Vorticity and Vortex Dynamics 2011 Part 6 pps

242 5 Vorticity Dynamics in Flow Separation x = ξ, y = η, u = u 0 (ξ,η),v= v 0 (ξ,η)atτ =0. (5.85) The boundary conditions are essentially the same as in the Eulerian descrip- tion: (x, u)=(ξ,0), (y, v)=(0, 0) on η =0, (5.86a) x τ = u → U(x, t)asη →∞. (5.86b) Once the integration of (5.83) to (5.86) gives x = x(ξ, η, τ) at a subsequent time, one obtains y(ξ,η,τ) from (5.81b), and then velocities from (5.79). An inspection of this Lagrangian formulation reveals a key simplification: owing to the approximate nature of (5.73), the streamwise position x and velocity u can be solved independently from solving the normal position y and velocity v. Moreover, although a rigorous proof is not available, there has been strong evidence that the dynamic system (5.83–5.86) remains regular even after the singularity is formed (but the solution for t>t s may not be physically realistic). Accepting this as a hypothesis, then, the singularity develops solely from the continuity equation. In this sense, the theory is entirely within kinematics. In particular, (5.81a) indicates that the mechanism for the singularity to occur is similar to the formation of shock in gas dynamics due to the coalescence of characteristics. In fact, the fluid-element normal location y can be found by integrating (5.81b) along the curves x =const.inthe(ξ, η) plane. Let l be the arclength along such a curve with l = 0 at the wall η =0, then y =  l 0 dl |∇ ξ x| =  s 0 dl (x 2 ξ + x 2 η ) 1/2 . (5.87) Now, at the separation point u x should be unbounded; so if u ξ and u η is bounded then (5.82) implies that y ξ and/or y η must be unbounded. Thus, the mapping between (x, y) and (ξ, η) is singular, which in (5.87) manifests as ∇ ξ x =0 at (ξ,t)=(ξ s ,t s ). (5.88) This singularity condition has two effects. First, all infinitesimal deformations δξ of fluid element do not cause any change of the streamwise position of the element in physical space: δx = δξ ·∇ ξ x =0 at (ξ,t)=(ξ s ,t s ). (5.89) Namely, as fluid elements move along their pathlines, they are blocked and squashed at a vertical barrier at some x, and hence must extend unbound- edly along the normal as schematically shown in Fig. 5.18, resulting in the separation. Second, by (5.83b) we see at once that (5.88) implies the first part of the MRS criterion, (5.72a) or (5.77b). Therefore, when the fluid-element squashing process reaches the singular state, it reaches zero-vorticity state too. Because the Lagrangian description does not distinguish steady and unsteady flow, 5.4 Unsteady Separation 243 Shen (1978) points out that the same mechanism as sketched in Fig. 5.18 is also responsible for the Goldstein singularity in steady separation within boundary-layer approximation, and the MRS version of the Prandtl condition (5.1) is derivable from (5.87) that is “no more than a formalized expression of the Prandtl concept — that the boundary layer must break away when a packet of fluid particles are stopped in their forward advance along the wall.” The second part of the MRS criterion can also be derived from (5.88). In fact, denote the Lagrangian coordinates of the singularity point by ξ MRS ,of which the propagation speed is (a dot denotes d/dt), owing to (5.88), d dt x(ξ MRS ,t)= ˙x + ˙ ξ MRS ·∇ ξ x =˙x, (5.90) which is indeed the local streamwise velocity of the element, in agreement with (5.77a). Therefore, the MRS criterion is rationalized. Van Dommenlen and Shen (1982) conducted a numerical calculation based on the earlier theory for flow over impulsively started circular cylinder. As sketched in Fig. 5.19, the singular point was found to appear at θ = 111 ◦ and t =3.0045, which moves upstream with u = −0.52U. The separation location differs from the full Navier–Stokes