Program Burn Algorithms Based on Detonation Shock Dynamics

Transcription

1 Program Brn Algorithms Based on Detonation Shock Dynamics Discrete Aroimations of Detonation Flows with Discontinos Front Models J. B. Bdzil, D. S. Stewart and T. L. Jackson DX-division, Los Alamos National Laboratory, Theoretical and Alied Mechanics, University of Illinois and Center for Simlation of Advanced Rockets, University of Illinois In the design of elosive systems the generic roblem that one mst consider is the roagation of a well-develoed detonation wave sweeing throgh an elosive charge with a comle shae. At a given instant of time the lead detonation shock is a srface that occies a region of the elosive and has a dimension that is characteristic of the elosive device, tyically on the scale of meters. The detonation shock is owered by a detonation reaction zone, sitting immediately behind the shock, which is on the scale of millimeter or less. Ths, the ratio of the reaction zone thickness to the device dimension is of the order of / or less. This scale disarity can lead to great difficlties in comting three-dimensional detonation dynamics. An attack on the dilemma for the comtation of detonation systems has lead to the invention of sb-scale models for a roagating detonation front that we refer to herein as rogram brn models. The rogram brn model seeks not to resolve the fine scale of the reaction zone in the sense of a DNS simlation, instead the goal is to resolve the hydrodynamics in the inert rodct gases on a grid mch coarser than reqired to resolve a hysical reaction zone. We first show that traditional rogram brn algorithms for detonation hydrocodes sed for elosive design are inconsistent and yield incorrect shock dynamic behavior. To overcome these inconsistencies, we discss a new class of rogram brn models based on detonation shock dynamic (DSD) theory. This new class yields a more consistent and robst algorithm which better reflects the correct shock dynamic behavior. D. S. Stewart, Corresonding Athor

2 BDZIL STEWART AND JACKSON. INTRODUCTION In the design of elosive systems the generic roblem that one mst consider is the roagation of a well-develoed detonation wave sweeing throgh an elosive charge with a comle shae. At a given instant of time the lead detonation shock is a srface that occies a region of the elosive and has a dimension that is characteristic of the elosive device, tyically on the scale of meters. The detonation shock is owered by a detonation reaction zone, sitting immediately behind the shock, which is on the scale of millimeter or less. Ths, the ratio of the reaction zone thickness to the device dimension is of the order of / or less. This scale disarity can lead to great difficlties in comting three-dimensional (D) detonation dynamics. Assme (as we do for the rest of the aer) that the hysical roblem of modeling the dynamic roagation of the detonation and the motion of the reacted rodcts in the following flow is comletely described by a soltion to the comressible Eler eqations for a reactive flow, with a secified eqation of state for the elosive and reaction rate of the form e = e(, v, ), r = r(, v, ), where, v, are the ressre, secific volme and the rogress variable of chemical reaction. Note that = corresonds to nreacted elosive and = corresonds to comletely reacted elosive. The rediction of the detonation dynamics can be achieved in rincile by a direct nmerical soltion (DNS) of the Eler eqations. In order to get a high qality soltion to the reactive Eler eqations, it is essential to have enogh oints in the reaction zone. Unfortnately even with modern algorithms, as many as - cells in the streamwise direction may be reqired to resolve the detonation reaction zone to sfficient accracy so as to comte the detonation seed. When one then considers the conseqences of sch a fine scale for the reaction zone, combined with the reqirement for global temoral and satial accracy in the meter-sized domain of the engineering device, hge comtational resorces are reqired [] (even given todays TeraFlo arallel comting resorces) for DNS of a detonation wave sweeing throgh a system. The comtational barrier to D design of elosive systems throgh direct simlation of the reactive Eler eqations is not newly discovered, and dates back to the se of comters to design elosive systems that started systematically shortly after WWII. A dilemma of sorts resents itself. One needs to try to make redictions in engineering systems bt one cannot overcome the stiff comtational reqirements needed to comte on the engineering device scale. One cold comte DNS simlations that are resolved for very small dimensions, bt those are at a minimm at least two orders of magnitde smaller than the engineering system scale. The dilemma osed above associated with trying to solve a hysically correct bt comtationally intractable model is similar in sirit to direct simlation of trblence on engineering device scales. In that disciline the need to resolve the hysics of trblence on larger engineering scales has led to the invention of classes of sb-scale models for trblence and most recently to large eddy simlation. An attack on the dilemma for the comtation of detonation systems has lead to the invention of sb-scale models for a roagating detonation front that we refer to

