Vorticity and Vortex Dynamics 2011 Part 4 pdf

142 4 Vorticity Dynamics boundary vorticity flux. Equations (4.22)–(4.27) still hold, applicable to any deformable solid wall or interface of two fluids including free surface. Since a and ω n are continuous across B (Sect. 2.2.4), if the wall is rigid and has angular velocity W (t), in (4.17) we may replace ω by the relative vorticity ω r = ω −2W with n·ω r ≡ 0. Meanwhile, by (4.26a), the tangent component of σ vis is reduced to the sole contribution of wall skin-friction τ w = µn × ω r via curvature: σ πvis = νω r · K = 1 ρ (τ w × n) · K, (4.28) where τ or ω r is in turn a temporal-spatial accumulated effect of the entire σ as will be demonstrated in Sects. 4.1.4 and 4.2.3. On a boundary B, σ n represents a kinematic tilting of the vorticity lines on B toward the normal direction. This mechanism can be very significant at a solid wall as seen in tornado-like vortices (Fig. 3.5a), and is an ingredient of three-dimensional flow separation (Chap. 5). Figure 4.5 shows a pair of σ n - peaks with opposite signs on a channel wall and a hairpin vortex above the wall in the sublayer of a turbulent flow, indicating the correlation between the σ n -pair and hairpin vortex. For a flow with given wall acceleration and external body force, σ a and σ f in (4.24) are known. As the measure of vorticity generation rate, these two boundary vorticity-flux constituents can be viewed as the roots of the vorticity field in the flow. In contrast, the stress-related constituents σ p and σ vis are the result or footprints of the entire flow and boundary condition. But once they are established through the momentum balance, they become z x y Fig. 4.5. A local plot of instantaneous σ n (contours) on the wall and vorticity lines right above the wall in a turbulent channel flow. The two dark spots on the wall are a pair of σ n peaks of opposite signs. From a direct numerical simulation of Zhao et al. (2004) 4.1 Vorticity Diffusion Vector 143 the root of the vorticity field. Distinguishing σ a and σ f from σ p and σ vis is very important for understanding the force and moment acting to the wall (Chap. 11) and for near-wall flow control (Zhu 2000; Zhao et al. 2004). For example, if in a conducting fluid an imposed near-wall electromagnetic field can effectively control the vorticity generation through a Lorentz force f (Du and Karniadakis 2000), then by (4.23) the on-wall effect of f can in principle be replaced by an equivalent wall tangent acceleration so that the control could be applied to nonconducting fluid (Zhao et al. 2004). But it will be hard (if not impossible) to impose a distributed σ p as control means. The relative magnitudes of the four constituents of the boundary vorticity flux vary from one specific problem to another. For an incompressible flow over a three-dimensional stationary body without body force, only σ p and σ vis exist. Naturally and as will be seen in the following sections, they are of the same order when Re  1, but σ p becomes much stronger and the most fundamental mechanism of vorticity creation when Re  1. In particular, for a two-dimensional flow in the (x, y)-plane over a stationary wall with σ = σ p e z , (4.24b) and (2.172b) form a pair of Cauchy-Riemann relations: µ ∂ω ∂n = − ∂p ∂s , ∂p ∂n = µ ∂ω ∂s . (4.29a,b) Lighthill (1963) was the first to interpret (4.29a) as the measure of vortic- ity creation and emphasize the role of tangent pressure gradient. His pioneer insight was followed by many workers who added other constituents and ex- tended the theory to two-fluid interface, see the review of Wu and Wu (1996). The physical implication of (4.29a) can be easily understood from Fig. 4.6. Replacing the pressure gradient by a wall acceleration from right to left or a body force from left to right, the mechanisms of σ a and σ f can also be easily understood. Notice the difference of Figs. 4.6 and 3.2. The mechanism of vorticity creation should not be confused with that of boundary vorticity ω B . We stress that although (4.23) is derived for viscous flow with acceleration adherence, the form of (4.29a) shows that the amount of σ is independent of viscosity. Thus, as µ → 0 there must