Vorticity and Vortex Dynamics 2011 Part 2 doc

40 2 Fundamental Processes in Fluid Motion due to the Gauss theorem and (2.98a). Thus, there must be u ⊥1 = u ⊥2 and hence ∇φ 1 = ∇φ 2 . This guarantees the uniqueness of the decomposition and excludes any scalar potentials from u ⊥ . Moreover, (2.89a) and (2.98b) form a well-posed Neumann problem for φ, of which the solution exists and is unique up to an additive constant. Hence so does u ⊥ = u −∇φ. Collecting the earlier results, we have (Chorin and Marsdon 1992) Helmholtz–Hodge Decomposition Theorem. A vector field u on V can be uniquely and orthogonally decomposed in the form u = ∇φ+u ⊥ ,where u ⊥ has zero divergence and is parallel to ∂V . This result sharpens the Helmholtz decomposition (2.87) and is called Helmholtz–Hodge decomposition. It is one of the key mathematic tools in ex- amining the physical nature of various fluid-dynamics processes. Note that from scalar φ one can further separate a harmonic function ψ with ∇ 2 ψ = 0, such that ∇ψ is also orthogonal to both ∇(φ−ψ)andu ⊥ .Thus, strictly, u has a triple orthogonal decomposition. The harmonic part belongs to neither compressing nor shearing processes, but is necessary for φ and ψ to satisfy the orthogonality boundary conditions and thereby influences both. For example, if a vorticity field ω has zero normal component on boundary so that ω = ω ⊥ , there can be ∇×ω =(∇×ω) ⊥ if the former is not tangent to the boundary. In this case we introduce a harmonic function χ, say, and write ∇×ω =(∇×ω) ⊥ + ∇χ, (2.99) where ∇ 2 χ =0 in V, (2.100a) ∂χ ∂n = n · (∇×ω)=(n ×∇) · ω on ∂V. (2.100b) The second equality of (2.100b) implies that χ is not trivial once ω varies along ∂V , of which the significant consequence will be analyzed in Sect. 2.4.3. 2.3.2 Integral Expression of Decomposed Vector Fields In the special case where the Fourier transform applies, we have obtained the explicit expressions of u  and u ⊥ in terms of a given u as seen from (2.96). This local relation in the spectral space must be nonlocal in the physical space after the inverse transform is performed. Indeed, comparing (2.87) with identity (2.86), it is evident that if we set u = −∇ 2 F then the Helmholtz potentials in (2.87) are simply given by φ = −∇·F and ψ = ∇×F . Computing these potentials for given u amounts to solving Poisson equations, and the result must be nonlocal. Without repeatedly mentioning, in what follows use will be frequently made of the generalized Gauss theorem given in Appendix A.2.1. Let G(x)be the fundamental solution of Poisson equation 2.3 Intrinsic Decompositions of Vector Fields 41 ∇ 2 G(x)=δ(x) (2.101) in free space, which in n-dimensional space, n =2, 3, is known as G(x)=      1 2π log |x|, if n =2, − 1 4π|x| , if n =3. (2.102) The gradient of G will be often used: ∇G = x 2(n − 1)π|x| n . (2.103) Now assume u is given in a domain V and u = 0 outside V . We define a vector F by F (x)=−  V G(x − x  )u(x  )dV  , where dV  =dV (x  ). On both sides we consider the Laplacian, which does not act to functions of x  but to G(x − x  ) only. By (2.101) we have −∇ 2 F =  V δ(x −x  )u(x  )dV  =  u if x ∈ V, 0 if x /∈ V. Thus, replacing u in identities (2.86) and (2.87) by F , and denoting the gradient operator with respect to the integration variable x  by ∇  so that ∇G = −∇  G and ∇ 2 G = ∇ 2 G, we obtain, when x is in V , φ = −∇ · F = −  ∇  G · u dV  =  V Gϑ dV  −  ∂V Gn · u dS  , (2.104a) ψ = ∇×F =  ∇  G × u dV  = −  V Gω dV +  ∂V Gn × u dS  , (2.104b) where the second-line expressions were obtained by integration by parts. When x is outside V , these integrals vanish. Therefore, we have constructed a Helmholtz decomposition of u: u = ∇φ + ∇×ψ for x ∈ V, 0 = ∇φ + ∇×ψ for x /∈ V. For unbounded domain, the Helmholtz decomposition is still valid provided that the integrals in (2.104) converge. This is the case if (ω,ϑ) vanish outside 42 2 Fundamental Processes in Fluid Motion some finite region or decay sufficiently fast (Phillips 1933; Serrin 1959). 