Vorticity and Vortex Dynamics 2011 Part 3 pptx

3.4 Vortical Impulse and Kinetic Energy This section establishes direct relations between vorticity integrals and twofundamental integrated dynamic quantities: the total momentum and kin

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 atinfinity, 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

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 = 0 on ∂V there

are two thin vortex filaments C1and C2, with strengths (circulation) κ1 and

κ2 respectively, away from which the flow is irrotational C1and C2 must be

both closed loops Suppose C1 is not self-knotted, such that it spans a piece

of surface S1without intersecting itself, and that the circulation along C1 is

In the present situation, Γ1 can only come from the contribution of the

fila-ment C2 Therefore, if C1and C2are not tangled (Fig 3.9a) then Γ1= 0; but

if C2goes through C1 once (Fig 3.9b) then Γ1=±κ2, with the sign

depend-ing on the relative direction of the vorticity in C and C More generally, C

Fig 3.10 Decomposition of a knotted vortex filament

can go through C1an integer number of times (Fig 3.9c), so that Γ1= α12κ2,

where α12 = α21 is a positive or negative integer called the winding number

of C1 and C2

By inserting one or more pair of filaments of opposite circulations, a knotted vortex filament can always be decomposed into two or more filamentswhich go through each other but are not self-knotted Figure 3.10 shows thedecomposition of a triple knot, for which we have

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)

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 position (2.115), there is

decom-ω · u =  ijk (φλ ,j µ ,k),i −  ijk (φλ ,j µ ,ik ),

where the second term vanishes Hence

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 nected (connectivity is also a topological property)

con-A flow also has its topological structure When a flow structure is a terial curve like a vortex filament, the state of its knotness or tangledness isits topological property Some new progress in the study of this property has

ma-(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) are the same as a

torus

94 3 Vorticity Kinematics

been reviewed by Ricca and Berger (1996) Later in Sect 7.1 we shall meetthe topological structure of a vector field, which is a powerful tool in studyingseparated vortical flows For fluid mechanics these topological properties are

of qualitative value; in fact, just because quantitative details are beyond itsconcern, the topological analysis is generally valid

3.4 Vortical Impulse and Kinetic Energy

This section establishes direct relations between vorticity integrals and twofundamental 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 andvortex 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 sincerelevant integrals are merely conditionally convergent To avoid this difficulty,

one appeals to the concept of hydrodynamic impulse (impulse for short) and angular impulse The potential 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

As ∂V encloses the entire vector field i(x), the surface integral vanishes since

i = 0 there by assumption This proves that

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

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)

is an alternative definition of the angular impulse, see (3.6) Comparing (3.81a)

and (3.81b), for n = 3 there is

so L
= L if n · ω = 0 on ∂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 face integrals, which can then be simplified owing to the derivative-moment transformation (A.25) and (A.28a,c):12

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 nally unbounded flow, by (3.49) (with Γ ∞ = 0 when n = 2) the convergence

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

∂VR

φn dS = −1

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 = 3 or n = 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 R2-divergence:

C

x2n × u ds = R2

2 Γ R e z .However, these discussions are of mainly academic interest What entersdynamics 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 momentumand 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

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)

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) wehave

3.4.2 Hydrodynamic Kinetic Energy

Lamb (1932) gives two famous formulas for the total kinetic energy in a

in terms of vorticity Here the flow is assumed incompressible with ρ = 1 The

first formula is based on the identity

q2= u · (∇φ + ∇ × ψ) = ∇ · (uφ + ψ × u) + ω · ψ, (3.91)

where φ and ψ are the Helmholtz potentials given by (2.104) with ϑ = 0 now.

