Vorticity and Vortex Dynamics 2011 Part 15 ppsx

698 A Vectors, Tensors, and Their Operations The particular importance of the permutation tensor in vorticity and vortex dynamics lies in the fact that ω i =  ijk u k,j =  ijk Ω jk , (A.11a) due to (A.3) and (A.6). Inversely, by using (A.9b) it is easily seen that Ω jk = 1 2  ijk ω i . (A.11b) This pair of intimate relations between vorticity vector and spin tensor show that they have the same nonzero components and hence can represent, or are dual to, each other. Note that (A.11b) also indicates that the inner product of a vector and an antisymmetric tensor can always be conveniently expressed as the cross-product of the former and the dual vector of the latter; for instance, a ·Ω = 1 2 ω × a, Ω ·a = 1 2 a ×ω. (A.12a,b) Obviously, we also have ∇·Ω = − 1 2 ∇×ω. (A.13) A.2 Integral Theorems and Derivative Moment Transformation The key result of tensor integrations is the extension of the fundamental the- orem of calculus in one dimension  b a f  (x)dx =  b a df(x)=f(b) −f(a) to multidimensional space. We state two classic theorems without giving proof (e.g., Milne-Thomson 1968), which are then used to derive useful integral transformations. A.2.1 Generalized Gauss Theorem and Stokes Theorem First, let V be a volume, having closed boundary surface ∂V with outward unit normal vector n and ◦ denote any permissible differential operation of the gradient operator ∇ on a tensor F of any rank. Then the generalized Gauss theorem states that ∇◦FdV must be a total differentiation, and its integral can be cast to the surface integral of n ◦FdS over the boundary surface ∂V of V , where n is the unit outward normal vector:  V ∇◦FdV =  ∂V n ◦FdS. (A.14) A.2 Integral Theorems and Derivative Moment Transformation 699 In particular, if F is constant, (A.14) yields a well-known result  ∂V n dS =  ∂V dS = 0, (A.15) i.e., the integral of vectorial surface element dS = n dS over a closed surface must vanish. Moreover, for the total dilatation and vorticity in V we have  V ϑ dV =  ∂V n ·u dS, (A.16a)  V ωdV =  ∂V n ×u dS. (A.16b) One often needs to consider integrals in two-dimensional flow. In this case the volume V can be considered as a deck on the flow plane of unit thickness. Then (A.14) and some of the volume-integral formulas below remain the same in both two and three dimensions, but care is necessary since in n-dimensions δ ii = n is n-dependent. Some formulas for n = 3 need to be revised or do not exist at all; see Sect. A.2.4 for issues special in two dimensions. Next, let S be a surface with unit normal n, then without leaving S only tangential derivatives can be performed and have chance to be integrated out, expressed by line integrals over the boundary loop ∂S. The tangent differential operator is naturally n ×∇, and the line element of ∂S has an intrinsic direction along its tangent, dx = t ds, where t is the tangent unit vector and ds the arc element. The directions of the normal n of S and t obey the right-hand rule. Then, as the counterpart of (A.14), the generalized Stokes theorem states that on any open surface S any (n ×∇) ◦FdS must be the total differentiation, and its surface integral can be cast to the line integral of dx ◦Falong ∂S:  S (n ×∇) ◦FdS =  ∂S dx ◦F. (A.17) Thus, if F is constant, (A.17) shows that the integral of element dx over a closed line must vanish  t ds =  dx = 0; (A.18) and if F = x, since [(n ×∇) ×x] i =  ijk  jml n l x k,m = −2n i , we obtain the well-known formula for the integral of a vectorial surface  S n dS = 1 2  ∂S x ×dx (A.19) 700 A Vectors, Tensors, and Their Operations with (A.15) as its