Vorticity and Vortex Dynamics 2011 Part 2 doc

Thus, while for Newtonian fluid the stress tensor T is uniquely given by 2.45, fs has infinitely many tensor potentials, among which the above T with only three independent components is t

due to the Gauss theorem and (2.98a) Thus, there must be u ⊥1 = u ⊥2 andhence∇φ1=∇φ2 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∇2ψ = 0, such that ∇ψ is also orthogonal to both ∇(φ−ψ) and u ⊥ 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

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 = −∇2F 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

not act to functions of x
but to G(x − x
) only By (2.101) we have

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

some finite region or decay sufficiently fast (Phillips 1933; Serrin 1959).10Therefore, (2.104) provides a constructive proof of the global existence of theHelmholtz decomposition for any differentiable vector field Moreover, (2.104)indicates that the split vectors ∇φ and ∇ × ψ can be expressed in terms of

dilatation and vorticity, respectively:

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 It does

not care any boundary condition for ∇φ and ∇ × ψ In order to obtain

the unique Helmholtz–Hodge decomposition, the simplest way is to solve thescalar boundary-value problem (2.89a) and (2.98b) To see the structure ofthe solution, we use Green’s identity

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

and hence u ⊥ = u − ∇ φ is the unique transverse vector.

The Helmholtz–Hodge decomposition is also a powerful and rational toolfor analyzing numerically obtained vector fields, provided that effective meth-ods able to extend operators gradient, curl, and divergence from differentialformulation to discrete data can be developed For recent progress see, e.g.,Tong et al (2003)

10For the asymptotic behavior of velocity field in unbounded domain see Sect 3.2.3

2.3.3 Monge–Clebsch decomposition

So far we have been able to decompose a vector field into longitudinal andtransverse parts It is desirable to seek a further intrinsic decomposition ofthe 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:

The variables φ, ψ, and χ 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

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)

=∇2χ ∇ψ − ∇2ψ ∇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 theHelmholtz–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

and thereby produce a Helmholtz–Hodge decomposition, the problem maynot be solvable Also note that neither (2.110) nor (2.111) satisfies the gaugecondition (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

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

2.3.4 Helical–Wave Decomposition

An entirely different approach to intrinsically decompose a transverse vectorand giving its two independent components clear physical meaning, free fromthe mathematical limitation of Clebsch variables, can be inspired by observinglight waves A light wave is a transverse wave and can be intrinsically splitinto right- and left-polarized (helical) waves.11 Mathematically, making thissplitting amounts to finding a complete set of intrinsic basis vectors, whichare mutually orthogonal in the sense of (2.90), and by which any transversevector 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) = λkφλ(k, x) in D,

n · φ λ (k, x) = 0 on ∂ 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 φλs can only be found in complex vector space Their orthogonality

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 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

indepen-an axial vector has only one independent variable (e.g., Lugt 1996)

12Additional condition is necessary in a multiple-connected domain

For example, for an incompressible flow with un= ωn= 0 on ∂ D, one can

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

k,λ

λ2u 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

where λ = ± 1, and e1(k), e2(k), and k/k form a right-hand Cartesian triad.

In this case (2.117a) is simplified to

Therefore, as we move along the z-axis, the locus of the tip of φλ(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 vectorintrinsically to its finest building blocks

Fig 2.7 A helical wave

The simple Fourier HWD basis cannot be applied to domains other thanperiodic boxes To go beyond this limitation, we notice that the curl of (2.116)

along with itself leads to a vector Helmholtz equation

∇2φ λ + k2φ 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 theHelmholtz 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

Morse and Feshbach (1953, pp 1764–1766) have shown that this can indeed

be realized in and only in Cartesian, cylindrical, spherical, and conical dinates Specifically, a transverse solutions (not normalized) of (2.124) can bewritten as

coor-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

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

which is the resultant surface force per unit volume and contains most of the

kinetic properties of flows Assume we have decomposed fs to

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

of fs Obviously there must be∇ · (T − T) = 0 Any other tensor, say T
, can

also be a tensor potential of fsprovided that T− T
is divergenceless Thus,

while for Newtonian fluid the stress tensor T is uniquely given by (2.45), fs

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 fs.

