Quasimolecular modelling of the cavity problem on a vector computer Donald Greenspan

Mathematics Department, University of Texas at Arlington, Arlington, TX 76019, USA (Received May 1987; revised November 1987)

A quasimolecular model is developed for the simulation of a prototype fluid flow problem, the cavity problem. The approach is formulated math- ematically as an n-body problem that incorporates gravity and classical molecular-type forces. Systems of 2576 second-order, ordinary differential equations are generated and solved numerically on a Cray X-MP/24. Ex- amples are described and discussed, as are the fundamental mechanisms for *~ortex development.

Keywords: vortices, particle model, vector computer

Introduction The availability of vector and parallel computers has motivated the development of n-body molecular models in science ~ and technology, 2 the motivation being, in part, that continuum models cannot incorporate the necessary singular potentials. Our purpose here is to apply a molecular type of fluid modeling 3 to the study of a prototype fluid flow problem, the cavity problem, formulated a s f o l l o w s : 4 determine the motion of a fluid that fills a square basin, or cavity, when the upper side, or lid, is in uniform horizontal motion.

Quasimolecular modeling Our fundamental physical assumption is that gross fluid behavior is the result of molecular interaction? Our foundational modeling assumption is that molecular interaction can be approximated by grouping mole- cules into larger units, called quasimolecules or par- ticles, and then applying suitably adjusted molecular dynamical formulas to the resulting n-body system. Of course, the process of lumping molecules into particles was known to both Boussinesq and Prandtl. 6'7

Quasimolecular modeling, also called particle modeling, utilizes coupled, second-order systems of nonlinear differential equations

Fi=mi~i i = 1,2 . . . . . n (1)

in which the F; are chosen in the following special way. Any fluid consists of a large, but finite, number of molecules. These molecules are acted upon by two types of forces: (i) long-range forces (e.g., gravity), and (ii) short-range, or local, forces (e.g., classical mo- lecular-type interactions). For short-range forces, each molecule P is acted upon only by its immediate neigh- bors, and, typically, these forces have magnitude F given by

G H F = - - - + - - G_>0 H->0 q>p>7 (2)

r p r q

where r is the distance from P to a neighboring molecule. To approximate actual molecular interaction by

quasimolecular interaction, we would, in general, al- low each Fi in (I) to consist of two parts, a long-range component and a local component, and to compensate for the grouping of molecules into quasimolecules, we adjust the parameters in local force formula (2) appro- priately. System (1) would then be solved numerically from given initial data by any of the currently popular numerical techniques.

lllustrative example Let us begin by discussing in complete detail an illus- trative example. For this purpose, consider a triangular mosaic of 2576 points arranged within and on the square

© 1988 Bunerworth Publishers Appl. Math. Modelling, 1988, Vol. 12, June 305

Quasimolecular model l ing of the cavity problem: D. Greenspan

of the square, it was reflected back symmetrically across that side with a velocity damping factor of 0.9. How- ever, when a particle had moved across the side AB of the square in Figure !, that is, whenever the particle had collided with the lid of the square, the particle was reflected as indicated and then the constant V was added to its x-component of velocity. For now let V = -1 0 .0 .

In the usual notation t~ = k At, k = 0, I, 2 . . . . . the system was solved numerically to t2s(,mo. The re- sults are described as follows.

Figure 2 shows the velocity field after 6000 time steps, that is, at t60oo. The figure is relatively mean- ingless because the field is dominated by Brownian- type motions, which reflect the strong molecular-type interactions. A smoothing or filtering process is there- fore required to clarify the gross fluid motions, and this is implemented simply as follows.

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Figure I T h e s q u a r e c a v i t y r e g i o n

ABCD shown in Figure 1. The side of the square is 12.5 units. The coordinates of the points are given by (x~, y~), where

x~ = - 6 . 2 5 y~ =6 .25 x s 2 = - 6 . 1 2 5 y s 2 = 6 . 0

X i + I = X i + 0 . 2 5 Y i + ~ = 6.25 i = 1,2 . . . . . 50

xi+ , = xi + 0.25 Yi+! = 6.0 i = 52, 53 . . . . . 100

X i = X i - i 0 i Y i = Y i - Io~ - - 0 . 5

i = 102,103 . . . . . 2576

This point set is symmetrical about both axes and the origin. The 51 rows contain, alternately, 51 and 50 points. In each row the distance between two adjacent points is 0.25. The distance between two consecutive rows is also 0.25.