3 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS herein as rogram brn (PB) models. The rogram brn model seeks not to resolve the fine scale of the reaction zone in the sense of a DNS simlation. The goal of a PB simlation (PBS) is to resolve the hydrodynamics in the inert rodct gases on a grid mch coarser than that reqired to resolve a hysical reaction zone. Ths a PBS mst deosit a rescribed amont of energy (and more generally mass and momentm) into a very few nmber of comtational cells behind a re-calclated shock front. The effective reaction zone in a PBS is always the region behind a recalclated shock front where sorce terms are added to accont for the deosition of energy. For ractical reasons, the effective reaction zone is always constrained to be a finite nmber of cells thick (between one and for say). The region where the sorce terms contribte, in the limit of zero cell thickness, limits to a shar front across which there are jms in the deendent state variables. The rogram brn sorce doses, while historically rescribed rely by the rescrition of the discrete algorithm sed in a articlar code, mst limit to a delta fnction sorce centered at the location of the shar front, which is then eternally rescribed by re-calclating the shock location. The delta fnction sorce terms mst, of corse, be reresented in the artial differential eqations that reresent the rogram brn model, indeendent of its discretization and the algorithms sed to solve it. One thing is clear from this discssion, the soltions of the reactive Eler eqations are not soltions of the eqations of the rogram brn model. In this aer we consider the following roblem: How does one make consistent and robst discrete aroimations of hysical detonation flows with a finite length reaction zone as modeled by the reactive Eler eqations, with a discrete aroimation to a rogram brn model for which the reaction zone and shock is collased entirely to a single discontinos front? The whole scheme where a Program Brn model have soltions that are in some sense close to those of the Eler eqations for a reactive flow deends very mch on the accracy of the aroimate theory in regards to the shock dynamics. This isse mst be decided irresective of nmerics. In Section we briefly resent direct nmerical simlations (DNS) of the reactive Eler eqations that are to be sed as the benchmark calclations for the rest of the aer. The geometry considered will be either lanar, cylindrical or sherical. For cylindrical/sherical geometry, crvatre of the lead shock is resent. In Section we comare the soltions obtained from DNS to the recent asymtotic theory of detonation shock dynamics (DSD), a key ingredient of the more modern imlementations of rogram brn. In Section we regress somewhat by resenting the traditional ressre-based rogram brn (TPB) model; sch a descrition is essential for nderstanding the rest of the aer. In addition, some nmerical calclations are resented showing the strengths and weaknesses of the model. Section resents resents varios models aimed at imroving the weaknesses inherent in the TPB model which crvatre is resent. This new class of models will be refered to as the modified ressre-based rogram brn models (MPB). There does not eist any reference which describes MPB, and so is resented here for the first time. Soltions obtained from MPB are comared with soltions obtained from TPB and DNS. The essential difference between TPB and MPB is that TPB ses a Hygen s constrction for the shock roagation rle (shock roagates with the Chaman-Joget seed), while if crvatre is

4 BDZIL STEWART AND JACKSON resent, MPB ses a roagation rle based on DSD. Finally, conclsions are given in Section 6.. DIRECT NUMERICAL SIMULATIONS In this section we resent the reactive Eler eqations that will be sed as benchmark calclations to be comared with the rogram brn models resented in sbseqent sections. We therefore assme that the DNS calclations are eact, and that any differences in soltion strctre will be de to the varios aroimations inherent in the rogram brn models themselves. For the DNS calclations, the conservative formlation of the reactive Eler eqations are given by where U t + F = G + qr, () U =[,, E, ] T, () F = [, +, (E + ), ] T, () G = j [,,(E + ), ] T, q =[,,, ] T, () where is the density, the ressre, the velocity, E the total energy defined by E = (e + ), () e the secific internal energy, and the mass fraction of the deficient comonent ( = for nreacted material, = for comletely reacted material). The geometric sorce terms from the flow divergence are reresented elicitly by G. The choice of j determines the geometry; j = for lanar, j = for cylindrical, or j = for sherical geometry. If one assmes a cylindrical/sherical shock, the shock total crvatre is related to the radis from the center of the coordinate system by = j/. To close the system, constittive laws for the internal energy and the reaction rate mst be given. For illstration roses, we take the eamle of a condensed hase elosive considered in [] and sed as a test roblem in [] and []. The eqation of state is taken to be that of an ideal gas e = (γ ) Q, (6) where γ is the ratio of secific heats and Q is the heat of reaction for the detonation. The reaction rate is given by r =.7µs ( ) /. (7)

