344 7 Separated Vortex Flows Thus, in (7.21), for the term with t we still have (7.17a,b), while for the terms with e θ we have, noticing ∂ s t = κn with κ being the curvature of C s , (e θ ×∇) ·ω =(n∂ s − t∂ n ) ·(tω s + e θ ω θ )=κω s − ∂ n ω s = u s (κ −∂ n log u s ) dΓ dψ − ru 2 s d 2 Γ dψ 2 . Therefore, (7.21a) yields d 2 Γ dψ 2  C s r 2 u s ds − dΓ dψ  C s r(κ − ∂ n log u s )ds =0, where the integrals are functions of ψ only. It thus follows that dΓ dψ = Ce  ψ g(η)dη ,g(ψ)=  r(κ − ∂ n log u s )ds  r 2 u s ds . (7.23) But, as C s shrinks to the core center as assumed, since n points toward the center there must be u s → 0 + with ∂ n u s < 0andκ → +∞. Hence, Γ would be singular if C = 0. The permissible solution is thus simply Γ = constant, implying that ω s =0andv = C/r (C = 0 if the bubble flow extends to the z-axis). On the other hand, this removes the e θ -component in (7.21b), making it the same as (7.18) and recovering (7.14). The proof of the theorem is therefore completed. We make two remarks on the theorem and closed bubble flow. First, the formation mechanism of these bubble flows is very different from that of con- centrated vortices by the rolling up of free vortex layers. For the former the viscous effect has to take sufficient time to fully act on the motion, sending the vorticity from the sheet to the interior, which finally reaches an equilibrium steady state. The sheet vorticity is supplemented by the outer flow. But for the latter there is no sufficient time for diffusion to reach equilibrium state. Goldstein and Hultgren (1988) have pointed out that in this case the vorticity can have variable distribution in closed streamlines. Second, the physical explanation of the theorem is simple. For steady two- dimensional flow, the viscous vorticity equation u ·∇ω = ν∇ 2 ω is a diffusion equation. If ω changes across streamlines, there must be an inward or outward vorticity diffusive flux. But at the center of closed stream- lines there is no vorticity source or sink (Sect. 4.1); so in steady state this diffusion cannot exist. The only possibility is, therefore, a constant vorticity, which implies no vorticity diffusion. In contrast, for rotationally symmetric flow, in cylindrical coordinates (x, r, θ) by (7.15) one finds a uniform axial viscous force ν(∇×ω)=−2ανe x , (7.24) which is balanced by a uniform pressure gradient. Thus, the inviscid velocity distribution is not altered. Both this physical interpretation and the proof 7.2 Steady Separated Bubble Flows in Euler Limit 345 procedure of the theorem stress the key role of vorticity diffusive flux σ = ν∂ n ω defined by (4.17) in the formation of the bubble’s core region. In fact, that flux was precisely introduced from examining the line integral of ν∇×ω, and can well replace ν∇×ω in the invariant conditions. To see this, we use the notation of (4.23) so that on a surface S −νn ×(∇×ω)=σ − σ vis . Then by (7.11) one easily obtains (σ − σ vis ) ×n =(u ·∇u) π + ∇ π p =(ω ×u) π + ∇ π H, (7.25) where again suffix π denotes tangent components. Now let S be a stream surface with u n = 0. Then for two-dimensional flow with both σ vis and (ω × u) π vanishing, along an open segment of a streamline C s we simply have H B − H A =  B A σ ds (7.26) at any Reynolds number, indicating that the difference of stagnation en- thalpy at ends points is solely due to vorticity diffusion across the streamline (Chernyshenko 1998). This is why in the Euler limit H becomes constant along the streamline. The corresponding invariant condition for closed streamlines is, evidently,  C s ∂ω ∂n ds =0, (7.27) which is an alternative form of the two-dimensional version of (7.16). For axisymmetric flow, from (7.20) it follows that r(n × e θ ) ·(σ − σ vis )=u s ∂Γ ∂s , (n ×u) ·(σ − σ vis )=u s ∂H ∂s , which are the alternative form of (7.21a,b) and yield, along a segment of a projected streamline, Γ B − Γ A =  B A r u s t ·(σ − σ vis )ds, (7.28a) H B − H A = −  B A e θ · (σ − σ vis )ds +  B A v u s t ·(σ − σ vis )ds, (7.28b) which explains the physical source of the variation of Γ and H along C s at any Reynolds numbers. The invariant conditions alternative to (7.22a,b) are obvious. We just note that in the Euler limit with ω n = 0 (4.25) gives 1 ν σ visπ = ω · K = −ω ·  t ∂ ∂s + e θ 1 r ∂ ∂θ  n = −ω ·  ttκ − 1 r e θ e θ cos φ  = tκω s − e θ ω θ r cos φ. Thus the preceding component results can be recovered. 