A study of compression waves produced in water by thsnon-uniform expansion of a cylindrical piston of non-zero initial radius is made by the artificial viscosity method of von Neumann &amp; Richtmyer. It is found that the damping dect introduced by the cylindrical geometry is much less pronounced than that of the spherical geometry

Pandey, B. C.

Prasad, Phoolan

English

Defence Science Journal

CYLINDRICAL PISTON PROBLEM IN WATER I. AvtiJicial Viscosity Method Indian Institute of Science, Bangalore (Received 1 November, 1968 ;revised 15 March, 1969) A study of compression waves produced in water by thsnon-uniform expansion of a cylindrical piston of non-zero initial radius is made by the artificial viscosity method of von Neumann & Richtmyer. It is found that the damping d e c t  introduced by the cylindrical geometry ie much less pronounced than that of the spherical geometry. In the present paper, we have studied compression waves produced in water by the expansion of a cylindrical piston. If the piston expands with uniform speed and its initial radius is zero, then the motion is self-similar and the eolution of this problem can be wily found out by the numerical integration of ordinary differential equations following Taylor'. Lighthill2 has discussed this problem for small mach number of the piston and has deduced that the strength of the shock, which forms the wave front, is of the order of the fourth power of the piston mach number. When the expansion of the piston is not usifor.3 or the initial radius of the piston is non-zero, the fluid ~ o t i 6 n  is not self-si.r.ilar and in this case we can solve the prob1e.n only by the numerical integration of partial differential equations. Sinoe we wish to take non-uniform expansion speed and POD-zero iritial radius of the piston, we have utilised the well-known artificial visoosity method of von Neumann & Richtmye9. It has been shown in reference (5 referred to a,s paper 1 in the sequel), that artificial viscosity term of von Newnann & Richtmye9 can be used even if the medium is water though the thickness of the transition region depends on shock strength, In paper 1, the correspondjng spherical piston problem hm been discussed i~ detail. E Q U A T I O N  O F  N O T I O N  We have used here the ~otations of von Neumarn & ~ i i h t . _ r ~ e r ?  with the differenoe that X And a: represent Euleriar and Lagrangian distmoes respectively from the axis of the cylbder. We assume the path of the piston to be given by the hypabola where rn, represents the final mymptotic speed of the piston, l / d j m  is the time required by the piston to attain half of the final asymptotic speed and X,' is the initial radius of the piston. The equations governing the flow with cylindrical symmetry in Lagrangian coorrli? nates are \ 2 DEF.~CI. J., VQL,~O, $ A ~ A B Y  1970 sv I(:,) rb +;:,- - -   at Po aa with the equation of state4 A' I p = - - B  vv , - Y = 7 for water and with the arti6cial Yisassi* term aU q = - 2V (6) We noh-dimensiollslise the set of equations (2) to (6) with the help of the m&sGurbed state values po, p,, a. of pressure, density, sowd speed and a ohwaoteri~tic length equal to the distance travelled by the souad wave in unit tirne and write the carreapding aysten ddifferenee equations. - n + l  A' - R -I- 4 = - B  (7;:: jP From (1) we obtab no~-dime&ionsl velocity of the position 4 PAI~DEY &~ e c w a  &a~t : ~yliu$ri~)ei Piston problem 3 r P c -- m l m t  u p  = 3 f (12) (l+rt.bP) The isitial aad boudwy conditions me for la> 1 In (7) to (14), the bar used for lion-dimensiolml variables has been dropped fronl the flow variables, but the non-dimensional parameters carry the bars. D I S C U S S I O N  A N D  R E M U L T S  We have carried out the computations with the following values of the constants asd mesh size : - - c = 2, Y = 7, B = 3000, A' = 3001, AX = 0.00001. - - nt Total time up to Yistorl Ceee ti,' mi rn - which ~esult Ax is obtained - -- ------- -- Cylindrical Piston I. 0.0001 0.28601 loo00 0.4 .0000 2. 0.0001 0.97212 30000 0.1 9 9 -00028 *Spherical PistonG 3. 0.0001 0.97212 -30000 0.15 .00011 I,  4. 