Three-Dimensional Disturbances in Flow Between Parallel Planes

Three-Dimensional Disturbances in Flow Between Parallel PlanesAuthor(s): J. WatsonSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 254, No. 1279 (Mar. 8, 1960), pp. 562-569Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2413870 .

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Three-dimensional disturbances in flow between parallel planes

BY J. WATSON

The National Physical Laboratory, Teddinyton, Middlesex

(Communicated by G. B. B. M. Sutherland, F.R.S.-Received 8 October 1959)

The close connexion between the stability of three-dimensional and two-dimensional disturbances in flow between parallel walls has been examined and this has led to the formation of a three-dimensional stability diagram where 'stability surfaces' replace stability curves. The problem which has been investigated is whether the most highly amplifying disturbance at any given Reynolds number above the minimum critical Reynolds number is a two-dimensional or a three-dimensional disturbance. It has been shown that the most unstable disturbance is a two-dimensional one for a certain definite range of Reynolds number above the critical. For Reynolds numbers greater than this no definite general answer has been found; each basic undisturbed flow must be treated separately and a simple procedure has been given which, in principle, determines the type of disturbance which is most unstable. Difficulty arises in following this procedure because it requires knowledge of the two-dimensional stability curves in a certain region where this knowledge is very scanty at the moment. Although this difficulty arises, in Poiseuille flow the calculations available indicate very strongly that the most unstable disturbance at any given Reynoldsnumber above the critical is two-dimensional. Further, it is believed that this result holds for all other basic flows.

A second result is that if the wave number (a) in the flow direction is specified, as well as the Reynolds number, then for a in a certain range, the most unstable disturbance is three- dimensional.

1. INTRODUCTION

In most mathematical investigations into the stability of flow between parallel walls the disturbance is assumed to be a wave of infinitesimally small amplitude travelling in the direction of the undisturbed flow. The stream function for this wave has the form 0(y) exp {ix(x - ct)}, with c = cr + ici, where X6 is the wave number, cr is the speed and acci is the time amplification factor of the wave. Squire (1933)

examined the stability of a more general type of disturbance consisting of a wave travelling at an angle to the direction of the basic flow but remaining parallel to the walls. The stream function for this wave has the form 0q(y) exp {i(acx + /z - ct)}, where ,? is the wave number in the cross-stream direction. He discovered that the stability characteristics of a given three-dimensional disturbance are directly related to the characteristics of a two-dimensional disturbance and he used the relation to prove that, at the minimum critical Reynolds number for two-dimensional disturbances, only a two-dimensional disturbance will be maintained, all three-dimensional ones damping out with time.

Further work on this subject was done by Jungclaus (I957) who showed that, for suitable choice of Reynolds number above the critical and of the wave number, a', it is possible to have a three-dimensional disturbance which is more unstable than the two-dimensional disturbance.

The aim of this paper is to generalize the result given by Squire (I 933) in a manner suggested by the work of Jungclaus (I 957). By suitable choice of axes, two-dimensional stability curves in the oa-oaR plane (where R is the Reynolds number of the

[ 562 ]



Three-dimensional disturbances between parallel planes 563

undisturbed flow) become three-dimensional stability surfaces of revolution in ac-acR-/ space, formed by rotating the curves in the ac-acR plane about the acR-axis; while all disturbances at a given Reynolds number are represented by points on a plane (R1 constant). Hence the curves of intersection of the plane with the surfaces of revolution are 'stability curves' for three-dimensional disturbances at that particular Reynolds number, and it is shown that the most unstable disturbance at the given Reynolds number must correspond to a point on a certain curve in the plane. Investigation into the degree of amplification along this curve reveals that a two-dimensional disturbance is the most unstable at any given Reynolds number in the range above the critical with upper limit at the Reynolds number at which ci has its maximum value. For higher Reynolds numbers no definite result can be stated; each basic flow must be treated separately and the degree of amplification must be calculated along part of the curve in order to determine the most unstable type of disturbance. The two-dimensional stability diagram is all that is necessary to perform the calculations. However, very little numerical information about the stability curves is available in the region concerned. In the case of Poiseuille flow this information is sufficient to provide strong evidence that the most highly amplifying disturbance at any given Reynolds number above the critical is a two- dimensional one. It is believed that this result holds for all other basic flows.

A second result of some importance is that, if the wave number (x) in the flow direction is fixed as well as the Reynolds number, the most unstable disturbance is a three-dimensional one when a lies in a certain range (defined at end of ?2). This point was first made by Jungclaus but the restrictions on oa for which it is true are derived in this paper.