solution (Fig. 5.15) since the former is for Re →∞asymptotically rather than at a finite Reynolds number. The separation location is also different from that of the Goldstein singularity for steady flow, θ = 104.5 ◦ . After the singularity is formed, the upper part of the boundary layer turns to a free separated vortex layer. On top of Fig. 5.19 are the profiles of velocity and vorticity (normalized by wall vorticity) close to separation, from which it is evident that as the bifurcation tears the boundary layer apart the irrotational region in between is enlarged. We now introduce local scales in the neighborhood of (ξ s ,t s ) so that the singularity can be removed. Assume t s is the first time for a singular boundary- layer separation point to form. Since x(ξ,t) is a regular function of ξ and t around (ξ s ,t s ) one can perform a Taylor expansion of x and form the deck structure thereby. Meanwhile, (5.88) should also be expanded to a Taylor series since it may not be satisfied anywhere for δt = t − t s < 0. To simplify the expansion, let δξ = ξ − ξ s , and make a proper shift and rotation of the previous arbitrarily chosen Lagrangian coordinate system to a new system (l 1 ,l 2 ,t) . The Jacobian J is invariant under the coordinate transformation, and has characteristics dl 1 dy = −x ,22 l 2 + ···, dl 2 dy = 1 2 x ,111 l 2 1 +˙x ,1 δt + ···, (5.91a,b) which results in a singularity when both right-hand side expressions vanish. Although at t = t s the boundary-layer approximation blows up, at times shortly before t s a rescaled asymptotic expansion can be conducted to describe the flow field. After some algebra, it can be found that the proper scales are l 1 = |δt| 1/2 L 1 ,l 2 = |δt| 3/4 L 2 , (5.92a,b) ¯x ≡ x −x(ξ s ,t)=|δt| 3/2 X, y = |δt| −1/4 Y, (5.93a,b) 244 5 Vorticity Dynamics in Flow Separation x s x s x s U y S S w Fig. 5.19. Vorticity contours obtained from the Lagrangian boundary-layer equa- tion for impulsively started circular cylinder. t =3.0045. On top are the profiles of velocity and vorticity (normalized by wall vorticity) close to separation. Reproduced from Van Dommenlen and Shen (1982) O(|dt | 3/2 ) O(|dt | -1/4 ) O(1) y x O(1) Fig. 5.20. Scales of unsteady boundary-layer bifurcation at δt before singularity is formed. Reproduced from Cowley et al. (1990) where L 1 ,L 2 ,X,Y = O(1). These scales at a δt < 0 are shown schematically in Fig. 5.20. Then, integrating (5.91) yields an analytical solution for Y , and a further O(1) transformation (L 1 ,X,Y) → (L ∗ 1 ,X ∗ ,Y ∗ ) similar to (5.55) can scale out all coefficients. In terms of the variables with asterisk, the analytical solution takes canonical form 5.4 Unsteady Separation 245 Y ∗ ∼  L ∗ 0 −∞ dL ∗ (2X ∗ − 3L ∗ − L ∗3 ) 1/2 ±  L ∗ 0 L ∗ 1 dL ∗ (2X ∗ − 3L ∗ − L ∗3 ) 1/2 , (5.94) where L ∗ 0 is the real root of the cubic polynomial in the square root of the denominator. The solution (5.94) can be cast to elliptic integrals of the first kind. The signs of the square roots and the limits of integration are determined by the topology of the lines of constant X ∗ that consists of three segments shown in Fig. 5.20. Leaving the mathematic details aside, the scaled vorticity contours of the nearly separated boundary layer is shown in Fig. 5.21, which also shows the sudden thickening of the boundary layer. Finally, similar to the steady case where the scaling is closed by find- ing the relation of the lower-deck thickness δ and Re, we now need to close the theory by finding the relation of δt and Re. Once again, since the MRS criterion implies the shearing is vanishingly small near S, the only possible mechanism to balance the normal extension of fluid elements is the normal pressure gradient ∆p y ∼ ∆p x in an irrotational upper deck. In the separation zone