special case since a closed surface has no boundary. In general, (A.17) implies  S (n ×∇) ◦FdS = 0 on closed S. (A.20) The most familiar application of (A.17) to fluid mechanics is the relation between total vorticity flux through a surface and circulation along the boun- dary of the surface. Since (n ×∇) · u = n · (∇×u), there is  S ω · n dS =  ∂S u ·dx. (A.21) A.2.2 Derivative Moment Transformation on Volume The Gauss and Stokes theorems permit the construction of useful identities for integration by parts. In particular, we need to generalize the one-dimensional formula  b a f(x)dx = bf(b) − af(a) −  b a xf  (x)dx, which expresses the integration of f(x)bythex-moment of its derivative, to various integrals of a vector f over a volume or surface, so that they are cast to the integrals of proper moments of the derivatives of f plus bound- ary integrals. We call this type of transformations the derivative moment transformation. We first use the generalized Gauss theorem (A.14) to cast the volume integral of f to the moments of its divergence and curl. Since (f i x j ) ,i = f j + f i,i x j ,  ijk  jlm (f m x k ) ,l =  ijk ( jlm f m,l )x k +  ijk  jkm f m , where x is the position vector, by (A.14) we find a pair of vector identities  V f dV = −  V x(∇·f)dV +  ∂V x(n ·f)dS, (A.22)  V f dV = 1 n −1  V x ×(∇×f)dV − 1 n −1  ∂V x ×(n ×f)dS, (A.23) where n =2, 3 is the space dimension. The factor difference comes from the use of (A.9c) for n = 3 and (A.10b) for n = 2. Note that for two-dimensional flow, (A.22) still holds if f is on the plane (e.g., velocity), but becomes trivial if f is normal to the plane (e.g., vorticity). 2 2 This can be verified by considering a deck-like volume of unit thickness. When the vector is normal to the deck plane, one finds 0 = 0 from (A.22). A.2 Integral Theorems and Derivative Moment Transformation 701 Then, we need to cast the first vector moment x×f to the second moments of its curl, say F = ∇×f. When n =3,F has three second moments x 2 F , x ×(x ×F ), and x(x ·F ), related by x ×(x ×F )=x(x · F ) −x 2 F . Note that x(x ·F )=0 for n = 2. Then one finds 2  V x ×f dV = −  V x 2 F dV +  ∂V x 2 n ×f dS, n =2, 3, (A.24a)  V x ×f dV =  V x(x ·F )dV −  ∂V x(n ×f) ·x dS, n =3, (A.24b) 3  V x×f dV =  V x×(x×F )dV −  ∂V x×x×(n×f )dS, n =3. (A.24c) Here, (A.24c) is the sum of (A.24a) and (A.24b). The trick of proving the first two is using (A.14) to cast the surface integrals therein to volume integrals first.Insodoingn becomes operator ∇ which then has to act on both f and x. If we make a Helmholtz–Hodge decomposition f = f ⊥ +f  , see (2.87) and associated boundary conditions (2.98a) or (2.98b), then we can replace f by f  on the right-hand side of (A.22). Namely, the integral of a vector is expressible solely by the derivative-moment integrals of its longitudinal part. However, (A.23) is not simply a counterpart of this result in terms of the transverse part of the vector. Rather, as long as n ×f  = 0 on ∂V , a boundary coupling with the longitudinal part must appear. For some relevant discussions see Wu and Wu (1993). A.2.3 Derivative Moment Transformation on Surface By similar procedure, we may use the Stokes theorem (A.17) to cast surface integrals of a vector to that of its corresponding derivative moments plus boundary line integrals. To this end we first decompose the vector to a normal vector φn and a tangent vector n×A, since they obey different transformation rules. Then for the normal vector we find a surface-integral identity effective for n =2, 3  S φn dS = − 1 n −1  S x ×(n ×∇φ)dS + 1 n −1  ∂S φx ×dx. (A.25) And, for n =3only, the integral