Kidder, R.E.

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This is a preprint of a paper intended for pub lie i t ion in a tuurnal or proceedings, Since changes may be made before publication, this preprint is made available with the understanding that it w i l l not be cited or reproduced without the permission of the author. UCRL- 7 6 2 W-PREPRINT LAWRENCE UVERMORE LABORATORY University otCalilornia/Li<&rmore.Calitomia FLUID DYNAMICS AND STABILITY OF COMPRESSIBLE IMPLODING SHELLS MASTER Kay E. Kidder October 10, 1974 - NOT ICE-Thb report was prepared as an account o f work sponsored by the United States Government. Neither the United States not the United States Energy Research and Development Administration, nor any of their employee*, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or impl ied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. This material was prepared for presentat ion a t the Conference on the Physics of Plasma Induced by l a se r s and Electron Beams, Arad, I s r a e l , December 29, 1974 - January 3 , 1975. n r . u r rncUMENl UNLIMITED OlSTRiEUT.QN OF iK l * COCJMU FLUID DYNAMICS AMD STABI. TY OF COMPFESSIBLE IMPLODING SHELLS Fay E. Kidder October 10, 1971 This naterial wss prepared for presentation at the Conference on the Physics of Plasma Induced by Lasers and Electron Beams, Arad, Israel, December 29, 1974 - January 3, 1975. 1 1. INTRODUCTION The laser-driven implosion of solid OT pellets has been treated theo­retically in a previous paper. In particular, the absorbed optical power P. required to achieve a given measure pR of inertial confinement was found to be expressible by the simple (approximate) relation: P.(terawatts)-v 300 [pR(gm/cm2)]2 . (1.1) This result is based on a model of self-regulating pellet ablation by hot electrons of the pellet corona, and assumes that the compression is accomplished isentropically. It follows from Eq. (1.1) that nearly 3000 terawatts of optical power 2 (2) are required to achieve the p8-product of 3 gm/cm believed to be necessary in the application of laser fusion to commercial power production. This power requirement is unoomfortably high, and can pertiaps be significantly reduced if thin hollow shells are imploded instead of solid pellets. In this case there will be an additional multiplication of hydrodynamic power that results when the kinetic energy imparted to the imploding shell over a relatively long time is converted,in part, into compressive work during the shorter time in which the shell decelerates at the center of convergence. In the present paper we shall compute the mechanical power P„ required to achieve a given pR-product by the implosion of thin sheiis, and compare it with the results obtained previously for solid pellets. We will assume that the shell is gently accelerated from rest by an externa], pressure that 2 increases initially with time to produce a homogeneous isentropic compression, and then makes a smooth transition to a (slower) exponential rate of increase. This specification provides a convenient one-parameter family of pressure-time relations, the exponential rate of increase having been found empirically (3) to be near-optimum in a series of detailed computer calculations. Having determined the power required to implode thin shells as compared with solid pellets, we then treat the Rayleigh-Taylor instability of both solid pellets and thin shells undergoing homogeneous isentropic compression. An important feature of this treatment is that it takes into account the com­pressibility of the fluid raedium, a property that increases the growth-rate of long-wavelength perturbations in the case of thin shells. We shall begin by extending the elementary theory of homogeneous isen­tropic compression to the case of thin shells. 