the vortex roll-up problem using lamb vortices for the elliptically loaded wing

Computers & Fluids Vol. 18, No. 1, pp. 139-150, 1990 0045-7930/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ~ 1990 Pergamon Press plc

T H E V O R T E X R O L L - U P P R O B L E M U S I N G L A M B

V O R T I C E S F O R T H E E L L I P T I C A L L Y L O A D E D W l N G t

CHARLES DALTON a n d XUEGENG W A N G

Department of Mechanical Engineering, University of Houston, Houston, TX 77204, U.S.A.

(Received 5 October 1988; in revised form 25 January 1989)

A ~ t r a e t - - T h e classical roll-up of the vortex sheet trailing the elliptically loaded wing is examined here. Three factors, each of which contributed in a different, yet influential, manner were included in the anlaysis: use of Lamb vortices as the discrete vortices, use of a variable time step, and a redistribution of the initial vortex discretization as a variant to the traditional equal-spacing or equal-strength models. The latter two factors are used in the analysis of this problem for the first time in this study. A smooth spiral roll-up was obtained for dimensionless times which exceed those of all but one previous investigators by at least a factor of ten without an increase in computat ion time for similar computers. In addition, very good comparison with the results of previous investigators is obtained for similar parameters. The redistribution of the initial discretization was found to prevent an early concentration of vortices in the spiral as the integration occurs. The spiral was found to form with an equal number of turns as in other calculations, but with less tightness which we attribute to the fact that Lamb vortices were used. Double precision arithmetic was necessary to obtain meaningful results.

I N T R O D U C T I O N

The roll-up of vortex sheets emanating from airplane wings has been of interest to computational fluid dynamicists since at least 1931. The traditional approach is to represent the continuous vortex sheet by an array of point vortices which are of equal strength or spacing. Various investigators have followed this approach, each with their individual variations. Other approaches have used the vortex-lattice method or the cloud-in-cell method. This study will concentrate on the point vortex method.

Rosenhead [1] was one of the first investigators to consider the vortex roll-up problem. In order to study the nonlinear development of a perturbed vortex sheet, Rosenhead replaced it by an array of two-dimensional, irrotational line vortices. He considered a number N of discrete vortices and studied their response to a sinusoidal disturbance. Rosenhead calculated a smooth rolling up of the vortex sheet into concentrated clusters of vortices with a spacing equal to the wavelength of the original perturbation. The Rosenhead calculation (1931) was done using a desktop hand calculator with large time steps. The inherent difficulties of this approach were obvious in later years with the advent of the high speed digitial computer.

A later study of the Rosenhead approach was made by Birkhoff and Fisher [2]. Their study used more vortices (22 vs 12) per wave length and much smaller time steps. They found that paths of the individual vortices were irregular and often crossed over one another. This cross-over phenomenon is physically impossible and, therefore, represents an unrealistic simulation of the vortex sheet.

Another follow-up of the Rosenhead study was by Hama and Burke [3] who used the same number of vortices per wave length as Rosenhead but shortened the time step. The results obtained by Hama and Burke extended the Rosenhead study by unequal spacing of the vortices; this approach yielded a much smoother roll-up of the line vortices into vortex clusters. However, many stability questions remained unanswered.

Another important early application of the discrete vortex method was by Westwater [4] in 1935 and was applied to the rolling up of the vortex sheet shed by a finite-span lifting wing into the trailing vortex pair. Use of the discrete-vortex model for this problem allows the vortex strength to vary along the wing span according to the wing loading. Westwater's idea was to compute the time development of the vortex sheet. This assumes that the vortex sheet is two-dimensional

";'This paper was originally presented in the 1988 National Fluid Dynamics Congress in Cincinnati. The paper number is AIAA 88-3746.

139

140 CHARLES DALTON and XUEGENG WANG

upstream and downstream of the wing and the effect of the wing itself is neglected. Westwater's results were consistent with what was expected physically; the ends of the vortex sheet rolled up into two equal, opposite-signed vortex concentrations, starting from the wingtips and rolling upward and inward to form the expected spiral.