5 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS. FIG.. Plot of the strctre for the case j = (cylindrical) at time t =µs. The vales Q =mm /µs and γ = are taken, with stream conditions o =, o =g/cc and o =. These vales give a Chaman-Joget detonation seed of D CJ =8mm/µs, and a steady-state one-dimensional reaction zone length of mm. To carry ot the DNS, these eqations are solved by a high-resoltion Eler solver, namely, a third-order TVD Rnge-Ktta scheme with a fifth-order WENO satial scheme, [6], [] and []. The grid is assmed niform with =.mm, which ts roghly grid oints in the reaction zone. Reslts for twice the nmber of grid oints, and hence has 8 grid oints in the reaction zone, gives essentially the same reslts. In all cases the CFL nmber was taken to be.. Wave strctres are resented in Figre. for the case of cylindrical geometry and in Figre for the case of sherical geometry.. DSD ASYMPTOTIC THEORY AND COMPARISON TO DNS In this section we briefly state the asymtotic theory of detonation shock dynamic (DSD) theory, a key ingredient of the rogram brn model that will be resented in sbseqent sections. We also comare certain flow featres between DSD theory and the DNS calclations of the reactive Eler eqations resented in the revios section... DSD Theory Detonation shock dynamic (DSD) theory is an asymtotic theory which describes the motion of the detonation shock by means of a relation between the normal shock velocity D n, the shock crvatre, and their time derivatives. For a throgh review of the theory, its assmtions and limitations, see [7]. For or roses here we shall

6 6 BDZIL STEWART AND JACKSON. FIG.. Plot of the strctre for the case j = (sherical) at time t =µs. only focs on the qasi-steady, one-dimensional theory. The relevant eqations, consistent with the reactive Eler eqations resented in a nearly integrable form that reflects the conserved first integrals of the governing eqations if the flow were steady and lane, are (U n ) n + (U n + D n )=, (8) (U n + ) n + U n (U n + D n )=, (9) ( e + v + ) n U n =, () n = U n (r), () where n is the coordinate normal to the detonation front, and U n = n D n is the relative normal velocity in the shock-attached frame. An alternative form of the energy eqation, dbbed the master eqation, is fond by sing the chain rle on e(,, ) in (), sing the mass eqation to sbstitte for the satial derivative of, and then sing the momentm eqation to sbstitte for

7 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS D n 6 *..... *.... *.. FIG.. Plot of the shock seed D n and the star states as a fnction of. the satial derivative of the ressre. With the standard definition of the sond seed by for an ideal EOS, c = γ/, one obtains (c U n) U n n = Qr(γ ) c (U n + D n ). () The generalized CJ conditions follow from the master eqation. When the flow is locally sonic and the velocity gradient is finite it follows that when η c U n =, () the right hand side of () mst also be zero, i.e., Φ Qr(γ ) c (U n + D n )=. () The first condition is the sonic condition, while the second is the thermicity condition. These conditions hold for detonations that travel near or at the CJ detonation velocity. The simltaneos reqirement that the sonic and thermicity conditions be satisfied reqire that there is a relationshi between and D n. For sch soltions one can find the sonic, or star (*), states. The soltion of this nonlinear eigenvale roblem can be done nmerically if desired and the star states can be fond as a fnction of the local crvatre. A lot of the star states is shown in Figre for the condensed hase eamle of the revios section. Note that for =, the star states are the CJ states, and D n = D CJ.

8 8 BDZIL STEWART AND JACKSON D n *.... *. * FIG.. Plot of D n and the star states as a fnction of for DSD (solid) and DNS (circles). Cylindrical geometry... DSD-DNS Comarisons Comarisons of DSD theory with direct nmerical simlations (DNS) have been carried ot in [] for the cases of detonation along a two-dimensional rate stick, in a converging channel, and a diverging channel. In all three cases the shock front locations as comted from DSD theory and from DNS were comared and good agreement between the two was fond. Similar comarisons can be fond in [] and []. However, a simle and direct comarison between DSD and DNS can be carried ot by considering cylindrical or sherical geometry where the crvatre is elicitly known and the eqations are essentially one-dimensional. To comare with DSD theory, we show in Figres and the shock seed D n and the star states as a fnction of crvatre for cylindrical and sherical geometry, resectively. In each figre, the solid crve corresonds to DSD theory, and the circles corresond to the DNS calclations. The wave front was determined to be the vale at which the reaction rogress variable was.; the seed is then the time derivative. Note the good agreement for both cylindrical and sherical geometries as the crvatre goes to zero, i.e., the long time soltion. For large vales of the crvatre, the agreement between the two diverge, either de to the transient effects of the DNS calclations at the earlier times or de to the first order and qasi-steady aroimation of DSD theory where the time derivatives have been ignored.