be ∂ω/∂n →∞to ensure the momentum High pressure Low pressure u Fig. 4.6. Schematic illustration of vorticity generation by pressure gradient and no-slip condition 144 4 Vorticity Dynamics balance and no-slip condition. At this asymptotic limit the newly created vorticity forms a vortex sheet adjacent to the wall. Corresponding to the boundary vorticity flux, we also have boundary en- strophy flux η defined by (4.20). In terms of this scalar flux the flow boundary B can be divided into three different parts: B 0 , where η = 0 due to the absence of boundary vorticity and/or its flux; B + , where η>0; and B − , where η<0. Thus  B η dS =  B + |η|dS −  B − |η|dS. (4.30) It can then be said that B + (or B − )isavorticity source (or sink), where the existing vorticity is strengthened (or weakened) by the newly created one. 4.1.4 Unidirectional and Quasiparallel Shear Flows In this subsection we illustrate the basic physics of vorticity diffusion and generation from boundaries by some simple unidirectional and quasiparallel viscousshearflows. Unidirectional Flow Driven by Pressure Gradient and Wall Acceleration Consider a flow on the half plane y>0 with ρ =1and u =(u(y, t), 0, 0), ω =(0, 0,ω(y,t)),ω(y, t)=− ∂u ∂y . (4.31) The fluid and boundary are assumed at rest for t<0, and at t = 0 let there appear a tangent motion of the boundary with speed b(t), and a uniform, time- dependent pressure gradient ∂p/∂x = P(t). In this case, the Navier–Stokes equation and vorticity transport equation are linearized: ∂u ∂t = −P (t) − ν ∂ω ∂y = −P (t)+ν ∂ 2 u ∂y 2 , (4.32) ∂ω ∂t = ν ∂ 2 ω ∂y 2 . (4.33) Applying (4.32) to the wall gives σ = db dt + P (t)aty =0, (4.34) which is evidently independent of viscosity. Equation (4.33) under the New- mann condition (4.34) has solution ω(y, t)=  t 0 − σ(t  )  πν(t − t  ) exp  − y 2 4ν(t − t  )  dt  . (4.35) 4.1 Vorticity Diffusion Vector 145 The flux σ can be regular or singular. If at t = 0 there is an impulsive P (t)and db/dt, they will cause a suddenly appeared uniform fluid velocity U =(U, 0, 0) and wall velocity b 0 , respectively. This yields σ(t)=−(U − b 0 )δ(t)=γ 0 δ(t)for0 − ≤ t ≤ 0 + , (4.36) where γ 0 = −(U − b 0 ) is the initial vortex-sheet strength. Separating this singular part from (4.35) yields ω(y, t)= γ 0 √ πνt exp  − y 2 4νt  +  t 0 + σ(t  )  πν(t − t  ) exp  − y 2 4ν(t − t  )  dt  . (4.37) With a finite ν, the initially singular vorticity in the sheet γ 0 is soon diffused into the fluid as reflected by the first term of (4.37). This problem is referred to as the generalized Stokes problem. Setting y = 0 in (4.37) gives ω B (t)= γ 0 √ πνt + 1 √ πν  t 0 + σ(t  ) √ t − t  dt  , (4.38) indicating clearly that in this example ω B is a temporal accumulated effect of σ. On the other hand, by (4.37) one may verify that the rate of change of total vorticity is d dt  ∞ 0 ω(y, t)dy = σ(t), (4.39) which confirms the physical meaning of σ. Two special cases of (4.37) were first studied by Stokes. The Stokes first problem or Rayleigh problem is that the flow is entirely caused by an impulsive start of the wall from rest, with P = 0 for all t and σ =0fort ≥ 0 + . Hence ω(y, t)= b 0 √ πνt exp  − y 2 4νt  . (4.40) The Stokes second problem is that the wall makes a sinusoidal oscillation, say b = b 0 cos nt (or b 0 sin nt, the difference being that the former contains an impulsive start). The full solution has been studied by Panton (1968). If only the transient boundary vorticity is considered, with b = b 0 cos nt the integral in (4.38) can be carried out analytically: ω B (t)= γ 0 (πνt) 1/2 + b 0  2n ν  1/2  S( √ nt)cosnt − C( √ nt)sinnt  , (4.41) where S(x)andC(x) are Fresnel’s integrals (e.g., Abramowitz and Stegun 1972). As t →∞this solution degenerates to a stationary oscillating state ω B (t)=b 0  n ν  1/2 cos  nt + π 4  . (4.42) 146 4 Vorticity Dynamics At this stage, inside the fluid the vorticity field also has a stationary oscillation, which is a viscous transverse wave propagating along the y-direction: ω(y, t)=b 0  n ν  1/2 e −y/δ cos  y δ − nt − π 4  ,δ= k −1 r =  2ν n  1/2 . (4.43) The length scale δ = k −1 r , with k r being the real part of the complex wave number, characterizes the diffusion distance