The later discoveries about Rosenhead's treatment of the discrete vortex method prompted others to repeat Westwater's calculations. Takami [5] increased the number of vortices and found that the details of the spiral were quite different from what Westwater found. Takami found a "strong randomization" of the vortices in the spiral roll-up and noted that the vortex equations are unstable with respect to the initial data. Moore [6] used four times Westwater's number of vortices and a very accurate time integration. Moore found a very contorted vortex path near the ends of the vortex sheet. This contortion is due to the propensity of close vortices to rotate about one another. The vortices formed a sheet which crossed over on itself, a physical impossiblity.

One way to eliminate the mutual circling of pairs of vortices is to eliminate the infinite velocity which occurs at the center of a point vortex. Chorin and Bernard [7] did this by using Chorin's vortex blob approach; the tip roll-up was accomplished smoothly and the tendency for crossover was eliminated.

Another approach is to limit the induced velocities on a pair of vortices. If the velocity limit is attained, the two offending vortices are merged into one vortex at the center of vorticity of the pair. Clements and Maull [8] used this approach and found that merging procedure increased the separation of vortices and reduced the maximum mutually induced velocities.

Fink and Soh [9] cast the discrete-vortex representation of the vortex-sheet problem in a new light. They recognized that the discrete-vortex method is essentially a technique for dividing a continuous vortex sheet into short segments. Each segment has a certain fraction of the strength of the entire sheet. The discrete vortex for each segment is located at the center of the vorticity for the segment and has a circulation equal to the circulation of the segment. A single vortex has no self-induced velocity; however, a single segment of the vortex sheet would induce a velocity on itself. Based on this, Fink and Soh hypothesized that optimum results were obtainable if the discrete vortices representing the sheet were equidistant from one another. This can be accomplished by a rediscretization after each time step. The result of one time-step calculation is a vortex sheet with a new set of nonequidistant positions. These nonequidistant discrete vortices were rearranged to give equidistant positions with the vortex strengths varied to give the desired sheet strength at any position. The calculations were repeated for the next time step after which the rediscretization procedure was repeated. This process was continued until the computations were stopped. This method of equal spacing did not yield the unnatural orbiting of the vortices near the tip of the sheet. It did, however, lead to increased computing time due to the greater complexity of the calculation.

Moore [10] reconsidered his earlier attempt at investigating the roll-up of the vortex sheet. His revised approach consisted of replacing the inner portion of the spiral by a single vortex located at the center of the spiral. Moore felt that a part of the earlier trouble both he and Takami had experienced was due to the inadequacy of a finite number of discrete vortices scattered about on the spiral to represent the spiral effectively. He found through extensive numerical computation that the random, chaotic motion near the tip was eliminated and that the rest of the vortex sheet was accurately modelled.

The difficulty that faced all of the previous investigators of the vortex roll-up problem was not numerical. In fact, Moore [6] showed that the chaotic behavior of the vortices would reverse itself to return to the original sheet if the time integration were reversed. This means that some other factor is influencing the calculations. In an effort to overcome the computational difficulties, investigators began to use the corrective procedures just discussed. These schemes were reasonably successful, but not quite satisfying since they were ad hoc artifices designed to make the computation fit the experimental results.

Later, Fink and Soh [11] claimed to have discovered the difficulty with the previous investiga- tions. The problem lay in the discretization process itself. Fink and Soh correctly point out that a continuous vortex sheet is not properly modelled by an array of discrete line vortices. They proposed a discretization method which includes consideration of Cauchy principal value integrals and higher order terms which are not present in discrete vortex models. Their expression for the

The vortex roll-up problem 141

induced velocity contains two terms not found in the usual discrete vortex approximation. Fink and Soh applied their ideas to the Westwater problem and found a tip roll-up consistent with what was expected physically.

Baker [12] reconsidered the analysis of Fink and Soh and showed that their test case was not adequate to demonstrate their method. The case of the roll-up of a vortex sheet shed by a ring wing was treated by Baker and their results were found to be unreliable. The reason for the breakdown in results remains unknown although Baker offered several possibilities.

Siddiqi [13] has done a recent review of the wake roll-up problem using the point vortex method. The Siddiqi approach redistributes the vorticity which gives better results while also improving the computational efficiency. The Siddiqi results will be discussed later and compared to the results from the analysis presented herein.