0.001 0.97212 15000 0.15 -0002 *These results have been taken from paper 1 for comparison (see reference.6). \ Figs. 1, 2 and 3 show the distribution of velocity, pressure, specific volr~~ne m ~ d  EuTerhl distance versus Lagrangia~ distrance a t  different cycles of time. In Fig. 1, we observe that the ve1wity a t  a fixed t h e  is greatest at  the piston aud it decreases as we move towamis the shook froat and ultimately it falls down to its zero value ahead of the shock. The velocity at  the piston incmes  with increasing n~umber of time cycles and it tends to its mymptotic value% ultimately. It is worthy of notice that only ia, w e  1 velocity-Lagran- gian distance curves (for a fixed time) are convex with xspeot to z axis after a certain time. This feature is nowhere present in the spherical owe discussed in wlier paperd We remark -3 Dm. SUI. Jet V?& 20, JAFTUAR~ 1950 L here that after a verylong time, when the ahmk % b&t;raveIled through a distance large cornpar- ed to the initial radius :the flow tends to be- come self-similar'. The shock strength in - C W  I case 1 is  very small cornparecl ta other cases --- .- C M  ',-..- - 111 due to low piston velocity XI. - - Iu Fig. 2, we observe that the pressure at  a fixed time is grestest at  the piston and it gradually decreases as we move towards the shock front and ultiwately it fdls dom to its value 1 of the undisturbed state. The choice of our scale f-m pressure gives an illusion that the pressure o w e s  end on p = 0; in fact they end on the line p = 1. Further we find that the pressure at the piston increases up to a certain ti:ne and thereafter it shows a decrease with time. It  is also evident fro? Fig. 2 that pressurn gradient throughout the wave region decreases with the increase of time and in case 1, the pressure curve is almost a straight line at  1% = 150. Fig. I.-Velocity versus Lagrangian distance for various cycles of time. 10- / I- 0 9 0  I !' ,so I f \ \ I I on- , , --c - --- ---------, J ------ =--ex 0 7&c--- - -  -   1 ----- c*SE \ C - C A Y  - f---- \, ---- - .- CAY IV ---pi----- - ----, 't \ ----- - 2 y . s - i  Fig. 2-(a) Specifio Volume versus Lagrangian distance at various cycles 04 time. f b )  Pressure versus Lagrangian distance at various cycles of time. Fig. L E u I e r i a ~ ~  distance oersus Lagrangian distance at various oycles of t h o .  Cases 2 and 3 blring out the difference in the flow pattern due to cylisdricd and: -- - spherical geometry sisoe x,, m,, m are same in the both cases. Cog-paring the curves of case 3 at the cycle 40 with that a t  cyle 60 of case 2 (cyli~&ical piston), we i%d that the velooity distribution given by the spherical motion behind the shock a t  each point except a t  the piston is less than that given by the cylindrical notion and sivilarly the pressure in case 3 is everywhere snall compared to that in case 2 (even a t  the piston), though both pistons have started with sa,ne initial radii and have moved for the same time with same acceleration. This brings out the faot that the damping effect of sphericity is stronge~ than that of the cylindrical geometry. The upper part of Fig. 2 shows the distribution of the specific volurrie a t  various oycles and suppo&s the above observations Fegarding pressure distribution as expected from the equation 5. ID oases 1, 2, 3 and 4 the shook has travelled about 6.5, 5.3, 1.5 and 0.27 times the initial pist011 radius respectjvely at the last oycle (i.e. n = 150, 280, 60, 100) in each case. This shows that the effect of cylindrical and spherical geoxet~y has been fully taken h t o  account in the f is t  three cases whereas in the fourth case the shock has not travelled suffi- ciently far to show full effect of spherioal geometry. Therefore, we may compare the results of the case 4 a t  n = 100 with those of case 2 a t  n = 90. We find that though the piston irelocity in case 2 is slightly greater than that in case 4, the shook is stronger in case 4. 'Fig. 3 shows the relation between Eulerian and Lagrmgian distances and we find that after 150 and 280 cycles the radii of the piston in cases 1 and 2 have inc~eased to 2.2 and 3.4 times the initial radii respeotively. Purther, we fkd that a t  a fixed time the dis- turbance due to cylindrical piston travmses a longer distance than that due to the spherical piston, the initial radii of both the cylindrical and spherical pistons being the same. A C K N O W L E D G E M E N T  'The authors wish to express theh gratitude to Prof. P. L. Bhatnagar for suggesting the problem and giving help and guidance th~oughout he preparation of this paper. 

CYLINDRICAL PISTON PROBLEM IN WATER

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CYLINDRICAL PISTON PROBLEM IN WATER

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