Each point will represent a particle, and the mass of each particle is taken to be unity. The particle with coordinates (x,-, y,-) is denoted by &. Thus, the sub- scripts of the Pi increase from left to right on any row, and the numbering begins on the top row and proceeds from any row to the next lower row.

Each particle is now assigned a small, randomly generated velocity vector whose speed is less than 0.002, so all initial data are now available.

Next , fix the gravity constant by g = -980 .0 and the local force parameters by G = 0, H = 100, p = 3, and q = 5, which allow the particles to move rel- atively freely, as is desirable in a fluid. Assume that gravity acts uniformly on all particles, but that local force interactions are restricted to pairs of particles whose distance of separation is less than 0.35.

The resulting system of 2576 second-order equations (1) was then solved numerically by the leapfrog formulas s with At = O.O001. Whenever a particle crossed a side

Defini t ion

For N a positive integer, let particle P~ be at (x;.~, y,.D at time t~ and at (X;.~-N, Y~.~- N) at time t~-N. Then Pi's average velocity VmN at time t~ is defined by

= ( X i , k ~ X i . k - N Y i , k - - Y i . k - N ~

Vi'k'N ~ N At ' N At }

In this paper we choose N = 1500, which was de- cided upon after several comparison runs with other values of N. In the remainder of the discussion, all velocity fields are average velocity fields.

Figures 3-11 show the motion of the fluid by dis- playing the velocity fields at times in the range ttsoo- t,_ooooo. The development of a primary vortex is clear, as is the motion of its core from the upper right to a more central location. These results are in agreement with experimental r e su l t s ) Most important, however, is the observation that the model reveals the mecha-

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Figure 2 I n s t a n t a n e o u s v e l o c i t y f i e l d a t t~ooo

306 Appl. Math. Modell ing, 1988, Vol. 12, June

Quasimolecular modelling of the cavity problem: D. Greenspan

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nisms of vortex development. Indeed, Figures 3-5 im- ply that there is compression in the upper left corner and partial vacuum in the upper right corner• (This will be established more clearly later in Figure •6.) The compression yields large repulsive forces, which result in motion downward• The repulsive forces between particles just below the partial vacuum result in upward motion to fill the void. Thus, rotational motion begins. The continued driving force of the lid results in an increase in the size of the vortex. However, when the

vortex has reached the size shown in Figure 8 at t~sooo, it changes relatively slowly thereafter. This is con- sistent with the usual assumption of a steady state, 9 which results when the energy being added to the sys- tem is dissipated at the same rate.

The figures reveal also other interesting aspects of the flow. Figure 6 accentuates a downward flow ad- jacent to the left wall. Figures 4, 7, and 9 show clearly the development of "arms" at the bottom of the pri- mary vortex that penetrate the relatively quiescent fluid

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Figure 4 A v e r a g e v e l o c i t y f i e l d a t tzooo

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Figure 6 A v e r a g e v e l o c i t y f i e l d a t tsooo

Appl. Math. Modelling, 1988, Vol. 12, June 31)7

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" ~ . . - ' ~ - ' T ~ ~ : " - ' - C - ' - , , , , ~.//..." ~ , , " / , . " / i / ~ , / ' ~ , , • .:.'~'~v~'-'~-"~_--"~--- -" ." " . ' . . . . . . . . . ~ . . / . / , , ~.~.,.~ . . . . . . . . . . . . . . / ,i..,//,.,,,,,

. . ~ ' ~ i ~ - , '¢'. - " > ~ . - " , / , " * . ' - ' 5 " - - ~ - ' . . ~ ~ " • . [ , f . " , "'/, % " ~ / , ' , ' / d " , , ~ ¢ , . " / " , . . . . . . _ - . . -..4.. < , . ' .,:.- ' ." . -~ / ' J / , , - , • ". ,.z& )> r / , ; 5 4 ( " - , " " " - - - . - ' _ - :W-~ ' - . - ' ' - . - c . . - , , , . , , , , . . ,'