Krasny [14] has reconsidered the elliptically loaded wing and has computed the vortex-sheet evolution through a desingularization of the equations defining the velocity of a Lagrangian point on the sheet. The desingularized equations are expressed in terms of a smoothing parameter which, when set to zero, allows the exact vortex sheet equation to be recovered. The smoothing parameter approach is similar to the vortex blob method of Chorin and Bernard [7] and does not correspond to a physical effect. The calculations of Krasny for small values of the smoothing parameter, which were also carried to relatively long times, provide a physically satisfying result and will also be compared later to the results of the present analysis.

We adopt the discrete vortex method using Lamb vortices for this analysis. Our purpose in this investigation is to reconsider the problem using the observations of the previous studies as guidelines. First, we will examine the roll-up process for a variable time step in an attempt to prevent the chaotic behavior which normally accompanies these calculations. Next, we will alter the circulation by changing the discrete vortex distribution while keeping the initial velocity distribution and the time step variable. Even though this second technique represents another ad hoc approach, it does provide results which are physically satisfying.

A N A L Y S I S

The problem which we seek to represent is the roll-up of the wingtip vortex which occurs on an elliptically loaded wing. At zero time, the continuous vortex sheet lies on the line y = 0 in the range ( - a ~< x ~< a). The initial vortex distribution can be expressed by

tn(x) = 2Ux(a 2 - x2) -1'2, (1)

where U is the instantaneous velocity in the negative direction at the initial origin. To discretize the vortex sheet, we choose 2N point vortices, N per side. The initial position of

the ith vortex is

~,.0 = x/a2 - [(.~a - ] - x~ + . ~ f~ - X~+l)/2] 2, (2)

and the initial circulation of the ith vortex is

F~.0 = 2U[x/~ 5 - x~ - ~ ] . (3)

We now define the following dimensionless variables:

Y,= x,/a, ( ,= ¢,/a, F,= r,/Ua. (4)

We choose a Lamb vortex to represent each individual vortex,

/~,(t) = F,.0{l - e x p [ - r~./4vt]}, (5)

where rjj is the distance between the center of the point vortex and the point influenced by it. A Reynolds number effect can be introduced by the following substitutions:

Y = r / a , f = U t / a , R e = U a / v . (6)

Thus, the time-dependent dimensionless circulation becomes

F,(t) = Fi.0 {1 - exp[ - Re F~,/4T]}. (7)


The Lamb vortex represents an exact solution to the Navier-Stokes equations. However, an array of Lamb vortices representing a vortex sheet no longer satisfies the Navier-Stokes equations because the linear combination of vortices does not properly represent the nonlinear (convective) terms in the Navier-Stokes equations. This issue has been addressed by Greengard [15], but a proper estimation of the error is still unclear.

The motion of the individual vortices is determined from their Lagrangian description. For an array of N vortices, we have the following velocity components:

and

[ ,=l. i~j \ rj . i / \ j,, / j

(8a)

: = - S r , - r , ' 2~: i= L i~y r j . , . \ ~ / J 4n~j'

where Fig. 1 shows the quantities ~, 5 , rT~, and fir with

e~ = ((j - (~)2 + (fir - Fh) 2, (9a)

and

17"~i = (4/+ (i)2.3ff (~]j _ •i)2. (9b)

The new positions of the vortices after a small time step are given by their first order Lagrangian approximation.

Clearly, the size of the time step is critical in determining accurate values of the new vortex positions. Thus, it is desirable to try to control the time step to prevent large position changes from occurring. The time step control is accomplished by allowing only small angular rotations of the line connecting two vortex centers. The definition sketch is shown in Fig. 2. We take

c~, / = (F , /2~6. , + Fj/Z~z6. ~)/~,, ,, (10)

We let A" and B" approximate B' and A' after a small time interval AL This assumption is subjected to the constraint that the angle of rotation, ~ = cS~. jAT, is small:

sin(e3~./Af)~e3~./Af and c o s ( @ j A / ' ) ~ l ,

or if the angle in radians is much less than 1,

05~./Af < 1.

_ q 7

Fig. 1. The locations of the ith and jth vortices and their images. Fig. 2. Relative motion of two vortices.