of the wave or the thickness of a shear layer in which the flow has significant transverse wave. This layer is known as the Stokes layer. The phase speed c and group speed c g of the transverse wave are c = n k r = √ 2νn, c g = dn dk r =2 √ 2νn > c, (4.44) which are frequency-dependent, so the wave is dispersive. Unidirectional Interfacial Flow As an extension of the Stokes first problem, we now insert a flat interface S at y = 1 into the preceding unidirectional flow at y>0 (Wu 1995). A flat interface of water and air may occur when the gravitational force is much larger than inertial force. Both flow 1 (e.g., the water) at y ∈ [0, 1] and flow 2 (e.g., the air) at y =(1, ∞) are governed by the same equations as (4.32) with P = 0 and (4.33), and the matching condition of two flows is velocity adherence and surface-force continuity (2.68), which yields an integral equa- tion for the unknown interface velocity u 1 = u 2 = v at y = 1. The only surface force on S is the shear stress µω × n, which by (2.68) implies a vor- ticity jump ω 1 /ω 2 = µ 2 /µ 1 . Thus, the impulsively started bottom wall drives flow 1, which drives flow 2 that in turn reacts to flow 1. The velocity profiles in water and air at different times are shown in Fig. 4.7a. The interface vorticity is initially zero, then increases to a posi- tive peak due to the diffusion of ω 1 > 0 (entirely generated at t =0)toS, and then decreases to zero as it diffuses into fluid 2, see Fig. 4.7b. In addi- tion to the singular generation of ω 1 at the wall, at S there also appears a boundary vorticity flux on both sides: σ 1 = ν 1 ∂ω 1 ∂y = − dv dt = −σ 2 = ν 2 ∂ω 2 ∂y at y =1. (4.45) Initially there is σ 1 = 0. When ω 1 > 0 is diffused to S to induce a tangent interface acceleration dv/dt, σ 1 starts to become negative, reaching a peak value and then returns to zero, see Fig. 4.7c. σ 2 follows a similar trend but with opposite sign and different magnitude (not shown). Since ρ 2 /ρ 1  1, the effect of the air motion on the water can be ignored and the interface problem can be simplified to a free-surface problem. Then the interface vorticity will be identically zero and flow 1 alone can be solved. A remarkable difference of this free-surface model and interface flow is that, 4.1 Vorticity Diffusion Vector 147 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.005 0.004 -0.04 0.003 -0.03 0.002 -0.02 0.001 -0.01 0 0.2 0.4 0.6 0.8 1 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 (a) (c)(b) Y t =10 20 30 40 w 1 on interface s 1 on interface U T T Fig. 4.7. Generalized Stokes’ first problem for interacting water–air system. (a)Ve- locity profiles at different times. (b) Time variation of water vorticity on the inter- face. (c) Time variation of vorticity flux at the interface. Solid lines are water–air coupled solution and dash lines are obtained by free-surface approximation. Quan- tities are made dimensionless by the bottom-plate initial velocity, the water depth, and water density. Reproduced from Wu (1995) for the former the boundary enstrophy flux η defined by (4.20) is zero both at y =0fort>0 due to σ =0,andaty = 1 due to ω = 0, respectively. Consequently, the enstrophy of the water cannot “escape” out of the free surface at all, although it eventually decays to zero. This can only be explained by a σ ≤ 0atS found in both interface and free-surface flows, which generates opposite vorticity that diffuses downward to cancel that generated at the wall. Despite the difference of interface vorticity (Fig. 4.7b), the prediction of free- surface model on the velocity profile of flow 1 and its boundary vorticity flux on S agree very well with that of the interface. Acoustically Created Vorticity Wave from Flat Plate As an extension of Stokes’ second problem, we replace P (t) in (4.32) by a harmonic traveling pressure wave (a sound wave): p = Ae ik(x−ct) ,A= ρu 0 c, c = n k , (4.46) 148 4 Vorticity Dynamics where u 0 and ρ are constant. Then instead of (4.34) there is σ = 1 ρ ∂p ∂x = Re  inu 0 e ik(x−ct)  = −nu 0 sin(kx −nt), (4.47) which excites a Stokes layer of thickness δ defined in (4.43). Let ζ = y/δ be the rescaled normal distance. Then the velocity field inside the layer is u = u 0  1 − e (i−1)ζ  e ik(x−ct) , (4.48a) v = ku 0 δ  −iζ + 1 − i 2  e (i−1)ζ − 1   e ik(x−ct) , (4.48b) indicating that the flow is no longer unidirectional when k =0.Theω-wave produced by the p-wave is ω =(i−1) u 0 δ e −ζ e i(kx−nt+ζ) + O(k 2 δ), (4.49) where the contribution of ∂v/∂x is ignored. The associated boundary enstro- phy flux is η =  n 2  3/2 u 2 0 √ ν  1+ √ 2sin  2(kx −nt) − π 4  , (4.50) which has a positive average. Lin (1957) has shown that for any external flow (even turbulent), if the frequency is so high that δ is much smaller than the boundary-layer thickness, then inside the Stokes layer the linear approximation (4.48) and (4.49) still holds. A direct numerical simulation of channel-turbulence control by flexible wall traveling wave confirmed Lin’s assertion (Yang 2004). Sound-Vortex Interaction in a Duct According to the vortex-sound theory outlined in Sect. 2.4.3, the sound- generated vorticity in the earlier example 3 will in turn produce sound, which may have strong effect when the sound wave is confined in a duct. Thus, consider a weakly compressible flow with disturbance velocity u =(u, v)and dilatation ϑ = ∇·u in a two-dimensional duct, bounded by parallel plates at y = 0 and 2. Assume the mean flow has unidirectional velocity U (y) that satisfies the no-slip condition. Due to the nonuniformity of U , a sound wave having a plane front at x = 0, say, must be refracted towards the walls, and only a part of modes can reach far downstream. This is a closed-loop coupling between shearing and compressing processes as well as sound propagation by U(y) in the duct. Both processes should be solved simultaneously, governed by a pair of linear equations derived from (2.168) and (2.169): 4.1 Vorticity Diffusion Vector 149 (D 0 − ν∇ 2 )ω = U  v + U  ϑ, (4.51a) D 0 ϑ + ∇ 2 p = −2U  ∂v ∂x , (4.51b) under boundary conditions (4.29). Here, D 0 ≡ ∂ t + U∂ x and (·)  =d(·)/dy.In example 3 the pressure wave is specified and only (4.51a) was used; while the inviscid problem (4.51b) alone with homogeneous boundary conditions (an eigenvalue problem) has also been well studied (e.g., Pridmore-Brown 1958; Shankar 1971). But now the fully coupled problem is nonlinear. A simplified approach was given by Wu et al. (1994a), who split this closed-loop interaction into two subprocesses and solved them sequentially. First, an inviscid refracted pressure field was computed by (4.51b) as an eigenvalue problem, which then produces a vorticity wave by (4.51a) and (4.29a). Second, (4.51b) was cast to a linearized vortex-sound equation in terms of the total enthalpy H as a special case of (2.170). With the mean-flow Mach number M = U/c,the dimensionless H-equation reads ∇ 2 H − (D 2 0 H + MM  D 0 v)=M  u − 2M  ω − M ∂ω ∂y , (4.52) from which the p-field due to the acoustically created ω-wave can be calculated and added to the initial inviscid p-wave solution. In solving (4.52) a viscous boundary condition derived from (4.29b) has to be imposed even though the equation is inviscid. For a parabolic mean flow M(y)=M ∞ (2y−y 2 ), the amplitude of vorticity wave produced by the refracted p-wave obtained by this sequential approach is shown in Fig. 4.8a, and the wall sound pressure level (SPL) at different wave number k and Reynolds number Re is shown in Fig. 4.8b. Note that Fig. 4.8b shows that the effect of viscosity may be nonmonotonic. When a 1.0 0.8 0.6 0.4 0.2 0.0 -100 -75 -50 50 -25 250 3.0 3.5 4.0 w log 10 (Re) 20 10 10 1.0 0.5 5 0 k=20 Inviscid limits (a) (b) ϱ SPL y Fig. 4.8. Sound–vortex interaction in a duct at M =0.3. (a) The amplitude of sound-generated vorticity wave at x = 20, k =5,andRe = 1,000 (real part: solid line; imaginary part: dash line). (b) The wall SPL at x = 20 and different k and Re. From Wu et al. (1994a) 150 4 Vorticity Dynamics p-wave excites an ω-wave through the no-slip condition, it loses some kinetic energy; but, (4.51a) indicates that there is an interior unsteady source U  v for disturbance vorticity, which makes the acoustically created ω-waveableto absorb enstrophy from the mean flow and becomes a self-enhanced source of sound. 