346 7 Separated Vortex Flows 7.2.2 Plane Prandtl–Batchelor Flows This subsection discusses the Euler limit of two-dimensional flow over a sta- tionary body with steady separated vortex bubble of area S bounded by C. 4 By (7.13), outside and inside the bubble the vorticity is zero and constant, respectively. We have Bernoulli integral not only outside the bubble but also in the core region of the bubble, since in (7.12) there is ω × u = ∇(ω 0 ψ), yielding 1 2 (∇ψ) 2 + p ρ + ω 0 ψ = p c ρ + ω 0 ψ c in the core region, (7.29) where the suffix c denotes the bubble center where q 2 = |∇ψ| 2 = 0. Therefore, let ψ = 0 along C with ψ>0andn>0 in the bubble as in Fig. 7.13, the Euler limit of the flow is a solution of the following problem: ∇ 2 ψ =0 for ψ<0, (7.30a) ∇ 2 ψ = −ω 0 for ψ>0, (7.30b)  |∇ψ| 2  =2[[H]] = c o n s t . along ψ =0, (7.30c) ∇×(ψe z )| |x|→∞ = U ∞ , (7.30d) where [[·]] = ( ·)| ψ=0 − − (·)| ψ=0 + denotes the jump across C. While in this inviscid formulation ω 0 and [[H]] (or the separation point) are arbitrary pa- rameters, only the relevant Euler solution is our concern, which is the limit of the true viscous solution as  ≡ Re −1 → 0 and in which ω 0 and [[H]] ar e specially determined. Here the Reynolds number is defined based on the body size R and U ∞ . As said in the beginning of this section, this is achieved by returning to viscous analysis, where the boundary vortex sheet is replaced by a cyclic vortex layer of finite thickness. This issue will be addressed later after discussing some general properties of the solution of (7.30). Let U e = ψ ,n | ψ=0 − be the potential velocity at C, by (4.118a) the vortex sheet strength γ = u −U e is given by γ = − [[ H]] U m ,U m = 1 2 (U e + u). (7.31) The stream function can be written as ψ = U ∞ y + ψ 1 + ψ 2 , where the first term is the stream function of uniform oncoming flow, and by the Biot–Savart law ψ 1 (x)= ω 0 2π  S log r dS(x  ), ψ 2 (x)= 1 2π  C γ(s) log r ds(x  ),r= |x −x  |, (7.32) 4 Analyses of axisymmetric Prandtl–Batchelor flow are relatively rare. One example is the Hill spherical vortex with swirl (Sect. 6.3.2). 7.2 Steady Separated Bubble Flows in Euler Limit 347 are the contributions of ω 0 and γ, respectively. Moreover, since ω 0 depends on U ∞ , we need a compatibility condition U ∞ =  2[[H]] − ∂ ∂y (ψ 1 + ψ 2 ) at the upstream end. (7.33) Then the inviscid problem amounts to finding the shape and strength of the sheet for given ω 0 and [[H]], including the separation and re-attachment points. A closed vortex bubble carried by a body will produce a lift. Let S and ω 0 be the dimensionless area and vorticity of the bubble, respectively, scaled by R and U ∞ . Then by the well-known (dimensional) Kutta–Joukowski lift formula L = ρU ∞ Γ (for more discussion see Chap. 11), with Γ = ω 0 S being the total vorticity or circulation of the bubble , the additional lift coefficient due to the bubble (not including the lift caused by vortex sheet γ) simply reads ∆C l = ∆L 1 2 ρU 2 ∞ R =2ω 0 S. (7.34) The small region near the separation point A needs special consideration. As argued in Sect. 4.4.2, in order to have a shedding vortex sheet, at A there must be [[H]] = 0 and the external potential flow must be tangent to the surface, see Fig. 7.14 later. Thus, in a small neighborhood of A, in terms of local Cartesian coordinates (x, y) the sheet equation y = f(x) must satisfy f(0) = f  (0) = 0, i.e., y = o(x). The flow inside the cusp varies mainly along the y-direction, so that (7.30b) is reduced to ∂ 2 ψ/∂y 2 −ω 0 , of which the solution satisfying ψ = 0 along y = f(x)is ψ − 1 2 ω 0 y[y −f (x)],x 1. Thus, as x → 0wehaveq = o(x) and by (7.30c) U e =  2[[H]] + o(x 2 ). Therefore, near the separation point the fluid in the cusp is indeed stationary as we inferred from the Kutta condition in Sect. 4.4.3. This being the case, the local flow is nothing but a Kirchhoff free-streamline flow, which gives f(x)=ax 3/2 + bx 5/2 + ···,x 1. (7.35) BA U y m 2L w Fig. 7.14. Sadovskii flow 348 7 Separated Vortex Flows -0.1 -0.1 0.1 0 0.1 0.3 0.5 0.7 0.9 1.1 Fig. 