The Kinematics of Vorticity

VORTICITY

INTRODUCTION

All real fluid motions are rotational. Even in nearly irrotational flows the relatively small amount of vorticity present may be of central importance in determining major flow characteristics, and even some of those whose interest in fluid dynamics is only of the practical sort are now beginning to learn that the hitherto largely neglected questions of vorticity must at last be faced.

The more deeply one penetrates the general character of fluid motions, the more apparent it becomes that the dynamical properties of fluids in the main are but names, interpretations, and methods of measuring purely kinematical quantities, and that in general the flow of a fluid, whether perfect¹ or viscous, may be defined by purely kinematical conditions.² It is no accident that the greatest contributions to practical fluid dynamics were preceded by kinematical analyses which in themselves belong to pure mathematics rather than to mechanics or physics; while the work of Stokes, Helmholtz, and Kelvin is familiar, it is less well known that Euler headed his several successive presentations of the general equations of perfect fluids by increasingly detailed and accurate investigations of the possible motions of any deformable continuum, and that the same Zhukovski who discovered the artifice by which perfect fluid theory can be turned to practical use in plane wing theory began his career with a long memoir on the kinematics of continuous media.³

In the realization that the kinematics of rotational motions contains the essence of fluid dynamics the present essay was conceived. Many a theorem generally regarded as dynamical will here be found in a purer form, presented at its proper station in a consecutive development. In particular, classical hydrodynamics may be characterized by the kinematical statement of Kelvin’s circulation theorem, and in this way all the general properties of barotropic flows of inviscid fluids subject to conservative extraneous force (properties which necessarily hold equally for a special class of flows of viscous liquids) will appear as special cases of certain purely kinematical theorems valid for arbitrary media. Let no one contend, however, that I have merely derived the old results in a new way. Rather, circulation-preserving motions afford but the simplest and most elegant applications of some parts of the general theory, a theory constructed in the hope that it will prove useful in understanding the behavior of complicated media whose dynamical response is more elaborate than that represented by the classical laws of viscosity. All dynamical statements I have relegated to parenthetical sections, appendices, or footnotes, not in a foolish attempt to diminish their physical importance, but rather to let the argument course freely, uninterrupted by merely interpretative remarks, and to leave the propositions free for application to such special dynamical situations as may be of interest either now or in the future—for I cannot too strongly urge that a kinematical result is a result valid forever, no matter how time and fashion may change the laws of physics.

¹ In describing the theory of perfect fluids the physicist Rowland [1880, 8, p. 262] remarked that the d’Alembert-Euler vorticity equations (94.1) contain the whole dynamics of the subject ....

² An exception is constituted by the beautiful general theorems of gas dynamics, where kinematics, mechanics, and thermodynamics truly co-operate. This field is presented in another monograph [1952, 2].

³ [1876, 3]. Cf. [1947, 7].

Chapter I. GEOMETRICAL PRELIMINARIES

1. Algebraic operations with vectors and tensors. In this monograph I employ regularly the single cross operation of Gibbs,¹ both for vectors and for tensors. My principal reason for adopting it is that my subject is exclusively Euclidean and three-dimensional, since a vorticity vector, vortex-lines, and vortex-tubes cannot exist in two-dimensional or (for general motions) in four-dimensional spaces. While for any dyadic Σ in a space of any number of dimensions we have

only in a space of three dimensions do we have the Gibbsian decomposition:²

whereby the skew part of the dyadic is expressed in terms of the axial vector Σx, and it is the consequences of this identity which make three-dimensional kinematics a peculiar discipline and give rise to the vorticity vector.

Let us now make our stand quite clear by explaining our symbolism, including that used in the two foregoing equations. First, the reader is assumed to be familiar with the elements of tensor analysis³ and with the notations for vectors commonly in use in works on mathematical physics. When we denote a vector by the bold face letter a, we shall mean that a is to be regarded as a short name for the set of contravariant components ai or covariant components ai, whichever is appropriate. Thus the single Gibbs equation

stands for both the equivalent explicit equations

Latin indices i, j, . . . are always assumed to range over the values 1, 2, 3. It goes without saying that the covariant and contravariant components of any given field are related by ai = gijaj ai = gijai, where the gii and gii are the covariant and contravariant components of the metric tensor. By a dyadic Σ we shall mean any one of the sets of covariant, contravariant, or mixed components of a tensor of the second order. Thus the one Gibbs equation

stands for all the equations

while

stands for

The symbol I denotes the dyadic whose components are gij,δii, gij, i.e., the metric tensor itself.