The second formula is the direct consequence of (3.74) for three-dimensional

2n − 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| → ∞, for n = 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 indealing with impulse and angular impulse indicates that for unbounded flow(3.92) can be written as a double volume integral

K = 12π

 

Gω · ω
dV dV
, (3.94)

where G is given by (2.102) Hence, in three dimensions there is

K = − 18π

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, whichcan be symbolically expressed by

K = K V (α) + K S (α) , (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

k(2)V (x) ≡ (ω × u) · x = (x × ω) · u, (3.97b)

do not represent the local kinetic energy density q2/2 They are even

not positively definite However, like in many other formulas from

inte-gration by parts, only k V (α) , α = 1, 2, have net volumetric contribution (positive or negative) to K, with more localized support but containing more information on flow structures than q2/2 In this sense, k V (α) can be

viewed as the net kinetic-energy carriers (per unit mass) As tion, Fig 3.14 compares the instantaneous distribution of q2/2 and ωψ/2

illustra-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, q2/2 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 S(1) vanishes; but the appearance of x in K S(2)

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 V(2) 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 q2/2 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 Thisfact is a reflection of the nonlinearity in vortical flow advection

Fig 3.14 Instantaneous distribution of (a) q2/2 and (b) ωψ/2 in a two-dimensional

homogeneous and isotropic turbulence, based on direct numerical simulation tesy of Xiong

Cour-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 doesaffect 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 q2/2 It is this mechanism

that is explicitly reflected by Lamb’s first formula For a similar mechanisminvolved in the total disturbance kinetic energy and its relation to flow stabilitysee 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 thesequantities 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 thevorticity form of the material acceleration, (2.162), and the result can beexpressed in a few equivalent forms:

fication, known as the Beltrami equation:

where the first two equalities imply that this term is a coupling of strain-rate

tensor and vorticity tensor , while the last equality implies that ω ·∇u−ϑω =

−ω · B with B = ϑI − (∇u)T being the surface deformation tensor, see thecontext of (2.27) This leads to another compact form of (3.98c):

Dω

The unique kinematic property of vorticity evolution in three-dimensional

flows is reflected by the key term ω · B Let dS = n dS be a cross surface element of a thin vorticity tube so that ω = ωn Then

material vorticity tube, measured by the rate of change of n and cross area

dS of the tube, respectively These mechanisms do not create vorticity from

an irrotational flow but alter the existing vorticity distribution, leading tovery far-reaching results (Sect 3.5.3, Chap 9and 10) In two-dimensional flow

ω · B = ωϑ, only the compressibility may change the cross area of a vorticity

tube

Associated with the tilting and stretching of vorticity tube, a vorticity line

is also subjected to tilting and stretching To see this, we assume∇ × a = 0,

so (3.99) indicates that the equation for ω/ρ has exactly the same form as

the rate of change of a material line elementδx given by (2.17) Suppose δx

is a segment of a vorticity line and at t = 0, δx = ω/ρ Then there is

ness theory for the initial-value problem of these equations ensuresδx = ω/ρ

for all later t (e.g., Whitham 1963) Therefore, once ∇ × a = 0 each line of ω/ρ remains the same material line at any time This is the second Helmholtz

vorticity theorem to be discussed later.

We now take the inner product of (3.99) and the gradient of an arbitrary

tensor S (we use a tensor of rank 2 in the algebra but the result holds for any



ω i ρ

ρ(∇ × a) · ∇S (3.104)

Assigning S with different quantities may lead to a variety of results Taking

S = x simply returns to (3.99) The major applications of (3.104) is when S

A special case of (3.105) is that S is a conservative scalar φ The scalar (ω/ρ) ·

∇φ is known as the potential vorticity introduced by Rossby (1936, 1940) and

Ertel (1942), governed by (the name will be explained in Sect 3.6.1)

Lagrangian invariant, an important example of generalized potential vorticity

∇ X x defined by (2.3) The vectors in (3.107) are defined in the reference space spanned by X, indicating that we have obtained the vorticity trans-

port equation in reference space Then, since in the Lagrangian description

D / Dt = ∂/∂τ , where τ = t is the time variable, (3.107) can be integrated

once Set x = X and ∇X = I at t = 0, we obtain

the inner product of both sides with F from the right, (3.108) is transformed

back to the physical space:

Truesdell (1954) calls (3.109) the fundamental vorticity formula Its two

terms have distinct physical sources and behavior Recall that F measures

the relative displacement of fluid particles that were in the neighborhood of

X at t = 0 At any t, F depends only on the displacement of the particles

from their initial position but independent of the history of their motion.