The above results are obtained from linearized theory, but it is of some interest to speculate on their relevance to the (possibly non-linear) mechanisms whereby three-dimensionality is produced in experiment. For example, in the experiments of Klebanoff & Tidstrom (I959), as described by Schubauer (I958), it seems likely that the wave number, a, in the flow direction is fixed by the frequency of the vibrating ribbon. A possible mechanism which then operates to make the flow three-dimensional as observed is that a three-dimensional disturbance to the wave arises with a spanwise wave number (fl) corresponding to a more unstable disturbance for the same wave number a. It should be noted that a combination of two disturbances with spanwise wave numbers /3 and -f/ produces a flow periodic in the spanwise direction and with zero wave velocity in that direction; this is what is observed to happen and it remains to be seen whether a mechanism of the kind proposed in this paragraph really does operate.

A further point to note is that the assumption, on linearized theory, that the most highly amplifying disturbance present will develop and prevail can only be proved, if indeed it is rigorously true, by reference to non-linear mechanisms.

2. GENERAL THEORY

Consider the steady, two-dimensional flow of an incompressible fluid between parallel walls. The origin, 0, is taken at some point midway between the walls, the x-axis in the direction of flow, the y-axis normal to the walls and the z-axis to com-

36-2



564 J. Watson

plete the right-handed set of axes. Now (cf. Squire I 93 3), the equations of motion are reduced to a non-dimensional form and so is the undisturbed velocity of the flow which becomes vi(y). A small disturbance with velocity components,

u(y) exp {i(ax + /z - xct)},

v(y) exp {i(acx + Jz - cct)}, (c, / > 0), (241)

w(y) exp {i(xx + /z - act)},J

is superposed on the undisturbed flow. Then the equations of motion may be linearized if the amplitude of the disturbance is sufficiently small and ultimately the equation for v, L(a2+ 2,' R,c)v = 0, (2.2)

is obtained, where

L~cz2/5)2c~B~) = - ~fd2 21d2-i L ( 2 + -x R, c) =[(,U c) {d2(a _ + 4) d2 L Cdy 2 (2+JJdy2 x

(d4 2(a2 + fl2) -y + (yC 33)

together with the homogeneous boundary conditions to be satisfied by v. If this eigenvalue problem is solved for v then u, w are the solutions of simple equations (Squire I933). It was noticed by Squire that if (C2 ?/52) is replaced by y2 and aR by yR* in equation (2.2) and the boundary conditions, then the problem reduces to a two-dimensional problem at a lower Reynolds number (R*). He deduced that if the basic flow is unstable to a three-dimensional disturbance at a particular Reynolds number then it will also be unstable to a two-dimensional disturbance at a lower Reynolds number. It is proposed to extend this result.

Suppose that the two-dimensional problem has been solved. Then this eigenvalue problem leads to a relation of the form

F(y2, yR*, c) = 0, or F(72, aR, c) = 0, (2.3)

where c = c, + ici is complex in general. Or, on eliminating c, between the equivalent two real equations, we obtain the real equation

G(y2, aR, c,) 0, (2.4)

or _2+/22 = H(xR, c,). (2.5)

Consider first two-dimensional disturbances (/3 0). Then the relation (2.5) becomes

-- = H(cR, ci). (2 6)

Now the stability curves are given by (2.6) for each constant value of cu. Although these curves are usually drawn in the ca-R plane it is much more convenient for our purposes to suppose them drawn in the ce-acR plane. For if axes /3, acR, as are drawn in space, then, for each constant value of ct, (2.5) represents the surface (or rather that part of the surface lying within the primary octant) generated by rotating the curve (2.6) about the acR-axis (see figure 1). Each surface represents the locus of points (disturbances) which have the same value of ci . A curve, C, in the ck-R plane is




defined by the points on the stability curves where the tangents to the curves are parallel to the a-axis. The significance of this curve is that (for any given value of acR) as a varies, the maximum value of ci occurs on the curve C, the value of ci decreasing monotonically on either side of this point. This curve C is shown in figure 2, together with the point Pm where ci assumes its greatest value, cim, in the entire ca-acR plane.

0

13

FIGURE 1. A stability surface for three-dimensional disturbances.

be~~~P

/ gY~~~~~~~~R

FIGURE 2. Curves of constant ci in the c-acR plane: --, curve C; +, point Pm;

planes R = constant.

Now consider three-dimensional disturbances. Each disturbance may be represented by a point on one of the surfaces of revolution mentioned earlier. Consider the set of disturbances with the same values of af and R. If af and R are such that the point (e, acR) in the ac-acR plane lies on the side of the curve C away from the origin then, with af and R fixed, as ,8 increases it is easily seen that the value of ci for the corresponding three-dimensional disturbance decreases monotonically as the suc- cessive surfaces of revolution (ci constant) are cut. So also will the value of the time- amplification factor, oxch. Hence the most unstable of these disturbances is the two-dimensional one. On the other hand, if af and R are such that the point (e, acR) lies

36-3



566 J. Watson

on the same side of C as the origin then as ,8 increases so also does ci until the point is reached where the line af = const., R = const. cuts the surface S formed by rotating C about the (XR-axis; as ,8 increases from this value, ci decreases monotonically. Hence the most unstable of these disturbances is the three-dimensional one corresponding to the point where the line through (e, acR) parallel to the fl-axis cuts the surface S. (This is a generalization of Jungclaus's (I 957) result mentioned in ? 1.) These properties will be used in the investigation of disturbances at a given Reynolds number. The Reynolds number range is split into two parts, that between the critical and the value corresponding to the point PI(R,2) and that beyond this value.