Let g det gij. Only those co-ordinate systems in which g > 0 are admitted in this work. Let eijk and єijk have their usual significance as permutation symbols. Then, given two vectors a and b, the equations

define covariant components c, of a vector c, whose contravariant components are

We write both (1.9) and (1.10) in the single form

For the magnitude of the vector b we shall use the customary symbols:

The dot or scalar Σ. of a dyadic Σ is the trace of the matrix of its mixed components:

Its cross or vector Σx is the axial vector field whose covariant components are defined by

The existence of this type of vector field, peculiar to a space of three dimensions, we shall later find to be the foundation of the doctrine of vorticity.

It is easy to verify the interesting identity

The cross products a × Σ and Σ × a are defined as the dyadics Ψ, Θ whose covariant components are given by

We may write our first identity (1.1) in the form

In a space of three dimensions, however, we have

which is the covariant form of the Gibbs decomposition (1.2). Its rather awkward appearance when written in tensor notation obscures the importance, evident from the direct form (1.2), of the axial vector Σx.

While it is easily possible, besides being logically preferable, to define the Gibbs operations directly, rather than (as we have done) merely regarding each equation in the Gibbsian symbolism as an abbreviation for a number of tensorial equations, we shall not attempt to use any one system of notation exclusively. Thus we shall not use Gibbs’s symbols,

in many expressions involving tensors of order greater than one we shall employ the common indicial notations of tensor analysis.

We shall find that in a few cases it makes for clarity to employ the physical components of vectors and tensors, provided the co-ordinate system be orthogonal. These components, which are explained in detail elsewhere,⁴ we shall denote by indices set in the middle of the line:

2. Two algebraic theorems. Another special property of three-dimensional space, discovered in principle by Cauchy,¹ concerns transformations leaving one co-ordinate fixed: x-1 = x¹, x-a = x-a (x², x³), α = 2,3. Let Σ be any three-dimensional dyadic, i.e., let 2W, i, j = 1, 2, 3 be coniravariant components of a tensor field of second order with respect to transformations x-i = x-i(x¹, x², x³). Let g' = det gaβ where α, β = 2, 3, and consider only those co-ordinate systems in which g' > 0. Then the

quantities

are mixed components of a two-dimensional dyadic 1Σ, which we may call the skew projection of Σ onto the direction 1. In proof of the theorem just stated we need only observe that the quantities Σγ α, γ, α = 2, 3, are contravariant components of a tensor field of second order with respect to transformations x-α = x-α (x², x³). The dyadic 1Σ is an axial dyadic, just as Σx is an axial vector. Connecting the two there is a singular identity, also discovered in principle by Cauchy:

the magnitude of the i component of the vector Σx is proportional to the scalar iΣ. of the skew projection. iΣ of Σ onto the i direction. In the case in (2.2) may be cancelled, provided we refer Σx to its physical components:

By the proper values of a dyadic Σ we shall mean the roots of the characteristic equation

We shall require two results from algebra, the familiar theorem of Cauchy² that the proper values of a real symmetric dyadic are real, and the principal directions corresponding to two distinct proper values are orthogonal, and its converse, given in principle by Kelvin & Tait:³ if all the proper values of a real dyadic be real, and if the principal directions corresponding to any pair of distinct proper values be orthogonal, then the dyadic is symmetric.

3. Differential operations on vector and tensor fields. The symbols grad x, will be employed in their usual senses. More generally, by grad b we denote the tensor whose covariant components Ψij are given by

the comma denoting covariant differentiation. Thus the equation

stands for

and all equations obtainable from it by the raising and lowering of indices. Thus we have

Similarly

stands for

so that

represents

Since our three dimensional space is assumed to be Euclidean, we have the familiar identities

which the reader shall use, along with several others found in textbooks on vector analysis, without explicit reference.

An extremely important example of the use of the cross product is furnished by the identity

4. Special notations. In the foregoing sections we have explained our interpretations for certain symbols already in more or less common use. We now introduce a few symbols peculiar to the present study.