shown in Fig. 5.20, as a fluid element moves past a streamwise extent O(|δt| 3/2 ) but climbs up a thickness (in global scale) O(Re −1/2 |δt| −1/4 ), it experiences a upwelling velocity v of O(Re −1/2 |δt| −7/4 ). The balance in the normal momentum, ∂v/∂x = −∂p/∂y, together with the fact that x ∼ y in the upper deck, indicates that the locally induced pressure reads ∆p ∼ v ∼ Re −1/2 |δt| −7/4 . So the pressure gradient is ∆p x ∼ Re −1/2 |δt| −7/4 ·|δt| −3/2 = Re −1/2 |δt| −13/4 . Then an unsteady triple-deck interaction (strictly, it is a quadruple structure) appears if ∆p x is of the same order as the acceleration x tt = O(|δt| −1/2 )in the expanding central region. This balance occurs when Re −1/2 |δt| −13/4 = |δt| −1/2 , i.e. |δt| = O(Re −2/11 ). (5.95) -10 0 0 5 10 10 X * Y * 20 Fig. 5.21. Canonical vorticity contours near separation. From Cowley et al. (1990) 246 5 Vorticity Dynamics in Flow Separation By (5.93b), at this time the scaled boundary-layer displacement thickness has grown to O(Re 1/22 ). The earlier scalings are in agreement with the analysis in terms of Eulerian description (e.g., Elliott et al. 1983) as well as some numerical tests. How- ever, the very small power of Re may lead to large difference between theory and experiment at moderate Re. A more fundamental problem of unsteady boundary-layer separation theory, in either Lagrangian or Eulerian descrip- tion, is that the unsteady triple-deck structure itself turns out to terminate at yet another finite-time singularity. While it is possible to rescale the vari- ables at times close to this new singularity in an even shorter time-scale, the rescaling process may have to go on as a cascade. It remains an open issue on whether this situation reflects the physical cascade process in tranasition to turbulence associated with successive instabilities at a series of decreasing scales, or simply due to the limitation of the matched asymptotic theory itself. 5.4.3 Unsteady Flow Separation We now turn to generic unsteady separation. Although some of the results of Sect. 5.2 are equally applicable to unsteady flow, a complete, general, and local unsteady separation theory had not been available until a very recent work of Haller (2004), who obtained an exact two-dimensional theory for both incompressible and compressible unsteady flow with general time dependence, applicable to arbitrary stationary or moving wall. The theory is essentially of kinematic nature, in which the separation point (to be defined later) can be either fixed on the wall or moving along the wall. The consistency of the the- ory with both Prandtl’s theory for two-dimensional steady separation and the Lagrangian theory unsteady boundary-layer separation has been confirmed. The theory has been further improved by Haller and coworkers, and extended to three dimensions (Kilic et al. 2005; Surana et al. 2005b,c). Therefore, we de- vote this subsection to an introduction to Haller’s unsteady separation theory based on Haller (2004) and Kilic et al. (2005), focusing on the simplest case. Namely, we assume the flow is incompressible with ρ = 1, and the separation point is fixed to a no-slip wall ∂B at y = 0, referred to as fixed separation. Its results turn out to apply to any unsteady flow with a mean component, including turbulent boundary layers and flows dominated by vortex shedding. The assertion made in Sect. 5.4.2, that flow separation is essentially a ma- terial evolution process, can be clearly demonstrated by the time evolution of a fixed separation and reattachment for an analytical periodic flow model shown in Fig. 5.22 (see (5.111) later). Similar to Fig. 5.19, a set of mater- ial lines initially aligned to the wall evolves to form an upwelling, then a singular-looking tip, and then a sharp spike. More crucially, there appears a distinguished material line, which attracts fluid particles released from its both sides and ejects them into the main stream. This special material line signifies the separation profile, of which a rational identification is the key 5.4 Unsteady Separation 247 (a) (b) (c) (d) (e) (f) Fig. 5.22. Time evolution of material lines and streamlines for a periodic separation bubble model (5.109) with circular frequency 2π.