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Unformatted text preview: 276' w. w. HSlEH AND v. T. BU'CHWALD Middleton. J. H., “Low frequency trapped Waves on a wide. reeffringed continental
5 shelf." J. Phys. Oceanogr., in press (1983). Middleton, J. H. and Cuhnin—gham, A., “Windforced continental shelf waves from a
' geographical origin," Continental V‘Sh'elflResq submitted (1983). Mysak, L. A., "Recent advances in shelf wa've' dynamids," Rev. Geophys. & Space Phys.
18, 211—241 (I980). ~‘ ' Noble, 8., Methods Based on the Wiener—Hop] Technique, Pergamon Press (1958). Wang, D.P., "Diffraction of continental shelf waves by irregular alongshore
geometry," J. Phys. Oceanogr. 10, 1187419911980). Wolanski, E. and Bennett, A. F.,_“Continental shelf waves and their inﬂuence on the
circulation around the Great Barrier Reef."‘Aust. J. Mar. Freshw. Res. 34, 2347
(1983). ..  ,  ‘ Wotanski, E. and Ruddick, 3., "Water circulation and shelf waves in the northern _ Great Barrier Reef lagoon," Aust. J. Mar. Freshw. Res. 32. 72~740 (l98l). ', Geophys. Axlrnphys. Fluid Dynamics. l984. Vol. 28. pp. 277—JM ' 0309 I929/H4/ZHU40277 S  3.50/0
(t) Gordon and Breach Science Publishers lnc.. I934
Printed in Great ltritnin The Generation andDecay’of
Vorticity B. R. MORTONT National Center for Atmospheric Research, Boulder, Colorado, U. S.A.1 (Rr't'r'im‘rf Dr't‘t'm/u'r III. I981) Vorticity: although not the primary variable of ﬂuid dynamics, is an important derived variable playing both mathematical and physical roles in the solution and understanding of problems. The following treatment discusses the generation of
vorticity at rigid boundaries and its subsequent decay. It is intended to provide a
consistent and very broadly applicable framework within which a wide range of
questions can be answered explicitly. The rate of eneration of vorticit is shown to
be the relative tangential acceleration of ﬂuid and boundar without takin viscosil
into account and the cnerating mechanism therefore involves the tan entittl ressure
gradient within the ﬂuid and the external acceleration of the boundar' onl. The
mechanism is inviscid in nature and independent of the nosi condition at the
min ury, at oug Viscous illusion acts immeditttcl after cncrntion to 3 read
vorticity outward from boundtirics. Vorticity diffuses neither out of boundaries nor ‘——_——__——"————“' . . . . . 1 . .
into them, and the only means of decay IS by crossthﬂusuvc annihilation Within the
ﬂuid. 1. INTRODUCTION The Helmholtz vorticity equation for an incompressible ' homogeneous ﬂuid, aw/az +(v . Vfw=(w  V)v + szw, TOn leave from the Department of Mathematics. Monash University, (.‘Inylon.
Victoria 368, Australia. Ile National Center for Atmospheric Research is sponsored by the National
Science Foundation. 277 ‘ k h > t . ' eh 278 B. R. MORTON includes the processing term (wV)v which describes the elTects of local ampliﬁcation (or concentratien) of Vorticity by vortex ﬁlament stretching and local turning. of ﬁlaments, and the term sza) representing the spread of vort1c1ty. due to vrscosn, where ='(§,11,C) is the vorticity. It contains no true generation term that
would cerrespond with the creation of fresh vorticity where none
existed before, and it has long been recognized that all sources of vorticity in homogeneous ﬂuids must lieat the boundaries of ﬂuid regions. Vorticity may be generated at interior points of
inhomogeneous ﬂuids, but is also generated at their boundaries,
presumably by the same mechanism as Operates in homogeneous
ﬂuids We may therefore restrictvthe following discussion of the
generation of vorticity at boundaries to the case of homogeneous
ﬂuids, although we shall later show that the same generation
mechanism operates universally. The complete determination of a Vorticity ﬁeld requires boundary
conditions as Well as a differential 'eqUation: for example, the
Helmholtz equation for the Rayleigh problem (Section 4.1) Of a semi
inﬁnite region of ﬂuid bounded by a plane boundary set impulsively
in motion with steady velocity 1n its own plane is represented by the reduced equation
ace/(3t = vVZn), and to obtain a unique solution we require a condition on the
creation of vorticity at the boundary. .We ﬁnd a curious situation
here, for although most of the many texts on ﬂuid mechanics at least
introduce the Helmholtz vorticity equation, very few so much as
mention boundary conditions. One of the very few exceptions is
Batchelor (1967, p. 280) who notes that the boundary condition on
vorticity is provided in effect by the noslip condition, though this
seems scarcely a satisfactory condition on vorticity which is a
physically distinct quantity with different dimensions from velocity.