of tangent vector can be cast to  S n ×A dS = −  S x ×[(n ×∇) ×A]dS +  ∂S x ×(dx ×A). (A.26) In deriving these identities the operator n ×∇should be taken as a whole for the application of (A.17). Setting φ = 1 in (A.25) returns to (A.19). In fact, 702 A Vectors, Tensors, and Their Operations (A.25) is also a special case of (A.23) with f = ∇φ. Note that the cross- product on the right-hand side of (A.26) can be replaced by inner product  S n ×A dS =  S x(n ×∇) ·A dS −  ∂S x(A ·dx), (A.27) where (n ×∇) · A = n · (∇×A). Then, for both n = 2 and 3, the integral of the first moment x × nφ can be transformed to the following alternative forms:  S x ×nφ dS = 1 2  S x 2 n ×∇φ dS − 1 2  ∂S x 2 φ dx (A.28a) = −  S x[x ·(n ×∇φ)] dS +  ∂S φx(x ·dx) (A.28b) = − 1 3  S x ×[x ×(n ×∇φ)] dS + 1 3  ∂S φx ×(x ×dx). (A.28c) Finally, to cast the surface integral of x ×(n ×A)=n(x ·A) −A(x ·n) to the second-moment of the derivatives of A, we start from two total deriv- atives  ijk (n ×∇) j (x 2 A k )=x 2  ijk (n ×∇) j A k + A k  ijk (n ×∇x 2 ) j = x 2  ijk (n ×∇) j A k +2(x i n k A k − n i x k A k ),  ljk (n ×∇) j (x i x l A k )=x i x l  ljk (n ×∇) j A k + A k  ljk (n ×∇) j (x i x l ) = x i x l  ljk (n ×∇) j A k +3x i n k A k − A i x k n k . Here, since what matters in x ×(n ×A) is only the tangent components of A, we may well drop its normal component n k A k . Hence, subtracting the second identity from 1/2 times the first yields an integral of x×(n×A). Using (A.17) to cast the left-hand side to line integral then leads to the desired identity  S x ×(n ×A)dS =  S S ·[(n ×∇) ×A]dS −  ∂S S ·(dx ×A), (A.29a) where S = 1 2 x 2 I −xx or S ij = 1 2 x 2 δ ij − x i x j . (A.29b) is a tensor depending on x only. As a general comment of derivative moments, we note that, since in (A.22),(A.23), and (A.25)–(A.27) the left-hand side is independent of the choice of the origin of x, so must be the right-hand side. In general, if I A.2 Integral Theorems and Derivative Moment Transformation 703 represents any integral operator (over volume or surface or a sum of both), than the above independence requires I{(x 0 + x) ◦F}= I{x ◦F} for any constant vector x 0 .Thus,wehave x 0 ◦ I{F} + I{x ◦F}= I{x ◦F}, which implies that, due to the arbitrariness of x 0 , there must be I{F} =0. (A.30) Namely, if we remove x from the right-hand side of (A.22), (A.23), and (A.25)– (A.27), the remaining integrals must vanish. It is easily seen that this condition is precisely the Gauss and Stokes theorem themselves. A.2.4 Special Issues in Two Dimensions The preceding integral theorems and identities are mainly for three-dimensional domain, with some of them also applicable to two-dimensional domain. A few special issues in two dimensions are worth discussing separately. In many two-dimensional problems it is convenient to convert a plane vector ae x +be y to a complex number z = x+iy by replacing e z × by i = √ −1 (Milne-Thomson 1968), so that e y = e z × e x =⇒ ie x (A.31a) and hence ae x + be y =(a + be z ×)e x =⇒ e x (a +ib). (A.31b) Then the immaterial e x can be dropped. Thus, denoting the complex conju- gate of z by ¯z = x − iy, for derivatives there is ∂ x = ∂ z + ∂ ¯z ,∂ y =i(∂ z − ∂ ¯z ), (A.32a) 2∂ z = ∂ x − i∂ y , 2∂ ¯z = ∂ x +i∂ y , (A.32b) so by (A.31) ∇ =⇒ 2e x ∂ ¯z , ∇ 2 =⇒ 4∂ z ∂ ¯z . (A.33) The replacement rule (A.31) cannot be extended to tensors of higher ranks. If in a vector equation one encounters the inner product of a tensor S and a vector a that yields a vector b, then (A.31) can be applied after b is obtained by common real operations. For example, consider the inner product of a trace-free symmetric