10 Therefore, (2.104) provides a constructive proof of the global existence of the Helmholtz decomposition for any differentiable vector field. Moreover, (2.104) indicates that the split vectors ∇φ and ∇×ψ canbeexpressedintermsof dilatation and vorticity, respectively: ∇φ =  V ϑ∇G dV  −  ∂V (n · u)∇G dS  , (2.105a) ∇×ψ =  V ω ×∇G dV  −  ∂V (n × u) ×∇G dS  . (2.105b) This result is the generalized Biot–Savart formulas to be discussed in detail in Chap. 3. The formulas not only show the nonlocal nature of the decomposition but also, via (2.103), tells how fast the influence of ω and ϑ at x  on the field point x decays as |x − x  | increases. It should be stressed that for bounded domain the earlier results only provide one of all possible pairs of Helmholtz decomposition of u.Itdoes not care any boundary condition for ∇φ and ∇×ψ. In order to obtain the unique Helmholtz–Hodge decomposition, the simplest way is to solve the scalar boundary-value problem (2.89a) and (2.98b). To see the structure of the solution, we use Green’s identity  V (G∇ 2 φ − φ∇ 2 G)dV =  ∂V  G ∂φ ∂n − φ ∂G ∂n  dS along with (2.98b) to obtain φ =  V Gϑ dV −  ∂V  Gn · u + φ ∂G ∂n  dS. (2.106) Compared with (2.104a), we now have an additional surface integral with unknown boundary value of φ. To remove this term we have to use a boundary- geometry dependent Green’s function  G instead of G, which is the solution of the problem ∇ 2  G(x)=δ(x), ∂  G ∂n =0 on ∂V. (2.107) This gives a unique ∇  φ =  V ϑ∇  G dV −  ∂V n · u∇  G dS, (2.108) and hence u ⊥ = u −∇  φ is the unique transverse vector. The Helmholtz–Hodge decomposition is also a powerful and rational tool for analyzing numerically obtained vector fields, provided that effective meth- ods able to extend operators gradient, curl, and divergence from differential formulation to discrete data can be developed. For recent progress see, e.g., Tong et al. (2003). 10 For the asymptotic behavior of velocity field in unbounded domain see Sect. 3.2.3. 2.3 Intrinsic Decompositions of Vector Fields 43 2.3.3 Monge–Clebsch decomposition So far we have been able to decompose a vector field into longitudinal and transverse parts. It is desirable to seek a further intrinsic decomposition of the transverse part into its two independent components. A classic approach is to represent the solenoidal part of a vector, say v ≡ u −∇φ = ∇×ψ,by two scalars explicitly, at least locally: v = ∇ψ ×∇χ. (2.109) The variables φ, ψ,andχ are known as Monge potentials (Truesdell 1954) or Clebsch variables (Lamb 1932). For the proof of the local existence of ψ and χ, the reader is referred to Phillips (1933). Then, since ∇×(ψ∇χ)=∇ψ ×∇χ, for the Helmholtz vector potential of u we may set (Phillips 1933; Lagerstrom 1964) ψ = ψ∇χ, (2.110) or more symmetrically (Keller 1996), ψ = 1 2 (ψ∇χ −χ∇ψ). (2.111) Both ways cast the Helmholtz decomposition (2.87) to a special form u = ∇φ + ∇ψ ×∇χ, (2.112) where the vector stream function ψ is replaced by two scalar stream functions ψ and χ. Accordingly, the vorticity is given by ω = ∇×(∇ψ ×∇χ) = ∇ 2 χ∇ψ −∇ 2 ψ∇χ +(∇χ ·∇)ψ − (∇ψ ·∇)χ. (2.113) The Monge–Clebsch decomposition has proven useful in solving some vor- tical flow problems (Keller 1998, 