: / , l l / | { t (,/. ~..\~ "~ i \C-- - - - - - - - . -~_.~ - - . / ' - ' : . ' . . ' I / , " ", ; ' ; , : ' - : •/, I , J . . . . . ,, "<.,", .". ~ x x ' - - ~ - ~ - " - = = - - - , . . % - - " -~ - ." , , - , "• " : - '

• / ) b Y , t , ' h '4 ~ ~ C % ' E _ . = - - - - - . ' - . ; ' d - C - - / - - : , : ' , I l l , ~ \ 1 - - ~ " - . ~ . - ~ _ ~ _ _ . , , . . . . , -

o [ , ' ~ , . . ' . , , , . - , - - . - - , : . ~ - - _ _ ~ , - - - $ . . . . . . . . . ' . : . . ' : . : , ' , ' , . . . . . . . ". ' . ".- " ' . - - . - " • - --.--- - - . - - 7 " [ . • . ' Z . ' . ' " . : . "" :

:::.::..........:•....`.•.....::::.........................................:...:

i l 7): !:(!!iii!;!:::!i:iiiii:i!i i i i!:ii :i:i:::ii::ii!!!:ii !:i i!i(!!:iiii!i! i!iiii!i :ii:i i i : ' " , - , , • . ' . ' . " , , - . - , . . . • , . , , . . . - , " ' " • • " • " . ' , ' . ' . ' , ' . " " . ' . " " ' . ' • ' . ' . " . ' • '. : .' . " .

F i g u r e 7 A v e r a g e v e l o c i t y f i e l d a t hosoo

D . G r e e n s p a n

,', ~ - ' = = - - ~ ' ~ ~ ~. ~ , I I l l l . j , I t i l l

~ . . . , ~ ' ~ = - . . . . _ ' z - ~ ' m ," ~ ' / ' ; " ~ ~ / . ' ~ . . t / / / i ' ,

~ / t t ? - - - I I I i I - ~ i l I l l / i I l

, ' , I : : :

: , ' , , " ,~' ,Y 6.~ XL'i'\.f,}",';'~-- . . . . . . - - ; s ;,,.:.$4-~,'; ' , , "4 - : - - . ' : " ' , ' , ", 'l\~k ~ I t l t ~ \ . , .~ . \ ' . , , ' x , -7 . 'z : -c -~ : l - / 1 , , , . , " , , ", ",. : ' " . ' . " • , ~ ' 1 , , ~ , , , ~ . , ' . , \ : , ' ¢ . . \ , . " ~ - ~ " ~ y • , , # , , , . . . " " . . . ' , , 1 ' ~ ~ I X . x ,,~X \ \ , . , \ ~ - ~ . ~ ~ . s . . . . . . ' , . . ',. , ' J t , g ~ , t \ , ~ , : , . - : . . ' . " . . - ' - _ - " : - ' - - , ' , ' , " . "- ' . • . • • • • , , " I i, ~ ~ , , ' , ' 2 ~ " c - . " \ " > £ _ - ~ ' - - ~ . - - ' - , " , ' , . , " " . ' • ' • " • " • ' . • : ' . I • "1 ~ ~.1 \ X x t % . , • ~-. ~_- - ~ - - - . ~ . . ~ - I ~ ~ , " • . - . " . " . ' . ' . .

. . , * _ , ~ , l ~ t . ~ . ~ x . ' , - - : ' - ~ - - . - _ - , ' . . , , , . . . . . . . . . . . . . • " " " " ' t ~ ~ x X l ' ~ , " x . ' - : ' - . - - - ' - - - . - - ~ , " . • " ' . ' , ' • " . "

• . , . . . ,~ . , , , , . , . \ % . - , - - _ . . - _ : _ - - - . - . . . . . . . . . . . . . - . . : . : : ,, . : . , ~,,. ? . : . - ' . , \ : : _ - z : _ - - . - . ' : z - : . "." .-: ..." . . : : : . " . . : . . : • : . ' , , , ' , ' , ' . ' , ' - - ' - - . . ' . 7 . ' . . - ' , ' " " . . . . . . . ' . ' . ' . ' . ' . . ' . : " - ' . "

: : . ' . • . - . . . " " . . . . ' . ' , ". " . ' . . . . , . • ' . ' . ' . ' . ' . , . ' . ' , ' . . , ' . ' , . . . . • . . , ' , " , . . ' . ' . ". " . . , . . ' , - . • . • : :

: . ' • ' . " • . . • " , . • , . . ' • ' . ' , • • ' • ' • • • . • • • • • • • " ' , • . ' • . . . ' . ' . . ' ." • . . . . • . ' . ' . • • ' • ' " . . . . . - . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . , . . . . . . . . . . .

F i g u r e 9 Average velocity field at t 3 5 o o 0

area below the vortex. Figure 7 reveals the beginning of the development of a dead zone near the right wail.

Since there is interest in the motions in the upper corners, we have shown in Figures 12-15 at every 500 time steps from to to t~ooo the dispersive mixing of var- ious upper-left-corner particles. Figure 12 shows the motion of PI-P4, Figure 13 shows Ps2-P.~s, Figure I4 shows Pio2-P~o~, and Figure 15 shows PI53--PL~. Fig- ures 12-15 reveal that P~, Ps2, P~02, and P m exhibit small oscillations during this initial period, but remain

relatively stationary. Figm'e 12 reveals an erratic mo- tion for P=, and Figure 13 reveals large motions toward the interior for P~.~-P.~.~. Figure 14 shows motion along the wall for Pto.~. Figure 15 reveals, initially, strong backward motions for Pts~ and PL~.~.

Secondary vortices were not identified easily, pri- marily because a moving vorl~ex is not readily recog- nizable and because small motions may require N to be larger than 1500 for proper display. However, two examples of secondary vortex development are given

~ _ _ ~ " = - 2 ~ - - - - , ~ - . \ ~ ' X x . . . . , I I x , I I ' l / ~ , t t / ,

_-.---.-,,. , ., , , , ' , ; I / , . s / , , , . " 1 ~ 1 1 / I ~ - - . - - ' l , I l l I I I l l l I I [ / + . - - - - I ~Z l l " 'J< ' , / '_ , ,~,"a '7~-- ' - , ;~ ' - , • , ' . , ' , . , • 2 ' " ~ / I / I / 7 ~ ; ~ / , / ~ ' , , " " t ' r /~. .-~. ~ . / / - ~ / " f f~ - / " . " • • . " . ~ ~ - " ' , ~ , • I J " , / I 1 ~ / Y I " ' ' "

" ? i ; / . / . " F / ~ / . , r . ~_ , . , . ~ / ~ . . . , . / . z . , - I i / . ~ / ~ , , , .

• W ~ i , b ~ l x ~, ; /6 , , ' , -~ , ' , - ' - - . .= . -~ ,.7_'70"2; ~< " , ' , ' ' : ' , ' .' : • ,1,~,'~ I , V . k ~ . ~ , L \ ' \ , , t ' , - " , - - - - - ' - / , . ' , # - , - ' ; . ' , . . • . : - i i . o y l ,~D~' L~ .x - t X ~,.:.~.~. - - ~ - - - : ~ - S . ' - " , . I , - . ' - ; . , . . . . , : . : t'l ' , / ' , . I ~',',7 ~ ¢.~ .',<~--;:-~---~'~S-,//< : ; . , ' - . , , ' , ' , . ' . - . :

• ' I ~ , C t " ~ - " X \ , L " \ ' : " ' X . . - ~ x " - - ~ ' ~ " - - - ~ - ~ / " ~ " " " - " " • " . " , , ~ ~ , \ , , ~ - \ . , \ , - - ' - . , - - - _ - , , ' , . . ~ . . . ' . ' . . . . . .

: . . - . ; , , - : . . : . : , , - - . - . ' . ' . . ' . ; . - . , . - . . . . . . . . . . . . . . . . : . . : . : . . : . .

!:i i::::::::::::::::::::::::::::::::::::::::;:;: ::>:: >:: ::::::::;! :i i i • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . , . . . . . • . , . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . , . • . . . . . . . . . . . . . . . . .

