Vorticity and Vortex Dynamics 2011 Part 3 pptx

3.3 Lamb Vector and Helicity 91  V (ω ×u + ϑu)dV =  ∂V  n · uu − 1 2 q 2 n  dS, (3.69)  V x × (ω × u + ϑu)dV =  ∂V x ×  n · uu − 1 2 q 2 n  dS. (3.70)  V x · (ω × u + ϑu)dV =  ∂V x ·  n · uu − 1 2 q 2 n  dS + 1 2 (n − 2)  V q 2 dV, q = |u|. (3.71) The n = 3 case of (3.71) is evidently relevant to the total kinetic energy of incompressible flows with uniform density (Sect. 3.4.2); while (3.69) and (3.70) are relevant to the evolution of vortical impulse and angular impulse (Sect. 3.5.2). In particular, in an unbounded incompressible fluid at rest at infinity, the surface integrals in these identities can be taken over the surface at infinity where u = ∇φ, which must vanish by (3.49). Therefore, it follows that:  V ∞ ω × u dV = 0, (3.72)  V ∞ x × (ω × u)dV = 0, (3.73)  V ∞ x · (ω × u)dV = 1 2 (n − 2)  V ∞ q 2 dV. (3.74) If the fluid has internal boundary, say a solid surface ∂B, to use (3.69) to (3.74) one may either employ the velocity adherence to cast the surface integrals over ∂B to volume integrals over B, or continue the Lamb vector into B. Both ways form a single continuous medium although locally ω is discontinuous across ∂B. The integral of helicity density ω · u is called the helicity. Moffatt (1969) finds that this integral is a measure of the state of “knotness” or “tangledness” of vorticity lines. We demonstrate this feature for thin vortex filaments (thin vorticity tubes). Assume that in a domain V with n · ω =0on∂V there are two thin vortex filaments C 1 and C 2 , with strengths (circulation) κ 1 and κ 2 respectively, away from which the flow is irrotational. C 1 and C 2 must be both closed loops. Suppose C 1 is not self-knotted, such that it spans a piece of surface S 1 without intersecting itself, and that the circulation along C 1 is Γ 1 =  C 1 u · dx =  S 1 ω · n dS. In the present situation, Γ 1 can only come from the contribution of the fila- ment C 2 . Therefore, if C 1 and C 2 are not tangled (Fig. 3.9a) then Γ 1 = 0; but if C 2 goes through C 1 once (Fig. 3.9b) then Γ 1 = ±κ 2 , with the sign depend- ing on the relative direction of the vorticity in C 1 and C 2 . More generally, C 2 92 3 Vorticity Kinematics C 1 C 2 (a) (b) (c) C 1 C 2 C 1 C 2 Fig. 3.9. The winding number of closed vortex filaments C 1 and C 2 .(a) α 12 =0, (b) α 12 = −1,(c) α 12 =2 C 1 C 2 C = Fig. 3.10. Decomposition of a knotted vortex filament can go through C 1 an integer number of times (Fig. 3.9c), so that Γ 1 = α 12 κ 2 , where α 12 = α 21 is a positive or negative integer called the winding number of C 1 and C 2 . By inserting one or more pair of filaments of opposite circulations, a self- knotted vortex filament can always be decomposed into two or more filaments which go through each other but are not self-knotted. Figure. 3.10 shows the decomposition of a triple knot, for which we have  C u · dx =  C 1 u · dx +  C 2 u · dx =2κ. In general, if there are n unkotted vortex filaments, then the circulation along the i th closed filament is Γ i =  C i u · dx = n  j=1 α ij κ j , where α ij is the winding number of C i and C j . Multiplying both sides by κ i , we get (repeated indices imply summation) κ i Γ i =  C i κ i u · dx = α ij κ i κ j . Now, observe that since the filaments are sufficiently thin, κ i dx is nothing but ωdV for the ith vortex filament; thus α ij κ i κ j =  V ω · u dV, (3.75) 3.3 Lamb Vector and Helicity 93 which is precisely the helicity. Therefore, the helicity measures the strengths of vortex filaments and their winding numbers. A remark is in order here. If we express the velocity by the Monge decom- position (2.115), there is ω · u =  ijk (φλ ,j µ ,k ) ,i −  ijk (φλ ,j µ ,ik ), where the second term vanishes. Hence  V ω · u dV =  ∂V n · ωφ dS, (3.76) which is zero by assumption, conflicting (3.75) if the filaments are knotted. This apparent paradox comes from the local effectiveness of (2.115). Brether- ton (1970) has pointed out that for knotted filaments the potential φ cannot be single-valued and hence the argument leading to (2.115) (Phillips 1933) does not hold. The knotness or tangledness, characterized by the winding number, is known as the topological property of a curve. A topological property of a geometric configuration remains invariant under any continuous deformation. Thus, configurations in Fig. 3.11a have the same topological property. To re- tain the continuity during the deformation process, no tearing or reconnection is allowed; thus the patterns in Fig. 3.11a are topologically different from those in Fig. 3.11b. The former is simply connected, but the latter is doubly con- nected (connectivity is also a topological property). A flow also has its topological structure. When a flow structure is a ma- terial curve like a vortex filament, the state of its knotness or tangledness is its topological property. Some new progress in the study of this property has (a) (b) Fig. 3.11. Topological property of geometric configurations. Topologically, the con- figurations in (a) are the same as a sphere, and those in (b)arethesameasa torus 94 3 Vorticity Kinematics been reviewed by Ricca and Berger (1996). Later in Sect. 7.1 we shall meet the topological structure of a vector field, which is a powerful tool in studying separated vortical flows. For fluid mechanics these topological properties are of qualitative value; in fact, just because quantitative details are beyond its concern, the topological analysis is generally valid. 3.4 Vortical Impulse and Kinetic Energy This section establishes direct relations between vorticity integrals and two fundamental integrated dynamic quantities: the total momentum and kinetic energy of incompressible flows with uniform density ρ = 1. The results suggest that almost the entire incompressible fluid dynamics falls into vorticity and vortex dynamics (complemented by the potential-flow theory of Sect. 2.4.4). 3.4.1 Vortical Impulse and Angular Impulse It has long been known that the total momentum and angular momentum of an unbounded fluid, which is at rest at infinity, are not well defined since relevant integrals are merely conditionally convergent. To avoid this difficulty, one appeals to the concept of hydrodynamic impulse (impulse for short) and angular impulse.Thepotential impulse has been introduced in Sect. 2.4.4, and we now consider the impulse and angular impulse associated with vortical flow, i.e., the vector field i(x) in (2.178), which is nonzero in a finite region. Since ω = ∇×i, integrating i and using the derivative-moment identity (A.23) in n-dimensional space, we obtain  V i dV = 1 n − 1  V x × ω dV − 1 n − 1  ∂V x × (n ×i)dS, n =2, 3. (3.77) As ∂V encloses the entire vector field i(x), the surface integral vanishes since i = 0 there by assumption. This proves that  V i dV = I ≡ 1 n − 1  V x × ω dV, (3.78) which defines the total vortical impulse I, already introduced by (3.42) for n = 3 and (3.45) for n = 2. Evidently, due to (3.18), I is well defined and finite. A similar argument on the instantaneous angular momentum balance, using (A.24a), shows that  V x × i dV = L ≡− 1 2  V x 2 ω dV (3.79) which defines the total vortical angular impulse. Now, by applying the same identities to the integral of u and x × u,we immediately obtain (Thomson 1883) 3.4 Vortical Impulse and Kinetic Energy 95  V u dV = I − 1 n − 1  ∂V x × (n × u)dS, (3.80)  V x × u dV = L + 1 2  ∂V x 2 n × u dS, (3.81a) = L  − 1 3  ∂V x × [x × (n × u)] dS, (3.81b) where L  ≡ 1 3  V x × (x × ω)dV (3.82) is an alternative definition of the angular impulse, see (3.6). Comparing (3.81a) and (3.81b), for