We shall use a bold face x to denote the three co-ordinates x¹, x², x³. Thus x does not denote a vector field. It is not to be confused in general with the field of radius vectors r, since only in a rectangular Cartesian system do the two symbols represent the same quantities x, y, z. Thus e.g. in a spherical co-ordinate system x stands for the three co-ordinates r, θ, ϕ, while the components of r are r1 = r¹ = r1 = r, r2 = r² = r2 = 0, r3 = r³ = r3 = 0. By dx we shall denote the vector field whose contravariant components are dxi.

When in an identity the symbol Ø occurs, it shall represent a scalar, vector, or tensor of any order, providing only that the resulting formula have a sense. The reader may easily formulate definitions of b.ø, ø.b, b × ø, ø × b extending (1.6) and (1.16). By grad ø we mean the tensor whose covariant components are given by

by div ø we mean the tensor whose contravariant components θij...k are given by

while by curl Ø we mean the tensor whose mixed components Ξil...m are given by

With these definitions we have always as a consequence of the Euclidean character of space the identities

We shall require some special notations for projections. Let n be a unit vector. Then the normal projection Øn and the tangential projection Øt of Ø onto the direction of n are defined by

We shall usually employ these notations in the case when n is the unit normal to a surface. For the directional derivative n· grad we shall often write d/dn) for c·grad, d/dc, etc.

In many theorems we shall wish to state conditions concerning the vanishing of certain integrals at infinity. Consider a region of space wholly or partially extending to infinity. Let there be described a sphere of radius r about the origin and let be that portion of the surface of this sphere which lies in the interior of the given region (Fig. 4.1).

FIG. 4.1

If

we shall write

For (4.7) to hold it is evidently sufficient that the numerical magnitude of each component of Øn shall be o(r –m–²) as r → ∞, this fact serving as motivation for the symbol (4.7). Similarly, if

we shall write

sufficient for this relation to hold being that the numerical magnitude of each component of Øt shall be o(r-m-2). Finally, if

we shall write

sufficient again being the stronger condition that each component of Ø shall be o(r-m-2).

The symbol ō may be read smaller mean order than.

The nth power b(n) of a vector b is defined as that tensor whose contravariant components Ψi1...in are given by

This definition is extended and put into inductive form as follows:

The symbol {b(n)ø} shall denote the symmetrized expression¹

Thus e.g.

5. Assumptions of smoothness. The terms curve, surface, and region will be used freely in this work. We tacitly assume sufficient smoothness that the transformations of Green (§7) and Stokes (§8) be valid for the usual class of fields and that existence and uniqueness of the solution to the Dirichlet problem hold for the usual class of boundary values. Thus a surface possesses a continuously turning tangent plane except upon a set of surface measure zero, and a curve possesses a continuously turning tangent except upon a set of linear measure zero; these exceptional points will henceforth in most cases be tacitly disregarded, since they may be omitted in the formation of line and surface integrals, respectively.

Precise definitions and sufficient conditions of smoothness are available in the literature.¹ It is not from mere carelessness that we do not state a particular set of them here. Rather, the quality of this work is for the most part formal. Questions of regularity are foreign to its purpose, but furnish a constant subject of research in pure analysis; whenever conditions sufficient for the truth of the fundamental theorems of analysis are weakened as a result of one of these researches, the range of validity of the results in this work is thereby automatically extended. In the few places where degree of smoothness is a matter of interest, special attention will be given to it.

6. Circulation, flux, total, and moments. Let Ø be an arbitrary integrable tensor field. The line integral

along a curve c is called the flow of Ø along c. A closed curve is called a circuit, and the integral (6.1) is then called the circulation of ø around¹ c and is written

Two curves which can be continuously deformed one into another while remaining in a given point set are reconcileable² in that set. A

circuit which can be continuously shrunk down to a point and in the process remain in a given set is said to be reducible in that set.

The surface integral

over a surface is called the flux of Ø across is a closed surface the integral is called³ the flux of Ø out of and is written

it being understood that ds points outward.

A closed surface which can be shrunk down continuously to a point and in the process remain within a given region is said to be reducible in that region.

The volume integral

is called⁴ the total Ø in b. More generally, employing the notation (4.12) we define the nth moment Ø(n) of Ø with respect to the origin by

The total Ø is thus the zeroth moment Ø(o).