1999), but it is not as powerful as the Helmholtz–Hodge decomposition since unlike the latter it may not exist glob- ally. Thus, if one wishes ψ and χ satisfy boundary condition (2.98a) or n · (∇ψ ×∇χ)=0 on ∂V (2.114) and thereby produce a Helmholtz–Hodge decomposition, the problem may not be solvable. Also note that neither (2.110) nor (2.111) satisfies the gauge condition (2.88) although both contain only two independent variables. Instead of the solenoidal part of u, one can also represent the vorticity ω in the form of (2.109). In this case we set u = ∇φ + v, v = λ∇µ, (2.115) so that ω = ∇λ ×∇µ and ∇×(u − λ∇µ)=0. This is the original form of the Monge decomposition, also called the Clebsch transformation (Lamb 1932; Serrin 1959). But in general λ∇µ is not a solenoidal vector and (2.115) does not represent any Helmholtz decomposition. 44 2 Fundamental Processes in Fluid Motion 2.3.4 Helical–Wave Decomposition An entirely different approach to intrinsically decompose a transverse vector and giving its two independent components clear physical meaning, free from the mathematical limitation of Clebsch variables, can be inspired by observing light waves. A light wave is a transverse wave and can be intrinsically split into right- and left-polarized (helical) waves. 11 Mathematically, making this splitting amounts to finding a complete set of intrinsic basis vectors, which are mutually orthogonal in the sense of (2.90), and by which any transverse vector can be orthogonally decomposed. Recalling that the curl operator retains only the solenoidal part of a vector, and observe that the sign of its eigenvalues may determine the right- and left- polarity or handedness. We thus expect that the desired basis vectors should be found from the eigenvectors of the curl. Indeed, denote the curl-eigenvalues by λk, where λ = ±1 marks the polarity and k = |k| > 0 is the wave number with k the wave vector. Then there is Yoshida–Giga Theorem (Yoshida and Giga 1990). In a singly-connected domain D, the solutions of the eigenvalue problem ∇×φ λ (k, x)=λkφ λ (k, x)inD, n · φ λ (k, x)=0on∂D, λ = ±1, (2.116) exist and form a complete orthogonal set {φ λ (k, x)} to expand any transverse vector field u ⊥ parallel to ∂D. 12 These φ λ s can only be found in complex vector space. Their orthogonality (normalized) is expressed by φ λ (k, x), φ ∗ µ (k  , x) = δ λµ δ(k −k  ),λ,µ= ±1, (2.117a)  k φ λi (k, x)φ λj (k, x  )=δ ij δ(x − x  ),λ= ±1, (2.117b) where the asterisk means complex conjugate and repeated indices imply summation. Using this basis to decompose a transverse vector is called the helical-wave decomposition (HWD). For neatness we use ·, · to denote the inner-product integral over the physical space, then the HWD of F ⊥ reads 11 A transverse vector, which can be constructed by vector product or curl operation, is an axial vector or pseudovector. It is always associated with an antisymmetric tensor (see Appendix A.1) and changes sign under a mirror reflection. And, like polarized light, an axial vector is associated with certain polarity or handedness.A true vector, also called polar vector, does not change sign by mirror reflection and has no polarity. In an n-dimensional space the number of independent components of a true vector must be n, but that of an axial vector is the number of the indepen- dent components in its associated antisymmetric tensor, which is m = n(n−1)/2. Thus, only in three-dimensional space there is m=n, but in two-dimensional space an axial vector has only one independent variable (e.g., Lugt 1996). 12 Additional condition is necessary in a multiple-connected domain. 