I f we select ~ to be 5 °, then we have

AT = x/(3605,.j) = n ~ , / [ 1 8 ( r i +/~j)]. (11)

Since there are N vortices on each side of the x-axis, A~" must be selected to satisfy the requirements of every pair of vortices:

J" (12) A f = min F "

EQUAL STRENGTH AND EQUAL SPACING RESULTS

We begin by considering the two obvious cases, both of which have received prior treatment: the equal spacing case and the equal strength case.

In the equal spacing case, we divide the length a into N segments to give a zero-time spacing of Ay = 1/N between the point vortices. The initial dimensionless location of the ith vortex is given by

(,.0 = ~/1 - ¼[~/1 - (i - 1)2/U z + ~ /1 - ( i /U)2] 2, (13)

and the initial dimensionless circulation is

F,,0 = (2/N)[ ~ / N 2 - (i - 1) 2 - - ~ 1 . (14)

In the equal strength case, each point vortex is given the same dimensionless circulation:

F,.o = Fo = 2 / N , (15)

In terms of the dimensionless variables, the vortex segment endpoint and initial position are

xi+l = [ 1 - - ( ~ _xi-2 _ l /N)2]l/z (16a)

and

(i.0 = [1 - (~/1 - :~ - 1/2N)2] '/2. (16b)

Examining the results of either the equal spacing or equal strength cases shows that the ratio M of Reynolds number to the number of vortices, M =- R e / N , has a significant effect on the ability of the sheet to roll up smoothly. First, we will examine the equal spacing results. Figure 3 shows the roll-up at a dimensionless time of t = 4 for M = 1. The roll-up is smooth for this and later

0 R e : I00 N : I00

y/a - I

D ~ I I I I I I I

0 0.4 0.fl 1.2 1.6

x/a

Fig. 3. Vortex roll-up with M = 1 and equal spacing, stable.

C A . F 18 I--J

144

y / a

CHARLES DALTON a n d XUEGENG WANG

0.2

-0.2

-0.6

-I.0

Re : 200 N : I00 T : 1.0

- I . 4 ' ' ' ' ' ' ' 0 0 . 4 0 . 8 1.2 1.6

x/a

Fig. 4. Vortex roll-up with M = 2 and equal spacing, crossover.

times. However, the calculation for M = 1 is impractical at large Re because of the large number of vortices which are required. For M ~> 2 and t = 1, as shown in Fig. 4 for M = 2, the vortices have begun to cross over in the core of the spiral; the crossover becomes more chaotic with increasing M. The equal strength case produces better results because the spiral rolls up without crossover for values of M up to about seven and for t as high as 6 as shown in Fig. 5. The reason for the smooth roll-up is because of the greater concentration of vortices in the tip region. We attribute the smooth roll-up to the fact that the adjacent vortices in one ring of the spiral influence each other to a greater extent than the vortices in another ring of the spiral. When the vortices in one spiral ring are too far apart, they are influenced by the vortices in adjacent rings and crossover occurs. For low values of M, the roll-up is not chaotic, but the spiral contour becomes wavy. The reason for the waviness in the spiral is due to the relatively heavy concentration of vortices away from the spiral.

y / a

- 0 . 4

- I . 0

- I . 6

- 2 . 2 0 1.6

Re = 6 0 0

~ N : I 0 0

T : 6

( i

0 . 4 0 , 8 1,2

xla

Fig. 5. Vortex roll-up with M = 6 and equal strength, stable.


and

These calculations were initially attempted with point vortices; this effort produced crossover in the computation process, even when using the variable time step. Lamb vortices were then substituted for the point vortices and the early crossover was eliminated, producing the afore- mentioned results. It was also found early that double-precision arithmetic was necessary for the calculations to be done on a VAX 11/750 machine.

VARIABLE SPACING AND STRENGTH ANALYSIS AND RESULTS

From the numerical results just discussed, we find that, with small viscous diffusion in the individual vortex cores, the vortex sheet tip rolls up very quickly even for small dimensionless time. This causes a problem because there are not enough vortices in the vortex sheet tip for a smooth rollup. If we increase the number of vortices in the tip region, the increase is accompanied by a sharp increase in CPU time. However, we also find that it is not necessary to concentrate many vortices away from the tip region. Thus, it seems that this difficulty can be eliminated by adjusting the density of the vortex distribution while retaining the sheet description given by eqn (1). The procedure to accomplish this redistribution is strictly an ad hoc approach, the details of which are unique to this particular problem. We feel that this type of redistribution is unique to the elliptically loaded wing because of the sharp increase in circulation (or vorticity) as the wing tip is approached.