(a) t =0,(b) t =8.2, (c) t =9.95, (d) t =15.0, (e) t =18.65, (f ) t = 25. The time-dependent curve initially cutting the material lines but then serving as their approximate asymptotic line is the separation profile (up to quadratic order) to be identified later. From Haller (2004) of a general separation theory. Note that Fig. 5.22 shows that instantaneous streamlines are irrelevant when the separation is unsteady. This being the case, we start from the dynamic system (5.5): ˙x = u(x, y, t), ˙y = v(x, y, t), (5.96) which due to the no-slip condition and continuity can be cast to, similar to (5.31), ˙x = yA(x, y, t), ˙y = y 2 C(x, y, t), (5.97) where A(x, y, t)=  1 0 u y (x, sy, t)ds, C(x, y, t)=  1 0  1 0 v yy (x, sqy, t)q dq ds. (5.98) The incompressibility further requires A x +2C + yC y =0. (5.99) Now, denote the material line signifying the separation profile by M(t), which as seen in Fig. 5.22 is “anchored” to the fixed separation point (x, y)=(γ,0) for all t by the no-slip condition. In dynamic system terms, M(t)isanunstable manifold for a fixed point on the wall, locally described by a time-dependent path x = γ + yF(y,t). (5.100) 248 5 Vorticity Dynamics in Flow Separation While generic material lines emanating from the wall converge to the wall as t →−∞, M(t) is an exception, with the following properties: 1. it is unique, i.e., no other separation profile emerges from the same bound- ary point; 2. it is transverse, i.e., does not become asymptotically tangent to the wall in backward time; 13 and 3. it is regular up to nth order (n ≥ 1), i.e., M(t) admits n derivatives that are uniformly bounded at the wall for all t. Then, substituting (5.100) into (5.97), one finds that M(t) satisfies a par- tial differential equation (the separation equation) F t = A(γ + yF,y,t) − yC(γ + yF,y,t)(F + yF y ), (5.101) from which unsteady separation criteria can be deduced. By (5.100), approx- imate separation profile can be expressed by series expansion x = γ + f 0 (t)y + f 1 (t)y 2 + 1 2 f 2 (t)y 3 + 1 6 f 3 (t)y 4 + ···, (5.102) where f 0 (t)andf 1 (t) are the slope relative to the y-axis and curvature of M(t)at(x, y)=(γ,0), respectively. Consider separation criteria first. Setting y = 0 in (5.101) yields a linear equation ˙ f 0 (t)=a(t), thus (t 0 is an arbitrary reference time) f 0 (t)=f 0 (t 0 )+  t t 0 A(γ,0,τ)dτ. (5.103) Since by the earlier property (2) M(t) cannot become asymptotically tangent to the wall, f 0 (t) must be uniformly bounded. By (5.98) and u y = −ω on the wall, therefore, a necessary separation criterion is lim t→−∞ sup     t t 0 u y (s, τ)dτ    = lim t→−∞ sup     t t 0 ω(s, τ)dτ    < ∞, (5.104) where and below s denotes the separation point (γ, 0). For steady separation the integral becomes ω s (t−t 0 ), so (5.104) is reduced to Prandtl’s first criterion (5.1a). Then, as the generalization of (5.1b), by using v yy = −u yx = ω x on the wall and after some algebra, it can be proved that the second necessary separation criterion is  −∞ t 0 u xy (s, τ)dτ = −  −∞ t 0 ω x (s, τ)dτ = ∞, (5.105) 13 All other material lines that start to be transverse remain so for any finite time, but become tangent to the wall as t →−∞(G. Haller, 2005, private communi- cation). 