7.15. Streamlines for airfoil with a trapped vortex. ω 0 = −20, [[H]] = 0 .53. Adapted from Bunyakin et al. (1998) Several Prandtl–Batchelor bubble flows with the above common features have been studied. Sadovskii (1971) was the first to solve the problem for the flow of Fig. 7.14, where a pair of Prandtl–Batchelor vortices of length L are symmetrically in touch, known as the Sadovskii flow. He derived a pair of integral equations for the sheet shape f(x) and strength γ(x) with a given constant [[H]]. The equations were solved numerically. The computation was improved by Moore et al. (1988) who gave a complete set of the solutions for Sadovskii flow. Other investigated Prandtl–Batchelor flows include corner flows (e.g., Chernyshenko 1984; Moore et al. 1988) and flow over a flat plat with a forward-facing flap (Saffman and Tanveer 1984). The latter was motivated by the concept that if at a large angle of attack a stationary vortex can be captured, then the lift will be greatly enhanced. The most interesting config- uration along this line is to capture a vortex by an airfoil with a cavity on its upper surface, studied by Bunyakin et al. (1996). Owing to (7.34), such an airfoil may have additional lift but avoid early separation under strong adverse pressure gradient on the upper surface, where the original solid wall is replaced by a free shear layer like a flexible moving belt. 5 The authors found that due to the structural instability 3 of closed bubble flow, the bubble shape is very sensitive to the given ω 0 and [[H]], and only in a certain range of these parameters can a meaningful solution be obtained. Figure 7.15 plots the configuration and flow pattern. We now turn to the cyclic viscous vortex-layer. Squire (1956) was the first to exemplify that a matched asymptotic expansion can be applied to fix the flow within the layer as well as ω 0 and [[H]]. We use the same intrinsic frame (t, n) as in Fig. 7.13, and let quantities be made dimensionless by body size R, density ρ,andU ∞ . Assume now Fig. 7.14 represents a bubble on a flat wall. Let s move from A (s =0)toB (s = s B ) along the separated vortex sheet 5 As remarked in Sect. 7.1.2, the vortex in the cavity is much weaker than that formed by the rolling-up of a free vortex layer. The latter was proposed by Wu and Wu (1992). Such a strong vortex must have axial flow and is expected to be more stable as well. 7.2 Steady Separated Bubble Flows in Euler Limit 349 (in free shear layer) and returns A (s = s A >s B ) along the wall (in attached boundary layer). Let u(s, ψ) be the streamwise velocity in the vortex layer such that dψ = udn. Both free vortex layer and attached boundary layer are governed by a boundary-layer type of equation: u ∂u ∂s = U e ∂U e ∂s +  ∂ 2 u ∂n 2 ,= Re −1 . Introduce the well-known Mises transformation (e.g., Rosenhead 1963) ∂ 2 u ∂n 2 = dψ dn ∂ ∂ψ  dψ dn ∂u ∂ψ  = 1 2 u ∂ 2 u 2 ∂ψ 2 and a rescaled stream function Ψ = Re 1/2 ψ, and denote the total enthalpy inside the viscous vortex layer by g(s, Ψ ) ≡ p s + 1 2 u 2 (s, Ψ). Then the above boundary-layer equation reads ∂g ∂s = u ∂ 2 g ∂Ψ 2 . (7.36) Assume the inviscid flow in the core region has been solved, so that at s =0 − there is a known potential flow with g = g 0 (ψ), and on the wall there is aknownp(s). Then the upstream condition, periodic condition, and wall condition for (7.36) are, respectively, s =0,Ψ < 0: g = g 0 (ψ); (7.37a) Ψ>0: g(0,Ψ)=g(s A ,Ψ); (7.37b) Ψ =0,s B <s<s A : g = p(s). (7.37c) Besides, since ψ =0 + corresponds to Ψ →∞in the Euler limit, (7.30c) gives the matching condition Ψ →∞: g → H(ψ)| ψ=0 + = H| ψ=0 − − [[ H]] < ∞, (7.37d) which ensures the uniqueness of the solution. Generically, the boundary-layer approximation cannot be applied to re- gions near A and B, referred to as turn regions. But when a turn region has a cusp, only the normal variation is important, and the inviscid cusp flow away from the viscous free vortex layer and boundary layer is stationary. In fact, the characteristic flow rate Q in the turn region must be of the same order as that in a boundary layer: Q turn ∼ Re −1/2 , so that locally there is Re turn ∼ U turn L turn Re ∼ Q turn Re ∼ Re 1 2 →∞. 