Moreover, boundaries in homogeneous ﬂuids are thesour'ce of all
vorticity, and we shall clearly need to consider what are the
appropriate boundary conditions. Some authors have suggested that. the generation of vorticity in a
region of homogeneous ﬂuid is related to its diffusion out of solid
boundaries, and its decay to diffusion into other boundaries. If, aw GENERATION AND DECAY OF VOR'flClTY 279 however. vorticity\is a physical entity relating to ﬂuid rotation, we
may reasonablyrask what is the physical effect on a‘boundary
suffering continuous loss, of vorticity, and what on a boundary
steadily gaining vorticity? Plane Poiseuille ﬂow has been interpreted
as a steady motion resulting from the generation of vorticity at one
boundary and its equal loss at the other; and as vorticity provides a
measure of rotation in a ﬂuid we might look for an effect of torque
on an experimental channel, although none has been reported.'
Moreover, a mere reversal of the coordinate frame would
interchange the boundary of generation and that of loss of vorticity.
There Can be no doubt that vorticity is a genuine physical entity,
corresponding with: the particular constituent of ﬂuid motion in the
neighborhood of a point associated instantaneously with rotation
about an. axis through that point. We shall, therefore, need to
consider more critically the notion'that vorticity may diffuse out of
or into boundaries. One certainphysical effect at boundaries is the
wall stress exerted by moving ﬂuid and the equal and opposite
tangential stress exerted by the wall on nearby moving particles of
ﬂuid. In rigid body dynamics tangential forces exert torques which
generate angular acceleration, and” we must determine whether wall
stress generates vorticity and indeed what roles are played by torque
and angular momentum in ﬂuid dynamics. We recall that ﬂuid dynamics is a branch of mechanics and that
ﬂuid motion is fully represented by the Navier—Stokes equation (a
form of Newton‘s equation of motion), together with a continuity
equation to keep track of mass and an energy equation if heat is
important in addition to mechanical energy. All flow problems can,
in principle if seldom in practice, be solved from these equations ' without introducing vorticity; but although we do not need vorticity for the description of fluid motions it satisﬁes approximately a
number of simple and farreaching conservation relationships and
provides a powerful alternative physical basis for the discussion of
complex threedimensional ﬂows. One further advantage of the use ‘ of vorticity which isoften emphasized is the absence Of pressure from the Helmholtz equation, a fact that has led many to assert that
pressure plays no role in vorticity dynamics, an assertion that we
shall question. We may reasonably argue that vorticity is a welldeﬁned variable
having a clear relationship with the physical concept of rotation in 230 "a. R. MORTON ﬂuid motion, satisfying a known differential equation and of widely
accepted value in the description Ofﬂuid ﬂows. It is therefore 'quite
unsatisfactory that so many who iwork with ﬂuids remain uneasy
over its use, and in particular . that. there "should persist such
widespread uncertainty ”as to “the behavior of vorticity near
boundaries, exempliﬁed by responses to the following questions: i) what are the boundary conditions'o'n vorticity and why are they
so generally overlooked; ii) does pressure playany role in vorticity dynamics;
iii) is vorticity generated by wall stress;
iv) what, if any, is the. role of torque in~ﬂuid dynamics; v) what is the physical mechanism _or mechanisms for the
generation of vorticity’at boundaries: and vi) what are the mechanisms for loss of vorticity, and in particular
can vorticity be lost by diffusion to boundaries? 2. PREVIOUS TREATMENTS OF THE GENERATION
AND DECAY OF VORTlClTY Few authors have seriously discussed either the generation Of