5.4 Unsteady Separation 249 which for steady flow becomes (t 0 + ∞)ω x = ∞ and hence ω x > 0, equivalent to (5.1b). In particular, for periodic flow with period T , the integration interval in (5.104) can be replaced by (0,T); while in (5.105) one splits the integrand into a mean and an oscillating part, with the former having to be negative. Thus, the two necessary separation criteria are simply reduced to  T 0 ω(s, t)dt =0,  T 0 ω x (s, t)dt>0. (5.106a,b) In general, criterion (5.104) can be expressed in a form more suitable for computations. Recall that any material lines emanating from any wall points near s will align with the wall as t →−∞, which by (5.103) is possible only if, for sufficiently small |x − γ|,  −∞ t 0 u y (x, 0,τ)dτ =  +∞ if x>γ, −∞ if x<γ. Thus, the backward integral of u y = −ω at s admits a sign change arbitrarily close to s for sufficiently large |t −t 0 |. Then, since the integral i t (x) ≡  t t 0 u y (x, 0,τ)dτ (5.107) is a continuous function of x at any t, it must have at least one zero that approaches s as t →−∞. Therefore, we may define an effective separation point γ eff (t, t 0 )by  t t 0 u y (γ eff , 0,τ)dτ = 0 such that γ = lim t→−∞ γ eff (t, t 0 ), (5.108) see Fig. 5.23. The reattachment point can be similarly defined. Moreover, while criteria (5.104) and (5.105) permit weak separation by which particles near s may turn back towards the wall for a finite period of time, a slight revision of (5.105) can give a sufficient condition for stronger monotonic separation by which particles near s move away monotonically from the wall without turning back. Haller (2004) proves that this is simply ensured by −u xy (s, t)=ω x (s, t) >c 0 > 0, (5.109) of which the physical implication is obvious (cf. Fig. 4.12). Haller (2004) has used the earlier theory to derive explicit general formulas for the time-dependent coefficients f 0 (t),f 1 (t), of (5.106) up to quadratic order. In particular, for steady flow the slope of M reduces to f 0 = − u yy (s) 3u xy (s) = − p x (s) 3τ x (s) , 250 5 Vorticity Dynamics in Flow Separation i t (x) g eff (t 1 ,t 0 ) g eff (t 2 ,t 0 ) g t = t 2 >t 1 t = t 1 x Fig. 5.23. The convergence of γ eff to γ in agreement with (5.30) where φ is the angle of the separation line relative to the x-axis. The second equality uses the Navier–Stokes equation as we did in Sect. 5.2, except which all the earlier results are evidently kinematic; use was made of only the continuity equation. The fixed separation conditions (5.104) and (5.105) have been improved by Kilic et al. (2005), assuming that the unsteady velocity fields under con- sideration admit a finite time asymptotic average in time. After some lengthy algebra, the authors show that (5.104) and (5.105) can be replaced by lim T →∞ 1 T  t 0 t 0 −T ω(s, t)dt =0, (5.110a) lim T →∞ 1 T  t 0 t 0 −T ω x (s, t)dt>0, (5.110b) which are a direct generalization of (5.106) to aperiodic flow. As an analytic example, consider a periodic separation bubble model de- rived by Ghosh et al. (1998), u(x, y, t)=−y +3y 2 + x 2 y − 2 3 y 3 + βxy sin nt, v(x, y, t)=−xy 2 − 1 2 βy 2 sin nt. (5.111) Substituting this model into (5.108) yields (γ 2 − 1)T =0and2γT > 0, T =2π/n. Thus, the fixed separation point is at γ = −1 and the reattach- ment point at γ = +1, as shown in Fig. 5.22 for n =2π and β =3,which is in agreement with the numerical observation of Ghosh et al. (1998). The expansion coefficients f 0 (t), , f 3 (t) can also be derived, which gives the ap- proximate separation profile also shown in Fig. 5.22. The preceding unsteady separation theory indicates that the final results on the separation definition and criteria are fully Eulerian that do not require 5.4 Unsteady Separation 251 the advection of fluid particles. This is a unique advantage of the theory. Like the general three-dimensional steady separation theory of Sect. 5.2, this un- steady