2. THEORY OF HOMOGENEOUS ISOTROPIC COMPRESSION AS APPLIED TO HOLLOW SHELLS The theory of homogeneous isentropic compression has been applied to solid pellets in Ref. (.'4). The assumptions of this theory are that changes of state are isentropic, the entiopy of the fluid is spatially uniform, and the fluid is describable as a perfect (y = 5/3) gas. The homogeneity of 3 the compression means that each element of volume d x in the Lagrangian description of the fluid, in which x(t) denotes the instantaneous position vector of a mass point, decreases at the same rate, i.e., d x = h (t)d XQ (2.1) 3 where x = x(t = 0), and h(t) is a dimensionless scale factor. Writing —0 — Eq. (2.1) in spherical coordinates, we immediately obtain r(r ,t> = rhtt) , (2.2) u(rQ,t) = r = rQh(t) From the law of mass conservation we may write (2.3) 2 2 dm = pr dr = dniQ = p 0r 0dr 0 (2.4) which, together with Eq. (2.2), implies that p(rQ,t) = p 0<r Q)/h 3(t) . (2.5) The equation of motion is given by u = - r2(dp/dm) = - (r2/pQrp)(dp/dr0) . (2.6) We wish to write dp/dr- in terms of dpg/drg. For this purpose, we invoke the relation between pressure p and gas density p along an isentrope of a perfect gas: p(r0,t) = a(r0)p'(rQ,t) p 0(r Q)/h 3 Y(t) . (2.7) Making use of these relations, we can write the equation of motion as h 3 y" 2h = - <l/p 0r 0)dp 0/dr 0 - - 1/t2 = const. (2.B) where the right-hand side of Eq. (2.8) is a function of r n alone and the left-hand side is a function of t alone. The constant t arising from th separation of variables has the dimensions of time. 2.1. TTME-DEPENIOICE OF THE SCALE-FACTOR AND THE BOUNDARY PRESSURE Restricting ourselves to the case in which the fluid is initially at rest, we find the time-dependence of the scale factor h to be h 2 r l _ T 2 , T = t/t . (2.9) c The physical significance of t is therefore the time at: which collapse of the fluid to zero radius occurs. It follows from Eqs. (2.7) and (2.9) that the pressure applied to implode the shell is given by: p(R2,t) = p(R2,0)/(l - T V 2 , (2.10) where R, = hR 7 f ) denotes the outer radius of the hollow shell. The pressure 5 at the inner radius of the shell R, = hR, 0 is taken to be zero, i.e., p(Ri,t) = 0 . (2.11) The subscripts (1) and (2) will always refer to the inner and outer radii of the shell, respectively, and the subscript (0) will refer to the initial state of the fluid. 2.2. RADIAL DEPENDENCE OF THE INITIAL FLUID PRESSUFE, DENSITY, AND TEMPERATURE Assuming, for simplicity, that the initial fluid state is one of uniform entropy <da/drQ = 0, R,Q <_ r Q <_ R 2n^' t h e solution o f the spatial part of Eq. (2.8), subject to the condition that the pressure vanish at the inner radius of the shell, becomes: p 0(r Q) : p 2 0[(r£ - R*0>/CR*0 - 4 0 ) ] 5 / 2 , (2.12) provided that the collapse time t, is defined as l c = ( R2 20 " ^ 0 ) / 3 c 2 0 ' ( 2 - 1 3 ) where c Q denotes the initial adiahatic speed of sound at the outer radius H,,Q of the shell, i.e., C20 * YP20 / p20 (2.14) The radial dependence of the density and temperature of the fluid is given by p Q(r Q) = P2QUvl - 4 0)/(R 2 2 0 - RJ[ 0>] 3 / 2 , (2.15) W -- T20 C ( r0 " R10 ) / ( R20 " 4 o ) ] * ( 2 - i 6 ) Henceforth we shall restrict our attention to the case of thin shells, that is, shells for which the "thinness number" £, defined as C = \/W2 - Rj) (2-175 is large compared with unity. In this case the spatial dependence of the in i t ia l pressure, density, and temperature can be conveniently written as P 0 (V * Wo ' ' 2 ' <VV = *2&V > V V * T2oV ( 2 ' 1 8 ) where 0 i *0 -' (r0 - V / ( R 20 ' V i l ' ( 2 - 1 9 ) The collapse time t can be written t c = ( ^ Q / C ^ ) / ? ^ = v5S7p~^ , (2.20) where Hmn is the total mass of the shell., m being related to t. according to m = (2/5)(p2QR^0/^){l + (17/14)?"1 + (SS/126)?"2 + ...} . (2.21) 7 2.3. MACH NUMBER It was shown in Ref. (<*) that homogeneous isentropic compression of solid pellets was impossible if the Mach Number of the flow anywhere exceeded the value /3 1. For thin shells, however, the least value of the Mach Number occurs at the outer surface cf the shell (it is always infinite at the inner surface of the shell because the sound speed is zero there), and is given by M 2 ( T ) = /3T/2"T . (2.22) The Mach Number M„ approaches the limiting value of /3^7? as t approaches the collapse time t . We conclude that thin (C>>1) shells, in contrast with solid pellets, can be imploded isentropically to very high Mach Numbers by application of the driving pressure defined by Eq. (2.10). 