Hence, so is the first term of (3.109) The evolution represented by this term

is purely kinematic In contrast, since the kinetics of shearing process is fully

reflected by∇ × a, we now see that through the time integral in (3.109) the

shearing kinetics yields an accumulated effect on the vorticity at all points

through which a fluid particle X goes This makes the final state of vorticity

of the fluid particles inherently rely on their history, which is precisely thecharacteristics of kinetics Following Truesdell (1954) but adding an adjective,

we call∇ × a the vorticity diffusion vector and ρ ư1(∇ × a) · Fư1 the material vorticity diffusion vector, to which the study of vorticity kinetics is thereby

attributed: what causes the vorticity to diffuse and how much the diffusionwill be This is the topic of Sect 4.1

3.5.2 Evolution of Vorticity Integrals

LetV be a material volume and V a fixed control volume The integral form

where n·(ωuưuω) = ưn×(ω×u) Applying these integrals to an unbounded

fluid at rest at infinity, by (3.18) there is

We proceed to consider the rate of change of some vorticity-related

inte-grals First, (3.110) is a special case of the time-evolution of a general

nth-order moment (3.17b), of which the derivation is longer but straightforward;the result reads:

of the Stokes theorem (A.17), we have obtained the Kelvin circulation formula

(2.32) Noticing that the circulation of the potential part of a, say ư∇φ ∗, mustvanish if φ ∗ is single-valued Thus, denoting the rotational part of a by a ,

scalar, Γ only reflects the strength of a vortex tube but not its orientation,

nor the vorticity magnitude at a point

Next, consider the evolution of helicity defined as the material volume

integral of helicity density Its rate of change can be derived from (3.99) byusing (2.41):

Finally, the rate of change of vortical impulse I and angular impulse L

defined by (3.78) and (3.79), respectively, are closely related to the integrals

of the Lamb vector l ≡ ω × u and its moment Instead of directly calculating the time derivative of I and L, we start from making the derivative-moment transformation of acceleration a by using (A.23) for any fluid domain D:

where k = n − 1 and n = 2, 3 is the spatial dimensionality, and ∇ × a is

expressed by (3.98a) We than use (A.23) again to transform the integral of

∇×l (l = ω×u denotes that Lamb vector) back to the integral of l, obtaining

k x × ω ,t + l

dV −1k



∂ D x × [n × (a − l)]dS (3.117b)

Hence, if the flow is incompressible andD is a material volume V, by (2.35b),

comparing (3.117a) and (3.117b) yields

In particular, if the fluid is unbounded both internally and externally, and

at rest at infinity, then since by (3.72) and (3.73) the integrals of the Lambvector and its moment vanish, we simply have

of a, we may apply (3.117a) to ar only, so that the second term of (3.120)

is cast to surface integrals of ψ ∗ and ar, which vanish as |x| → ∞ (e.g., for viscous incompressible flow ψ ∗ =−νω) The same can be similarly done for

the right-hand side of (3.121) Hence, the vortical impulse and angular impulse

of an unbounded fluid is time invariant.

3.5.3 Enstrophy and Vorticity Line Stretching

Owing to the F¨oppl theorem (3.15), the integral of vorticity vector ω cannot

tell the total amount of shearing in a flow domain Similar to the kinetic

energy, we use the enstrophy ω2/2 for such a measurement and now examine

its time evolution

The inner product of ω with (3.101) yields

D

Dt

1

2ω

2

=−ω · B · ω + ω · (∇ × a). (3.122)Unlike circulation, in (3.122) the stretching effect is retained but tilting effect

is removed Write ω = tω, we have

−ω · B · ω = αω2, α ≡ −t · B · t = t · D · t − ϑ. (3.123)

The scalar α is the stretching rate of an infinitely thin vorticity tube or a

single vorticity line Namely, there is

D

Dt

1

106 3 Vorticity Kinematics

showing that the magnitude of vorticity will be intensified by a positivestretching

In Lagrangian description (3.124) or (3.125) can be integrated with respect

to time Again let τ be the Lagrangian time and denote

Figure 3.15 shows a numerical example due to Siggia (1985), who

com-puted the evolution of a vorticity loop which is initially elliptic on the (y, plane Let L be its total length and σ be the cross-sectional radius, and assume that σ2L is time-invariant At time t = 0, σ0, and L0 were taken as 0.2 and

z)-∼10, respectively, and the ratio of the axes of the initial elliptical ring was