F i g u r e 8 Average velocity field at t~sooo

- - _.~--~.~-. . -~----~-" - -~ ' - - "~ " ~ ' ~ " ~ - ~ . ~ ' ~ ~ x , ~ \ I I I I . L f / / l . l . / ~ - - ~ - ~ - - - - ' - --------'~---~-~ -~ - - ' - - ' -~__~-~- ' - \ ~ ~ I t , . I. / P, I r / 7 ,,

~--'-~_-2""~.- c---,"-...---~-~--'-- -:-" ._~ - C, ,',' , I/.:.~/7..//i,/. l~ t t :

f , C l , ) I ~z,,"~, '7..- -, > " , " • " -" - . . . . " - "." ~ ~,1 ? f / d l ; I, ~' ' • • , . , , / i 1 . / ~ , t i - - ~ _ ~ . - ~ ' . ~ " . . . s . 1 1 1 , / ' / j i i i , , . • , ' , t : l l , v ; / . / f x / > / . ,., ,~ . . . . : - - _ - _ - - - : - : - C ; ~ / , :~, $ , ~ , - " , . . . : ~ yl, h V l ' , / r , ~ } u d Y, ,~" ~ , ' - - - - ~ z ' . . - _ - . ' : ; - J ~ ; , ' < - , , ' , : , ~ :

. , . / ~,,i ~ \ _ - - - . . . . _-~, , . • , , . . . . . . . • ,, ....... ., .,,, ....

. , - , AA' . . 'X\ ' , '1, , ~ , ' ~ . ' - , . : _ - _ ~ _ . - - - - = - . : , , / . . . . . . . . . . . • , i t "" '. - d~X.'~'X~ ~\'\'~',~-"-'~ - ' - " = - - - . v - , , , , . . . . . . . . . . : , : . . . ' - ,",, ".~ X,",', ,~x4\\ ,~ ~ \ ' . ~ - ' - ' , % ~ _ - _ , z - $ - - , ' , ' , . ' , . : . . . . . :

, . , " , . " , ' , , , . . . X , , ' , ; , \ N , . X . - " , . - " ~ - - . - - " " , , , " - • • . . , - , ,~ , , , . ~ - k ~ . , . \ , ' , - . $ _ . _ - - - - _ - , . , . . . . ..

• , ; , ; • . ' . . . . : ' . , , . . - . : - . ~ - - - _ - . . . . - . . . , . - : : , . . . . - . - . . . . : . . . . .

: : ",...'... : . : . . . . . . . . . . , . - . : . - . - : : . . . . . . . . . . : . , - : . - : , . . . . . . . . : : . : . : : . : : : : : :

, . , , . • . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . , , . • . . . . . .

F i g u r e 10 Average velocity field at tlooooo

3 0 8 A p p l . M a t h . M o d e l l i n g , 1 9 8 8 , V o l . 1 2 , J u n e

Quasimolecular modelling of the cavity prob/em: D. Greenspan

- '=~-: . .Z:~_-~-~- '~-~-:~__~ ~ , ~ " ~ ~ , ,~ l ~ l ~ ~-.--,---~-'=-~-'~='----Z-_'--~-~'"~'-'~_ "~' ,,," ~ r / ' , ~ - ~ / / t AI.t I '~

" / ~ 1 ~ \ ; ~ " - . . e ' ~ ~ - ~ . - e " . . * . " " 1 ; / / I / / / ' 1 " / - - t ' " " , l l f ~ p l t ' [ , ¢ ~ ' / " t / ~ I ~ * , , * I . ~ , / 1 / , 7 ~ 1 1 1 / 1 I 1 , 1 ' * " • t * g l / , / , / t ¢ l ~ ' . ' - ~ r _ I t . , . ' ' . I . / . ~ / l ' ~ / ; ~ I t , I / / ' t t

• ' , ' l ( ( ' d ; / ~ ) , t ) f ' ~ " , ' . . . . " ' " . ' - > " , ; < . 5 ¢ ~ ' , , , " " : , t/th l , ' x , ~ ; i / "A¢; , ; : - ' - ' ; ' - ' - , - - - Z . ' . ' ; ~ 7 - ' ", ' . : ' :

". ' , l I ' , ~ h ~ / X \ ' , \ \ " , \ - - - - - - - ~ / / - C ' - " , ' , , ; ; " . ' : . " : ', I.t~ ~ ~ ,~ ' \ / W A . ' , d , \ , ,%,z-- . . - . .___.- , ; ; , , % ; ; , , ,, . ' : . , • .. : - ' ,~ ~ .~,~,~,~I , '~, '~:~\ ' , \ ' , -Z-L-:L~d.---"-~/., ; , ' , ' " • : : " '