9 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS 9 D n *.... *. * FIG.. Plot of D n and the star states as a fnction of for DSD (solid) and DNS (circles). Sherical geometry.. TRADITIONALLY IMPLEMENTED PROGRAM BURN MODELS In this section we discss the basic ideas behind the imlementation of rogram brn as it has traditionally been imlemented in design hydrocodes sed for elosive engineering. Althogh several versions eist, we shall discss only one model, the traditional ressre-based rogram brn model (TPB). The other models have similar strengths and weaknesses, and only one model is sfficient to clarify the discssion. Program brn was first osed as a nmerical algorithm, not as a differential system. One of the earliest blished references to an algorithm of this tye is fond in [9]. The algorithm has the following ingredients: i) There is a re-determined, comtational grid and a chosen algorithm for the inert hydrodynamics. The grid defines the domain of the elosive and the algorithms are sed to solve the Eler eqations for the (inert) elosive rodcts. ii) A graded set of brn-times, t b, are assigned to each comtational cell on the grid. The brn-times are the times that the detonation shock front crosses the coordinates of the initial osition of the comtational cell. The traditional way to comte the brn-times is to select the nreacted elosive geometry, ick the locs of an initial Chaman-Joget (CJ) detonation, and then comte the motion of the detonation shock emanating from the initial locs by means of a Hygen s constrction. The Hygen s constrction roagates the shock normal to itself at the constant CJ wave seed, D CJ. iii) A cell-based algorithm either adds energy to designated brning cells or modifies the eqation of state in cells dring the interval of the shock assage over the cells, as dictated by the re-calclated brn-times. The eqation of state adjstment

10 BDZIL STEWART AND JACKSON has been done in varios ways throgh either increments in the ressre or secific volme. In what follows we give a descrition of a traditional ressre-based rogram brn algorithm which modifies the eqation of state in the brning cells. The definitions of the brn-fraction, the brn-time field, re-calclated shock motion and modification of the eqation of state are key ingredients of the model.... Brn-fraction Based on a reviosly calclated assignment, each cell is assigned a brn-time, t b. If the resent time of a comtational cell is below the brn-time, t<t b, then the cell is not brning and the brn-fraction Y is assigned zero. If t>t b, then the brn-fraction mst be calclated. The brn-fraction is sally assigned to be the volme fraction of the ndistrbed cell that has been crossed by the detonation shock at that time, and hence has a comted vale, <Y <. The details of the comtation deends on the secific grid and algorithm and whether the brn-times are stored at cell centers or at the nodes. If the whole cell has been crossed the brn-fraction is simly Y =.... Brn-time field Once the brn-fraction algorithm is selected, the discrete field of brn-fractions can be re-calclated from the discrete field of brn-times. While (as the grid is resolved) the brn-times limit to a iecewise continos field in the domain of the nshocked elosive, the discrete brn-fraction field mst limit to a singlar Heaviside fnction which is attached to the contors of the brn-time field (i.e., the re-calclated shock osition). The brn-time field is re-calclated and the traditional way to do this is to se a Hygen s constrction. Ths, once the nreacted elosive geometry is selected, the initial locs of an initial CJ detonation is icked, and the motion of the detonation shock that emanates from the initial locs is comted by means of a Hygen s constrction.... Shock srface motion and the limits of discrete fields The way to eress these ideas mathematically is as follows. Let the brn-time field, which eists as a iecewise continos field with a discrete reresentation on a grid which covers the domain of the nreacted elosive, be given by t b ( ). Then, at a fied time t, the shock locations are the contors of the brn-time field = s : t = t b ( s ). The limit of the discrete brn-fraction field at a time t as the mesh is resolved is reresented by the Heaviside fnction H(( s (t )) ˆn), where ˆn is the normal to the shock that oints in the direction of roagation.

11 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS As an eamle, consider a one-dimensional detonation wave roagating with constant ositive seed D CJ. Then, according to Hygen s constrction, we have d s dt = D CJ, () where s is the location of the detonation front at time t. Integrating we get the motion rle for the front s (t) = o + td CJ, (6) where o is the initial osition. The domain < o is assmed to be comletely reacted, and is nreacted for > o. This relationshi can be inverted to yield the brn-time field t b ( s )= s o D CJ. (7) For the discrete aroimation, let the nmerical grid have a niform mesh, i, with grid sacing. Then the discrete version of the brn-field can be written as t b ( i )= i o D CJ, for i > o. (8) Note that the brn-time is not defined for i o, which indicates that this region of the flow field has already reacted. Also note that the brn-time is iecewise continos in the nreacted domain. For the rescrition of the brn-fraction, which we shall denote by Y i, we date Y i according to the rle i > s, Y i = s i s < i < s, (9) i < s. This articlar descrition of the brn-fraction is defined over a single cell. In the limit as, we see that the brn-fraction aroaches a Heaviside fnction. Figre 6 shows a sketch of the shock osition as a fnction of time and a sketch of the brn-fraction Y. The se of the brn-fraction Y is described in more detail in the following section.... Modification of the eqation of state and aarent weak detonation strctre In the traditional ressre-based rogram brn algorithm one assmes an eqation of state for the inert rodcts e rodcts (, v) e(, v). Since condensed elosives are being considered, the initial ressres (one barr) are etremely small comared to the detonation ressres behind the lead shock

