C. Hancock, E. Lewis and H.K. Moffatt- Effects of inertia in forced corner flows

8/3/2019 C. Hancock, E. Lewis and H.K. Moffatt- Effects of inertia in forced corner flows

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J . Fluid Mech. (1981), vol. 112, pp. 316-327Printed in Great Britain 315

Effects of inertia in forced corner flowsBy C. HANCOCKT, E. LEWIS

AN D H. K. MOFFATTSchool of Mathematics, University of Bristol

Department of Applied Mathematics and Theoretical Physics,University of Cambridge(Received24 October 1980 and in revised form 12 June 1981)

When viscous fluid is contained in the corner between two planes intersecting at an0 angle a, flow may be forced either by relative motion of the two planes keeping aconstant (the paint-scraper problem- Taylor 1960) or by relative rotation of theplanes about their line of intersection (the hinged-plate problem -Moffatt 1964) . Ineither case, a similarity solution is available describing the flow sufficiently near thecorner, where inertia forces are negligible. In this paper, we investigate the effects ofinertia forces, by constructing regular perturbation series for the stream function, ofwhich the leading term is the known similarity solution. The first-order inertial effectis obtained analytically, and, for the Taylor problem with a = Qn, 5 terms of theperturbation series for the wall stress are generated numerically. Analysis of thecoefficients suggests that the radius of convergence of the series is given by r 1 U ( v z 9.1,where r is distance from the corner, U is the relative speed of the planes, and v is thekinematic viscosity of the fluid. For the hinged-plate problem, discussed in 5 5 , theunsteadiness of the flow contributes to an inertial effect which is explicitly incorpo-rated in the analysis.For both problems, streamline plots are presented which indicatethe first influence of inertia forces at distances from the corner at which these becomesignificant.

1. The corner problem of Taylor (1960)When incompressible viscous fluid is contained in the corner between two planes0 = 0, a figure l ) ,and when one of the planes (0 = 0 ) s movedparallel toitself withsteady velocity U , he flow that is generated is dominated by viscous forces sufficientlynear to the corner. In this region, the stream function $ 8 ( r , 0 ) ,which satisfies thebiharmonic equation, is given by the similarity solution (Taylor 1960, 1962)where = r U f i ( 0 )= rU(Bsin0+CBcos0+D8sin0), ( 1 . 1 )

(1 .2)- 2 sin2a a-sinu cosaa2- in2 1 09- in2a a2- in2a *B = C = D =The streamlines = constant are shown (fora = in) y the solid curves of figure 1 ;the flow is, in this approximation, reversible, so that, if U is replaced by - U , thearrows on the streamlines are reversed, but the streamline pattern is unaffected.

t Present address: Bath College of Higher Education, Newton Park,Bath BA2 9BN.I1 FLM I12

www.moffatt.tc


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316'1c

rnV

5

-0.1\

-1

V

+0.1-

C . Hancock, E . Lewis and H . K . Moffatt

U(> 0)FIGURE. Streamlines for the Taylor corner problem with U = in; he solid curves are thestreamlines of the Stokes flow (= $ J v ) = constant ; the dashed lines are the streamlines+ a = constant, including the first inertial correction (obtained analytically).


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Ejfects of inertia in forced corner flows 317The above solution breaks down a t a distance of order v/I U / rom the origin (where v

is the kinematic viscosity of the fluid) due to the increasing importance of inertia asr increases.? In order to analyse inertial effects, we consider the exact equation for the

and we seek to solve this, subject to the boundary conditions(1 .4 )I= o , a $ / a e = - r U on 8 = 0 ,$ = 0, a$/ae = o on e = N.

We may anticipate the asymptotic behaviour$ - $$(r ,O) as r -+ 0. (1 .5 )

The natural procedure is simply to seek the solution in the form of a regular pertur-W

bation expansion$1 ~ . VI: + n ( r , 0 )) where $n r+)nfn(e) ) (1 .6)1

with fl(0) given by ( 1 . 1 ) .Substitution in ( 1 . 3 )yields a sequence of equations

and this in turn yields a sequence of linear inhomogeneous equations for the f n ( e ) ,viz