n = 3 there is L  − L = 1 6  ∂V (2xx + x 2 I) · (n × u)dS = 1 6  V (2xx + x 2 I) · ω dV = 1 6  ∂V x 2 x(n · ω)dS, (3.83) so L  = L if n ·ω =0on∂V . Each of these vortical impulses differs from the total momentum and angular momentum only by a surface integral. While identities (3.80) and (3.81) hold for any volume V , an important situation is that V contains all vorticity so that on ∂V the flow has acyclic potential φ (see Sect. 2.4.4). Then we can replace u by ∇φ in the above sur- face integrals, which can then be simplified owing to the derivative-moment transformation (A.25) and (A.28a,c): 12 − 1 n − 1  ∂V x × (n × u)dS =  ∂V φn dS, (3.84) 1 2  ∂V x 2 n × u dS = − 1 3  ∂V x × [x × (n × u)] dS =  ∂V x × φn dS. (3.85) Recall the definition of potential impulse and angular impulse I φ and L φ given by (2.179) and (2.180), we see that the total momentum and angular momentum in V with ρ = 1 are reduced to I + I φ and L + L φ , respectively. As observed in Sect. 2.4.4, if V extends to infinity as in the case of exter- nally unbounded flow, by (3.49) (with Γ ∞ = 0 when n = 2) the convergence 12 The derivative-moment transformation is a set of integral identities in two- and three-dimensional spaces, which express the integral of a vectorial function to that of the moment of its derivatives, plus a boundary term. The details are given in Appendix A.2. 96 3 Vorticity Kinematics property of I φ and L φ are poor. This unpleasant feature is evidently given to the volume integrals of u and x × u (e.g., Batchelor 1967; Saffman 1992; Wu 1981). Take the far-field boundary shape as a large sphere (n =3)or circle (n = 2) of radius R →∞. We can then estimate the surface integral in (3.84) by using (3.49). This yields (for n =2,Γ ∞ has no contribution to the integral)  ∂V R φn dS = − 1 n I. (3.86) Thus, no matter how large R could be, there is always I/n being communi- cated to the potential flow outside the sphere or circle. This apparent paradox, that a potential flow can carry a part of vortical impulse, is explained by Lan- dau and Lifshitz (1976) as due to the assumption of incompressibility. Once a slight compressibility with constant speed of sound c is introduced, then at time t the momentum inside the sphere R = ct is (n − 1)I/n and the “lost” momentum I/n is transmitted by a spherical pressure wave front R = ct. In contrast, the surface integral in (3.85) is simple when n =3orn =2 with Γ ∞ = 0, since over the sphere or circle x × φn = Rφn × n = 0. But for n = 2 with Γ ∞ =0,φ is not single-valued and it is better to apply (3.50b) to the surface integral of (3.81a). This yields an R 2 -divergence:  C x 2 n × u ds = R 2 2 Γ R e z . However, these discussions are of mainly academic interest. What enters dynamics is only the rate of change of these integrals, for which the diver- gence issue does not appear at all (Sects. 2.4.4 and 3.5.2; Chap. 11). In two dimensions, the simplest vortex system with finite total momentum and angular momentum is a vortex couple of circulation ∓Γ e z (Γ<0) located at x = ±r/2, respectively, see Fig. 3.12. Then by (3.78) there is I = e y Γr. (3.87) The fluid in between is pushed downward by the vortex couple. I y G -G x Fig. 3.12. The impulse produced by a vortex couple with Γ<0 in two dimensions 3.4 Vortical Impulse and Kinetic Energy 97 x O I G C t ds Fig. 3.13. The impulse produced by a vortex loop in three dimensions In three dimensions, the simplest vortex system is a closed loop C of thin vortex filament of circulation Γ , see Fig. 3.13. In this case (3.78) is reduced to, owing to (A.19) I = Γ 2  C x × t ds = Γ  S dS = Γ S, (3.88) where dS = x × t ds/2 is the vector surface element spanned by the triangle formed by x and dx = t ds,andS is the vector surface spanned by C. Note that |S| is the area of the minimum surface