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

ttt(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  ttt is feasible when one considers the volume integral of fs:

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, ttt) is convenient only when fshas 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 anybody force to a resultant surface force Whether a force is a body force

or surface force should be judged by physics rather than mathematics; asurface 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 duced the intrinsic decomposition of vector fields, we can now gain a deeperinsight into the roles of compressing and shearing processes in the kinematicsand dynamics of a Newtonian fluid Our main concern is the splitting andcoupling of the two processes in the Navier–Stokes equation (2.47) As just

intro-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

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 sure of the shearing process remains to be the vorticity ω (multiplied by the

mea-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 indeinde-pendent 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 2.3.5and 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 DT= D but ΩT=−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

so that by (2.27) there is an intrinsic triple decomposition of D and ∇u (Wu

and Wu 1996):

Thus, the velocity gradient and rate of strain consist of a uniform

expan-sion/compression, a rotation, and a surface deformation Here, B has most

complicated structure among the three tensors on the right-hand side, andneeds a further examination

As seen from (2.26) and (2.29), the primary appearance of B is in the form

of n · B for the rate of change of a surface element dS = n dS, which involves

the kinematics on the surface only Denoting the tangent components of any

vector by suffix π, it can then be shown that (Appendix A.3.2)

n · B = (∇ π · u)n − (∇ π un+ u · K), (2.137)where

is the symmetric curvature tensor of dS consisting of three independent

tan-gent components Then (2.137) can be written as

W π =−n × (∇ π un+ u · K). (2.140b)Return now to the triple decomposition (2.136) As a special case and

a fundamental application, we consider its form on an arbitrary materialboundaryB with unit normal n and velocity u = b The resulting kinematic

knowledge (Wu et al 2005c) is indispensible in studying any fluid-boundaryinteractions Due to the adherence, we may write

∇u = n(n · ∇u) + ∇ π u = n(n · ∇u) + ∇ π b on B.

Since (2.136b) gives

n · ∇u = nϑ + ω × n − n · B,

we obtain

∇u = nnϑ + n(ω × n) − n(n · B) + ∇ b on B. (2.141)

Here, the tensor∇ π b depends solely on the motion and deformation of B but

independent of the flow To see the implication of (2.141), we first assumeB

is rigid to which our frame of reference can be attached, so the last two termsvanish:

∇u = nnϑ + n(ω × n).

Then, taking the symmetric part immediately yields the Caswell formula

(Caswell 1967):

2D = 2nnϑ + n(ω × n) + (ω × n)n, (2.142)

where by (2.140a) ϑ can be replaced by u n,n since rs= 0

We now extend (2.142) to an arbitrary deformable surfaceB (to which no

frame of reference can be attached) For this purpose we only need to treatthe tensor∇ π b in (2.141) In general, it consists of a tangent–tangent tensor

(∇ π b) π and a tangent–normal tensor

(∇ π b · n)n = (∇ π bn− b · ∇ π n)n = −(W × n)n

due 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 = nnu n,n + n(ω × n) − [n(W × n) + (W × n)n] + (∇ π b) π (2.143)

Note that rsin 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 = 2nnu n,n + n(ωr× n) + (ωr× n)n + 2S, (2.144)

2Ω = n(ω × n) − (ω × n)n + 2A, (2.145)

where ωr≡ ω−2W is the relative vorticity with n·ωr= 0, see (2.69) Tensor S

has three independent components and can be called surface-stretching tensor.

In contrast, A has only one independent component, which must be ωn Notethat on a rigid boundary (2.144) reduces to (2.142), but viewed in a frame ofreference fixed in the space

The generalized Caswell formula (2.144) contains the key kinematics on

an arbitrary boundary B It represents an intrinsic decomposition of D with

respect to one normal (N) and two tangent (T) directions, along with theirrespective physical causes Namely: the N–N component caused by the normalgradient of normal velocity; the N–T and T–N components caused by the rela-tive vorticity; and the T–T components caused by the surface flexibility With

known b(x, t)-distribution and surface geometry, (2.143) and (2.144) express

a basic fact that the motion and deformation of a fluid element neighboringB

are described by only three independent components of velocity derivatives.The form of (2.144) suggests a convenient curvilinear orthonormal basis