(n = 2 , 3 ... . (1 .8 )Since v$l already satisfies the boundary conditions (1.4))equations (1 .8 ) are to besolved subject to the homogeneous boundary conditions

f n = f h = O on 8=O,a ( n = 2 , 3 ... . (1 .9 )2. Possible 'eigenfunction' contributionsBefore carrying out this procedure, there is a possible complication that needscomment. At the lowest order in the expansion (1.6), the solution $, = is notunique (regarded as a solution of the biharmonic equation satisfying the boundaryconditions (1 .4 ) ) but may be supplemented by the general solution $c(r , 0) of thehomogeneous problem

V4$c = 0, $c = a$,/atI= 0 on 0 = 0,a. (2 .1 )t It also breaks down in an immediate vicinity of the corner, since it implies an (unphysical)pressure singularity p = O(r -1 ); n practice, this means that there is always a small leakage of

fluid through a small gap between the planes - an important practical consideration, which isnot, however, treated in this paper.1 Note added in proof. Dr E. Erdogan has drawn our attention to the interesting paper oflnouye (1973) in which the particular case a = in has been studied both by a boundary-layertechnique (which seems to be applicable only when U >0) and by the expansion techniqueadvocated here. Inouye obtained the first correction $2, but he did not attempt to obtainsubsequent terms or to establish the radius of convergence of the series.11-2


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318 C . Hancock, E . Lewis and H . K . Moflatt2 5

20

15

10

5

I I I I I I I I I I 1 I I I 1 I I I 1 In14 *I 2 3n/4 (Y nFIGURE. Exponents of dominant eigenfunctions as functions of a ; ee the inequalities(2.6)for the Taylor problem and (5.5) for the hinged-plate problem.

This has the well-known form (Moffatt 1964, Lugt & Schwiderski 1965, Weinbaum1968, Liu & Joseph 1977) m@ c ( r , 6 )= Re Anrhfign(6) , (2.2)

(2.3)(2.4)

n = lwhere the A, are the roots (with positive real part) ofordered so that

sin ( A n - 1)a= ( - 1)" (An- l )s ina,0 c Reh, < ReA, < ...

(complex solutions occurring in complex conjugate pairs), The functions g,(8) aregiven by( n odd)( n even)

cosAn(O-#a) cos (A , - ) ( 6- *a)-cos #An a cos $ ( A n- 2)asinAn(o-#a) sin (A , - 2) (e-#a)-sin BA, a sin BA, a

(complex solutions occurring in complex conjugate pairs). The functions g,(8) aregiven by( n odd)

- ( n even)

cosAn(O-#a) cos (A , - ) ( 6- *a)sin An(8-#a) sin (A , - 2) (8-#a)

sin BA, a sin BA, a

-cos #An a cos $ ( A n- 2)a


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Effects of inertia i n forced corner flows 319and the (complex) constants A , are determined in (in principle) by conditions far fromthe corner. The apparent freedom here simply reflects the fact that additional con-ditions a t infinity must be imposed on @ in order to make the solution of the problem(1 .3 ) , (1.4)unique, even in the inertia-free limit. It is clear from (2.5) hat odd/evenvalues of n correspond to flows that are antisymmetric/symmetric about 8 = &a.Thecomputed variation of ReA, and Re A, with a s shown in figure 2.

The solution (2.2) should clearly participate in nonlinear (inertial) interactions,leading to additional terms in the general solution of the problem (1 .3 ) ,(1.4).However,the dominant behaviour for r IU / v < 1 is given by the first N terms of the expansion(1.6)where

(2.6)< ReA,(a) < N + 1,subsequent terms being dominated by eigenfunction contributions. For example(from figure 2) when a = $n, n, in,we have N = 3, 6, 1 1 respectively. For smalla,N becomes large, and the solution becomes progressively less sensitive to eigen-unction contributions. For all values of a n the range 0 < a < n, he first inertialcorrection, given by @ in (1.6),dominates over any eigenfunction contribution, andmay therefore be regarded as having absolute significance, independent of conditionsat infinity. (The limiting case a f n is special, in that A,(n) = 2; this case will betreated separately below.)