We divide the N vortices into three groups with the number of vortices N1, N 2 - N 1 , and N - N2 as shown in Fig. 6. From the case of equal strength, we have

Fo = F,o,ai IN, which is the horizontal line in Fig. 6.

If we increase the circulation of the individual vortices in the region {N1, N2}, then the number of vortices in this region is decreased to retain the original sheet strength for the region. We also decrease the individual vortex circulations in the region {N2, N} with a corresponding increase in the number of vortices, again to retain the original sheet strength for the region. A simple way to do this is by replacing the constant distribution by a variable distribution. Conditions to be maintained at the juncture of the regions are equal values and equal slopes of the circulation distribution. We have found that a sine wave variation such as shown in Fig. 6 will allow these conditions to be met. Thus, we can express the circulation of the ith vortex by the following relationships:

re.0=2/N, l<<,i<~N1 (17a)

[ ( ; " ')1 F, .o=Fo+AFI I - c o s 2r~N2_N ! , N I<i< .N2 (17b)

I / i - N 2 y ] /~i.0=/~0-AF2 1 -cos t rc~- i - - -~ ) . J , U2<i<~U (17c)

This analysis is shown in Fig. 6 where the physical distance along the vortex sheet is increased in the region {N 1, N2}. The number of vortices in the new region remains unchanged even though the end of the region has been stretched from if(N2) to ~'(N2). Since the total circulation of the sheet remains unchanged, the increase of circulation in the region {M1, M2} should equal the decrease in the region {N2, N}, that is

AF,(N2 - N1) = Ar2(N - N2). (18)

Let us define E = A/~,/it0, then we get

Fi.o=2/N, l~<i~<N1 (19a)

[ ( /-,<l l ' , .0=2/N I+E--ECOS 2 ~ ] i f T i - ] . j , NI~<i~<N2 (19b)

and

F~o = 2/NI 1 / N 2 - N I ' ~ I 'N2-NI'~ ( n i - N 2 ) ] - ' t , c°s N2 < i ~< N. (19c)

146 CHARLES D A L T O N a n d XUEGENG W A N G

I

x ' ( N I ) I I l X

~ ( N 2 ) ~ ( N ) = I

"x(Nl) ~

X ( N 2 )

I ~ I I

NI N2 N

y / a

1.0

0.5

0.0

- 0 . 5

- 1 . 0

- 1 . 5

0.0

- 0 . 5

- 1 . 0

- 1 . 5

- 2 . 0

- 2 . 5

- 3 . 0

- 3 . 5

- 4 . 0 - 2

(a) t = 2 . 0

e % * ° e , •

0.0

- - 0 . 5 • • .

- l O . :.':,~:.:::.. • o., o,.. • . . : , ~ . .

- 1 .5 . * ** ~.g*'-**Y* ° o:s~ ".-'::

- 2 . 0

- - 2 . 5

(b) t = 6 . 0

• ~ (c) = 10.0

• ..;~.....t., ee e eee e • -...~.']'.~ • e e e e e ~ • . t i t •

I I I I -1 0 1 2

x / a

Fig. 6. Redistribution of vortices. Fig. 7. Vortex roll-up with M = 20 and vortex redistribution for three different dimensionless times, stable. Re = 2000,

N = 100.

The actual values representing N1, N2, A,r~, and Aft2 are determined from numerical experimen- tation of their variations for a given N and F0. Careful evaluation of these changes shows that the values of N1 and N2 which produce the best results are NI = 0.1N and N2 = 0.7N. The value of A/~ which produced the smoothest roll-up was AF~ = 0.15F0. The variable spacing and strength results still correlate with the parameter M. Figure 7 shows results for M = 20 (Re = 2000, N = 100) at three different times, i.e. as the computed spiral tip evolves. The tip is found to roll-up smoothly. The left-hand side of the figure represents individual vortex locations while the right-hand side shows the contour of the vortex sheet. The entire spanwise structure of the sheet moves downward, but the slight inward drift of the spiral center noted by Krasny is not obvious in Fig. 7. However, if the results for t < 2 are also included, the inward drift is quite evident. At t = 10, the sheet has actually moved into the spiral; this represents a situation which appears physically unreal but may be explained by recognizing that at such advanced times the circulation is concentrated in the spiral and the remainder of the sheet has little circulation.