tensor S = e x e x S 11 + 1 2 (e x e y + e y e x )S 12 + e y e y S 22 ,S 11 + S 22 =0, 704 A Vectors, Tensors, and Their Operations and a vector a = e x a 1 + e y a 2 =⇒ e x (a 1 +ia 2 ). After obtaining a · S = S ·a by real algebra, we use (A.31) to obtain 2a ·S =⇒ e x (a 1 − ia 2 )(2S 11 +iS 12 ),S 11 + S 22 =0. (A.34) In this case S appears as a complex number S 11 +iS 12 /2 but a appears as its complex conjugate a 1 − ia 2 . Now, if F = f(x, y) is a scalar function, (A.17) is reduced to e z ×  S  e x ∂f ∂x + e y ∂f ∂y  dS =  S  e y ∂f ∂x − e x ∂f ∂y  dS =  ∂S fdx. Hence, by (A.31) this formula becomes i  S  ∂f ∂x +i ∂f ∂y  dS =  ∂S f dz, which by (A.32) is further converted to  ∂S f(z,z)dz =2i  S ∂f ∂z dS, (A.35a)  ∂S f(z,z)dz = −2i  S ∂f ∂z dS, (A.35b) the second formula being the complex conjugate of the first. Milne-Thomson (1968) calls this result the area theorem. The two-dimensional version of the derivative-moment transformation on surface, i.e., the counterpart of (A.25) and (A.26), also needs special care. We proceed on the real (x, y)-plane. Let C be an open plane curve with end points a and b,ande s and n be the unit tangent and normal vectors along C so that (n, e s .e z ) form a right-hand orthonormal triad. Then since ∂x ∂s = e s , n ×∇φ = n ×  e s ∂ ∂s + n ∂ ∂n  φ = e z ∂φ ∂s for any scalar φ and tangent vector tA, there is  C nφ ds = −e z × (xφ)| b a −  C x ×(n ×∇φ)ds, (A.36)  C tA ds =(xA)| b a −  C x ∂A ∂s ds. (A.37) The transformation of the first-moment integral of a normal vector nφ has been given by (A.28). But that of a tangent vector, say x × e s A, cannot be similarly transformed at all, because x ×e s A = ∂ ∂s (x ×xA) −e s × xA −x ×x ∂A ∂s A.3 Curvilinear Frames on Lines and Surfaces 705 simply leads to a trivial result. What we can find is only a scalar moment  b a x ·At ds = − 1 2  b a x 2 ∂A ∂s ds + 1 2 x 2 A    b a . (A.38) A.3 Curvilinear Frames on Lines and Surfaces In the above development we only encountered Cartesian components of vectors and tensors. In some situations curvilinear coordinates are more conve- nient, especially when they are orthonormal. Basic knowledge of vector analy- sis in a three-dimensional orthonormal curvilinear coordinate system can be found in most relevant text books (e.g., Batchelor 1967), where the coordi- nate lines are the intersections of a set of triply orthogonal surfaces. But, the Dupin theorem (e.g., Weatherburn 1961) of differential geometry requires that in such a system the curves of intersection of every two surfaces must be the lines of principal curvature on each. While the concept of principal curvatures of a surface will be explained later, here we just notice that the theorem excludes the possibility of studying flow quantities on an arbitrary curved line or surface and in its neighborhood by a three-dimensional ortho- normal curvilinear coordinate system. These lines and surfaces, however, are our main concern. Therefore, in what follows we construct local coordinate frames along a single line or surface only and as intrinsic as possible, with an arbitrarily moving origin thereon. A.3.1 Intrinsic Line Frame If we are interested in the flow behavior along a smooth line C with length element ds, say a streamline or a vorticity line, the intrinsic coordinate frame with origin O(x)onC has three orthonormal basis vectors: the tangent vector t = ∂x/∂s,theprincipal normal n (toward the center of curvature), and the binormal b = t × n, see Fig. A.2. This (t, n, b) frame can continuously move along C and is known as intrinsic line frame. The key of using this frame is to know how the basis