vorticity at boundaries or its subsequent decay, with two notable
exceptions: Lighthill and Batchelor, who have resolved parts of the
problem, without resolving it as a whole. Lighthill (1963), in an elegant and wideranging introduction to
boundary layer theory, espoused the use Of. vorticity as an effective
means of solution for aerodynamic problems. He noted that at
almost all points of theboundary there is a nonzero gradient of
vorticity along the normal, with ﬂux density of “total,vorticity"
having xcomponent —v(6€/6z),=0 “out of the solid surface”, where z
is normal distance from the boundary, assumed locally plane. By
applying theNavier—Stokes'equation at a stationary plane boundary 2:0,
— “(35/62, arr/dz, 0): =0 = pT 1(ap/ay’ _ a17/0", 0): =0! which he took as the local strength of a distribution of vorticity ‘9 _..__3 ”Wm... GENERATION AND DECAY'iOF VORTlClTY 28! sources spread over the solid boundary. it follows that tangential
vorticity must be created at the boundary in the direction of the
surface isobars at a rate proportional to the tangential pressure
gradient. What does not immediately follow is why this should be so. Although Lighthill described the boundary as a distributed source
of vorticity, similar to a' distributed source of heat, it is clear from his wider discussion of ﬂow development that he envisaged the
generation process as taking place at the boundary surface. £1
discussing twodimensional boundary layers he then invoked
vorticity sources in a region of falling pressure along the boundary
an vort1c1ty sin 5 at w 10 vort1c1ty IS a stracte at the surface) in
a following region of rising pressure. At this stage the situation does
not appear .to be fully resolved; in the absence of a physical
mechanism for the generation of vorticity, the relation between
vorticityfﬂux density and tangential pressure gradient does not
distinguish between the outward diffusion of positive vorticity and
the inward diffusion of negative vorticity. Nor is it clear whether
vorticity can be lost by diffusion to boundaries. We cannot,
therefore, say whether vorticity of a particular sense "is being first
generated at and subsequently lost to the boundary as the pressure
falls and rises again, or whether there is continuous generation Of
vorticity first of one sense and then of the other as fluid moves along
the boundary. Indeed, it is not clear whether there is any meaningful
distinction between these two. The difficulty with the vorticity ﬂux density relationship as a
boundary condition for the Helmholtz equation in calculating ﬂow
fields past aerofoils is that it involves the pressure field. Lighthill
sidestepped this implicitly by considering an adjustment process in
which an initial inviscid ﬂow field is used to determine the freeslip
velocity at the boundary and the production of vorticity is inferred
from changes in slip velocity; from this point our interests diverge. Lighthiil made two cautionary statements to which we shall need
to return. That although vorticity relates instantaneously and locally
to the angular momentum Of infinitesimal spherical ﬂuid particles,
the ﬂow Of vorticity cannot be viewed as diffusion of angular
momentum as it is continuously transported to fresh ﬂuid elements
which suffer rearrangement in the ﬂow. Hence there is no continuing
axis about which it is meaningful to calculate vorticity~related
angular momenta, and it is angular velocity and not angular J, 282' V B.  R. MORTON momentum to which vorticity should be related. Secondly. the
Ilillusion of voIticity Iclutea to the diffusion ot'(iIIeIII) IIIIIIIIcIItuIII lo
which we choose the interpretation, that vorticity diliuses in effect
because linear momentum diffuses. ! I. Batchelor gives a careful discussiOn of vorticity in his Introduction
to Fluid Dynamics (1967), but although he devotes a substantial