2.3 Intrinsic Decompositions of Vector Fields 45 F ⊥ (x,t)=  k F λ (k,t)φ λ (k, x), (2.118a) F λ (k,t)=F (x,t), φ ∗ λ (k, x). (2.118b) Note that F −F ⊥ , φ λ  =0. For example, for an incompressible flow with u n = ω n =0on∂D, one can expand u(x,t)=  k,λ u λ (k,t)φ λ (k, x), ω(x,t)=  k,λ λu λ (k,t)φ λ (k, x). (2.119) Here the term-by-term curl operation on the infinite series converges. However, as seen from (2.99) and (2.100), although ∇×ω is solenoidal, only (∇×ω) ⊥ can have HWD expansion on which the term-by-term curl operation converges. Therefore, the result is ∇×ω =  k,λ λ 2 u λ φ λ + ∇χ, (2.120) where χ is determined by (2.100). The specific form of HWD basis depends solely on the domain shape. In a periodic box (2.116) is simplified to ik ×φ λ (k, x)=λkφ λ (k, x), from which the normalized HWD eigenvectors can be easily found (Moses 1971, Lesieur 1990): φ λ (k, x)=h λ (k)e ik·x , h λ (k)= 1 √ 2 [e 1 (k)+iλe 2 (k)], (2.121) where λ = ±1, and e 1 (k), e 2 (k), and k/k form a right-hand Cartesian triad. In this case (2.117a) is simplified to h λ (k) ·h ∗ µ (k)=δ λµ . (2.122) If the Cartesian triad is so chosen such that k = ke z , then φ λx (k, x)= 1 √ 2 cos kz, φ λy (k, x)=− λ √ 2 sin kz, φ λz (k, x)=0. (2.123) Therefore, as we move along the z-axis, the locus of the tip of φ λ (k, x) will be a left-handed (or right-handed) helix if λ = 1 (or −1), having a pitch equal to wavelength 2π/k, see Fig. 2.7. In other words, each eigenmode with nonzero eigenvalue is a helical wave. This explains the name HWD. A combined use of the Helmholtz–Hodge and HWD decompositions permits splitting a vector intrinsically to its finest building blocks. 46 2 Fundamental Processes in Fluid Motion k Fig. 2.7. A helical wave The simple Fourier HWD basis cannot be applied to domains other than periodic boxes. To go beyond this limitation, we notice that the curl of (2.116) along with itself leads to a vector Helmholtz equation ∇ 2 φ λ + k 2 φ λ = 0. (2.124) Unlike (2.116), now the three component equations are decoupled, each rep- resenting a Sturm–Liouville problem. Then in principle one can use the Helmholtz vectors to construct the HWD bases. Since a transverse vector field depends on only two scalar fields, say ψ and χ, a simplification may occur if both scalars are solutions of the scalar Helmholtz equation ∇ 2 ψ + k 2 ψ =0. (2.125) Morse and Feshbach (1953, pp. 1764–1766) have shown that this can indeed be realized in and only in Cartesian, cylindrical, spherical, and conical coor- dinates. Specifically, a transverse solutions (not normalized) of (2.124) can be written as a ⊥ = M + N , M = ∇×(ewψ), N = 1 k ∇×∇×(ewχ), where e can be three Cartesian unit vectors, the unit vector along the axis in cylindrical coordinates, or that along the radial direction in spherical and conical coordinates, but none other. the scalar w in the first two cases is 1, while in the others is the radius r. In particular, when ψ = χ there is ∇×(M + λN )=λk(M + λN ); thus, one can write down the HWD basic vector (not normalized) φ λ = ∇×(ewψ)+ λ k ∇×∇×(ewψ). (2.126) Chandrasekhar and Kendal (1957) have given the HWD basis in terms of spherical coordinates. For vorticity dynamics the basis in terms of cylindrical coordinates (r, θ, z) is of interest. Assume that along the z-axis we can impose periodic boundary condition. Then a scalar Helmholtz solution that is regular 2.3 Intrinsic Decompositions of Vector Fields 47 at r =0is ψ = J m (βr)e i(mθ+k z z) ,k z = πn L , β 2 = k 2 − k 2 z ,m=0, 1, 2, , n= ±1, ±2, , where J m (βr) are Bessel functions of the first