(e , e , nnn) on B Let e2 be along ω ,nnn = −n (towards fluid), and hence e1is

Fig 2.8 Principal directions of the strain-rate tensor on a rigid boundary

along ωr× nnn It will be seen below (Sect 2.4.2) that for Newtonian fluid e1

is the direction of shear stress τ ; thus we call this frame the τ -frame Then

We have seen in Sect 2.1.2 that the first and third eigenvalues λ 1,3 and

associated principal directions of D reflect the maximum stretching/shrinking

rates and directions, respectively On a boundaryB they have dominant effect

on the near-boundary flow structures and their stability Let θ1,3be the angles

between the stretching/shrinking principal axes and e1 of the τ -frame If B

is rigid with S = 0 and if the flow is incompressible, from (2.146) it follows

at once that λ1,3 =± ωr/2, λ2 = 0, and θ1,3=± 45 ◦, as sketched in Fig 2.8.

But the situation can be very different ifB is deformable, as demonstrated by

Wu et al (2005c)

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

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

nat-ural Helmholtz tensor potential of the surface force In other words, for flow

with constant µ, the surface deformation process does not affect the local

momentum balance (Wu and Wu 1993) This explains the origin of (2.134)

and permits introducing the reduced stress tensor and reduced surface stress

solely in terms of compressing and shearing processes:



ttt = n · T = −Πn + µω × n. (2.149)The conceptual simplification brought by T can also lead to significant compu-

tational benefit For example, Eraslan et al (1983) have developed a Navier–

Stokes solver where some components of T were found never useful and hence

simply dropped Consequently, for incompressible flow the solver reached abig saving of CPU in stress computation

Evidently, although for constant µ the surface-deformation stress does not

contribute to the differential momentum equation nor integrated surface forceover a closed boundary, it must show up in more general circumstances Here

we consider two simple examples; in Sect 4.3.2 we shall see the crucial role ofthe surface-deformation stress in free-surface flow

First, on any surface element of unit area in the fluid or at its boundary,

we have a triple decomposition of the surface force:

t =  ttt + ts, ts=−2µn · B = −2µ(nrs+ W × n), (2.150)

where ts is the viscous resistance of the fluid surface to its motion and

defor-mation, which has both normal and tangent components Thus, denote the

orthogonal decomposition of the stress by t = −  Πn+τ , by (2.135) the general

normal and tangent components are



Π = Π + 2µrs= p − λϑ − 2µu n,n , (2.151a)

On a solid wall τw = −τ is the skin-friction stress, always determined by

the relative vorticity Note that ts = 0 only if the surface performs uniform

translation; even a rigid rotation will cause a nonzero ts due to the viscous

resistance to the variation of n.

Second, using the generalized Stokes theorem (A.17) and its corollary(A.26), from (2.150) and (2.29) we find the general formulas for integrated

force and moment due to ts on an open surface S with boundary loop C:

In particular, on a closed boundary ∂V of a fluid volume V , by the generalized

Gauss theorem (A.14), the total moment due to tsis proportional to the total

vorticity in V (Wu and Wu 1993):

Corresponding to (2.147), a triple decomposition can also be made for the

dissipation Φ, which brings some simplification too We start from a kinematic

explicitly revealed Note that the B-part of Φ can be either positive or

nega-tive More interestingly, we have

t · u = −Πn · u + µω · (n × u) − n · (2µB · u);

substituting this and (2.155) into (2.52), we see that the B-part of Φ and that

of t · u are canceled Explicitly, at each point the work rate per unit volume

2q2

is the reduced dissipation, again due only to compressing and shearing processes.

Wu et al (1999a) notice that the characteristic distribution of scalars ϑ2, ω2,

and Φsin (2.157) may serve as good indicators for identifying different tures in a complex high Reynolds-number flow However, it should be warned

struc-that it is the full dissipation Φ rather than  Φ that causes the entropy

pro-duction in (2.61) Otherwise a fluid making a solid-like rotation would have a

dissipation µω2, which is of course incorrect

Finally, for constant µ, from (2.155) the total dissipation over a volume V

If ∂V extends to infinity where the fluid is at rest, then only the reduced

dissipation Φ contributes to the total full dissipation.