section (Section 5.4) to the source of vorticity in motions generated from rest, his conclusions remain general and do not result in an expression for generation rates Again he identiﬁes the free slip. velocity in the inviscid solution for flow past a= body set in motion as
the effective source of vorticity in real ﬂoWs, and hence solid
boundaries as vorticity sources. He takes the further step, however
of explicitly identifying the _no s—lip condition at the boundary In real
ﬂuids as a mechanism for the production of vorticity, though he uses
this only in a broadly descriptive manner. Heargues also (Section
5 2) that vorticity cannot ' be destroyed in the interior of a
homogeneous ﬂuid and this appears to lead to the concept of loss of
vorticity by diffusion to boundaries. It is perhaps unfortunate that
the expressions “diffusion 'of vorticity out of”, “from’, “’to’ and
“across” solid boundaries have become established in the literature
without any very cleardilferentiation. There‘is little doubt that the generation process takes place at the‘boundary and neither within the wall nor within the ﬂuid, but it is less clear what is intended
when vorticity is regarded as lost by diffusion to a wall, as for
instance when Batchelor describes steady motion as due to the
steady flux of vorticity out of one solid boundary being balanced by
an equal steady flux into another boundary (1967, p. 281), These are
matters that will be discussed more precisely below In these two treatments Lighthill has concentrated on the role of
pressure gradients over stationary surfaces and Batchelor on surfaces
accelerated or set impulsively to motion. Many relevant matters have
been raised but they do not seem to have come into focus sufficiently
for us to answer the questions posed in the foregoing section The
following analysis is intended to clarify these aspects of the behavior
of vorticity; much of it is interpretive, but the questions are so
fundamental to our effective use of the concept of vorticity and the
confusion they engender is so widespread among those using the
methods of fluid dynamics that no further excuse seems to be needed. GENERATION AND DECAY 'DF VORTlClTY 283 3. VORTlClTY AT AND NEAR BOUNDARIES \
For motion near a plane boundary take origin 0 in the boundary with 02 normal and n the unit normal vector (Figure l). The
following discussion may be restricted to twodimensional flow in
(x,z) planes without loss of generality. Our state of understanding
makes it difﬁcult to specify boundary conditions on the vorticity to
directly, but these can be derived from the condition on velocity, v=(u,w)=0 on z=0, for all x and t. O x FIGURE 1 The coordinate system for the neighborhood of a boundary. It folIOWS that on z=0,
a(u, w)/ase=o, 62w, w)/ax2=o, for all x, t;
and from thecontinuity equation for an incompressible fluid
au/dx + aw/az = 0,
6w/6z=0 on 2:0, for all x, t. The remaining component of.av/6z relates to the tangential
boundary stress r=(rx,Q) through the relation TX = [l (in/62,
and is nonzero except at a separation point. Hence, =(€,Cr1,)={O(6u/6z)—(0w/6x),0} G.A.F.D. E 234 ‘ B. R. MORTON ‘ takes the value on 2:0,
. (D,
"(0 =(0) ,1 era 0)) so that coo10:0. Thus vortex ﬁlaments are tangential to a
stationary boundary and inclined at angle +7r/2 to the wall stress or
skin friction t, a result which holds also in three dimensions. This
boundary condition on w appears again to be of limited value
because I is unknown at this stage It is, however, a result of some importaneetand the first indication that {wall stress cannot generate vorticity; indeed, wall stress and vorticity are closely related velocity gradients. 3.1 The flux density of vorticity at a stationary plane
boundary At the boundary z=0 the Navier—Stokes equation reduces to
0 = * P ' ‘(VP)o + V(V2V)o, and hence on 2:0, 52 ) l 5 0
~—— w =—— —— —— .
622% u 6x'dz p Thus,
(aw/(32)0 = [6(0, I], O)/0z]0 = {0, dZu/OZZ, 0}0 = {0, it“ ap/ax,0}o=#‘ 1n X (Vplo, with a corresponding result for are/ax on z=0.