The variable time step defined in eqn (12) started as a very small value, 7~t = 0.0001, and increased, subject to the small angle constraint, as the calculation continued to a value of 0.02. The increase in 7~t allowed the total dimensionless time to increase to a relatively high value. An earlier study which also used a variable time step was that of Higdon and Pozrikidis [16]. This earlier study started with a relatively large nondimensional time step of 0.02 and then let the dimensionless time step decrease to 0.000003 as the calculation proceeded. The problem studied by Higdon and Pozrikidis was the circular vortex ring which is the same example chosen by Baker [12] to demonstrate the incompleteness of the Fink and Soh method [11]. The time step of Higdon and Pozrikidis decreased because of the increased number of marker points as the computation progressed. The larger number of marker points required smaller and smaller values of time step to be used in the computational scheme.


COMPARISON TO OTHER CALCULATIONS

The results shown in Fig. 7(c) for M = 20 (M = Re~N) and t = 10 represent a calculation at a far greater dimensionless time than was obtained by others (except Krasny [14]) who have done similar calculations. These results are obtained for CPU times comparable to those presented in the summary table of Siddiqi[13]. However, our results in Fig. 7 are for more vortices (N = 100) and a far greater value of dimensionless time than Moore[10] and for the same number of vortices and an even greater increase in dimensionless time than that of Fink and Soh Ill]. In fact, our results have been carried as far as t = 20 without chaotic behavior occurring; these results are not shown. The calculations of Krasny [14] were carried to a dimensionless time of 25 (using the definition of dimensionless time as defined herein). For small times (t ~< 2), Krasny used a very small value of the smoothing parameter (~ = 0.05) and 200 vortex blobs and obtained a smooth roll-up with multiple spiral turns. The effect of decreasing the smoothing parameter in Krasny's study caused the number of spirals to increase. The smoothing parameter plays the same role in Krasny's calculation as the Reynolds number effect in our calculations. To obtain results at dimensionless times up to 25, Krasny increased the smoothing parameter to the relatively large value of 0.2. The larger value of smoothing parameter was necessary to obtain solutions for reasonable CPU times at large dimensionless times, still with 200 blobs. The number of turns in Krasny's spiral is approximately the same as the number of turns produced in our calculations. The difference in the spiral centers for both our calculations and those of Krasny is indistinguish- able through large times, although this result is not shown. As noted earlier, all calculations herein were double precision in order to get meaningful results.

Figure 8 shows our results for N = 160 vortices at a dimensionless time of 2. Moore's [10] results are also shown at t = 2 and N = 60 with amalgamation (13 vortices) of the spiral into a single-core vortex. The comparison is quite good. Krasny's [14] results are also shown in Fig. 8 where very good agreement is shown up to the completion of about one turn in the spiral where Krasny's results begin to produce a tighter spiral; this tightness continues as the simulated vortex sheet continues to roll-up. Figure 9 shows a comparison between the results of Fink and Soh [11], Krasny, and ours. The spiral of Krasny rolls up faster than the Fink and Soh spiral which also rolls up faster than ours. The reason for the looser spiral in our results is the viscous diffusion present in the Lamb-vortex model which we used. No further comparisons with Krasny are reported because of the different spiral behavior. Our results are also compared to those of Siddiqi [13] in Fig. 10. Our calculations are for 100 vortices while Siddiqi used only 20. The calculation

0.2

0.0

- 0 . 2

- 0 . 4

y/a - 0 . 6

- 0 . 8

- I . 0

- I . 2

- I . 4 . 0

Re = .500

&

I I I I I I I I I I I I I I I

0.4. 0 .8 1.2 1.6

X/0

Fig. 8. Comparison of present calculations with those of Moore [10]. A, Moore, 60 vortices with amalgamation, At = 0.002; O Krasny, 200 vortices, At = 0.01; + present, 160 vortices, At varies.