So far we have assumed a constant µ More generally, with µ = µ(T )

the natural Helmholtz decomposition of the momentum equation (2.134) nolonger exactly holds Rather, there will be (Wu and Wu 1998)

number P r = µcp/κ and S µ are of O(1) Because q is along the normal of

isothermal surfaces, the extra term in (2.160a) is proportional to the viscous

resistance of isothermal surfaces to their deformation In most cases this effect

is small

2.4.3 Internal and Boundary Coupling of Fundamental Processes

For vorticity dynamics (and sometimes “compressing dynamics” as well), theNavier–Stokes equation expressed for unit mass is more useful than that forunit volume But, once the fluid density is a variable, such an equation with

constant µ,

Du

Dt = f −1

where ν = µ/ρ is the kinematic viscosity, is no longer a natural Helmholtz

decomposition To discover the underlying physics of the two fundamentalprocesses and their coupling most clearly, we need a different decomposition

to maximally reveal the natural potential and solenoidal parts.

First, on the left-hand side of (2.161) it is the nonlinear advective

accel-eration u · ∇u that causes all kinematic complexity of fluid motion, never

encountered in solid mechanics Thus we decompose it first Write

∇u = ∇u − (∇u)T+ (∇u)T= 2Ω + (∇u)T,

2q2

Therefore, the advective acceleration consists of two parts One is the gradient

of kinetic energy, evidently a longitudinal process, implying that the ation increases as fluid particles move toward higher kinetic-energy region

acceler-The other is the Lamb vector ω × u already encountered in (2.85), which

is analogous to the Coriolis force observed in a rotating frame of reference(see Sect 12.1.1) and drives the particles move around the vorticity direction

Generically ω × u is neither solenoidal nor irrotational, and hence appears in

the evolution of both longitudinal and transverse processes

Next, on the right-hand side of (2.161) there is an inviscid term−∇p/ρ,

which by (2.64) equals −∇h + T ∇s, where ∇h is also a natural

longitudi-nal process Thus, (2.161) can be cast to the Crocco–Vazsonyi equation (its inviscid version is the Crocco equation):

a basic viscous force due to vorticity diffusion, so that on the right-hand side

of (2.163) an explicit Helmholtz decomposition is recovered The extra term

η
originates from compressibility and thermodynamics, which in most cases

is negligible Therefore, the main complexity of (2.163) lies in its inviscid

nonlinear term called generalized Lamb vector :

which is usually dominated by the Lamb vector ω × u.

Now, in order to examine to what degree the two fundamental processescan be split and how they are coupled, we assume the external body force

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

∂t − ν0∇2ω = −∇ × (L − η
), (2.166)

∂ϑ

∂t +∇2H = −∇ · (L − η
). (2.167)While (2.166) represents the general vorticity transport equation which char-

acterizes all transverse interactions, (2.167) characterizes all longitudinalinteractions and suggests that the viscous effect is much weaker than that

in (2.166) Because the Helmholtz decomposition is a linear operation, the

nonlinearity in L − η
in both equations makes the coupling of both processesinevitable It is their intersection.

For weakly compressible flow the entropy gradient can be neglected and ν

is nearly constant Then (2.166) is reduced to the classic Helmholtz equation

for the vorticity to be studied in depth in this book:

∂ω

where the solenoidal part of the Lamb vector ω ×u is the only nonlinear term.

Meanwhile, under the same condition, (2.167) is reduced to

∂ϑ

where the potential part of the Lamb vector ω × u appears As a scalar

equation, the spatial structure of (2.169) is simpler than (2.168); but we have

to express one of the two scalars ϑ and H by the other, which complicates the final form of (2.169) The result is an advective wave equation for the total enthalpy H obtained by Howe (1975, 1998, 2003), with the Lamb vector and entropy variation being the source of the wave We illustrate the Howe

equation for inviscid homentropic flow Multiplying the inviscid version of

(2.163) by ρ and taking divergence, we have

Here, using (2.40) and the inviscid version of (2.65), there is

c2

DH Dt

,

where c = (dp/dρ) 1/2s is the speed of sound Therefore, it follows that (Howe,2003)