4:1 The self-induction is nonuniform, as can be inferred from (3.32) Thisinduction causes the vortex ring to deform quickly and the filament to stretchnonuniformly Figure 3.15a shows a sequence of the vorticity-loop shapes and

Fig 3.15b shows the growth of L in time, where the last three points imply

an exponential growing

As a nonlinear effect, vortex stretching is a crucial kinematic mechanism

in the entire theory of shearing process This mechanism and vortex tilting, aswell as the cut and reconnect of vortices due to viscosity (Sect 8.3.3), are thekey to understanding many complex vortical flows In particular, stretching

is responsible for the cascade process in turbulence, by which large-scale

vor-tices become smaller and smaller ones with increasingly stronger enstrophy

(Chap 10) In fact, turbulence may be briefly defined as randomly stretched

vortices (Bradshaw, private communication, 1992) The strain rate D that

causes stretching can be either a background field induced by other vortices,

or induced locally by the vortex itself The strongest stretching and

shrink-ing occur if ω is aligned to the stretchshrink-ing and shrinkshrink-ing principal axes of D, respectively Then the stretching rate α is the maximum eigenvalue of D Generically, ω is not aligned to any principal axis of D, and it is desired to

analyze the mechanisms responsible for α We now give two general formulas

for incompressible flow The first formula is local Similar to the intrinsicstreamline triad used before, we introduce an intrinsic triad 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 = (u s , u n , u b) Then by the Frenet–Serret formulas (A.39) there is

(b)

Time 1.51.0

10 100

x

y

z x

L

z x

y

y z

Fig 3.15 The self-induced stretching of a vorticity loop, starting from an elliptical

vortex ring on the (y, z) plane (a) The shape evolution of the loop, (b) the growth

of the length L of the loop From Siggia (1985)

108 3 Vorticity Kinematics

For small κ and τ , the ω-line stretching (or contracting) and tilting are

dom-inated by the increments of u s and (u n , u b ) along s, respectively (Batchelor

1967) However, a strong stretching may occur at a point of a vortex filament

even if ∂u s /∂s = 0, as long as κ 1 and u n < 0 (outward from the curvature center) Inversely, if u n > 0 (toward the curvature center) then the vortex fil-

ament will be significantly compressed and becomes much thicker Note thatthis curvature effect cannot be explained solely by the strain-rate tensor of abackground flow; the local self-induced velocity has to be included

The second formula is global, which expresses the stretching rate caused

by a distributed vorticity field rather than a thin vortex filament Assume theincompressible flow is unbounded and at rest at infinity The strain rate tensor

D has been expressed by vorticity integral in (3.37) Making inner product at

its both sides with the unit vector t(x) along ω(x), and writing ω(x
) = t
ω
,the desired formula follows (Constantin (1994)):

To see the implication of this result, assume the distribution of|ω(x)| = ω(x)

is given Then α(x) solely depends on the orientation of the three unit vectors

e, t
, and t Here, |e · t| ≤ 1 and the scalar in square brackets is the volume of

the prism formed by these three unit vectors This volume crucially depends

on the orientation of t
and t The local vorticity ω(x
) will have a strongest

contribution to the stretching rate α(x) if ω(x) and ω(x
) are perpendicular,

and if ω and r is neither parallel nor perpendicular Denote t ·t
= cos φ, then

|(e · t)(t
× t) · e| ≤ | sin φ|.

It should be stressed that, as seen from Fig 3.15, when the vortex ring isstretched it is also tilted A straight vortex cannot be stretched unboundedly:Chorin (1994) points out that by (3.95a) this would increases the induced

kinetic energy K unboundedly, but the total kinetic energy is conserved (or

decreasing due to dissipation) if in (2.52) no external force is imposed and thefluid is unbounded To offset the increase of kinetic energy due to stretching,therefore, there must be vortex tilting, see the first term of (3.102), which

may cause a partial cancellation of part of D Note that, contrary to the

stretching, (3.95a) indicates that the main contribution to K is from those vortex segments that are parallel to each other.