vectors change their directions as s varies. This is given by the Frenet–Serret formulas, which form the entire basis of spatial curve theory in classical differential geometry (e.g., Aris 1962): ∂t ∂s = κn, ∂n ∂s = −κt + τb, ∂b ∂s = −τn, (A.39a,b,c) where κ and τ are the curvature and torsion of C, respectively. The curvature radius is r = −1/κ with dr = −dn. The torsion of C measures how much a curve deviates from a plane curve, i.e., it is the curvature of the projection of C onto the (n, b) plane. For a plane curve τ =0andwehavea(t, n) frame as already used in deriving (A.36)–(A.38). 706 A Vectors, Tensors, and Their Operations b S n t O Fig. A.2. Intrinsic triad along a curve Now, let the differential distances from O along the directions of n and b be dn and db, respectively. Then ∇ = t ∂ ∂s + n ∂ ∂n + b ∂ ∂b , (A.40) which involves curves along n and b directions that have their own curvature and torsion. Then one might apply the Frenet–Serret formulas to these curves as well to complete the gradient operation. But due to the Dupin theorem we prefer to leave the two curves orthogonal to C undetermined. For example, if C is a streamline such that u = qt, then the continuity equation for incompressible flow reads ∇·u = ∂q ∂s + q∇·t =0, (A.41) where, by using (A.39), ∇·t = n · ∂t ∂n + b ∂t ∂b . (A.42) Similarly, there is ∇×t = κb +  n × ∂t ∂n + b × ∂t ∂b  . Here, since |t| = 1, it follows that: n ·  n × ∂t ∂n + b × ∂t ∂b  = t · ∂t ∂b = 1 2 ∂ ∂b |t| 2 =0, b ·  n × ∂t ∂n + b × ∂t ∂b  = −t · ∂t ∂n = − 1 2 ∂ ∂n |t| 2 =0. A.3 Curvilinear Frames on Lines and Surfaces 707 Therefore, the second term of ∇×t must be along the t direction, with the magnitude ξ ≡ t · (∇×t)=b · ∂t ∂n − n · ∂t ∂b . (A.43) The scalar ξ is known as the torsion of neighboring vector lines (Truesdell 1954). Thus, using this notation we obtain ∇×t = ξt + κb. (A.44) This result enables us to derive the vorticity expression in the streamline intrinsic frame ω = ∇×(qt)=∇q × t + q∇×t = ∇q × t + ξqt + κqb. Thefirsttermofis ∇q × t = ∂q ∂b n − ∂q ∂n b, so we obtain (Serrin 1959) ω = ξqt + ∂q ∂b n +  κq − ∂q ∂n  b. (A.45) Thus, ξ0ifω · u = 0. Note that in a three-dimensional orthonormal frame there must be ξ ≡ 0, so by the Dupin theorem a curve with ξ = 0 cannot be the principal curvature line of any orthogonally intersecting surfaces. A.3.2 Intrinsic operation with surface frame Derivatives of tensors along a curved surface S can be made simple by an intrinsic use of an intrinsic surface frame, which is more complicated than the intrinsic line frame since now there are two independent tangential directions on S. Covariant Frame At a given time, a two-dimensional surface S in a three-dimensional space is described by the position vector x of all points on S, which is a function of two independent variables, say u α with α =1, 2. Then r α (u 1 ,u 2 ) ≡ ∂x ∂u α ,α=1, 2, (A.46) define two nonparallel tangent vectors (not necessarily orthonormal) at each point x ∈ S, see Fig. A.3. 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Tensors, and Their Operations The particular importance of the permutation tensor in vorticity and vortex dynamics lies in the fact that ω i =  ijk u k,j =  ijk Ω jk , (A.11a) due to (A.3) and (A.6) the counterpart of (A.25) and (A.26), also needs special care. We proceed on the real (x, y)-plane. Let C be an open plane curve with end points a and b,ande s and n be the unit tangent and normal. in (A.22),(A.23), and (A.25)–(A.27) the left-hand side is independent of the choice of the origin of x, so must be the right-hand side. In general, if I A.2 Integral Theorems and Derivative Moment