kind. Substituting this ψ into (2.126) and multiplying the result by a constant iα λ (λ, n) ≡ iλk/k z , we obtain φ λ r =  J m+1 (βr) − m βr (1 + α λ )J m (βr)  e i(mθ+k z z) , φ λ θ =i  α λ J m+1 (βr) − λm βr (1 + α λ )J m (βr)  e i(mθ+k z z) , φ λ z =i β k z J m (βr)e i(mθ+k z z) . (2.127) This result has been applied to vortex-core dynamics, see Sect. 8.1.4. The HWD has also found important applications in the analysis of turbulent cascade process; e.g., Waleffe (1992, 1993); Chen et al. (2003), and refer- ences therein. For a bounded domain of arbitrary shape, finding the curl- eigenvectors is not an easy task and requires special numerical algorithms (e.g., Boulmezaoud and Amari 2000). 2.3.5 Tensor Potentials Before closing this section we take a further look at the Helmholtz potentials. Return to the Cauchy motion equation (2.44) and denote f s ≡∇·T = lim V →0  1 V  ∂V t dS  , (2.128) which is the resultant surface force per unit volume and contains most of the kinetic properties of flows. Assume we have decomposed f s to f s = −∇Φ + ∇×Ψ , ∇·Ψ =0. (2.129) Observe that −Φ ,i +  ijk Ψ k,j = −(Φδ ji +  jik Ψ k ) ,j , where  jik Ψ k ≡ Ψ ji = −Ψ ij (2.130) is an antisymmetric tensor. Thus, (2.129) can be written as f s = ∇·  T,  T ij ≡−(Φδ ij + Ψ ij ). (2.131) As a generalization of the concept of scalar and vector potentials φ and ψ in (2.87), we may view the stress tensor T and the tensor  T as tensor potentials 48 2 Fundamental Processes in Fluid Motion of f s . Obviously there must be ∇·(T −  T)=0. Any other tensor, say T  ,can also be a tensor potential of f s provided that T −T  is divergenceless. Thus, while for Newtonian fluid the stress tensor T is uniquely given by (2.45), f s has infinitely many tensor potentials, among which the above  T with only three independent components is the simplest one. We call it the Helmholtz tensor potential of f s . The value of introducing the Helmholtz potential lies in the fact that in (2.44) the six-component T plays a role only through its divergence. Therefore, once the expression of  T (or the Helmholtz potentials Φ and Ψ ) is known, in the local momentum balance T can well be replaced by the simpler  T.Thus we call  T the reduced stress tensor. However, on any open surface  T produces a reduced surface force  tt t (x, n)=n ·  T(x)=−Φn + n × Ψ , (2.132) which is generically different from the full surface force t given by (2.43). It is here that the extra part of T cannot be ignored. Nevertheless, the replacement of t by  tt t is feasible when one considers the volume integral of f s :  V f s dV =  ∂V t dS =  ∂V  tt t dS (2.133) due to the Gauss theorem. Remarks: 1. We know the divergence of any differentiable tensor of rank 2 is a vector. Now since for any vector field one can always find its Helmholtz potentials as proven by the Helmholtz–Hodge theorem, we see the inverse is also true. 2. Because the Helmholtz decomposition is a global operation, in applica- tions the replacement of (T, t)by(  T,  tt t ) is convenient only when f s has a natural Helmholtz decomposition, as in the kinematic case of (2.86). This important situation also occurs in dynamics as will be seen in the Sect. 2.4. 3. The body force ρf in (2.44) can also be expressed as the divergence of a tensor potential. But this in no ways means that one may cast any body force to a resultant surface force. Whether a force is a body force or surface force should be judged by physics rather than mathematics; a surface force is caused by internal contact interaction of the fluid. 