7. Green’s transformation. We shall employ Green’s transformation¹ in the forms

and continuously differentiable throughout the interiors of each of a finite number of regions, of which υ is the sum, and that the volume integrals are convergent.²

The verbal expression of (4.2) in the terminology of §6 is: the total divergence of a quantity in a region equals the flux of the quantity out of the boundary of the reaion.

Another Form of Green’s transformation is³

Here b and c are arbitrary continuously differentiable fields and n is an arbitrary non-negative integer.

8. Kelvin’s transformation (Stokes’s theorem). , bounded by the curve c. Then¹

subject to the usual right-handed screw convention connecting the sense of description of c with the sense of the normal field ds. We shall call (8.1) Kelvin’s transformation.² Its verbal expression in the terminology of §3 is: the circulation of a quantity around a closed circuit equals the flux of its curl across any surface bounded by the circuit.

A formal equivalent of (8.1) is

9. Vector-lines, vector-sheets, and vector-tubes. A curve everywhere tangent to a given continuous vector field c is a vector-line of c. Necessary and sufficient that a given curve be a vector-line of c is that it admit a representation x = x(l) satisfying the differential equation

A field which is continuous in a closed region possesses at least one vector-line through each interior point of the region; if moreover the field c satisfy a Lipschitz condition,¹ there is exactly one vector-line through each point where c ≠ 0.

Consider a region in which c ≠ 0, and let t, n, b be the unit tangent, principal normal, and binormal to the vector-line. The field t is especially useful in questions concerning the geometry of the vector-lines of c. Put

We have then

Since

we have

where d/ds denotes the derivative with respect to arc-length along the vector-line.

An intrinsic expression for curl c has been obtained by Bjørgum² in the following way. For any unit vector e and any vector k we have

identically

and hence in particular

where the last step follows by (9.7). Since

we may put (9.8) into the form

whence by one of the Serret-Frenet formulae follows

where K is the curvature of the vector-line. Hence

Putting this result and (9.3)2 into (9.7) yields the desired expression:

By putting c = 1 we obtain the corollary:

The quantity Ω which appears in the foregoing expressions is an important invariant of the system of vector-lines. We shall call it the abnormalityof area s, containing P and bounded by the circuit c, reducible on the surface. Form the circulation of t around c, by Kelvin’s transformation obtaining the identity

where N . Now we reduce c to the point P. Then s → 0, N → t, and by (9.2)3 we get

be one of these we get from (9.16) Ω = 0. Thus the value of Ω in general may be regarded as a measure of the departure of the field c from the property of having a normal congruence.

A surface everywhere tangent to the field c is a vector-sheet of c. If it be supposed that at least one vector-line of c passes through each point upon a certain curve c, these vector-lines sweep out a vector-sheet; when c is a circuit, the vector-sheet is called a vector-tube of c. In order that a surface f(x) = const. be a vector-sheet of c it is necessary and sufficient that

except at a set of points of surface measure zero. A field possessed of a unique vector-line through each interior point of a region is endowed with a unique vector-sheet through each sufficiently short curve interior to the region.

2 be two surfaces whose complete boundaries are c1 and c2, respectively (2, we may call

Fig. 9.1

its flux through these surfaces the strengths of the vector-tube at the respective cross-sections. 2. Then, since ds.c = 0 upon the vector-tube, we have

In 2 be taken inward, we have

the flux of c out of a surface formed by a vector-tube and two cross-sections equals the difference between the strengths of the tube at these two cross-sections.

3. If now c be continuous throughout the closure of υ and continuously differentible throughout the interiors of a finite number of regions of which υ is the sum, then by Green’s transformation (7.2) we have

it being assumed that the volume integral is convergent. That is, the total divergence of a continuously differentiable field c in a region bounded by a vector-tube and two cross-sections equals the difference between the strengths of the tube at these two cross-sections.

10. Solenoidal fields. I. Integral properties. An integrable vector field c in its region of definition is zero is called solenoidal.¹ For a solenoidal field it follows at once from (9.4) that the strength of any vector-tube must be the same at all cross-sections. This property constitutes the characterization of Helmholtz (1858):² a field continuous³ in a closed region is solenoidal if and only if the strength of every vector-tube be the same at