theory meets three highly desired requirements proposed, respectively, by Sears and Telionis (1975), Cowley et al. (1990), and Wu et al. (2000), and summarized by Haller: independent of our ability to solve the boundary- layer equations accurately; independent of the coordinate system selected; and expressible solely by quantities measured or computed along the wall. Summary 1. Phenomenologically, flow separation is a local process in which fluid ele- ments adjacent to a wall no longer move along the wall but turn to the interior of the fluid. In its strong form and at large Reynolds numbers, the process may evolve to boundary-layer separation where the whole layer breaks away and thereby significantly alters the global flow field. Physi- cally, flow separation is due to the boundary coupling of the two funda- mental dynamic processes. A near-wall adverse pressure gradient yields a boundary vorticity flux σ p , which creates new vorticity with direction dif- ferent from that of existing one, so the accumulation of the former in space and time causes a transition of the near-wall vorticity from being carried along by the wall to shedding off. Thus, a vorticity-dynamics description of separation is especially illuminating, which can be obtained from the conventional momentum considerations owing to the on-wall equivalence between the τ w -field and its orthogonal ω B -field, and that between the ∇ π p-field and its orthogonal σ p -field. 2. A general flow-separation process without any specification to its strength can be studies in an infinitesimal neighborhood of a separation point or separation line, by using a Taylor expansion of the continuity and Navier– Stokes equations. The criteria for separation zone and separation line at large Reynolds numbers can be formulated in terms of the earlier two pairs of orthogonal on-wall vector fields. For steady separation and in two dimensions, the criteria amount to those well-known ones due to Prandtl. In three dimensions, the separation zone is characterized by the strong converging of τ -lines or large positive on-wall curvature of ω-lines. If the separation starts at a fixed point of the τ -field (“closed separation”), a generic separation line can be uniquely determined. But at large Re a significant separated free shear layer may start to form and/or cease to shed off at ordinary points of a τ -line; for which the separation line may be approximately identified as the line with maximum ω-line curvature in the separation zone. 3. Boundary-layer separation at large Re involves the flow behavior in the whole layer and its interaction with external flow in a small but finite zone. Although this process is governed by the Navier–Stokes equation, the matched asymptotic expansion has contributed an elegant triple-deck [...]... ∇ · l to ∇2 φl , by using (6. 7) we find ∞ φl = vω dr + C log r, r where there must be C = 0 Thus, from the r-momentum equation u ∂u v 2 ∂p − =− ∂r r ∂r (6. 36) 266 6 Typical Vortex Solutions and using (6. 4c), we can obtain l and then l⊥ from (6. 36) 1 l = −v(r)ω(r)er = −∇ H − w2 , 2 (6. 37a) 1 1 ∂Γ eθ , l⊥ = − γrω(r)eθ = − γ 2 2 ∂r (6. 37b) Γ = rv(r) While l is independent of γ and has the same form as nonstretched... coordinates a Stokes stream function 1 (6. 51) Ψ = − U r2 + ψ 2 with u and ωθ being solely determined by ψ through (6. 8) and (6. 9c) Dynamically, (6. 5b) is reduced to ωθ u Dωθ ωθ = + ν ∇2 ωθ − 2 Dt r r (6. 52) of which a neater form is (6. 10b): Df 2 ∂f = ν ∇2 f + Dt r ∂r , f≡ ωθ r (6. 53) If in addition the flow is generalized Beltramian, then (3 .63 ) and (3 .67 ) hold While (3 .67 ) and (6. 4) imply a Bernoulli equation... function Ψ and boundary ∂A of the core cross-section A such that ∂2 1 ∂ ∂2 + 2 − ∂r2 r ∂r ∂z and Ψ (r, z) = −Ωr2 0 in A, outside A, (6. 