12 BDZIL STEWART AND JACKSON t s (A) Y= s (B) Y= i- i- i- i i+ o FIG. 6. (A) Sketch of the shock location s(t) as a fnction of time. (B) Sketch of the brn-fraction Y on a discrete grid. (hndreds of Kilo-bars) sch that the ressre ahead of the shock in the nreacted elosive can be considered to be zero. This is similar to the strong shock aroimation. In a PBS in the brning cells, where the brn-fraction Y is between zero and one, the eqation of state is modified by relacing with /Y to obtain e rodcts (/Y, v). This is eqivalent to relacing the ressre with a artial ressre which is redced by the brn-fraction for that cell. When Y =, the eqation of state for the rodcts is recovered. Finally, in the nbrnt cells in the nreacted elosive one mst give an energy that is consistent with the heat of detonation. This is done in the following way. One considers the standard Rankine-Hgoniot relations for a gasdynamic discontinity for a steady Chaman-Joget discontinity traveling at laboratory seed D CJ. One then sets the energy datm e in the nreacted elosive consistent with that algebra. The eqation of state for TPB can ths be written as e = e { H(( s (t))) ˆn} + H(( s (t)) ˆn) e rodcts (/Y, v). () An eamle of selecting e is resented in the following section. If we consider the ressre variation across the shock dring a PB, the ressre starts ot from zero and is broght to a high vale near the CJ-ressre. Indeed, when the brn-fraction Y is zero then the ressre is necessarily assmed to be zero, in fact the scheme comtes the ressre based on an assmed eqation of state and therefore the nderlying hydrodynamic algorithm increments the ressre in sch a way that the internal energy is assmed to be finite. A simle conclsion is that the effective reaction zone strctre of traditional rogram brn starts at the nreacted state at the ambient ressre, and not at the shock state. If the rogram brn algorithm can be interreted in terms of an effective distribted rate law, then the corresonding detonation strctre looks like a weak detonation, and not a strong detonation. Note that the hysically-based argment against a weak detonation strctre is absent in a PBS, since the re-calclated shock motion rovides the seqenced brn-times for the cells that trigger the change in the eqation of state in the vicinity of the shock. An alternative interretation is that the PB scheme is a catring scheme which intends to catre states that are near or at the steady state eqilibrim CJ-vales and hence ct off, or do not reresent in any way, a hysical reaction zone strctre from the inert nreacted shock state (the von Nemann sike) to the sonic oint that normally wold be comted as art of the reaction zone in a DNS.

13 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS.. Eamle: Ideal EOS To illstrate the traditional imlementation of rogram brn, we will start with an eqation of state (EOS) for the detonation rodcts, e(, v) and be even more secific by sing the gamma law eqation of state e(, v) = γ.... CJ states To comte the CJ-states, we first assme that the nbrnt stream state (with the strong shock aroimation) ahead of the wave is given by = o, =, =, e = e o, () with e nsecified at this oint bt will be chosen in the corse of the analysis. Let[]=() o () b denote the jm in a qantity across the interface from the o-state (Y = ) to the comletely brnt state (Y = ) denoted by a b-sbscrit. The normal jm conditions across the interface moving with seed D n are given by [( n D n )] =, () [ n ( n D n )+] =, () [E( n D n )+ n ]=, () where E is the total energy defined earlier. With the assmtion of the ideal EOS in the brnt rodcts, the algebra of the above jm conditions are redced to a qadratic eqation in the normal article velocity n, say. If we identify the seed D n as the CJ vale (D CJ ), the qadratic eqation for n can be solved to give n = D CJ ± D CJ (γ )e γ + The CJ state is associated with the zero of the argment of the radical and lead to the identification of either the D CJ in terms of the energy e or vice a versa. Since we generally regard D CJ as being given eerimentally, we choose to write the condition as. Then the CJ states are e = D CJ (γ ). () ( ) γ + CJ = o, CJ = odcj γ γ +, CJ = D CJ γ +. (6)

14 BDZIL STEWART AND JACKSON It also follows simly that the CJ state is locally sonic. Note that in working ot the Rankine-Hgoniot jm conditions across a rogram brn discontinity, from the nreacted elosive to the brnt elosive where the brn-fraction Y is set eqal to one, one obtains eactly the same Rankine-Hgoniot algebra as the reactive Eler eqation where is set eqal to one. Ths, the variation of a brn-fraction variable has no effect on the calclation of the CJ-states themselves. As an eamle, we take the condensed hase elosive fond in []. With γ =, o = and D CJ = 8, we get for the CJ states CJ = 8, CJ =, CJ =. (7)... Eqation of state with modified ressre and effects on the strctre In keeing with the notion that one relaces with /Y in the brning cells with <Y, the ideal EOS becomes e = γ Y. Again one assmes that in the fresh material one has the same initial secific internal energy e, and the role of e is the same as the heat of detonation. To frther analyze this strctre let U n = n D n be the relative normal velocity in the shock-attached frame. For a qasi-steady traveling wave, the RH-relations hold throghot the strctre, ecet now the internal energy has the deendence on the brn-fraction Y. As before, one can again solve the RH-relations U n = D n U n + = D n e + U n + = e + D n, with e = for a qadratic eqation in U n with soltions Y (γ ), U n = [+(γ )Y ]D n ± D n (γ )Y [+(γ )Y ]e +(γ )Y. (8) When Y = the ls root corresonds to the nreacted flow state, and hence to the starting oint for a weak detonation strctre, U n = D n, or n =. The root associated with the mins sign is athological and has U n =or n = D n, and corresonds to a finite ressre bt infinite density. In contrast, the standard strong shock state U n = (γ )/(γ +)D n is achieved if the eqation of state e = v/(γ ) is sed instead of the modified eqation of state e =(/Y )v/(γ ). The isse is which state is selected, and we trn to the acostic character of the distribted strctre net. From the fndamental definition of the sond seed,