. . . . . , , , ~ , ~ , , ~ , . . - . _ _ _ - _ - _ - . . : , . , : . . . . . . . . . . . . : . . . "

• : : , , - . . . . : ,_--',~,-.~.:. , . - . ' . : . : . - - . . . . - : , : - ' . , : . - , . . : . : . : . : - : : . . . . , . , . . - . - . , , ' ; : , , " ; . , - - , - ; - : . : , . . ' . , . . . . . . , - . : . : . . . . : . . . . . . : . , . : - - . " , - . , , . , , . , . . : . . , . ; . . . . . . . . . . : . . . . : . - . . . . . . , . . . . . : . - : : , . . , . : . ' , , , . . . . , . ' : , . ' . . ; : : . : : : : . : . : . . ' . : . : . : . .

. . . . . . . . . . . . . . . . . . • . . . • . . . - • . . . . . . - . . . . . . . . . . .

: : : : . . . . : . ' : .-. ' : . . . ' . ' . ' . - . . . . . . . . . . . . . " - . . . . . . . - . . . . . . . . . . • : . ' . . ' . ' . " .... - . - . . . ' . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . - . . . . . . . . .

• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 1 Average velocity field at t2ooooo

in Figures 16-17: one in the lower left corner at q5oooo and one in the lower right corner at t2ooooo. For display purposes, the velocity vectors required magnification. If CTV represents the magnification factor, then CTV=25 was used for Figure 16, and CTV= 15 was used for Figure 17. An interesting additional benefit of these two figures is that they reveal quite clearly the compression in the upper left corner and the partial vacuum in the upper right.

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6

Figure 13 Initial mot ion of P62-Ps5

Additional examples

The number and variety of examples that can be ex- plored are unlimited, since one need only vary g, p, q, G, H, V, N, the number of particles, the masses, the distance of local interaction, and the wall reflection protocol. We therefore describe only three of these studies in which V was varied.

Figure 12 Initial mot ion of P1-P4 Figure 14 Initial mot ion of P~oz-Plos

Appl. Math. Modelling, 1988, Vol. 12, June 309

Ouasimolecular modelling of the cavity problem: D. Greenspan

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l!.]iiiiiiiiilllllllli!i!||| ']illllllllllllll/iiiiiiil; ""2 EEfiEEfiEfifiEfififi

i i ii ! ili i

i i !!i • . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . .

Figure 17 L o w e r r i g h t c o r n e r s e c o n d a r y v e r t e x

Figure 15 I n i t i a l m o t i o n o f P~s3-P~se

For V = - 2 . 5 , no vor t ex ever develops. Instead, there is an undula t ion through the fluid, which resem- bles a compression wave. For V = - 7 and V = - 13, the resul ts at tL~ooo are shown in Figures 18 and 19, respectively. Relative to the results for V = - 10 shown in Figure II, Figure 18 shows a smal ler pr imary vor t ex , and Figure 19 shows a larger one , which are not unexpected.

Comparisons

The cavity problem has been studied extensively from the continuum point of view (see, e.g., Refs. 8, 10-18 and the numerous references contained therein)• Nu- merically, both finite difference and finite element methods have been applied. Usually, the equations considered have been the two-dimensional, steady-state, Navier-Stokes equations in stream and vorticity vari- ables. Good results have been obtained. A shortcoming is that, invariably, incompressibility is assumed, whereas compressibility is required on the molecular level for identifying the mechanism of vortex generation.

( i i i •(•••• • • • :•••• • • : :

. . . , , . ( . ' . . . ' . . . , . • . "

w •

Figure 16 L o w e r l e f t c o r n e r s e c o n d a r y v e r t e x

T t t t ~ - " - " , . / ~ i t t . - . f . , . . . . . . . - . l * I / t l t i ,

x / , x t . , . - - • , - . " , c ~ I I t ~ • . i t _ , _ - , . , _ _ . ' - - J / a s t . .

. , ' I ,~, ~.~\ ,~ , - , , . . . - . - . - - . , - . . \ - - . _ _ _ - _ . , - ~ - , , , , . . .