4.2 Vorticity Field at Small Reynolds Numbers It has been asserted in Chap. 2 that the dominating parameter of the shear- ing process is the Reynolds number. The effect of this parameter on vortical flows is very complicated, as demonstrated by the well-known photographs of flow over a circular cylinder of diameter D at different R D = UD/ν (Van Dyke 1982; see also Fig. 10.42). If R D = O(1), we have the full Navier–Stokes equation and no simplification can be made. But both R D  1andR D  1 provide a small parameter, and the matched asymptotic expansion (e.g., Van Dyke 1975) can lead to approximate solutions. We take this convenience to discuss the behavior of incompressible vorticity field at small Reynolds num- bers in this section, and at large Reynolds numbers in the next two sections. Small Reynolds-number flows are called Stokes flows. The viscous length scale of a flow is ν/U, where U is the oncoming velocity. Compared to the body length scale D, R D  1 implies that Viscous length scale Body length scale  1. This occurs if either (a) U  1, or (b) D  1. To the leading order, case (a) implies that the inertial force can be ignored, while case (b) implies that the flow is almost uniform. These two views led to different approximate solutions studied by Stokes (1851) and Oseen (1910), respectively. They had not been unified until 1950s, when Kaplun (1957) realized that the Stokes solution is effective only near the body surface while the Oseen solution is effective for far field, and they should be matched to form a uniformly effective solution. We illustrate the situation by a steady incompressible flow U e x over a sphere of radius a, examined in the spherical coordinates (R, θ, φ) shown in Fig. 4.9. This problem has extensive applications in many fields of science and technology, such as artificial raining, air dust removing, boiling heat transfer, powder transportation, measurements of fluid viscosity and charge of electron, and the motion of blood cells, etc. From now on we use a to define the Reynolds number Re = aU/ν. When Re =   1, the steady governing equations read u ·∇u = −∇p −∇×ω. (4.53) 4.2.1 Stokes Approximation of Flow Over Sphere We first follow Stokes’ approach to simply set  = 0, which leads to four component equations from ∇×ω = 0 and condition ∇·u = 0 for three 4.2 Vorticity Field at Small Reynolds Numbers 151 f R q x x Fig. 4.9. Flow over a sphere at small Reynolds number unknown variables. To make the problem solvable we retain the pressure term by setting P ≡ (p − p ∞ ). Then by (4.53), the lowest-order approximation (the Stokes approximation)is ∇P + ∇×ω = 0. (4.54) Hence, both P and ω are harmonic: ∇ 2 P =0, ∇ 2 ω = 0, (4.55a,b) which and (4.29) indicate that in two-dimensional flow P +iω is a complex analytic function. Although the inviscid coupling of shearing and compressing processes via nonlinearity inside the flow field is absent, the viscous linear cou- pling is strong on the body surface via the adherence condition (Sect. 2.4.3). In the spherical coordinates, after scaled by a and U, the boundary con- ditions read u = e x ,P=0,R→∞, (4.56a) u = 0 at R =1. (4.56b) The flow occurs on the (R, θ) plane and is rotationally symmetric. As argued by Batchelor (1967), because the form of (4.55) is independent of the choice of coordinates, by inspecting the form of (4.56) one finds that P and ω can only depends on x, e x ,andR. P musttakeontheformx ·e x F (R), while ω must be along the e φ -direction and only depends on e x ×xF (R). Then since 1/R is a fundamental solution of Laplace equations (the origin is singular but outside the flow field), the proper P should be found in the series solution (cf. Sect. 3.2.3) P = ∞  n=0 C n  ∂ ∂x  n  1 R  . [...]... 2 L⊥X = L X (4. 100a) + O(δ 2 ) (4. 100b) Therefore, equating (4. 98) and (4. 100a) recovers (4. 85a), while equating (4. 97) and (4. 100b) recovers the Bernoulli integral (4. 83) outside the layer Note that although generically L = 0 at Y = 0, this is not so for L⊥ and L unless σ = 0 Indeed, (4. 100a) indicates that L⊥X = − ∂Ω ∂Y or ey × l⊥ = −ez ∂ω ∂y at Y = 0, (4. 101) which is the boundary vorticity flux... decomposition can be explicitly written because (4. 61) is linear For the unsteady version of (4. 