350 7 Separated Vortex Flows 0.2 0 0.04 0.08 0.12 3 S 2 1 0.3 0.4 0.5 0.6 R Fig. 7.16. Velocity profile as a function of ψ in the cyclic layer of Fig. 7.15 for the relevant Euler solution of flow over an airfoil with trapped vortex. Adapted from Bunyakin et al. (1998) Thus, to the leading order the flow is inviscid and Bernoulli integral holds. In this case it can be shown that when using (7.36) as the governing equation the turn region can be ignored (see Bunyakin et al. (1998) for references). Thus the formulation for the cyclic vortex layer is completed. For example, by solving (7.36) under condition (7.37), as well as the con- ventional boundary-layer equation for flow over a wall, Bunyakin et al. (1998) extend their inviscid solution shown in Fig. 7.15 to include the viscous attached and free vortex layers. Then the whole flow field is determined. Figure 7.16 shows the velocity profiles in the cyclic vortex layer. The airfoil was carefully selected to avoid any smooth-surface separation other than the fixed front and rear points of the cavity. To make the airfoil look more realistic, tangent blowing was introduced at three points of the cavity wall (arrows in Fig. 7.16). This requires an extension of boundary conditions (7.37) and is omitted here. 7.2.3 Steady Global Wake in Euler Limit The preceding discussion on plane Prandtl–Batchelor flows is for the situa- tion where the bubble size is of the same order of the body size. A different and more challenging problem relevant to the Prandtl–Batchelor flow is the asymptotic form of the entire vortical wake behind a bluff body. It will be seen in Sect. 7.4 that as the Reynolds number Re = UD/ν (based on diame- ter D) increases to about 50 the wake behind a circular cylinder starts to be spontaneously unsteady and vortex shedding occurs. However, the mathemat- ical existence of a steady but unstable wake cannot be excluded. Numerically, careful Navier–Stokes calculations (Fornberg 1985) which specifically elimi- nate the possibility of unsteadiness and asymmetry have shown that steady wake is a Navier–Stokes solution. Theoretically, such a mathematical solution has been obtained in the Euler limit. Roshko (1993) remarks that, while this 7.2 Steady Separated Bubble Flows in Euler Limit 351 solution is mainly of academic interest, “it is an intriguing and important one for theoretical fluid mechanics and it provides perspective on the ‘real problem’.” The classic Kirchhoff free-streamline wake, which is open at downstream end and the fluid therein is stationary, was criticized by Batchelor (1956b). The dilemma is: if the wake is open, how can the downstream boundary con- dition that the flow resumes uniform be satisfied? And, if the wake is closed, then the Prandtl–Batchelor theorem requires that the wake has a uniform vorticity rather than being stationary. Thus, Batchelor (1956b) proposed that the steady wake in the Euler limit is a closed bubble with ω 0 and [[H]] as para- meters. It has been found that the wake length increases linearly as Re;and its width increases initially as O(Re 1/2 ), but after Re > 150 turns to be O(Re) as well, see Fig. 7.17. Moreover, in such a big pair of separated bubbles the vorticity is basically constant as predicted by the Prandtl–Batchelor theorem; and at the outer boundary of the bubbles there is a thin vortex layer, which tends to vanish as the characteristic velocity increases toward downstream. After many researchers’ effort, a complete asymptotic theory of steady separated flow has been established and supported by numerical tests. For comprehensive reviews see Sychev et al. (1998, Chap. 6) and Chernyshenko (1998); a few major points are briefly outlined here. First, in the global bubble scale the flow is a uniquely determined inviscid Sadovski flow (where the cylinder shrinks to a point as Re →∞), of which the width-to-length ratio is h/L =0.300 and the area is S = αL 2 with α  0.44. The vorticity ω 0 is fixed such that, by the Bobyleff–Forsythe formula (2.159) and from (2.76), the total dimensionless dissipation rate and the total drag coefficient C d (nondimensionalized by ρU 2 ∞ R) are, respectively, C = ω 2 0 S  0.73, (7.38) C d = C Re . (7.39) Secondly, in addition to the global bubble-scale flow, special care is needed for the flow in turn