spanned by the loop, just like the area of a soap film spanned by a metal frame. It is very different from the area S of a cone with apex at the origin of x that depends on the arbitrarily chosen origin. Similarly, if the vortex loop is isolated, by (3.82) and (3.83) we have L = Γ 3  C x × (x × t)ds = 2Γ 3  S x × dS. (3.89) 3.4.2 Hydrodynamic Kinetic Energy Lamb (1932) gives two famous formulas for the total kinetic energy in a domain V , K =  V 1 2 q 2 dV, q = |u|, (3.90) in terms of vorticity. Here the flow is assumed incompressible with ρ =1.The first formula is based on the identity q 2 = u · (∇φ + ∇×ψ)=∇·(uφ + ψ × u)+ω · ψ, (3.91) where φ and ψ are the Helmholtz potentials given by (2.104) with ϑ =0now. The second formula is the direct consequence of (3.74) for three-dimensional 98 3 Vorticity Kinematics flow only. Thus, Lamb’s first and second formulas for kinetic energy read, respectively, K = 1 2  V ω · ψ dV + 1 2  ∂V u · (nφ + n × ψ)dS, n =2, 3, (3.92) K =  V (ω × u) · x dV +  ∂V x ·  1 2 q 2 n − uu · n  dS, n =3. (3.93) If there is u = ∇φ on ∂V , the surface integrals in both formulas are reduced to the potential-flow kinetic energy K φ given by (2.175). More specifically, as x = |x|→∞,forn = 3 the surface integrals in both formulas decay as O(x −3 ). For n = 2, by (3.46) and (3.47), if Γ ∞ = 0, then the surface integral in (3.92) decays as O(x −2 ). However, if Γ ∞ = 0, there will be |uφ|∼uψ =O(x −1 ln x) and the surface integral is infinity. Therefore, for unbounded two-dimensional flows Lamb’s first formula can be applied only if Γ ∞ = 0. We will be confined to this case. By taking a large sphere or circle, the preceding argument in dealing with impulse and angular impulse indicates that for unbounded flow (3.92) can be written as a double volume integral K = 1 2π  Gω · ω  dV dV  , (3.94) where G is given by (2.102). Hence, in three dimensions there is K = − 1 8π  ω · ω  |x − x  | dV  dV (3.95a) =  x · (ω × u)dV. (3.95b) Some general comparisons of the two formulas for any domain V can be made. They both consist of a volume integral and a boundary integral, which can be symbolically expressed by K = K (α) V + K (α) S , (3.96) with α =1, 2 denoting which of the two formulas is referred to. Then: 1. Since both formulas are obtained by integration by parts, the integrand of the volume integrals in (3.92) and (3.93), k (1) V (x) ≡ 1 2 ω · ψ, (3.97a) k (2) V (x) ≡ (ω × u) · x =(x × ω) · u, (3.97b) 3.4 Vortical Impulse and Kinetic Energy 99 do not represent the local kinetic energy density q 2 /2. They are even not positively definite. However, like in many other formulas from inte- gration by parts, only k (α) V , α =1, 2, have net volumetric contribution (positive or negative) to K, with more localized support but containing more information on flow structures than q 2 /2. In this sense, k (α) V can be viewed as the net kinetic-energy carriers (per unit mass). As illustra- tion, Fig. 3.14 compares the instantaneous distribution of q 2 /2andωψ/2 for a two-dimensional homogeneous and isotropic turbulence obtained by direct numerical simulation. We see that while ωψ/2 has high peaks in vortex cores and hence clearly shows the vortical structures, q 2 /2 distrib- utes more evenly with larger values in between neighboring vortices of opposite signs due to the strong induced velocity there. 2. While k (1) V directly reflects the vortical structures of the flow, k (2) V depends on the choice of the origin of x. Thus, when the flow domain is a periodic box, the surface integral K (1) S vanishes; but the appearance of x in K (2) S makes the boundary contribution to K from opposite sides of the box doubled. In a sense, by integration by parts, (3.93) shifts more net kinetic- energy carrier from the interior of the flow to boundary. 