ajf2 a+Ut lift per unit span = ,oOU/ y0(x, t)6(y)—6Fdxdy = ,0on (3.8.17) V a 3.9 The Joukowski airfoil The generation and release of the starting vortex removes the singular velocity at the trailing edge and eliminates the associated suction force. The corresponding pressure distribution on the airfoil — a combination of leading—edge suction and a sideways pres- sure force — produces a lift force in a direction precisely at right angles to that of the impinging mean flow. For the thin-plate airfoil, however, there remains a leading—edge flow with a nominally infinite maximum velocity, which is not realisable in practice and at which a real flow would immediately separate; this would create a region above the airfoil of relative low velocity and high pressure, filled with vorticity, that would ulti— mately cause the airfoil to ‘stall’. This is avoided by proper airfoil design. The leading edge must be sufficiently thickened and rounded and thereby furnished with a nonzero 176 IDEAL FLOW IN TWO DIMENSIONS high speed low pressure . Figure 3.9.1 radius of curvature (see Figure 3.9.1). The effect is that separation does not occur pro- vided the angle of attack or is sufficiently small (say, less than about 8°—10°). Indeed, the convergence of the streamlines in Figure 3.8.4(b) at the leading edge shows that flow over the airfoil forward of the stagnation point A is accelerated. This is also true for a rounded edge: The flow is accelerated into the low-pressure region above the airfoil and therefore tends to remain attached to the surface provided the radius of curvature at the leading edge is not too small. This would not happen at the trailing edge of Fig- ure 3.8.4(a), however, even when rounded, because the edge flow is directed towards the high—pressure stagnation point 1B, which retards the fluid and inevitably leads to separation. / 3.9.1 Streamline flow past an airfoil The Joukowski transformation (3.8.1) can be used to study airfoils with rounded nose profiles. Recall that the edges of the thin-plate airfoil 2': id correspond to the singular points of the transformation where rig/dz : 00. These points are at g“ = :l:1 in the 5“ plane. They lie inside a cylinder 9* '= ,Bele (—7r < (9 5 7r) when )3 > 1, which must therefore be mapped conformally into a smooth profile (without edges) in the 2 plane. According to the second of Equations (3.8.1) the profile in the ,2 plane is the elliptic cylinder (Figure 3.92) z x . y __ 1 1 i 1 . a a +10 _ 2 <fi+fi>cos®+§(fi—E>sm®, —7r <®57r. Suppose the circle |§| = B in Figure 3.9.2 is shifted a distance 6 to the left along the real axis until it just touches the circle 1;] :: 1 at § 2 1. Then [3 = 1 + 6, and the new circle (in Figure-3.9.3) is §+ 5 = (1 + (3)610, —n < o 5 It, a > 0. (3.9.1) Mapping (3.8.1) of this circle is nonconforinal at g“ = 1, and its image in the z plane has a sharp e'dge'at z = a. The leading edge remains smooth, however, because the other singular point at g“ = —1 is still an interior point. The profile in the z plane is the symmetric Joukowski airfoil illustrated in Figure 3.9.3 for 6 = 0.1. The airfoil extends along the x axis over the interval a 2 1. <1+26+1+25> <x<a ,7 3.9 THE JOUKOWSKI AIRFOIL 177 5“ plane 2 plane Figure 3.9.2 with chord 2a(1 + 6 )2 / (1 + 28), and the radius of curvature of the nose is 8a62(1 + 6)3 = ~ W- 7?. In the 4“ plane the complex potential of flow at speed U at angle at to the real axis is given by the following modification of (3.8.9): _Ua V 1 2 id w _ 7 +5)e—“1 + Li (5+5) The trailing edge of the airfoil corresponds to the point 5 = l, and the velocity will remain finite provided dw/d; = 0 there, which yields ' :1 + 1n(§ + 8). (3-9-2) l‘ = 271Ua(1 + 5) sina. When this condition is satisfied the velocity is finite everywhere on the airfoil. The airfoil experiences a lift force equal to pal" U per unit span, as in the case of the thin—plate airfoil. 5“ plane z plane Figure 3.9.3 l 7....» i. E l . l } 178 IDEAL FLOW IN TWO DIMENSIONS T poFU _,, U Figure 3.9.4 Figure 3.9.4 shows the typical streamline pattern when the angle of attack at = 10° (21 large angle for practical airfoils, chosen for the purpose of illustration), when the airfoil and flow are rotated through this angle so that the incident flow is horizontal. Symmetric airfoils of this kind are used for tailplanes and rudders when there is no preference in the desired direction of lift. For aircraft wings, however, improved lift characteristics at smaller angles of attack are obtained by use of cambered profiles, where the lower side of the airfoil is either flat or concave. We can generate such profiles by considering flow around a circle in the 4" plane that again passes through the singular point 4“ = 1, but with its centre shifted to g = —8 + i6’ (0’ in Figure 3.9.5), that is the circle g: —5+is’+,/(1 +5)2 +5269, —7r < o 5 Jr. The leading edge of the airfoil is rounded because g = —1 lies within the circle. The case shown in the figure corresponds to 6 = 0.1, 5’ = 0.2. The flow pattern can be calculated as before, but such details are left as an exercise for the reader. Z plane {—(—6+i6‘) = \/{(1+6)2+6'2}ei9 . Figure 3.95 ...
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Howe_conformal_mapping - 3.8 THE JOUKOWSKI TRANSFORMATION...

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