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Morton - 276 w w HSlEH AND v T BU'CHWALD Middleton J H...

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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. reef-fringed continental 5 shelf." J. Phys. Oceanogr., in press (1983). Middleton, J. H. and Cuhnin—gham, A., “Wind-forced 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/ZHU4-0277 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 no-s|i 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 cross-thflusuvc 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 3|68, 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 (w-V)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 lie-at 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 no-slip 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 certain-physical 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 far-reaching conservation relationships and provides a powerful alternative physical basis for the discussion of complex three-dimensional 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 well-defined 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 iwor-k 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 wide-ranging 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 non-zero gradient of vorticity along the normal, with flux density of “total,vorticity" having x-component —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 VORTlClT-Y 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 two-dimensional 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 free-slip 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 two-dimensional 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 non-zero 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 coo-10:0. 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 two-dimensional 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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