o. 2 Re = I000 N : ,00 / ~ x h T : 0500

0.0

y /a - 0 . 2

- 0 4

- 0 . 6 . . . . . . . . . . 0 0.2 0.4 0.6 0.8 1.0

x /o

Fig. 9. Comparison o f present calculations with those of Fink and Soh [11]. • Fink & Soh, 40 vortices with amalgamation, At = 0.00l for first 90 steps and At = 0.004 subsequently. G Krasny, 200 vortices,

At = 0.0ld; --present, 100, vortices, At varies.

times are comparable because our time steps are variable and our time step increases as the calculation progresses. Very good agreement is noted between our results and those of Siddiqi. The last comparison is made in Fig. 11 with the results of Moore [10] at a dimensionless time of 7. Moore used 60 vortices to represent the roll-up with 28 of them amalgamated in the vortex core. The center of our spiral is at essentially the same location as Moore 's vortex core.

Comparisons to other calculations can also be made on the basis of computer (CPU) time. Siddiqi [13] presented such a table for comparison of the various computat ions on which he reported. For our purposes, we adopt Siddiqi's method of reporting these times and include our own calculations and Krasny's in part of Siddiqi's table. These comparisons are shown in Table 1 where the time step shown for our calculations is the average of our variable time step over the duration of the calculations. The time step in Siddiqi's table has been modified to put the time steps on the same basis; Siddiqi's time step is rc times that of Moore [10], Fink and Soh [11], and the present study. Krasny's time step is twice that of ours. Examination of the results in Table 1 shows that the present calculations, with the variable time step, are as efficient as those reported by

0.0

y/o - 0 . 2

- 0 . 4 z3 Z~

Re : t000 N : 100 T : 0.375

0.2

- 0 . 6 . . . . . . . . . . 0 0.2 0.4 0.6 0.8 1.0

x /e

Fig. 10. Comparison of present calculations with those of Siddiqi [13]. A Siddiqi, 20 vortices, At = n x 10-3; --present, 100 vortices, At varies.

y / a

The vor t ex ro l l -up p r o b l e m

- 0 . 5 ,

- 1 . 0

- 1 . 5

-2 .0

- 2 . 5

l R e = 2000

N = 200

T = 7 . 0

- 3 . 0 J , I 0 0 . 5 1.0 1.5 2 . 0

149

x /a

Fig. l l . C o m p a r i s o n o f p re sen t ca l cu l a t i ons wi th those o f M o o r e [10]. + M o o r e , 60 vor t ices wi th co re a m a l g a m a t i o n , At = 2 x 10 -3. - - P r e s e n t , 200 vort ices , At varies.

Siddiqi, as is indicated by the Cost Factor entry and more efficient than those of Krasny. In addition, the present calculations are able to be carried to relatively very long times without the presence of vortex crossover in the spiral. We note that all of these calculations, including ours, were done without any attempt to optimize the parameters involved in the calculation. We also note that inviscid vortex theorems are not satisfied in this study due to our use of Lamb vortices.

C O N C L U S I O N S

The ability to calculate the roll-up of the vortex sheet at large dimensionless times is due to the avoidance of crossover in the spiral. This result arises from a two-step process involving three factors: use of Lamb vortices, use of a variable time step, and an initial redistribution of vortices within the sheet. First, we considered irrotational point vortices to represent the vortex sheet. Even with the variable time step, crossover was found to occur early in the computation. Lamb vortices were then introduced and the variable time-step computational process was repeated with reasonable results, but with a limit involving the ratio of Reynolds number to number of vortices. Analysis of these results lead to the observation that the computational problems occurred when

Table I. Computation speed comparisons

Computation Researcher CPU time Cost factor method and per integration Usable step CT x 2 and N computer step-CT size-dt dt x N

4th O PVM Moore [10] 0.26 2 X 10 -3 2.15 N = 60 CDC 6600

5 / 6 0 PVM Siddiqi [13] 0.4 ~z x l0 3 10.11 N = 13 VAX 11/780 UNIX

5 / 6 0 PVM same 0.64 7t x l0 -3 10.18 4th O PVM same 0.18 ~ x 10 3 2.86 5 / 6 0 PVM Siddiqi [13] 0.085 ~ x 10 3 0.901