3. The first inertial correction @,(r ,8 )With n = 2, (1.8)becomes

wheref, s given by ( 1 . 1 ) . Substitution givesF,(8) = ( ~ B C 2 - 2 ) sin 28- 2 ( ~C )D COS 28+4CD 8 sin 28+~ ( C Z 2 ) e COS 28,

(3.2)and the solution of (3 .1 )satisfying (1.9)(withn = 2) may then be found by elementarytechniques in the form

f2(8) E8 COS 28+Fe sin 28+GO2 COS 28+H82 sin 28+P + Qe+R OS 28+x in 28, (3.3)where E ,F, . S are constants, depending only on a,whose values are obtained in theAppendix,

The streamlines @1 +@ 2 = constant (fora = in) re shown by the dashed curves infigure 1a ( U < 0 ) and 1 b ( U > 0); in the former case, inertia tends to compress thestreamlines towards the fixed wall 8 = a, .e. we see the beginnings of a wall-jet type offlow on 8 = a,whereas in the latter case the flow has a sink-like character in theregion in 5 (35 in.


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320T h e case a = n

As noted above, this case needs special treatment, and indeed this is now evidentalso from the solution ( 3 . 3 ) , n which (see appendix) the constants P and R becomeinfinite as a + n ; n fact, writing a = 7r- , the asymptotic form of ( 3 . 3 ) s

C . Hancock, E. Lewis and H . K . Moffatt

1 38(1- OS 2 8 ) (8- 4 8 sin 2 8f,(4- -- 1 -cos 2 8 )- 16n2 +O ( ) . ( 3 . 4 )326 32n2The singular behaviour that occurs when B --f 0 can be resolved (cf. Moffatt & Duffy1980) by choosing A , (in ( 2 . 2 ) )so that

is finite. Now, for E < 1, h , - 2(1 + E l m ) from ( 2 . 3 ) ,and, from (2.5),2 e(COS 8 - 1 ) --sin 28 + O(62).l (e) nHence we require A , = - U 2 / 3 2 s v 2 , and the appropriate inertial correction whena = 7r is, from ( 3 4 ,

1 U rJ , = 32n2 (y) ( 2 n r - 8 ) ( 1- os2 8 )+ 2 8 ( 8- ) in ee l . ( 3 . 7 )4. Numerical determinationof higher terms in the expansion (1.6)

It is of some interest t o investigate the rate of convergence of the expansion (1.6),since this is the driven ingredient of the flow which is present irrespective of remoteconditions. We have integrated the equations for the particular case a = in, orn = 3 , 4 , .. 10, using a standard finite-difference procedure?; the number of steps Noin the 8 direction was variable, the maximum value used being 280. The partial sums

aare shown in table 1 , for 8 = in and for various values of p = r U / v and of N . Theresults suggest a radius of convergence pc somewhat less than 10. The streamlinesY, = constant are shown in figure 3 , and they are close to the streamlines Y,=constant for JpI c 5 , where the convergence appears to be strong.

5 6 8 101 - .080 - .296 - .728 - .1602 - 1.283 - .588 - .247 - .9715 - .397 - .808 - .891 - .50610 - .399 - .809 - .786 - .357

TABLE . Values of @ ~ ( p , = an),where p = r U / v .

t Full computational details may be found in Hanoock (1981) .


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Effects of inertia i n forced cornerJows 32 1

FIGURE. Streamlines obtained from the numerical solution for the Taylorproblem, with a = &r; -Y 1 1 - - - - Y ; * - * - . - * - Y3 (see (4.1)).


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Effects of inertia in forced corner flows 323Pc

5.4558.4478.6548-8409.1709.0929.094

Bln0.35560.41530.38740.40220.41190.41170.4118

TABLE . Poles zc = pce**@f the " I N ] Pad6 approximants for N = 1, 2, ...,9. (For N = 10,11, the Pad6 approximant is degenerate due to the near-coincidence of a zero of the numeratorwith z = 2,; for N = 12, the pole z = z, reappears at z, x 9.1e*0'41ni. )

of signs of the coefficients settles down to (+ + - - - . A model function exhibitinga similar pattern is

1 1 -cosp+z mf ( z ) = - ' ( - + - )ei8-x e-ZB - - 1- 2 cosp+22 = - =Oc x ~ c o s ( n + l ) p , (4.5)

with /3 = in; his function has simple poles at x = e%i8,and this suggests that the seriesCan n may have singularities near arg x = f Qn.