69a) Ψ and ∇Ψ are continuous across ∂A, Ψ = k on ∂A, 1 2 Ψ + U r → 0 as r2 + z 2 → ∞ 2 (6. 69b) 278 6 Typical Vortex Solutions (a) (b) 2 2 a=÷2 a=1.25 1 1 0 0 8 0 .6 0.4 a=1.2 1 a=÷2 1.2 1.0 0.8 0 .6 0.4 0.2 0.2 0 0 r 1 -1 r -1 -2 1 -2 Fig 6. 6 Fraenkel–Norbury vortex rings... z) = (6. 56) 4π R Using the notations defined in Fig 6. 4, (6. 56) takes the form ψ(r, z) = G(r, r , z − z ) = R= ωθ G(r, r , z − z ) dr dz , rr 4π 2π 0 cos β dβ, R β =θ−θ , (z − z )2 + r2 + r 2 − 2rr cos β (6. 57a) (6. 57b) (6. 57c) Let the moving point x be along a circle with varying θ and fixed (r , z ) By (6. 57c) and Fig 6. 4, in this circle r1 = (z − z )2 + (r − r )2 , r2 = (z − z )2 + (r + r )2 (6. 58)... 2 ) (6. 63a) 2 − k2 r0 k 2 E(k) , (6. 63b) − 2r 1 − k 2 2(1 − k 2 ) r0 r + (r + r0 )2 (6. 63c) In particular, at the ring center (r, z) = (0, 0) there is u0 = − Γ ez , 2r0 (6. 64) indicating that the circular line vortex moves with constant velocity along the −z direction without changing its shape After imposing U = Γ/2r0 to make the flow steady, the streamlines are shown in Fig 6. 5 Equation (6. 64) is... 4νt , (6. 26b) where M represents the total angular momentum about the axis ∞ M= 2πr2 vdr (6. 27) 0 This solution has zero total circulation (because ω changes sign once) and finite M Note that (6. 26) is nothing but the time derivative of (6. 25) The velocity profiles of Oseen–Lamb vortex and Taylor vortex are compared in Fig 6. 1c All higher modes with n > 1 in (6. 24) have zero total circulation and zero... u(r, t) is derived from (6. 3) The vorticity has only a z-component The flow can take this form only locally (r < ∞, |z| < ∞) From (6. 5c) and 264 6 Typical Vortex Solutions (6. 28), the vorticity equation reads: ν ∂ ∂ω = ∂t r ∂r r 1 ∂ω + γω + γr 2 ∂r ∂ω ∂r (6. 29) On the right-hand side, the second term is a radial advection, while the third term is a uniform stretching If at t = 0 a vortex element has unit... ∂2ψ r ∂z 2 (6. 9c) The role of Γ for ωr and ωz is exactly the same as that of ψ for u and w Contours of Γ and ψ on an (r, z)-plane are the intersections of vorticity surfaces and stream surfaces with the plane, respectively Then (6. 4b) and (6. 5b) can be cast to ∂ DΓ =ν r Dt ∂r D Dt 1 ∂Γ r ∂r ωθ r 2 ∂ r ∂r = ν ∇2 + + ∂2Γ , ∂z 2 ωθ r + 1 ∂Γ 2 , r4 ∂z (6. 10a) (6. 10b) which govern the azimuthal and meridional... z) = γz 1 − 3e−βr (6. 42b) 2 , (6. 42c) where x H(x) ≡ η exp −η + 3 0 0 1 − e−ζ dζ dη, ζ H(∞) = 37.905 (6. 43) Sullivan (1959) wrote down this solution without giving derivation The vorticity components of this Sullivan vortex are ωr = 0, ωθ = − ωz = (6. 44a) 2 3γ 2 rze−βr , 2ν γΓ0 exp −βr2 + 3 2νH(∞) (6. 44b) βr 2 0 1 − e−ζ dζ ζ (6. 44c) 268 6 Typical Vortex Solutions Like the Burgers vortex, u is not smooth... of z Then (6. 3) implies that u = C(t)/r, which would lead to a singularity at the vortex axis if C = 0 Hence (6. 16) follows Once (6. 16) holds, in (6. 5) for ω = (0, ωθ , ωz ), the viscous terms are solely balanced by the unsteady terms ν ∂ ∂ωz = ∂t r ∂r r 1 ∂ ∂ωθ =ν ∂t r ∂r ∂ωz ∂r r , ∂ωθ ∂r (6. 17a) − ωθ r2 (6. 17b) Note that although for a generalized Beltrami vortex (6. 5) is linearized and one has . separation control. Part II Vortex Dynamics 6 Typical Vortex Solutions In Chap. 4 we have studied attached and free vortex layers, and seen that the rolling up of a free vortex layer forms a vortex which. Batchelor (1 964 ). So the q -vortex is also called the Batchelor vortex. Evidently (6. 19) satisfies (6. 11). On the other hand, the vor- ticity has a Gaussian distribution, and hence the q -vortex is. Note that (6. 26) is nothing but the time derivative of (6. 25). The velocity profiles of Oseen–Lamb vortex and Taylor vortex are compared in Fig. 6. 1c. All higher modes with n>1 in (6. 24) have