(a) Disturbances at fixed R (R, < R < Bi)

All disturbances at a given Reynolds number, R, are represented by the points on the plane through the fl-axis which makes an angle tan-' 1/R with the 8-acR plane. This plane cuts some of the surfaces of revolution (ci constant) in a set of curves on each of which ci is constant. It is readily seen that this set of curves will have the

0 0 Q T r2

13

FIGURE 3. Curves of constant ci in plane R = constant (R < Rm): - - -, curve C1.

form as shown in figure 3. The curve of intersection, C1, of the plane with the surface S is also shown there. The r-axis is defined as the line of intersection of the plane with the ca-acR plane. Distance along the r-axis is proportional to a, being r = c(1 + R2)1.

The plane corresponding to the given Reynolds number touches one and only one of the surfaces of revolution (that which corresponds to ci = c* say) and the point of contact, T, lies in the c-xR plane. This will be the only point that the plane and the surface (ci = c*) have in common. This point is represented in figures 2 and 3. From figure 2 it is obvious that: (i) the point T is further from the origin than the point Q where the plane cuts the curve C, and (ii) there will be no curves of ci constant in figure 3 for ci > c*, since the plane cuts none of the corresponding surfaces of revolution. The following further properties of the ci constant curves in figure 3 are readily seen. Each curve cuts the r-axis in two points, one on either side of T and there the curves will be parallel to the T8-axis. For values of ci which are less than the value at the point Q, the curves bend towards the f8-axis as shown; while for ci greater than the value at Q they do not bend towards the fl-axis. As ci increases, the

ci constant curves converge towards the r-axis until the point T is reached where the curve degenerates into the point T. The curve of intersection, C1, of the plane with the surface S is also the locus of points in the f8-r plane where the tangents to the ci constant curves are parallel to the ,8-axis.




Consider any point on the r-axis on the origin side of the point Q. The straight line in the /3-r plane parallel to the f8-axis and passing through this point represents a set of disturbances and, from the properties described at the end of ? 2, the most unstable of these is that corresponding to the point where the line cuts the curve C and this represents a three-dimensional disturbance. Also as we move along CO towards Q the value of ci increases and so does r = ox(1 + R2) or ar (since R is constant). Hence of all the disturbances represented in the block on the origin side of the line through Q parallel to the fl-axis, the one with the largest amplification factor, ac, is that corresponding to the point Q. A similar argument holds for the remaining disturbances. Of the set of disturbances represented by the straight line parallel to the ,8-axis through any given point on the r-axis further from the origin than Q, the most unstable is the one corresponding to the point on the r-axis. Thus of all the disturbances at the given Reynolds number, satisfying Rl< < R?< Rm, the most unstable corresponds to a point on the r-axis; in other words the most unstable disturbance is a two-dimensional one.

0 ?t T Q OZ

13

FIGURE 4. Curves of constant ci in plane R = constant (R > Rm) ---, curve Cl; +, point Qm,.

(b) Disturbances at fixed R (R > Rm) The arguments of ? 2 (a) can be applied in this case and results of a similar nature

are obtained. The ci constant curves in the Reynolds number plane are readily seen to have the form as shown in figure 4, in which the curve of intersection, 01, of the plane with the surface S is also shown. The plane touches one and only one of the surfaces of revolution (corresponding to ci = c* say), the point of contact, T, lying in the cc-acR plane. Thus the curve ci = ci* in the ,-r plane cuts the r-axis in one and only one point, namely T. The curves for ci greater than c( appear in the plane as shown, none of them cutting the r-axis, and as ci increases the curves converge to the point Qrn where ci = cim. All the curves for ci less than ci cut the r-axis in two points, one on either side of T, and the curves are parallel to the fl-axis there. In this case T is nearer the origin than the point Q where the plane cuts the curve C; furthermore, it is obvious that Qm is nearer to the ,8-axis than T is. For ci less than the value at Q the points on the ci constant curves furthest from the fl-axis all lie on the r-axis; while for ci greater than the value at Q the points furthest from the fl-axis lie on the curve C,.