D

Dt

1

by unsteady vorticity field, known as vortex sound named by Powell (1961,

1964), who studied this topic extensively under far-field approximation (seethe review of Powell (1995)) By using singular perturbation to match the far-field and near-field solutions, Crow (1970a) was the first to prove rigorouslythat the principal source of sound at low Mach numbers is the divergence

of the Lamb vector In aeroacoustics, the variable describing the sound field

in a flow can be the fluctuating part of p, ρ, etc but it has been found that the most appropriate one is the disturbance total enthalpy H (Howe 1975; Doak 1998) Hence, (2.170) reveals that in a homentropic flow the mov-

ing vorticity is the only source of sound, or sound is a byproduct of vortex

motion

In particular, at low Mach numbers with

be replaced by their constant mean values c0 and ρ0, so that the equation is

reduced to the classic linear wave equation and can be solved by using the

Green’s function method In this case the sound-wave length λ is generically

much larger than the scale of the moving vortices, so that to an observer at

a far-field point x the emission-time difference at different points y in the

source region is negligible: one has r = |x − y|  |x| Then a general result

of interest is that the far-field acoustic pressure p
depends linearly on the unsteady vorticity alone (M¨ohring 1978, Kambe et al 1993, Powell 1994):

3y i(y × ω) j (t − x/c0)d3y, (2.171)

where x = |x|, β i = x i /x is the directional cosine of x in the observation

direction, and the integrand is estimated at an earlier time t − x/c0

The earlier coupling between the two processes dominated by the nonlinear

Lamb vector is inviscid in nature It no longer exists on a solid boundary ∂B

with known motion or at rest Due to the acceleration adherence (2.70), theleft-hand side of (2.161) is known, or simply vanishes in the frame of reference

fixed to the solid boundary However, there appears a different type of viscous

boundary coupling of which the effect may also reach the interior of the flow

field To illustrate the situation in its simplest circumstance, assume the flow

is incompressible and the solid wall is at rest Then the momentum balanceimplies

∇p + µ∇ × ω = 0 on ∂B,

mathe-later The absence of advection makes the coupling linear and hence a series

of formally analytical results are possible

From the preceding analysis we may also summarize the involvement

of thermodynamics in dynamic processes, which can be both inviscid andviscous The viscous involvement of thermodynamics has been encountered

in (2.160) Inviscidly, it appears in the generalized Lamb vector (2.165) If

the flow is baroclinic, i.e., with more than one independent thermodynamic

variables, thermodynamics enters both compressing and shearing processes

through (p, ρ) in (2.161), or through (T, s) in (2.163) Two independent

vari-ables determine a surface with a normal direction; and in a baroclinic flowthis normal is∇ρ × ∇p or ∇T × ∇s In contrast, if the flow is barotropic, i.e.,

with only one independent thermodynamic variables so that∇T ×∇s = 0, the

inviscid coupling of thermodynamics and dynamics occurs only in compressingprocess For more discussion see Sects 3.6.2 and 4.1.2 later

2.4.4 Incompressible Potential Flow

In the context of Helmholtz–Hodge decomposition we have seen that a scalar

function χ with ∇2χ = 0 is necessary, and in general at large Re a big portion

of a viscous flow field can be irrotational It is appropriate here to brieflyreview some basic issues of incompressible potential flow

Consider an externally unbounded fluid domain Vf which is at rest at

infinity and in which a moving body B causes a single-valued or acyclic (or noncirculatory) velocity potential φ,14 which is solved from the kinematicproblem

vor-where n = 2, 3 is the space dimensions, n = x/r, and A depends on the body

shape and velocity

First, by (2.173), the total kinetic energy of the flow is

Thus, the flow has no memory on its history but completely depends on the

current motion of boundary Since K = 0 implies ∇φ = 0 everywhere, if ∂Vf

is suddenly brought to rest then the entire flow stops instantaneously, which

is evidently unrealistic In reality a fluid flow without moving boundary must

be vortical and/or compressible.