3. MECHANICAL POWER REQUIRED TO COMPRESS HOLLOW SHELLS We consider the implosion of thin shells that are accelerated from rest by an external pressure p 0 that initially increases with time according to Eq. (2.10), and then makes a smooth transition to a (slower) exponential rate of increase at a time t (<t). That is , F ( T ) , 0 < T < T < 1 , ~ ' a C3.1) F(T )exp[5T ( T - T )/(1-T ')] , T > T ci o 3. a — 3. P 2(T)/P 2 0 = 8 where T = t /t , and a a c F(T) = ( i - T 2 r 5 / 2 . (3 We note that during the first phase of the implosion (t<t ) the thinness ratio t, remains constant, due to the homogeneous character of the compres­sion, while the shell is accelerated and isentropically compressed. During the second phase (t <t<t.) of the implosion (t denotes the time when the inner radius R, of the shell has decreased to zero) the driving pressure increases exponentially, further compressing and accel­erating the shell. The compression continues to be isentropic but is no longer homogeneous, the thinness ratio ? decreasing to zero. The final phase of the implosion consists of the deceleration and (non-isentropic) final compression of the ingoing fluid by an outgoing spherical shock, generated at the time t. of void-closure. The three phases of the implosion process are shown in Figs. 1, 2, and 3, respectively. These results were obtained with the computer program WAZER, and apply to the implosion of a hollow OT shell with a thinness ratio t of 2k. The initial state and driving pressure were specified according to Eqs. (2.12), (2.15); and Eqs. (3.1)., (3.2), respectively. Other specifics and results of this computer run are listed in Table I. The time-dependence of the mechanical power P„ delivered to the imploding shell is shown as the solid curve in Fig. 4. For comparison, 9 the mechanical power delivered to a solid pellet of equal mao-, to achieve the same value of pR by means of hctnogvjneous isentropic (4) compression (with a compression-rate parameter 8 of 0.34), is shown in Fig. 5. (Note that the time and distance scales of Fig. 5 are one-tenth, are? the power scale is ten-times, the corresponding scales of Fig. 4.) This comparison shows that the peak power re­quired to implode the shell is nearly an order of magnitude lower than that required to implode the solid pellet, although the energy supplied to the shell is somewhat greater as is indicated by the greater area under the power-time curve of Fig. 4 as compared with Fig. 5. The greater energy requirement results from the non-isentropic character of the final shock-compression. Additional computer runs were carried out for hollow shells of equal nass with thinness ratios of 24, 230, and 2200. For each thinness ratio, the maximum power and pR-product was determined for each of four selected values of the parameter T appearing in Eq. (3.1) describing the driving pressure. Decreasing T had the effect of decreasing the peak power delivered to the imploding shell. The results of these runs are shewn in Fig. 6, where the curves labeled A, B, C apply to shells with thinness ratios 24, 230, and 2200, respectively. The data of Fig. 6 show that the peak mechanical power required by achieve a given pR-product is greater than, but comparable with, (0.77) 2 (pR) for thm shells with thinness ranging over two orders of magnitude. That is, 10 PM(terawatts) > (G.77)[pR(gm/cm2):i2 , (24 < C < 2200) . (3 In Ref. 1 it is sha.-m that the peak mechanical pow°r P„ for homogeneous isentropic compression of a solid pellet is similarly 2 proportioral to (pR) , the factor of proportionality being roughly ten tines larger than that appearing in Eq. (3.3) above. Although the data of Fig. 6 apply to shells having a nonuniform initial state specified by Eqs. (2.12) and (2.15), substantially the same results are obtained if the initial pressure and density are uniform and equal to p 2 Q and p. , respectively. The thinness ratio is then 2.5-times greater for the same shell mass. 4. RAYLEIGH-TAYLOR INSTABILITY OF COMPRESSIBLE PELLETS AND SHE1LS The mechanical power requirements of thin shells have been shown to be roughly ten-times smaller than those of solid pellets, to achieve the same measure of inertial confinement pR in the final imploded state. The optical (laser) power requirements would be expected to be reduced by at least as great a factor, because the efficiency of conversion of optical energy into mechanical work would be expected to be somewhat higher: The intensity of the absorbed light is lower, resulting in a cooler "rocket exhaust" and more efficient propulsion. ine lcwer light intensity will also lead to reduced excitation of nonlinear optical effects and plasma instabilities, and improved core-corona coupling. It would appear, then, that thin shells are clearly superior to (7) the solid pellets initially proposed as targets for