regions, cusp, and at the body scale, as well as their matching. In the body scale, the velocity in the bubble is found to be much less than outside, so that one returns to the inviscid Kirchhoff free-streamline flow with drag coefficient C d =2k d [[ H]] , (7.40) Fig. 7.17. Schematic flow pattern of steady global wake behind a circular cylinder (the small semicircle at the left end of the plot). From Chernyshenko (1998) 352 7 Separated Vortex Flows where k d is drag coefficient in the Kirchhoff flow with the velocity magni- tude on the free streamline equal to unity, depending on the body shape and separation point. Comparing (7.39) and (7.40) yields [[ H]] = C 2k d Re . (7.41) Thirdly, to ensure the existence of the viscous solution in the cyclic vortex layer, there must be an equality among parameters (Chernyshenko 1988): [[ H]] = 1 2D 0  Cω 0 2Re , (7.42) where D 0 is a constant; for flow past an isolated body D 0  0.235. The key physics behind (7.42) is the vorticity balance. The vorticity diffuses toward the symmetry line where it vanishes, and also diffuses across the bubble boundary. This loss of vorticity must be compensated by that produced from the body surface and advected into the flow. In this problem one only needs the net effect of vorticity discharged from the body rather than the detailed diffusion and advection process; so it suffices to know the sum of vorticity diffusive flux σ and advective flux u n ω across any line segment, which is nothing but the end-point difference of the total enthalpy. Indeed, as an easy extension of (7.26), by applying (7.25) to any line segment there is (Chernyshenko 1998) H B − H A =  B A (u n ω − σ)ds. (7.43) This is why [[H]] enters (7.41), which also shows that the jump must vanish in the Euler limit. The four equations (7.38–7.40) and (7.42) then determine the four unknowns ω 0 , S, C d ,and[[H]], with only k d depending on the body shape. Namely, in addition to (7.39) for the drag and (7.41) for [[H]], there is ω 0 = 2CD 0 k 2 d Re ,S= k 4 d Re 2 4CD 4 0 ,L= k 2 d Re 2D 0 √ αC ,α 0.44. (7.44) 7.3 Steady Free Vortex-Layer Separated Flow Closed-bubble separated flows discussed in Sect. 7.2 are relatively rare in re- ality. The common situation is free vortex-layer separated flow, in which a separated vortex layer rolls into a vortex and the flows at both sides of the layer come from the same main stream. As said in Sect. 7.1.2, the free vortex- layer may come from both closed separation initiating at a saddle point of the τ w -field, for example at the apex of a slender delta wing as shown in Fig. 7.7b, and open separation initiating at an ordinary point of the τ w -field, as seen 7.3 Steady Free Vortex-Layer Separated Flow 353 in Fig. 6.1. A prototype of free vortex-layer separated flow is a pair of vortex sheets shed from a slender wing, which roll into vortices above the wing and greatly enhance the lift. Being steady and stable in a range of parameters, this kind of detached-vortex flow has become the second generation of aeronautical flow type in practical use (after the attached flow type over streamlined body; e.g. K¨uchemann (1978)). No general theory is available for free vortex-layer separated flow even in the Euler limit, because as seen in Sect. 4.4.4 the self-induced rolling-up process of a vortex sheet is inherently nonlinear. One has to appeal to approxi- mate theories or numerical simulation. The simplest theory in the Euler limit is fully linearized, in which the vortex-sheet rolling up is completely ignored so that the sheet location is known. This is the case in Prandtl’s classic lifting-line theory for a thin wing of large aspect ratio (e.g., Prandtl and Tietjens 1934; Glauert 1947; see also Chap. 11). But here we need to ad- dress the nonlinearity of the self-induction, with the expense that in some other aspects significant simplification has to be made. This is the case of the slender-body theory to be used throughout this section. 