As in ? 2 (a) it is deduced that, of all the disturbances represented in the block to the origin side of the line through Qwn, the one with the largest amplification factor,



568 J. Watson

acci, is that corresponding to the point Qm which has the amplification factor acjc", the value of cc being cmRm/]R, where am is the value of a at the point P.. in figure 2. Similarly, the most unstable of the remaining disturbances must correspond to some point either on that part of C, between Q,, and Q or on the part of the r-axis on the far side of the origin from Q. In this case, although ca increases as we move along C from Qm to Q, ci decreases. Consequently, no definite statement can be given as to whether a two- or a three-dimensional disturbance is the most unstable. Since the disturbance corresponding to Qm is unstable, while the disturbances corresponding to all points on the r-axis on the far side of the origin from 02 (one of the points where the curve ci = 0 cuts the r-axis) are stable, then the most unstable disturbance corresponds to a point which lies either on the part of C, between Qm and Q (in which case the most unstable disturbance is a three-dimensional one) or on the r-axis between Q and 02 (the most unstable disturbance being two-dimensional). A method will now be given to determine which type of disturbance is the most unstable. It depends on the fact that the characteristics of any given three-dimensional disturbance are given by those of a related two-dimensional disturbance.

Consider any point on C beyond the point Pm (figure 2), that is, at a higher Reynolds number than R.. Let this point, PO, correspond to a disturbance with ac = o, R = Ro, ci = cio. Then this point on C corresponds to the point

(caR, a) =jcot1j- )i ,oRo)

on the curve C, in the Reynolds number plane R = R say. This point on C, represents a disturbance with ci cio. The amplification factor of this three-dimensional disturbance is coRo ciol/R which can be calculated, for given R, from only the characteristics of the two-dimensional disturbance represented by the point P. in the cc-acR plane. Hence the amplification factors for the disturbances corresponding to that part of C1 between Qm and Q (figure 4) are found from the corresponding disturbances between Pm and Q in the cc-acR plane (figure 2). The set of disturbances corresponding to the part of the r-axis between Q and 02 in figure 4 are the two-dimensional disturbances corresponding to the part of the line R = -R in the oc-acR plane between Q and 02 (figure 2). Hence the maximum value of acRci/B as calculated on C between Pm and Q, together with ccci on the line R = R between Q and 02, will reveal whether the most unstable disturbance at R = R is two- or three-dimensional.

If the c? constant curves are given in the c-R plane then the following procedure can be used. First, the curve C must be drawn in this plane. Now the ci constant curves in the cc-acR plane are given by caR = f (c, ci) say, so that by definition the curve C is defined by the points where (afl/ac),j = 0. The ci constant curves in the cc-R plane are given by R = f (c, cj)/ac = q(cc, ci) say, and the curve C in this plane is defined by the points where

(af) (_ag__ 0, ta Ci \ ccO

that is where cc-+]? I=,




or where ( ?C

In figure 5 these are the points on the ci constant curves where

tan(2-0) = tan( - or where 0 = ~fr. This curve C will pass through Pm (where ci = cim). Having drawn that part of C between Pm and Q (where the line R = R cuts C), we calculate aRciJ/R corresponding to this part of the curve and cai corresponding to that part of the line R = R between Q and the point 02 above Q where the line cuts the neutral curve. If the greatest value of these occurs at a point on the line R = R then the most unstable disturbance at R = B will be two-dimensional; otherwise the most unstable disturbance is three-dimensional.

R=R R FIGURE 5. The curve C in the cvR plane; - --, curve C; +, point Pm.

This theory was applied to plane Poiseuille flow by means of the curves obtained by Shen (I 9 54). The result of ? 2 (a) shows that, at any Reynolds number between the minimum critical (5400) and Rm (56000) the most unstable disturbance is two- dimensional. There is insufficient information regarding the ci constant curves for much higher Reynolds numbers but the information available indicates that the variation of a and ci along the relevant part of C is small compared with the variation in R. This implies that the most unstable disturbance at anygiven Reynolds number above Rm is also a two-dimensional one.

The author is greatly indebted to his colleague, Dr J. T. Stuart, at whose sug- gestion this work was undertaken, for much valuable advice and constructive criticism.

The work described above has been carried out as part of the research programme of the National Physical Laboratory, and this paper is published by permission of the Director of the Laboratory.

REFERENCES

Jungelaus, G. I957 see Appl. Mech. Rev. 11, 630 (November 1958). Klebanoff, P. S. & Tidstrom, K. D. 1959 N.A.S.A. Tech. Note D-195. Schubauer, G. B. I958 Boundary layer research, I.U.T.A.M. Symposium, Freiburg 1957, p. 85.

Berlin: Springer. Shen, S. F. I954 J. Aero Sci. 21, 62. Squire, H. B. I933 Proc. Roy. Soc. A, 142, 621.



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Three-Dimensional Disturbances in Flow Between Parallel Planes

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