If one adds any disturbance u
to the velocity field with kinetic energy

K
> 0, such that u1=∇φ + u
, and if u
· n = 0 on ∂Vf, then

This is the famous Kelvin’s minimum kinetic energy theorem: Among

all incompressible flows satisfying the same normal velocity boundary tion, the potential flow has minimum kinetic energy.

condi-Next, the total force and moment acting to Vfcome only from the pressure

on ∂B The Crocco–Vazsonyi equation (2.163) can then be integrated once to yield the well-known Bernoulli equation

where C(t) can be absorbed into φ without affecting u = ∇φ Therefore, once

φ is known by solving (2.173), one can obtain the total force and moment

act-ing on Vf via the first equality of (2.71) and (2.72), respectively But in force

and moment analysis it is often convenient to employ the concept of

hydro-dynamic impulse and angular impulse (Lamb 1932; Batchelor 1967; Saffman

1992) These are the hypothetical impulsive force and moment that bring the

fluid from rest to the current motion instantaneously Suppose at t = 0+there

is a finite velocity field u(x, 0+), which is imagined to be suddenly generated

from the fluid at rest everywhere at t = 0 − by an impulsive external force

density F = i(x)δ(t) distributed in a finite region We integrate the

incom-pressible Navier–Stokes equation (say (2.161) with Π = p and constant ν)

over a small time interval [0− , δt] Since all finite terms in the equation

in-cluding advection and diffusion can only have a variation of O(δt), u(x, 0+)

must be solely generated by the infinitely large F at t = 0, which also causes

Hence, the momentum balance implies

From (2.178) it follows that∇2P = ∇ · i and ω = ∇ × i While the integral

of i leads to so-called vortical impulse to be addressed in Sect 3.4.1, in a

potential flow one simply has P = −φ The integrals of −nP and −x × nP

over ∂Vf = ∂B + ∂V ∞ are known as the potential impulse and angular impulse

of the fluid, respectively, denoted by Iφ and Lφ:

Dt (φni dS) = [(φ,t + φ,j φ ,j )ni − φφ ,ij n j] dS,

where by using (2.173a) the last term integrates to

where the integral over ∂V ∞ with constant p = p ∞ vanishes

Similarly, for computing dL φ /dt there is

D

Dt (ijk x j φn k dS) = ijk[xj (φ,t + φ,l φ ,l )nk + φφ,j n k − x j φφ ,kl n l]dS.

By casting the last two terms on the right to a volume integral over B and

simplifying the result, the volume integral can be transformed back to surface

integral and yields

Specifically, assume B moves with uniform velocity b = U (t) In this case,

since both (2.173a) and (2.173b) are linear, there must be φ = Uj φj, where



φ j is the potential caused by the same rigid body moving with unit velocity

along the j-direction Namely,  φ j satisfies (2.173a), while if ej (j = 1, 2, 3) are

the Cartesian basis vectors, (2.173b) becomes

n · ∇ φ j = n · e j = nj at ∂B, (2.185a)

n · ∇ φ j = 0 at infinity. (2.185b)Therefore, we can define a tensor solely determined by the body’s geometry:15

Now, substituting φ = U j φj into (2.183) and noticing the integral over

∂V ∞plays no role, we obtain

where ˙U = dU /dt Hence, the total force experienced by the body of mass

mB is

implying that Mij acts as a virtual mass (or apparent or added mass) as

one calls it Thus, in an acyclic potential flow a rigid body performing

con-stant translation experiences no force This is the classic D’Alembert

para-dox which will be revisited later in different contexts Likewise, (2.175) is

15

The approach can be generalized to rigid-body rotation, e.g., Batchelor (1967),but not to deformable body

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) isnot affected In this case the D’Alembert paradox can be said for drag only,

and a finite lateral force perpendicular to U is not excluded, say a lift (see

Sect 4.4 and Chap 11)

It should be stressed again that the result of this subsection can mately represent a real viscous flow only in a portion of the flow domain and

approxi-in the sense of effectively approxi-inviscid flow There is no such thapproxi-ing like ideal fluid,and a globally effective potential flow is an oversimplified model To quoteSaffman (1981):

“if ω = 0 everywhere in an incompressible fluid, then the fluid really ceases

to be a fluid; it losses its infinite number of degrees of freedom, which makes possible the infinite variety of fluid motion, and becomes a flexible extension

of the bodies whose movement generates the flow; bring the walls to rest and the fluid stops immediately.”