: . , I ~ t ~ , . ~, x , , _-. - - , , , - , , . , - _ - . - - , . , . , , . . . ' . . . - . . _ ~ \ , , , , - , - . - , - - . . . . . ' _ - - _ . - . . _ , - - . - . . . - .

] ! i i ! i :.:.:...-:::.]...:::i:: ::(::: ::::: ::: :: :::::::::::::::::::::::::::::::::::: i i : i : ! 7 - ) ) ) ; i : : i i :2 : :>> i ) : : } ) ; ) )21 ; : 2;:i:!-):! . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . * .

: , - . - . . . . . . . . . . - . - . - . . . - . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . ; . : : : : : : " " : ". " . ' . ' . ' . . ' . . ' . . ' . ' . ' . ' . ' . ' . ' . ' . ' . ' - ' . ' . ' . ' . ' - ' . ' . ' . ' . ' . . ' . ' - ' . ' . ' . ' . ' . ' . " • : - 2 - : :..- . . - . • . - . . •• : . - . - •.L.• • . . . . . - . . . . . . . • . • . . . . . • . U. . . ' . ' . " " : . ' . . . . : . - . - - : . : - : . . . . . . . . . . . . . . . . - . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . - • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •

Figure 18 V e l o c i t y f i e l d f o r V = - 7 a t tlsooo

318 Appl. Math. Modelling, 1988, Vol. 12, June

Ouasimo/ecular modelling of the cavity problem: D. Greenspan

~ ' ~ - ~ ' ~ - ~ ' ",\~.-~ ~,<. 1 , 'o , I" i l l ' / , / , t ,~1 r / t , ,

~ . ~ _ _ ~ , ~ - ~ ~ : - - ~ - , , / . ~?~ / . , , , ; , , , ~(/J, , . , . . ~ . , - ~ " L , " f . . G ~ ' ~ - ~ , , C - ~ - ~ 7 . . . . ,~.~,, t~ t . l l . / I I l ' t =,~' ' ,

:, ;71~,i,~ ~ I~ I (~ ,~ ,~ )S \~< , '~ : ").: , . :S;;,~, " , :'.,: : , ' , " : : . •:

• ...,, , . , , , . ' : ~ , ~ \ ~x,~, .x,¢ >--<_.,-,:, . ~ . ~ , " , _ , ; , : : - - . . . . - . . . . . . : , , , ,~ ~.~ ~ ~,.,~ \ ~ \ \ ~ . \ ~ \ _ .... , i ; . - . , , . - 7 - . - - ' - " , ' • " • , " ,~ ~ ' ~ . ~ ' \ ' ~ , ~ , ~ " ~ . ~ - - - ~ ~ 2 " ' - : - ' " . ". -- : . . . . • " . •

• . . ~, ~ , , ~ , ~ , , . . . . • . . . . . . . . . . , , , , , , , , , . . . . . . . . . . . . . . . . .

• :- , ' ," , ' ; ' , \ ' A ~ A , : , \ ~ - . ~ - - - . , > . ' , , ; ; - : - - " • - : " : - : - . - : - - . . , . , , , ~ , . , ~ . ~ , . . . . - . , ~ - , . , : - . x , , . . . . , . / , - _ . - . , . , . . . . . . . . :, ...-, .... ,': : . . ,, ,~:,, ~'"- ._~: ' .- .- : : . . . . . ; : : : : - . ' . . . , . : : : : : - . : . . . . . . . _ . , . , , . ' . ' . " . " , _ , - : " _ . . . . , 1 . . - . , . . , , , . . . . . . . . . . .

: ' ; ; - ' , . ' , " l , . , " , - ' . i . . ' . ' , ' . ' - ' . " ' . ' . - - ' - ; : . " ' . " . ' . ' ' , ' " . . ' , ~ . . . " . . . . . . - : . , . . . - , - - . ' , - " . . . • . . . . . " , . . " . . . . . . . . . . ~ . . ° ' - • - ~ o . . - , • . , ~ . . - - - ° . , . . • . . . . . . . . . . . . . . . . • -

. . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F i g u r e 19 Veloc i ty f ie ld for V = - 1 3 at tlsooo

The method for choosing reasonable parameters, such as those in the third section, has been described in detail elsewhere. 3 However, a shortcoming of the particle approach is that, in general, we do not as yet know how to choose the parameters appropriately when a particular fluid, such as water at 25°C, is specified a priori. Thus, the development in the previous three sections has been qualitative. The only quantitative particle modeling done thus far has been for stress wave propagation in thin aluminum bars.19

Nevertheless, the availability of two distinctly dif- ferent modeling techniques is of value from the sci- entific point of view, since it provides a check and balance system for modeling natural phenomena.