The vortex stretching also plays a key role in the mathematical aspects

of the Navier–Stokes equation It is especially relevant to a long-standingunsolved problem on whether a three-dimensional Navier–Stokes solution withsmooth initial condition can spontaneously develop a singularity at a finite

time t ∗ (and hence for t > t ∗the solution no longer exists) Beale et al (1984)have proved that only if the maximum of |ω| diverges as t → t ∗, can the

three-dimensional Euler solution blow up In other words, vortex stretchingdominates the regularity of flows (see also Majda 1986; Doering and Gibbon1995; Majda and Bertozzi 2002) But the final answer remains unknown

Finally, we mention that in two-dimensional flow without vorticity-tubestretching and tilting, the vorticity field may also evolve to some structuresand the transition to turbulence may still happen The underlying physicalmechanism is quite different, though, and will be addressed in Chap 12.

where C is any material loop Equation (3.130c) 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

Condi-tions in (3.130) define a special class of flows of significant interest, known as

circulation-preserving flows (Truesdell 1954).

In terms of the two fundamental processes, an overall physical ing can be gained for the very nature of circulation-preserving flows Because

understand-a understand-and ∇ × a are the bridges of kinematics and kinetics in the momentum

equation and the vorticity transport equation, respectively, we see at once that

(3.130a) implies that in a circulation-preserving flow the evolution of shearing process is purely kinematic Consequently, a series of important Lagrangian

invariant quantities or conservation theorems for the shearing process exist,which we present first On the other hand, (3.130b) implies that the kinetics

only enters the compressing process through the acceleration potential φ ∗,which suggests a possibility for the momentum equation be integrated once

to yield Bernoulli integrals This is our second topic Then, since the ity conservation and Bernoulli integral appear as the two sides of a coin, acombination of both may lead to a deeper understanding of this class of flowsand some further important theoretical results These can be best revealed inthe Hamiltonian formalism and is our third topic The study of this sectionnaturally paves a way to vorticity dynamics, which starts from Chap 4

vortic-3.6.1 Local and Integral Conservation Theorems

Almost all the results of this subsection come solely from (3.130a) The centrallocal conservation theorem 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.

is constant along a streamline On the other hand, under the same conditionsany 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)

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 andNaghdi 1991) Then by (2.1), (3.107), and (3.109), we have

physical space, of course ω keeps evolving, but is solely driven by the

deforma-tion gradient tensor F and independent of the history Evidently, any F(Ω)

is also conserved, and in reference space we have

if the flow is circulation-preserving.

Second, taking S = φ as a conserved scalar, from (3.131) follows that, if

Dφ

Dt = 0, for any φ and P ≡ ω

ρ · ∇φ, (3.135)then

DP

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 ther the flow is circulation-preserving or ∇φ is perpendicular to the vorticity diffusion vector.

ei-The dimension of P may not be the same as that of the vorticity ei-The

name “potential vorticity” came from the fact that, if the distance between

two neighboring iso-φ surfaces increases such that |∇φ| is reduced, then by

(3.136) the component of ω/ρ parallel to ∇φ must be enhanced If ρ varies

very weakly, what is changing must be the vorticity, and the stretching of

the distance between iso-φ surfaces has an effect similar to the vorticity-tube

stretching

The Ertel theorem has found most comprehensive applications in ical fluid dynamics to be examplified in Chap 12 As an easy application of

geophys-the Ertel geophys-theorem, take φ = z in two-dimensional flow Since Dz/Dt = w = 0,

∇z = e z , and ω · e z = ω is the only nonzero vorticity component, for

two-dimensional circulation-preserving flows there is

D

Dt



ω ρ

which also follows directly from the Beltrami equation (3.99) Similarly, for

rotationally symmetric flow, take φ as the polar angle θ in a cylindrical dinates Then since rDθ/Dt = u θ= 0 and∇θ = eee θ /r, we get

coor-D

Dt



ω θ ρr

In addition to these local conservation theorems, there are various gral conservation theorems of which the central one is the Kelvin circulationtheorem (3.130c) Except its extensive practical applications, this powerfultheorem has been used to prove other conservation theorems, including theCauchy potential flow theorem and the following theorem

inte-The Second and Third Helmholtz Vorticity inte-Theorems (Helmholtz

1858) If and only if the flow is circulation-preserving, a material vorticity tube will move with the fluid (the second theorem) and its strength is time- invariant (the third theorem).