6 We consider the slender approximation of vortex-sheet conditions first, then review methods for computing the self-induced evolution of leading-edge vortex-sheet and free wake vortex sheet. Finally, we analyze the stability of a class of slender free vortex-layer separated flow. 7.3.1 Slender Approximation of Free Vortex Sheet Consider a steady flow over a point-nose slender body shown in Fig. 7.18. In a body coordinate-system Oxyz with the body axis along the x-direction and z-axis vertical up, let the local angle of attack at x be α(x)=O()  1, so that the constant oncoming velocity U has (x, z) components U =(U cos α, U sin α)=(U, Uα)+O( 2 ), (7.45) and the disturbance velocity components are (u  ,v,w)=O(U ). Due to the slenderness, the x-wise disturbance of the body to the fluid is much smaller than those in cross directions. Consequently, a three-dimensional flow problem is reduced to a cross-flow Uα over two-dimensional sections of the body at different x, and away from the vortex sheet we only need to con- sider a two-dimensional disturbance velocity potential ϕ(y, z; x). The three- dimensionality of the flow lies in the x-dependent boundary condition, with 6 This section could be shifted to Chap. 11 on aerodynamics. We put it here for understanding the basic physics of free vortex-layer separated flow as the coun- terpart of closed-bubble separated flow. Although in engineering applications the slender-body theory has now been replaced by more numerically oriented meth- ods, it provides an opportunity to demonstrate how the general theory of three- dimensional vortex sheet dynamics is specified to concrete problems of significant practical value. [...]... and 1970s so we call it the RAE method In this method (7.54), (7.55), (7.57), and (7. 58) are all necessary input A major difficulty is that once the vortex sheet rolls into a vortex with distributed vorticity and spiral arms (Chap 8) , inside the vortex core the conformal mapping can no longer be used In the RAE method (Smith 19 68) the tightly rolled-up part of the sheet is replaced by a single line vortex. .. different t 7 In numerical approaches the vortex- sheet conditions (7.48a) and (7.48b) are automatically satisfied, but the Kutta condition and Biot-Savart formula or its equivalence remain necessary 360 7 Separated Vortex Flows 0.6 1.50 0.4 1.00 1.50 z/s 0.50 1.00 0.2 0.50 0.25 0.25 0 0.4 0.6 y/s 0 .8 1.0 Fig 7.20 Location of vortex sheet (solid curve) and line vortex (circle) over a slender delta wing... leading-edge vortex layer travels downstream to the wake, it must meet the trailing-edge vortex layer and merge to a single and complicated structure The rolling-up of wake vortex- sheet alone has been beautifully computed by Krasny (1 987 ) also within the same U = x/t approximation, as already exemplified by Fig 4.21; but the evolution of the merged leading- and trailing-edge vortex sheets is of particular... gradient tensor ∇u0 Thus, within the linearized theory, the vortex system will be stable (necessary 7.3 Steady Free Vortex- Layer Separated Flow 363 for nonlinearly stable), neutrally stable, and unstable (sufficient for nonlinearly unstable) if both λ1 and λ2 have negative real parts, are imaginary, and at least one of λ1 and λ2 has positive real part, respectively It is easily verified that 1 2 (7.65a)... free shear layer and its rolling up, vortex interactions, various shear instabilities, transition to three-dimensional flow and to turbulence, and unsteady turbulent separated flow After over a century of effort since Strouhal ( 187 8) observed that the frequency f of vortex shedding is proportional to U/D with the proportionality constant now being known as the Strouhal number St = f D/U , and K´rm´n a a... irregular regime no vortex pair Re < 4 vortex pair 4 < Re < 47 2-D periodic shedding 47 < Re < 180 oblique shedding mode n = 1, 64 < Re < 160 n = 2, ~70 < Re . (7.55), (7.57), and (7. 58) are all necessary input. A major difficulty is that once the vortex sheet rolls into a vortex with distributed vorticity and spiral arms (Chap. 8) , inside the vortex core. flow. Other investigated Prandtl–Batchelor flows include corner flows (e.g., Chernyshenko 1 984 ; Moore et al. 1 988 ) and flow over a flat plat with a forward-facing flap (Saffman and Tanveer 1 984 ). The latter was. first for airship aerodynamics, and then applied to attached flow over slender wing and body, wing-body combination, etc., based on (7.47); e.g., Nielsen (1960) and Ashley and Landahl (1965). In extending