2.4 Splitting and Coupling of Fundamental Processes Having reviewed the basic principles of Newtonian fluid dynamics and intro- duced the intrinsic decomposition of vector fields, we can now gain a deeper insight into the roles of compressing and shearing processes in the kinematics and dynamics of a Newtonian fluid. Our main concern is the splitting and coupling of the two processes in the Navier–Stokes equation (2.47). As just 2.4 Splitting and Coupling of Fundamental Processes 49 remarked, the splitting will be convenient when a natural Helmholtz decom- position manifests itself from (2.47). Remarkably, this is the case as long as µ is constant. By using (2.86) we have 2∇·D = ∇ϑ + ∇ 2 u =2∇ϑ −∇×ω, so (2.47) immediately yields a natural Helmholtz decomposition of the total body force (inertial plus external) as pointed out by Truesdell (1954): ρ Du Dt − ρf = −∇Π −∇×(µω), (2.134) where Π ≡ p − (λ +2µ)ϑ (2.135) consists of the pressure and a viscous contribution of dilatation. The appear- ance of pressure changes the dynamic measure of the compressing process to the isotropic part of T (per unit volume), which is Π; and the dynamic mea- sure of the shearing process remains to be the vorticity ω (multiplied by the shear viscosity). 13 The elegance of (2.134) lies in the fact that Π and ω have only three inde- pendent components, and three more independent components in the strain- rate tensor D do not appear. This fact deserves a systematic examination of its physical root and consequences, which is the topics of this section. 2.4.1 Triple Decomposition of Strain Rate and Velocity Gradient According to our discussion on tensor potentials of a vector in Sect. 2.3.5 and the Cauchy motion equation (2.44), the natural Helmholtz decomposi- tion in (2.134) implies that there must be a natural algebraic decomposi- tion of the stress tensor T for Newtonian fluid, able to explicitly reveal the Helmholtz tensor-potential part of T. This, by the Cauchy–Poisson equation (2.45), in turn implies that the desired algebraic decomposition must exist in the strain-rate tensor D. Therefore, we return to kinematics. Unlike the classic symmetric-antisymmetric decomposition (2.18), we now consider the intrinsic constituents of D in terms of fundamental processes. The result is simple, but has much more consequences than merely for rederiving (2.134). Since D T = D but Ω T = −Ω, we may write (∇u) T = D −Ω + ϑI − ϑI, 13 The dynamic measure of compressing process per unit mass may switch to other scalars, see later. [...]... into heat (Wu et al 1999a) Hence, instead of (2. 52) and (2. 53) we can simply write d dt ρ V ρ 1 2 q dv = 2 D Dt 1 2 q 2 V t · u dS, (ρf · u + pϑ − Φ)dv + (2. 156a) ∂V = ρf · u + pϑ + ∇ · (T · u) − Φ, (2. 156b) where T and t are given by (2. 148) and (2. 149), and Φ ≡ Φ − Φs = (λ + 2 ) 2 + µω 2 ≥ 0 (2. 157) is the reduced dissipation, again due only to compressing and shearing processes Wu et al (1999a) notice... × u, ts dS = 2 S x × ts dS = 2 S (2. 152a) C x × (dx × u) − 2 C n × u dS (2. 152b) S In particular, on a closed boundary ∂V of a fluid volume V , by the generalized Gauss theorem (A.14), the total moment due to ts is proportional to the total 54 2 Fundamental Processes in Fluid Motion vorticity in V (Wu and Wu 1993): x × ts dS = 2 ∂V n × u dS = 2 ∂V ω dV (2. 153) V Corresponding to (2. 147), a triple... to (2. 138) and (2. 140b) Substituting these into (2. 141) and using (2. 139) to express n · B, we obtain a general kinematic formula: ∇u = nnun,n + n(ω × n) − [n(W × n) + (W × n)n] + (∇π b)π (2. 143) Note that rs in the normal–normal component is canceled Then, let S and A be the symmetric and antisymmetric parts of (∇π b)π , from (2. 