Remark The first Helmholtz theorem (Sect 3.2.1) is universally true, since

it only relies on the spatial property∇·ω ≡ 0 The second and third Helmholtz

theorems are conditional since they involve dynamic assumptions.

The proof of the second and third theorems can be found in standardtextbooks We just recall that a proof of the second theorem has alreadyappeared following (3.103), where the same circulation-preserving condition

was used but applies to any single vorticity line, more general than the

sec-ond Helmholtz theorem Actually, the sufficient and necessary csec-ondition for avorticity line to remain a material line, and hence for the second Helmholtz

112 3 Vorticity Kinematics

theorem to hold, can be relaxed to (Truesdell 1954)13

i.e.,∇ × a is aligned to ω To prove this, consider a material line x = x(s, t)

with s being the parameter defining the line At a time t this line coincides

with a vorticity line if and only if

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 alignment of

∇ × (ω × u) and ω, which by (3.59) ensures the existence of Lamb surfaces

(Sposito 1997) Thus, the material surfaces forming any vorticity tubes areLamb surfaces

Next, by (3.115) we have the helicity conservation theorem (Moffatt

1969): The helicity of circulation-preserving flow in a domain is time-invariant

if ω · n = 0 and n × a = 0 on the boundary:

fil-satisfy this boundary condition, it cannot be circulation preserving near thewall

Finally, it is worth emphasizing again that, as shown in Sect 3.5.2, the

vortical impulse and angular impulse are invariant even for viscous flow if it

is incompressible, unbounded, and at rest at infinity (e.g Saffman 1992).

13

This argument is invalid for two-dimensional flow, since there (3.140a,b) is possible

im-3.6.2 Bernoulli Integrals

We now turn to the consequence of (3.130b) to seek Bernoulli integrals, first

in terms of the Eulerian description and then Lagrangian description We stay

with the acceleration potential φ ∗ in its general form until the most generalBernoulli integral of circulation-preserving flows is found The kinetic content

of φ ∗ will then be identified

Substituting (3.130b) into (2.162) yields

∂u

∂t + ω × u + ∇

1

2q

2+ φ ∗

Evidently, if in a region the flow is irrotational such that u = ∇ϕ, then (3.142)

can be integrated once to yield the most commonly encountered Bernoulliintegral, with (2.177) being a special case:

a line, a surface or in a volume the first two terms of (3.142) can be reduced

to a gradient of a scalar, then (3.142) can be integrated once on that line,surface or volume, yielding a corresponding Bernoulli integral

In the Eulerian description, when (3.130b) holds, the weakest condition

for the existence of a Bernoulli integral is that the flow is rotational but the vorticity is steady:

But this implies at once that (3.63) is satisfied, i.e., the flow is generalized

Beltramian If the potential χ of ω×u is known, say (3.67) for two-dimensional

or rotationally symmetric flow, then a volume Bernoulli integral exists:

The Bernoulli integrals appearing in most books represent various cations and simplifications of (3.145) or (3.146) But since the circulationpreserving is a Lagrangian property, the most general form of the Bernoulliintegral should be best revealed in the Lagrangian description, where the trou-blesome nonlinear Lamb vector in (3.142) is absent This integral follows from

appli-inspecting the X-space image of the acceleration, which reads (for derivation

see Appendix A.4)

∂U

∂τ = A + ∇ X

1

be attributed to pure kinematics even if the flow is circulation-preserving.This is what we asserted in the beginning of the section.

We now need to map (3.152) from the reference space to physical space

First, we make a Monge decomposition of u0 at τ = 0 (see (2.115)):

Taking inner product of both sides with ∇X then gives the counterpart of

(3.152) in physical space at any time:

con-Helmholtz second vorticity theorem Note that as remarked following (2.115),

Φ is not the full velocity potential and depends on history.

Finally, substituting (3.154a) into the acceleration formula (2.11), i.e.,

a = ∂u/∂t + u · ∇u, and using (3.154b), we obtain