3. Despite the above inconvenience of Lamb’s second formula, it has some unique significance. As seen in Sect. 2.4.3, the Lamb vector ω × u is at the intersection point of two fundamental processes. Moreover, (3.97b) indicates that k (2) V may be interpreted as an “effective rate of work” done by the “impulse density” x × ω. In particular, if we consider the rate of change of the local kinetic energy q 2 /2 by taking inner product of (2.162) and u, then evidently the Lamb vector has no contribution. But now it dominates the total kinetic energy as a net kinetic-energy carrier. This fact is a reflection of the nonlinearity in vortical flow advection. (a) (b) Fig. 3.14. Instantaneous distribution of (a) q 2 /2and(b) ωψ/2 in a two-dimensional homogeneous and isotropic turbulence, based on direct numerical simulation. Cour- tesy of Xiong 100 3 Vorticity Kinematics It is of interest to observe that, if we use (2.162) to compute the rate of change of the kinetic energy, then since (ω × u) · u = 0 the vorticity will have no local nor global inviscid contribution, see (2.52) and (2.53). Now, for incompressible flow Lamb’s second formula asserts that the vorticity does affect the total kinetic energy, but indirectly. In fact, through the Lamb vector, the vorticity as an analogue of the Coriolis force must induce a change of not only direction but also magnitude of u, and hence of q 2 /2. It is this mechanism that is explicitly reflected by Lamb’s first formula. For a similar mechanism involved in the total disturbance kinetic energy and its relation to flow stability see Sect. 9.1.3. 3.5 Vorticity Evolution We now examine the temporal evolution of vorticity and related quantities, including the rate of change of circulation, total vorticity and its moments, helicity, vortical impulse, and total enstrophy. In the evolution of all these quantities there appears a key vector ∇×a, where a =Du/Dt is the fluid acceleration which bridges kinematics to kinetics. Following Truesdell (1954), to keep the results universal we shall often stay with ∇×a in its general form. But it should be kept in mind that behind ∇×a is the shearing kinetics,which will be addressed in Sect. 4.1. 3.5.1 Vorticity Evolution in Physical and Reference Spaces The time-evolution of vorticity in physical space comes from the curl of the vorticity form of the material acceleration, (2.162), and the result can be expressed in a few equivalent forms: ∇×a = ∂ω ∂t + ∇×(ω × u) (3.98a) = ∂ω ∂t + ∇·(uω − ωu) (3.98b) = Dω Dt − ω ·∇u + ϑω. (3.98c) Introducing the continuity equation (2.39) into (3.98c) brings a slight simpli- fication, known as the Beltrami equation: D Dt  ω ρ  = ω ρ ·∇u + 1 ρ ∇×a. (3.99) Moreover, since ∇u = D + Ω and ω · Ω = ω × ω/2 = 0, there is ω ·∇u = ω · D = D ·ω = ω · (∇u T ), (3.100) [...]... is a direct consequence of (3. 130 ) and (3. 105) The Generalized Potential Vorticity Conservation Theorem Let S be any conservative tensor with DS/Dt = 0 and assume (∇ × a) · ∇S = 0 110 3 Vorticity Kinematics Then the generalized potential vorticity (ω/ρ) · ∇S of each fluid particle is preserved Namely, D ω · ∇S = 0, (3. 131 a) Dt ρ or ω ω0 · ∇S = · ∇S0 following particles (3. 131 b) ρ ρ0 For steady flow this... ∂s Dt where by (3. 140a,b) the right-hand side reads: fω × Dω − ω · ∇u Dt = f ω × (∇ × a) due to (3. 98c) Thus, (3. 139 ) is a necessary condition Conversely, if (3. 139 ) holds and if at t = 0 we have (3. 140a,b), then (∂x/∂s) × ω will vanish at t = 0 and have a vanishing material derivative; hence it must always be zero Thus (3. 139 ) is also sufficient Note that by (3. 98a), for steady flow (3. 139 ) implies the... potential vorticity is constant along a streamline On the other hand, under the same conditions any tensor function F of the generalized vorticity must also be a conserved quantity, and so is the material-volume integral of ρF owing to (2.41) ρF V ω · ∇S dv = const ρ (3. 