143) it follows that 2D = 2nnun,n + n(ωr × n) + (ωr × n)n + 2S, (2. 144)... split and how they are coupled, we assume the external body force 2. 4 Splitting and Coupling of Fundamental Processes 57 does not exist (to be discussed in Sects 3.6 .2, 4.1.1, and Chap 12) , and take the curl and divergence of (2. 163) Since ν0 ∇×ω is divergence-free, this yields ∂ω − ν0 2 ω = −∇ × (L − η ), ∂t (2. 166) ∂ϑ + 2 H = −∇ · (L − η ) ∂t (2. 167) While (2. 166) represents the general vorticity. .. (∇u)T + (∇u)T = 2 + (∇u)T , such that 2u · Ω = ω × u, u · (∇u)T = ∇u · u Then (2. 11) is cast to the vorticity form, very important in vorticity and vortex dynamics: ∂u Du = +ω×u+∇ Dt ∂t 1 2 q , 2 q ≡ |u| (2. 1 62) Therefore, the advective acceleration consists of two parts One is the gradient of kinetic energy, evidently a longitudinal process, implying that the acceleration increases as fluid particles move... as demonstrated by Wu et al (20 05c) 2. 4 .2 Triple Decomposition of Stress Tensor and Dissipation We move on to dynamics Substituting (2. 136a) into (2. 45) immediately leads to a triple decomposition of the stress tensor T = −ΠI + 2 Ω − 2 B, (2. 147) where Π is defined by (2. 135) Then since by (2. 34b) µB is a trace-free tensor, the first two terms on the right-hand side of (2. 147) form the natural Helmholtz... body 2. 4 Splitting and Coupling of Fundamental Processes 63 simplified to K= ρ Ui 2 ni φ dS = ∂B 1 Ui Mij Uj , 2 (2. 189) so by (2. 76), dK ˙ = UD = U · M · U, (2. 190) dt where D is the drag Note that without the assumed acyclic feature of φ we cannot derive (2. 183) and (2. 184), and hence neither (2. 187); but (2. 189) is not affected In this case the D’Alembert paradox can be said for drag only, and a... to r × u dV = V 1 2 ( 2 − r2 )ω dV, V so M /ρ is solely related to vorticity moments (3.8) 3 .2 Vorticity Integrals and Far-Field Asymptotics 71 Next, let be sufficiently small such that inside the sphere ω(x) = ω(x0 ) is constant Then by (2. 24) we may replace δu by ω × r /2 Substituting this into the left-hand side of (3.8) then yields M= 1 ω · J, 2 (3.9) where (r2 I − rr) dV J=ρ V or (r2 δij − ri rj )... implies ∇p + µ∇ × ω = 0 on ∂B, 2. 4 Splitting and Coupling of Fundamental Processes 59 so that n × ∇p = −µn × (∇ × ω), (2. 172a) n · ∇p = −µ(n × ∇) · ω (2. 172b) This pair of equations, especially (2. 172a), contain rich physical and mathematical information relevant to vorticity dynamics and will be fully explored later The absence of advection makes the coupling linear and hence a series of formally... kinematic identity (cf Truesdell 1954) 1 D : D = 2 + ω 2 − ∇ · (B · u), 2 (2. 154) where ω 2 ≡ |ω |2 is called the enstrophy Thus, for constant µ, (2. 54) yields the triple decomposition of dissipation rate: Φ = (λ + 2 ) 2 + µω 2 − ∇ · (2 B · u) (2. 155) Once again, the role of the three viscous constituents of stress tensor is explicitly revealed Note that the B -part of Φ can be either positive or negative . x)=h λ (k)e ik·x , h λ (k)= 1 √ 2 [e 1 (k)+iλe 2 (k)], (2. 121 ) where λ = ±1, and e 1 (k), e 2 (k), and k/k form a right-hand Cartesian triad. In this case (2. 117a) is simplified to h λ (k) ·h ∗ µ (k)=δ λµ . (2. 122 ) If. Then, let S and A be the symmetric and antisymmetric parts of (∇ π b) π , from (2. 143) it follows that 2D =2nnu n,n + n(ω r × n)+(ω r × n)n +2S, (2. 144) 2 = n(ω ×n) −(ω × n)n +2A, (2. 145) where. D = ϑ 2 + 1 2 ω 2 −∇·(B · u), (2. 154) where ω 2 ≡|ω| 2 is called the enstrophy. Thus, for constant µ, (2. 54) yields the triple decomposition of dissipation rate: Φ =(λ +2 )ϑ 2 + µω 2 −∇· (2 B ·