Secondly, the dominant singularities may be located using Pad6 approximants.Table 4 shows the poles2, = pceh@' of the " I N ] Pad6 approximants forN = 3,4, . 9(Van Dyke, private communication). The position rapidly stabilizes to x, x 9.1e*041ni;as expected, the angle 0.41n is near to the value &n uggested by the pattern of signs.The inference is that the series (4.2) (and so presumably the series (1.6))has radius ofconvergence

pc x 9.1. (4.6)

5. The hinged-plate problemConsider now the related problem sketched in figure 4: the plates 8 = & *a are

hinged at 0, and rotate about 0 with angular velocities f w (so that dald t = 20). Wesuppose for simplicity that w is constant; even so, the resulting flow is necessarilyunsteady due to the changing geometry; in the Stokes approximation, however, thestream function &(r)8) is instantaneously determined, and the similarity solutionanalogous to ( 1 . 1 ) is (Moffatt 1964)

sin 28- 8 cosasina -a osa *I-,= &r2f,(8, a ) = - wr2

Figure 4 shows the streamlines II-,= constant (solid curves) for a = in.Two simple properties of this stream function are worth noting in the presentc0ntext.t Firstly, whena = in, 2$s = 0 ;for this particular angle, it might be thoughtthat (5.1) provides an exact (irrotational) solution of the Navier-Stokes equations;however, there is an inertial correction due to unsteadiness (see(5.11 ) below). Secondly,when sina = a cosa, i.e. when a x 257') II-,is singular because (Moffatt& Duffy 1980)

t As for the Taylor problem, there is a pressure singularity ( pN In r ) a t r = 0, which wemake no attempt to resolve in this paper.


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324 C . Hancock, E . Lewis and H . K . Moffatt

FIGURE. The hinged plate problem; streamlines for a = jn; he solid curves are the stream-lines for the Stokes flow ( = @Jv) = constant (equation (5.1)); he dashed lines include thefirst-order inertial correction $hg.( a )w < 0; b )o > 0.an eigenfunction solution rAf (0) with h = 2 exists for this critical value of a; hesingularity can be resolved only when the eigenfunction solution is included, and thisleads to an r21nr behaviour. The same problem arises in a more acute form wheninertial corrections are considered- ee (5.8)below.For general values of a, nertia forces associated both with nonlinearity and withunsteadiness are important. We require to find a stream function $ satisfying


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Effects of inertia in forced corner jaws 325and subject to the boundary conditions

The series solution analogous to ( 1 . 6 )has the form

Again, eigenfunction contributions can also intervene in the general solution; here weare concerned only with flows symmetric about 8 = 0 (on the assumption that theremote conditions introduce no asymmetry), i.e. with even values of n in (2 .3) ; thefirst N terms of (5.4) dominate over all eigenfunction contributions where now

(5 .5)For a 2 45", the inertial correction $2 is always dominated by the eigenfunctionrAzg2(8),.e. for such large angles inertial effects are necessarily associated with the'remote' conditions. For a 5 145' however, a t least the term $2 is always physicallysignificant, and we therefore focus attention on this leading-order inertial effect inwhat follows.

Substitution of (5.4) into ( 5 . 2 ) leads to a succession of linear equations for thefn(e, ): he equation for f 2 (using the expression (5.1) forfi nd the fact thatreduces to

2 N < Reh, < 2 ( N + 1 ) .

afl /at = 2 w a f 1 / a 4

and the solution, subject to the conditionsis f 2 = af2/ae = o on e= 5 4 (5.7)

9 ( 5 . 8 )B sin 2 0 +D sin 4 8- sin3a( 8COS 2 8 COS a - 8 )24 sin3a(sina - cosa )22 =B = sin 2a(4 +cos 2 a ) -a( + 8 cos 2 a + cos22a),D = 5a cosa - in a ( 4 +cosa).where (5.9)

Note th at a new singularity appears when a = n; his singularity can be resolved byincluding appropriate eigenfunction contributions (as in $ 3 above), the ultimate resultbeing, for a = n,

(5.10)This result is, however, of dubious significance since, as noted above, Reh, < 4 fora 2 45"; in fact, for a = n, , = 3, and the correspondingcontributionto$,if present,will certainly dominate over (5.10) near r = 0.

When a = an (in which case V2$, = 0 as remarked above) the inertial correctionsimplifies to

1 wr2 ( 1 + COS 2 0 ) 5 sin 28In r +e ( 5 COS 28- )).' ii.2=S(Y)

1 wr2 2$2 = (%) (n in 20 -sin 4 e- e). (5.11)


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326 C. Hancock, E. Lewis and H. K. Moffatt(for a = in) s dashed curves for w < 0

(figure4a) and w =- 0 (figure 4b). In the former case, a jet-type flow tends to developas the fluid is squeezed out of the gap; in the latter case, inertia forces tend to spreadthe flow more evenly over the range of 8 (relative to the Stokes flow).