Summary

1 The vorticity and vortex dynamics for Newtonian fluid is based on thegeneral principles of fluid dynamics, especially the Navier–Skokes equa-tions with small viscosity The tangent continuity of velocity and surfaceforces across boundaries is of crucial importance for the vorticity genera-tion from boundaries, which accordingly excludes any globally ideal fluidmodel in the study of vorticity dynamics But at large Reynolds numbers

a big portion of the flow can be treated effectively inviscid, to which theEuler equation applies The viscous effect is significant only in thin layerswith extremely high concentration of vorticity or dilatation In the asymp-totic limit of infinite Reynolds number (the Euler limit), these layers aretreated as surfaces of tangent and normal discontinuities in an effectivelyinviscid flow

2 The mathematic tool for understanding the decomposition and coupling

of the fundamental kinematic and dynamic processes in a flow field is theHelmholtz decomposition and its modern development The Helmholtzdecomposition allows splitting a vector field into solenoidal and potentialparts It is sharpened by the Helmholtz–Hodge decomposition that en-sures the uniqueness and functional orthogonality of the split parts, i.e.,transverse and longitudinal vectors, respectively A transverse vector is

always an axial or pseudovector with two independent components, whichcan be expressed explicitly (and at least locally) by two Clebsch variables.The helical-wave decomposition further splits the transverse part into twointrinsic polarity states, and any transverse vector can be expanded by acomplete orthonormal set of curl-eigenvectors.

3 The fundamental processes in volumetric fluid motion are longitudinalcompressing and transverse shearing, with governing dimensionless para-meter being the Mach number and Reynolds number, respectively Thelatter is more complicated since it is a vector process There is yet a surfaceprocess due to fluid surface deformation The explicit coexistence of theseprocesses in the strain rate tensor, velocity gradient tensor, stress ten-sor, surface stress, and the dissipation roots in the very fundamental andsimple triple decomposition (2.136) The coupling and decoupling of thetwo volumetric processes and a surface process in the governing dynamicequations and boundary conditions can then be examined systematically

4 The representative variable for shearing process is always the vorticity

vector ω, governed by vorticity transport equation The representative

variable for compressing process varies in different situations and

formu-lations Kinematically it is the dilatation ϑ Dynamically the choice of compressing variable is not unique It can be the normal force Π, pressure

or density as in classic acoustics, or the total enthalpy as in vortex-sound

theory It can also be the velocity potential φ as in classic high-speed

aerodynamics The process is governed by the Howe equation

5 For a Newtonian fluid with constant shear viscosity µ, the compressing

variable Π and the shearing variable µω form a reduced stress tensor T,

having three independent components It is the Helmholtz tensor potential

of the resultant surface force per unit volume T can replace the full stress tensor T in the Cauchy motion equation and the kinetic-energy balance without affecting any result The remaining part of T comes from surface deformation rate tensor B This B-part enters the angular momentum

balance, its rate of work is always directly and locally transferred to heatand thereby affects the internal-energy increase only

6 The two fundamental volumetric processes are generically coupled throughtwo mechanisms In the interior of the flows the coupling is caused

by the nonlinearity in advection (mainly the Lamb vector) and thenonlinear involvement of thermodynamics especially in baroclinic flows

At a solid boundary a linear viscous coupling is caused by the tum balance, which occurs whenever the longitudinal and transverse vari-ables are not uniformly distributed on the boundary But the role of oneprocess in the evolution of another is not equally important In certainsituations the shearing process is dominant and the compressing process

momen-is a byproduct, e.g., vortex sound; while in some other situations theirrelative importance is opposite (cf Sect 4.1.2)

7 In some purified situations the two fundamental processes are decoupled

In a viscous incompressible flow there only exists shearing process and

inviscid coupling of thermodynamics and dynamics occurs only in compressingprocess For more discussion see Sects 3.6 .2 and 4.1 .2 later

2. 4.4 Incompressible Potential Flow... known by solving (2. 173), one can obtain the total force and moment

act-ing on Vf via the first equality of (2. 71) and (2. 72) , respectively But in force

and moment analysis... class="text_page_counter">Trang 22

Hence, the momentum balance implies

From (2. 178) it follows that∇2< /small>P = ∇ · i and ω = ∇ × i