Finally, we report on a computational problem in using the Cray X-MP/24. This computer has a 64-bit word length, but does not have double precision built into the hardware. Thus, though a 1500-time-step cal- culation required 11 rain in single precision, it required 8.6 h in double precision. Moreover, though the gross fluid motions were qualitatively the same, whether one used single or double precision, position and velocity calculations for individual particles often varied sig- nificantly. One would expect that double-precision cal- culations would be more desirable because of the sin- gular behavior of intermolecular forces, but the time required was prohibitive. Thus, as a compromise, the results in Figures 3-11 were calculated in double pre-

cision, whereas those through 125oooo were calculated in single precision. Comparison results using a Cyber and/or other numerical techniques for ordinary differ- ential systems would be of interest.

References 1 Karplus, M. and McCammon, J. A. The dynamics of proteins•

Sci. Amer• 1986, 254, 42 2 Dienes, G. J. and Paskin, A• Computer modeling of cracks. In

Atomistics of Fracture. Plenum, New York, 1983, p• 71 3 Greenspan, D. Computer studies in particle modeling of fluid

phenomen. Math. Modelling, 1985, 6, 273 4 Anderson, D. A., Tannehill, J. C., and Pletcher, R. H. Com-

putational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York, 1984

5 Feynman, R. P., Leighton, R. B., and Sands, M. The Feynman Lectures on Physics. Addison-Wesley, Reading, 1963

6 Prandtl, L. Uber die ausgebildet Turbulenz. ZAMM, 1925, 5, 136

7 Schlichting, H. Boundary Layer Theory. McGraw-Hill, New York, 1960

8 Pan, F. and Acrivos, A. Steady flows in rectangular cavities. J. Fluid Mech. 1967, 28, 643

9 Vemuri, V. and Karplus, W. J. Digital Computer Treatment of Partial Differential Equations. Prentice-Hall, Englewood Cliffs, 1981

10 Bulgarelli, U., Casulli, V., and Greenspan, D. Pressure Meth- ods for the Numerical Solution of Free Surface Fluid Flows• Pineridge, Swansea, 1984

I I Daikovskii, A. G., Polezaev, V. I., and Fedoseev, A. I. Cal- culation of bounding conditions for non-steady-state Navier- Stokes equations in stream function, vorticity variables (in Rus- sian). Cisl. Metody Meh. Splosn. Sredy 1979, 10, 49

12 De Vahl Davis, G., and Mallinson, G. D. An evaluation of upwind and central difference approximations by a study of recirculating flow. Computers and Fluids 1976, 4, 29

13 Ghia, U., Ghia, K. N., and Shin, C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a

multigrid method• JCP 1982, 48, 387 14 Nallasamy, M. and Prasad, K. K. On cavity flow at high Rey-

nolds number. J. Fluid Mech. 1977, 79, 391 15 Runchal, A. K. and Wolfshtein, M. Numerical integration pro-

cedure for the steady state Navier-Stokes equations. J. Mech. Eng. Sci. 1969, 11,445

16 Shestakov, A. I. Numerical solution for slightly viscous flow in a square cavity• UCRL-79289, Lawrence Livermore Labs., Livermore, Cal., 1977

17 Takemitsu, N. On a finite-difference approximation for the steady-state Navier-Stokes equations• J. Comp. Phys. 1980, 36, 236

18 Tuann, S.-Y. and Olson, M. D. Studies of rectangular cavity flow with Reynolds number by a finite element method. TR #19, Dept. Civil Eng., Univ. British Columbia, Vancouver, B.C., 1977

19 Reeves, W. R. and Greenspan, D. An analysis of stress wave propagation in slender bars using a discrete particle approach. Appl. Math. Modelling 1982, 6, 185

A p p l . M a t h . M o d e l l i n g , 1 9 8 8 , V o l . 1 2 , J u n e 3 1 1