It would of course be possible to obtain subsequent terms of the series (5.4)numerically (as done above for the Taylor problem). The limited additional infor-mation that this would provide, however, makes this an exercise of doubtful value.A more interesting problem might be to allow for unsteadiness in w , and hence forexample to study the development of the flow due to angular acceleration of the platesfrom a state of rest. We leave this problem to a future investigation.

Figure 4 shows the streamlines of

We are grateful to a referee of a previous version of this paper, whose penetratingcomments led to significant improvements; to Professor Milton Van Dyke for hissuggestions concerning the use of Pad6 approximants; t o Professor S.N. Curle fordiscussion on the topic of series improvement; and to the Science Research Councilfor supporting the work of one of us (C. H.) by a Research Studentship. 0Appendix

In equation (3.2)) etK, = 2BC+C2-D2,K, = 4CD,

K2= -2(B+C)D,K, = 2(C2- D2),

where B , C, D , are given by (1.2). A particular integral of (3.1)s thenf !f) = E8 COS 28+F 8 sin 28+G82 cos28+H82 sin 28,

where, as is easily verified,32E = 2Kl- 3K4,16G = K,,

32F = - 2K2- 3K3,16H = -K4.

The complementary function isf 2") = P +Q8+R COS 28+ X sin 28,

and, with f2 = f 8 )+ p), he boundary conditions (1.9) giveP + R = 0 ,

Q+2S+E = 0,P+Qa+Rcos2a+Ss in2a = -X(a),

Q- 2Rsin 2a +2 s cos2a = -X'(a),


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Ejfects of inertia in forced corner flows 327where X ( a ) = E a cosa+Fa ina+G a2cosa +Ha2sina and X ' ( a ) is the derivativeof X , regarding E , F , G , H as constants. Solving (A 5 ) , we find

2Ea sin2a- 2 X ( a ) in2a- ( E- ' ( a ) )(a- ina cosa)4 sin a(sina - a cosa)P = - R = 1

rEFGHPQR.s

2 X ( a ) osa- ' ( a ) ina-E sina2(sina- a cosa)( 2 E acosa- X ( a )cosa-E sin a+X ' ( a ) in a) in2a4(sina- osa)

& = 9S =

Hence f 2 ( S ,a) s completely determined.For the particular case a = in, he constants E ,F, ,R ,S take the values

164(n2- )'-

- 8(n2+ 1 2 )- 16n(n2+6 )64n16(n2-4)n(n4+ 1 4 ~ ' - 2 4 )- n2(n2+ 8 )- n(n4+ 14n2- 42(n2+ 6 ) (n2+ 4 )

The streamline plots in figure 1 are based on this solution.

R E F E R E N C E SHANCOCK,. 1981 Corner Flows. Ph.D. thesis, University of Bristol.INOUYE,. 1973 lhoulement d'un fluide visqueux dans un angle droit. J. de Mdcanique 12 ,LIIJ, C. H. & JOSEPH,. D. 1977 Stokes flow in wedge-shaped trenches. J.FZuluid Mech. 80,LUGT,H. J. & SCHWIDERSKI, . W. 1965 Flows around dihedral angles. I. EigenmotionMOFFATT,H. K. 1964 Viscous and resistive eddies near a sharp corner.J.FZuid Mech. 18,l-18.MOFFATT,H. K. & DUFFY,B. R. 1980 On local similarity solutions and their limitations.

J. FZuid Mech. 96, 299-313.TAYLOR, . I. 1960 Similarity solutions of hydrodynamic problems. Article in Aeronautics andAstronautics (Durand Anniv. Vol.), pp. 21-8. Pergamon.TAYLOR, . I. 1962 On scraping viscous fluid from a plane surface. In The Scientific Papersof Sir Geoffrey Ingram TayZor, vol. IV (ed. G. K. Batchelor), pp. 410-413. CambridgeUniversity Press, 1971.WEINBAUM,. 1968 On the singular points in the laminar two-dimensional near wake flowfield. J.Fluid Mech. 33, 39-63.

609-628.443-463.analysis. Proc. Roy. Soc. A 285, 382-412.

C. Hancock, E. Lewis and H.K. Moffatt- Effects of inertia in forced corner flows

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