CYBER 175 PVM present study 0.897* 1.18 x l0 -2. 2.53

N = 6 0 VAX 11/750 PVM Krasny [14] 4.5** 0.01 4.5

N = 200 VAX 8600

*Time step is variable, these figures represent averages over the entire range of calculations. **Value based on approximate CPU time reported in Krasny [14]. The computer ratings in MFLOPS are as follows: CYBER 175 = 1.8; VAX 8600 = 0.48; VAX 11/780 = 0.14; CDC 6600 = 0.49; VAX 11/750 = 0.12.


vor t i ces in one spiral l o o p were in f luenced t o o s t rong ly by vor t i ces in a d j a c e n t spira l l oops o r by

ad j acen t vor t i ces tha t were t o o c lose wi th in a g iven spira l loop . Th is last diff icul ty was e l im ina t ed

by a r ed i s t r i bu t i on o f vor t i ces to p r o v i d e a f a v o r a b l e ini t ia l c o n c e n t r a t i o n o f vor t i ces in the sheet.

Thus , the c o m p u t a t i o n was ab le to be ca r r i ed w i t h o u t c r o s s o v e r o r o t h e r diff iculty to a

d imens ion l e s s t ime o f 20; we c o n j e c t u r e tha t this c a l c u l a t i o n c o u l d h a v e been c o n t i n u e d indefini te ly .

H o w e v e r , large t ime so lu t i ons i nc lud ing the effects o f v iscos i ty c o u l d p r o d u c e d i f fus ion to the ex ten t tha t the resul ts w o u l d no l o n g e r r ep resen t a v o r t e x sheet.

R E F E R E N C E S

I. L. Rosenhead, Formation of vortices from a surface of discontinuity. Proc. R. Soc. A. 134, 170 (1931). 2. G. D. Birkhoff and J. Fisher, Do vortex sheets roll up? Rc. Circ. Mat. Palermo Ser., 8, 77 (1959). 3. F. R. Hama and E. R. Burke, On the rolling up of a vortex sheet. University of Maryland, TN No. BN-220 (1960). 4. F. L. Westwater, The rolling up of a surface of discontinuity behind an aerofoil of finite span. ARC R & M 1692 0935). 5. H. Takami, Numerical experiment with discrete vortex approximation, with reference to the rolling up of a vortex sheet.

Dept. of Aero. and Astronaut., Stanford University Rep. SUDAER 202 (1964). 6. D.W. Moore, The discrete vortex approximation of a finite vortex sheet. Cal. Inst. Tech., Rept. AFOSR-1804-69 (1971). 7. A. J. Chorin and P. S. Bernard, Descretization of a vortex sheet with an example of roll-up. J. Comput. Phys. 13, 423

(1973). 8. R. R. Clements and D. J. Maull, The respresentation of sheets of vorticity by discrete vortices. Prog. Aerospace Sci.

16, 129 (1975). 9. P. T. Fink and W. K. Soh, Calculation of vortex sheets in unsteady flow and applications in ship hydrodynamics. Tenth

Symp. Naval Hydrodynamics, Cambridge, MA (1974). 10. D. W. Moore, A numerical study of the roll-up of a finite vortex sheet. J. Fluid Mech. 63, 225 (1974). 11. P. T. Fink and W. K. Soh, A new approach to roll-up calculations of vortex sheets. Proc. R. Soc. A. 362, 195 (1978). 12. G. R. Baker, A test of the method of Fink & Soh for following vortex-sheet motion. J. Fluid Mech. 100, 209 (1980). 13. S. Siddiqi, Trailing vortex rollup computations using the point vortex method. AIAA paper no. 87-2479-CP (1987). 14. R. Krasny, Computation of the vortex sheet roll-up in the Treffetz plane. J. Fluid Mech. 184, 123 (1987). 15. C. Greengard, The core spreading vortex method approximates the wrong equation. J. Comp. Phys. 61, 345 (1985). 16. J. J. L. Higdon and C. Pozrikidis, The self-induced motion of vortex sheets. J. Fluid Mech. 150, 203 (1985).

the vortex roll-up problem using lamb vortices for the elliptically loaded wing

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