15 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS c = / e/ e/, we have c = [+(γ )Y ]. Net, if we se the energy eqation e + / + Un/ =Dn/ +e and se the definitions of e and the last reslt for c, we can eliminate / in favor of c and write an eression for the sonic arameter, η, as follows [ η c Un = e + ] (D n Un) (γ )Y Un. If the detonation wave starts ot on the weak branch, then at Y =,c =, and U n = D n, the sonic arameter η = D n <, and the wave is sersonic at the oint of the lead distrbance. In fact one can comte the sonic locs in a (U n,y) - lane by setting c = U n to obtain U n = γ (γ )e Y [+/(γ )Y ]. (9) The character of the strctre of the (weak) detonation can be characterized by lotting its trajectory in a (U n,y) - lane. The weak CJ soltion trajectory starts from the ndistrbed state, U n = D n and terminates at the sonic state. Figre 7 shows this trajectory for the secific case of D n = D CJ. Note the sqare root behavior in U n as Y, sggesting that the normal derivative has a sqare root singlarity. This is de to the fact that the thermicity condition in the master eqation does not vanish at the sonic oint. The other reqired ingredient for a weak detonation is a sersonic trigger. Ordinarily the sersonic trigger is regarded as ahysical. Bt for its alication as a nmerical algorithm, rogram brn assigns times at which the cell releases its energy. Secifically, the vale of the brn-fraction is changed from Y = to Y = in roortion to how mch of the article cell has been crossed by an assmed shock wave. Therefore the distribtion of times when the cell is crossed by a shock is know a-riori, and is sed to create the sersonic trigger. For steady, one-dimensional flow for a CJ detonation, the brn-times simly and eactly reflect the CJ detonation velocity. We note that the state variables do deend on the brn-fraction if the brnfraction were distribted in a discrete reresentation; i.e., not resolved to a Heaviside ste fnction. Then the brn-fraction distribtion on a finite mesh has the aearance of a sedo-reaction zone strctre. In the following discssion, for convenience, we will model this distribtion not by a difference based scheme, bt instead modeled by an effective rate law in the steady detonation frame, U n Y n = R(Y ), ()

16 6 BDZIL STEWART AND JACKSON 8 weak strctre 6 sersonic U CJ η= sbsonic...7 FIG. 7. U CJ,Y-lane showing the trajectory of the weak, CJ detonation. The dash crve corresonds to the sonic locs given by (9) and the solid crve corresonds to the weak strctre given by (8). (The strong strctre branch is not shown.) Y where R(Y ) is an effective rate. In actal ractice this rate is not given at all, rather the nmerical scheme that defines the brn-fraction merely makes an assignment for the increase in Y sch that it goes to Y = when the detonation shock crosses the comtational cell comletely and R(Y ) is inferred from the details of that assignment. Bt certainly R(Y ) is both grid and algorithm-deendent. Integration of (), with the weak-strctre relation between U n and Y and the condition that Y =at = (which is eqivalent to the secification of the triggering event at the rogram brn-time), leads to a distribtion fnction Y () which has the basic rofile shown in Figre 6. An imortant observation is that the thickness of the heat-release zone in the rogram brn reaction zone will be a fnction of the grid thickness and can be comted asymtotically as O( ), sch that as, the rogram brn reaction zone vanishes, as measred relative to any hysical length scale. Ths the effect of the nmerical algorithm that R(Y ) imitates is to aroimate a delta fnction, centered at the brn-times and saces on the grid as dictated by the brn table... Nmerical reslts of TPB and comarisons to DNS We resent some nmerical reslts comaring the soltions obtained sing the traditional ressre-based rogram brn model (TPB) to the soltions obtained from a DNS calclation. We se the condensed hase elosive described in [], [], and []. The eqations and nmerical scheme for the DNS calclations were resented in detail in Section. For the TPB model we solve the corresonding nonreactive Eler eqations with the EOS given by () and (). Althogh crrent codes se a second-order scheme, we choose to se the same high-order scheme that is sed for the DNS calclations to minimize errors reslting from different nmerical algorithms, ths isolating any differences between the two soltions as rising from the varios assmtions in the TPB model itself. To restate, we assme that the DNS calclations are eact, and that any differences in soltion strctre will be de to the varios aroimations inherent in the TPB model. A mesh which