61), then, the potential and rotational parts of the disturbance velocity represent longitudinal and transverse waves, respectively (Lagerstrom 19 64) 4. 2 Vorticity Field at Small Reynolds Numbers 155 4. 2.3 Separated Vortex and Vortical Wake From the point of view of vorticity dynamics, a remarkable feature of the... Fig 4. 12 Sketch of the profiles of velocity (a) and vorticity (b), and the variations of the fluxes of vorticity (c) and enstrophy (d) on the wall, for a flat-plate flow in a pressure gradient changing from favorable to adverse 158 4 Vorticity Dynamics It is conceptually useful to divide the vorticity field created by a moving body into two parts One part is dragged along by or attaches to the body, and. .. potential and vortical parts .4 Then the pressure is obtained from (4. 61), and the drag can be computed by considering the normal and shear stresses on the sphere (e.g., Milne-Thomson 1968) The result is ω=− 3 D = 6ρU 2 πa 1 + Re , 8 or CD = 12 Re 3 1 + Re , 8 (4. 67) which will be compared with (4. 59) in Fig 4. 14 below For more discussions see Proudman and Pearson (1957) and Chester (1962) 4 This decomposition... dS = 0 (4. 116b) S d dt for n = 2, S which are the extension of (3.15) and (3.16), respectively Therefore, if some vorticity moves out of V and appears to be lost, then it is really gained by the vortex sheet 4. 4 Vortex Sheet Dynamics We now proceed from the boundary-layer approximation at Re 1 to the asymptotic state of a viscous fluid as Re → ∞, which was called the Euler 4. 4 Vortex Sheet Dynamics. .. velocity us , and on each side of S there is an attached vortex sheet of strengths γ 1 and γ 2 , forming a sandwich structure Then γ 1 = n × (u1 − us ), γ 2 = n × (us − u2 ) (4. 126) 4. 4 Vortex Sheet Dynamics 175 This sandwich structure is often modeled by a single net vortex sheet inside S with strength γ = γ 1 + γ 2 If S is a lifting surface, the net vortex sheet is conventionally called a bound vortex. .. equation and permitting slip velocity at boundaries But, except within very thin sheets the flow is irrotational and indeed satisfies the Euler equation 1 74 4 Vorticity Dynamics n g u uu+ Fig 4. 18 A vortex sheet where the overline denotes averaged value Therefore, if [[ρ]] = 0 so that [[ρu]] = ρ[[u]], the left-hand side of (4. 121) will have vanishing normal component and hence so must the right-hand side,... of the 166 4 Vorticity Dynamics wall where there is a singularity In particular, for the Blasius solution, the boundary vorticity flux is zero for all x δ, implying no new vorticity is produced therefrom Actually, as the flat plate starts moving at zero tangent pressure gradient, the vorticity in the transient boundary layer is created solely by σ a defined in (4. 24a) and illustrated by (4. 34) But once... by (4. 118b) there is γ = n × ∇π [[φ]] = n × ∇π Γ, ∇π Γ = γ × n (4. 129a,b) Moreover, we define the vortex- sheet Lamb vector ¯ lγ ≡ γ × u P+ PS C Fig 4. 19 A loop across the vortex sheet once (4. 130) 176 4 Vorticity Dynamics ¯ ¯ Set u = uπ + un n, by (4. 129) and noticing that ∇Γ = ∇π Γ since ∂Γ/∂n = [[un ]] = 0, it follows that ¯ ¯ n(lγ · n) = γ × uπ = −n(uπ · ∇Γ ), n × (lγ × n) = γ × un n = un ∇π Γ (4. 131a)... must be un = 0 and hence lγ can only has normal component; but (4. 133) implies that if the sheet is free this component must vanish Therefore, the Lamb vector of a steady free ¯ vortex sheet must vanish, or γ and u must be aligned (cf Lighthill 1986b, p 2 04) In contrast, when the flow is unsteady, by (4. 131a) and (4. 133) one has ∂Γ ¯ (4. 1 34) γ × uπ = n ∂t Thirdly, since dynamics enters (4. 133) via the . (t) in (4. 32) by a harmonic traveling pressure wave (a sound wave): p = Ae ik(x−ct) ,A= ρu 0 c, c = n k , (4. 46) 148 4 Vorticity Dynamics where u 0 and ρ are constant. Then instead of (4. 34) there. of sound-generated vorticity wave at x = 20, k =5,andRe = 1,000 (real part: solid line; imaginary part: dash line). (b) The wall SPL at x = 20 and different k and Re. From Wu et al. (1994a) 150 4 Vorticity Dynamics p-wave. Zhao et al. (20 04) 4. 1 Vorticity Diffusion Vector 143 the root of the vorticity field. Distinguishing σ a and σ f from σ p and σ vis is very important for understanding the force and moment acting