We recall that Lighthill tl963l has already identiﬁed —[email protected][dz)0
as the diffusive flux density (or ﬂow er unit area er unit time of (positive) vorticity outwards from the wall and interpreted —v(aw/az)o= —p' '(n x W A.“ .\\ ,\. as a local boundary source of vorticity. This is obviously an / .
flatlj/ 1'
( EMS/I .51’/7Cro'~\/Ci ﬁe —O l (QXV/P/O J ,/
)) do] / Naif/n, ,y( 1/ //m2 )7“ ,3
‘5‘”? I \’/i(‘{//(/ ('“t’1( /(7‘A€ —5 (ler/% (‘r‘r 'IXVZ’D ~€ay+___.,.~_.,_s_ ___.____ ..___.__
‘i . GENERATION AND DECAY OF VORTlClTY 285 important result, but in the absence of any physical mechanism we
are unable to determine the balance of positive and negative
vorticity generated at the wall, lost by diffusion from or gained by
diffusion to the wall, or indeed whether the vorticity field might be
produced entirely by a pattern of generation and diffusive loss of a
single sense of vorticity to the boundaries. . 3.2 Conditions at a wall moving in its own plane We have seen already from Batchelor (1967) that boundary motion
must be included in our discussion, and we shall therefore generalize
the foregoing results to the case of a rigid boundary 2:0 moving in
its own plane with velocity V={U(I),0}. We consider tangential
motion only here, but will return to the differences between
tangential and normal motion of boundaries. At the boundary z=0, =(U,0),, for all x,1,
and as there is no spatial variation of boundary motion,
[ta/ax. 52/6x2)(u, Wu, = Continuity in an incompressible fluid involves only spatial variation
of velocity, and as before ' (dw/Oz)o=0, and (no: {041 ‘ ‘Ix,0}.
Substitution in the Navier—Stokes equation at the boundary yields
[0%, ail/62210 ={/1"(0p/<3X)+\" '(dU/dt),i1 " Eli/Mo,
and —— v(6w/(32)o ={0 fp = ——p— 1[(n x V)p]o~n x (dV/dl). (Zip/c x) +dU/dt], Oio 286 » B. R. MORTON . ' boundary acceleration; and partiCularly the symmetry involved in
the‘acceleration of ﬂuid by the tangential pressure gradient and the tangential acceleration of the boundary. 3.3 The relationship of vortex lines and streamlines at
the boundary We can expand the velocity in a Taylor series for the neighborhood
of the origin 0 on the boundary, ) 0v + a: +§2~a—21 +zx 62v
v(x,z ~v0+x $0 2 62 0 x 6x2 0 ' 026x 0 where the sufﬁx 0. refers to conditions at the origin and r2=x2+zz.
Using the results from the previous sections, v(x, z) =,u‘ ‘ {zr+zx(6r/5x) +§zz(Vp)o} + 0(r3).
This may be expressed wholly in terms of r=(r,, 0) using
(Vp)0={6Ix/az,0, ~6rx/6x}; or in terms of vorticity w=(0, r7, 0) using it x t=pwo.
The streamlines are given by dx/u=dz/w,
and for small values of z
dx/t, zdz/O; thus in the limit z—+O the limiting streamlines touch the boundary
and are in the direction of the wall stress. and hence orthogonal to
the limiting vortex lines which also touch the boundary. A stream function is available for twodimensional flows, and has vi GENERATION AND DECAY OF VORTlClTY ' 287 expansion valid in a neighborhood of the boundary origin 0. tax,z)‘=¢o+t22tau/az)o+0tr°>z¢o+tnozz. 4. CASE STUDIES OF EXACT SOLUTIONS Further insight can be gained into the signiﬁcance or these results by
reinterpretation of the small number of exact solutions of the
Navier—Stokes or Helmholtz equations These solutions are simple in
form and so familiar that we take them for granted, but like all exact ~ solutions they provide a source of relevant information whether or not we app...
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