laser fusion. However, we have not yet considered the effects of hydrodynamic in-(8 9) stabilities of the Fayleigh-Taylor variety ' where the advantages might be expected to be the other way around. This work was performed under the auspicies of the United States Atomic Energy Commission. REFERENCES Kidder, R. E., "Laser Compression of Matter: Optical Power and Energy Requirements", (to be published -̂ Nuclear Fusion). Nuckolls, J. H., Laser Interaction and Related Plasma Phenomena (Schwarz, H., and Hera, H., Eds.), Plenum, New York, 1974, Vol. 3B, p. 449. Nuckolls, J. H., Laser-Fusion Program Semiannual Report — July-December 1973, UCRL-50021-73-2, April 1974, p. 87. Kidder, R. F., Nucl, Fusion 14_, (1974) 53. Kidder', R. E., Barnes, W. S., WAZER, A One-Dimensional, Two-Temperature Hydrodynamic Code, UCRL-50583 (January 31, 1969). Kidder, R. E., Zink, J. W., Nucl. Fusion 12_, (1972) 32b. Nuckolls, J. et al., Nature 239_ (1972) 139. Taylor, G., Proc. Roy. See. A 201_ (1950) 192. Bodner, S., Phys. Rev. Letters 33_ (1974) 761. TABLE I 13 Specifics of Computer Run Initial Properties of Shell: Pressure (p2n) Density <P2n^ Thickness (R, '20 R 1 C ) Mass (turn) Thinness Ratio {0 0.15 Mbar 0.2 girt/ cm 50 um 75 wgm 24 Compression Parameters: Transition Time (t ) a Homogeneous Collapse Time (t ) Transition Parameter (T ) 17.06 nsec 17.77 nsec 0.96 Implosion Results Peak Mechanical Powei (P.,) Mechanical Energy Supplied (W„) Void-closure Time (t. ) Peak Inertia! Confinement (pR) Core Compression at Peak pR (n ) 10 9.H terawat ts 8.2 kJ 17.95 nsec 3.2 gm/cm . . i t 14 FIGURE CAPTIONS Figure 1. Trajectories of fluid particles during the implosion of a shell having a thinness ratio of 24. Radius r (in millimeters) of fluid particles versus time (in nanoseconds). Figure 2. tfagnified view of the snail rectangular region shown in Fig. 1. Radius r (in millimeters) versus time (in nanoseconds). Figure 3. fegnified view of the small rectangular region shown in Fig. 2. Radius r (in microns) versus time (in picoseconds). Figure 4. Time-dependence of the mechanical power P M delivered to imploding shell. F M (in terawatts, scale at left) versus time (in nanoseconds) to peak pR. The daslied curves show the inner and outer radii of the shell (in microns, scale at right) as it implodes. Figure 5. Time-dependence of the mechanical nower P M delivered to imploding pellet. P„ (in terawatts, scale at left) versus time (in pico­seconds) to peak pR. The dashed curve shows the radius of the pellet (in microns, scale at right) as it implodes. Figure 6. Peak mechanical power P M (terawatts) required to achieve a given o measure of inertial confinement pR (gm/cm ). Solid curves labeled A, B, C are based on computsr runs for shells with thinness ratios 24, 230, 2200, respectively. The dashed line represents the relation: P M = (0.77)(pR)2. i l ru re j . t(nr-cc) Tinirry 2 t(:v.ec) r i f t i re 3 S Figure 4 1.2 0.9 0.6 0.3 Time (nsec) t o Peak pR Figure 5 ICC Bu —pr n 60 i .. 20 r .... :|.: • ^ -i- : • : = •j ' : : ; • i ' .:;~ :::: ;;; ;: — iH: ~3. "1 : : ::.- :"::.T = ! Hi; Hi! hi; 1 ... | - : 1 - - -": :w- : ::i iili HH ; i / ::: • I • ' : • • t>! : ;i : :: • \ ; ' : h ; ;-:.-l. • ' . . . '•'• ! ' : • • ; ! -r- -, ::':': ; ;j :-\ . . j . . . • • • : | : : : ; ." : . : • • ; "• " 7 - ,i •': : : • • _.|_ I :i!: ~;|!:!l :!; :;;: 13 " : ' • . I ~' i I ; • • : : : :::=|=-:: : ; i - . _ ; .:;: .' :: ._j._. ,„J t -' :..: - ; • : • : ' ,:-;' '-•' :::: : • - . . I i • -y ••; 1 r r : — j — • | _ . j — — J. / • ^ .> — — ;"i ;! I ... J_ • - - —. ~. i „ i T — ~ i i i ™ t " i — 1 ""; • . ! • I j - ~T J J. I ! : _ i - • 1 J i - 1 _ 40 30 20 10 120 90 60 30 Time (psec) to Peak pR REFERENCE S. Kidder 15 T.I .D. 15 T-Division 3 Ihiv rcptifi * J - I1lv•r•"t•>, J ' J ! 1 J..'>um "1 «**U *jmti«itffj h> the Int l fJ Suii \ t,.>kftnph'ui Vtiln-i H»- 1 micd %uir* nm ihi- limird Suif- Ai.-tu. I iii-ifv ( "mm.»iiMi. nur any i.l their employer*, no JHi nl ihru HHIIM.MI* Hjtnnnirji.ti>i% w ihtii rmpluyrrv mtkn jny wjiunn ctpicum impltcil.ni JisumcN *nyttg*\ luhilny •>( ic%jn'n*ihiim |.-i <lnjmi(jv> vi.nipliU'iic*'. "t u\etiAtK\% "i *ny ifiliititijinin jppjf iiuv I'fmJini iti PHKL'.-. i lnjinftl. ot ttpirtcnlt lti.ii n> «v <*<>iM not infinite I'tujlcly-nmntd nfhtt " 

Fluid dynamics and stability of compressible imploding shells

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