132 ) This theorem has two important corollaries First, setting S = X in (3. 131 ) and define ω (3. 133 a) Ω(X, τ ) ≡ · F−1 ρ as the Lagrangian vorticity, ... following three equivalent conditions holds: ∇ × a = 0, d dt (3. 130 a) a + ∇φ∗ = 0, (3. 130 b) u · dx = 0, (3. 130 c) C where C is any material loop Equation (3. 130 c) comes from (3. 114) and is the well-known Kelvin circulation theorem: If and only if the acceleration is curl-free, the circulation along any material loop is time invariant Conditions in (3. 130 ) define a special class of flows of significant interest,... for any φ and P ≡ ω · ∇φ, ρ (3. 135 ) DP = 0 (3. 136 ) Dt This is the famous Ertel’s potential -vorticity theorem (Ertel 1942): The potential vorticity defined in (3. 106) is Lagrangian invariant if and only if either the flow is circulation-preserving or ∇φ is perpendicular to the vorticity diffusion vector 3. 6 Circulation-Preserving Flows 111 The dimension of P may not be the same as that of the vorticity. .. formula (2 .32 ) Noticing that the circulation of the potential part of a, say −∇φ∗ , must vanish if φ∗ is single-valued Thus, denoting the rotational part of a by ar , 104 3 Vorticity Kinematics (2 .32 ) can be written dΓC = dt C ar · dx = S (∇ × a) · n dS, (3. 114) which also implies (3. 16) Compared with (3. 98), (3. 99), or (3. 101), the evolution of circulation is independent of the effect of vorticity stretching... along a vorticity line, with (t, n, b) being the unit tangent, normal, and bi-normal vectors, respectively Let κ and τ be the curvature and torsion of the ω-line, and u = (us , un , ub ) Then by the Frenet–Serret formulas (A .39 ) there is 3. 5 Vorticity Evolution 107 z (a) y (b) 100 x z L y x z 10 1.0 Time 1.5 y x Fig 3. 15 The self-induced stretching of a vorticity loop, starting from an elliptical vortex. .. τ ) ≡ · F−1 ρ as the Lagrangian vorticity, which is the image of the physical vorticity in the reference space (see Appendix A.4 and a comprehensive study of Casey and Naghdi 1991) Then by (2.1), (3. 107), and (3. 109), we have ω ω0 ∂Ω = 0 or = · F, (3. 133 b) ∂τ ρ ρ0 known as the Cauchy vorticity formula Thus, the Lagrangian vorticity is stationary in the reference space, always equal to its initial distribution... tensor F and independent of the history Evidently, any F(Ω) is also conserved, and in reference space we have ∂ F(Ω)ρ0 d3 X = 0 (3. 134 ) ∂τ V Moreover, as a special case of (3. 133 b) we have Cauchy potential-flow theorem: every initially irrotational fluid element will always be irrotational if and only if the flow is circulation-preserving Second, taking S = φ as a conserved scalar, from (3. 131 ) follows... incompressible and D is a material volume V, by (2 .35 b), comparing (3. 117a) and (3. 117b) yields dIV + dt V l dv − 1 k x × uωn dS = ∂V 1 k V x × (∇ × a)dv (3. 118) A similar approach to the integral of x × a by using (A.24a) yields dLV + dt V x × l dv + 1 2 x2 uωn dS = − ∂V 1 2 V x2 ∇ × a dv (3. 119) In particular, if the fluid is unbounded both internally and externally, and at rest at infinity, then since by (3. 72) . = 0, (3. 130 a) a + ∇φ ∗ = 0, (3. 130 b) d dt  C u · dx =0, (3. 130 c) where C is any material loop. Equation (3. 130 c) comes from (3. 114) and is the well-known Kelvin circulation theorem: If and only. origin. Similarly, if the vortex loop is isolated, by (3. 82) and (3. 83) we have L = Γ 3  C x × (x × t)ds = 2Γ 3  S x × dS. (3. 89) 3. 4.2 Hydrodynamic Kinetic Energy Lamb (1 932 ) gives two famous formulas. setting S = X in (3. 131 ) and define Ω(X,τ) ≡ ω ρ · F −1 (3. 133 a) as the Lagrangian vorticity, which is the image of the physical vorticity in the reference space (see Appendix A.4 and a comprehensive