17 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS 7 has only one grid oint in the reaction zone ( =mm) is sed. The reason we choose this articlar grid size is that tyical imlementation of the rogram brn methodology ses only a fine enogh grid to resolve the hydrodynamics behind the wave front. The grid chosen here is ths tyical of that sed in engineering ractice; no attemt is made here to otimize nor stdy the effect of grid sacing on the soltion strctres. Figre 7 shows the strctre from the DNS (solid) and from the TPB (circles) calclations for lanar geometry. In each case, the soltions were stoed when the shock location reached s (t) = mm. The arrival times of the two calclations is seen to be aroimately the same (for DNS, t =.7µs; for TPB, t =.µs), the % relative difference being de to differences in the grid resoltions and to the modeling assmtions of the reaction zone by the TPB model. Note how well the rogram brn model catres the overall strctre. The only differences are seen in the density lot, where the DNS calclates a weak density jm downstream of the lead shock while the TPB calclations (with the coarser grid) does not, and in the shock region where the DNS calclations show a strong detonation rofile and the TPB calclations show a weak detonation rofile. We also ran long-time soltions, ntil the shock was located at s (t) = 9mm (Figre 9.). The arrival times of the two calclations have a relative difference of less than % (for DNS, t =.77µs; for TPB, t =.97µs). Again, note how well the rogram brn model catres the overall strctre. The major weakness of the TPB model, however, occrs when crvatre is resent. Figre shows the strctre from the DNS and from the TPB calclations for the case of cylindrical geometry. Since the TPB ses a Hygen s constrction to roagate the shock, we see that the arrival time of the shock to the location s = mm is mch qicker (t =.7µs) than that of the DNS calclations (t =.µs); this reresents roghly a % error in the arrival times. This large difference is not de to grid resoltion, bt rather to the TPB modeling of the shock seed sing a Hygen s constrction. Since Hygen s constrction over-estimates the seed of the shock when crvatre is resent, we also see noticeable differences in the soltion strctres downstream of the lead shock. As in the lanar case, we also ran long-time soltions, ntil the shock was located at s (t) = 9mm (Figre ); a close look at the strctre is shown in Figre ). The arrival times of the two calclations is seen to be converging (for DNS, t = 7.6µs; for TPB, t =.µs). In terms of the strctre, the rogram brn model does seem to catre rather well the overall strctre at the longer times. A closer look at the time behavior can be eamined by comaring the shock seed and the star states to those obtained from DSD theory (see Figre ). Note that the shock seed over-redicts the shock seed obtained from DSD, and that the star states are only asymtotic to the star states obtained from DSD theory. The above reslts illstrates the strength and weaknesses of sing the traditional rogram brn model to catre the hysics of real detonations. For the lanar case, the shock is roagated at the correct CJ seed, and the strctre is reresented well with only / th the nmber of grid oints. This reresents significant comtational savings. However, when crvatre is resent there are major differences in not only the shock location bt also in the strctre of the soltion. These differences are de to the fact that Hygen s constrction over estimates the

18 8 BDZIL STEWART AND JACKSON FIG. 8. Plot of the strctre for lanar geometry at s(t) = mm. Circles corresond to the TPB model (t =.µs), and the solid crve to DNS (t =.7µs) FIG. 9. Plot of the strctre for lanar geometry at s(t) = 9mm. Circles corresond to the TPB model (t =.97µs), and the solid crve to DNS (t =.77µs).

19 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS FIG.. Plot of the strctre for cylindrical geometry at s(t) = mm. Circles corresond to the TPB model (t =.7µs), and the solid crve to DNS (t =.µs) FIG.. Plot of the strctre for cylindrical geometry at s(t) = 9mm. Circles corresond to the TPB model (t =.µs), and the solid crve to DNS (t = 7.6µs).

20 BDZIL STEWART AND JACKSON FIG.. Blow- of the shock strctre shown in Figre for cylindrical geometry at s(t) = 9mm. Circles corresond to the TPB model (t =.µs), and the solid crve to DNS (t = 7.6µs). D n *.... *. FIG.. (circles). * Plot of D n and the star states as a fnction of for DSD (solid) and TPB

21 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS seed of the roagating shock. Since crvatre is resent in almost all engineering devices, it is essential to roerly take into accont effects de to crvatre. It is this weakness that we address in the sbseqent section of this aer.. MODIFIED PRESSURE-BASED PROGRAM BURN MODELS (MPB) In the revios section we have seen that when crvatre is resent, the traditional ressre-based rogram brn model is deficient in that the se of Hygen s constrction over estimates the seed of the detonation front, leading to significant differences of the shock location and the strctre between the DNS and the TPB simlations. A simle modification can be made by etending the theory to inclde a shock seed which is crvatre deendent, as is fond in DSD theory. In articlar, we modify the brn-times to inclde the crvatre deendence d s dt = D n(), () and comte D n according to DSD theory, given a articlar eqation of state. Once this modification has been made, the net natral qestion arises as to how best model the hysical reaction zone? In the following sbsections we resent varios models aimed at imroving the TPB model when crvatre is resent... Model I or MPB- In this model we modify the stream internal energy to accont for crvatre affects, namely e = D n (γ ), () where D n = D n () is the seed of the front with crvatre deendence determined from DSD theory (see Section for details). We refer to this modification of the brn-times and the stream internal energy sing DSD theory as the modified ressre-based rogram brn model I, or MPB-. Figre shows reslts for the condensed hase elosive described in [] in cylindrical geometry. Note that a simle change in the way the brn-times are comted and in the definition of the stream internal energy can lead to significant changes in the errors. As before, the soltions were stoed when the shock location reached s (t) = mm. The arrival times of the two calclations is seen to be aroimately the same (for DNS, t =.µs; for MPB-, t =.µs), the 7% relative difference being a major imrovement when comared to the % relative difference in the arrival times between TPB and DNS. Comaring Figres and, we see that the MPB- catres the overall hysics better than the TPB model. The long time soltion, when s (t) = 9mm, is shown in Figres and 6, and shold be comared to Figres and, resectively, from the TPB model. However, the MPB- fails to catre the correct sonic (or star) states. Catring the correct star states is an imortant indicator of how well a given scheme does since both the strong detonation and the weak detonation shold terminate at this oint. We lot in Figre 7 the star states as comted here from the nmerical

22 BDZIL STEWART AND JACKSON FIG.. Plot of the strctre for cylindrical geometry with =mm at s(t) = mm. Circles corresond to the MPB- (t =.µs), and the solid crve to DNS (t =.µs). simlations to the star states determined from DSD theory. There is still an naccetably large discreancy in the star states. This shows that althogh the correct seed can be modeled sing DSD theory, the overall strctre is still not correct. This is a major deficiency of the model. We show in Figre 8 the star states for a grid resoltion of =.mm. In these calclations the energy released is still over a single grid oint, so redcing redces the effective reaction zone. Alternatively, one cold kee the reaction zone fied so that redcing wold imly more oints in the reaction zone; we have not done this comarison bt lan to do so in the ftre. Note that there is better agreement in the star states. Also note that the oscillations in the shock seed D n observed in Figre 7 have been redced by grid resoltion. A frther refinement wold violate the sirit of the rogram brn model, and so no frther grid refinements were carried ot. One final comment. In the mid-99 s Bdzil and Stewart modified major TPB codes at Los Alamos National Laboratory to inclde crvatre deendence sing DSD theory (which we referred here as the MPB-). The reslts of their work was not blished at the time and to date, no other comarisons between TPB and MPB- to DNS have been blished. To or knowledge this is the first discssion of these models. Blow- of the shock strctre shown in Figre for cylindrical geometry with =mm at s (t) = 9mm. Circles corresond to the MPB- (t = 6.76µs), and the solid crve to DNS (t = 7.6µs).

23 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS FIG.. Plot of the strctre for cylindrical geometry with =mm at s(t) = 9mm. Circles corresond to the MPB- (t = 6.76µs), and the solid crve to DNS (t = 7.6µs) FIG. 6. Blow- of the shock strctre shown in Figre for cylindrical geometry with =mm at s(t) = 9mm. Circles corresond to the MPB- (t = 6.76µs), and the solid crve to DNS (t = 7.6µs).

24 BDZIL STEWART AND JACKSON D n *.... *. * FIG. 7. Plot of D n and the star states as a fnction of for DSD (solid) and MPB- (circles) with = mm. 9. D n 7 *... *. * FIG. 8. Plot of D n and the star states as a fnction of for DSD (solid) and MPB- (circles) with =.mm.

25 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS.. Model II or MPB- In this model we kee the secification of the stream internal energy e = D CJ (γ ), () bt modify the length scale of the rogram brn reaction zone to mimic the reaction zone thickness of the hysical roblem. That is, in the TPB model, the rogram brn reaction zone thickness is ket at one grid cell, indeendent of grid resoltion. Ths, as the grid size becomes smaller, so does the rogram brn reaction zone thickness. We modify this by re-assigning a rogram brn reaction zone thickness which has a length scale of aroimately the same size as the hysical roblem, sch that as the grid size becomes smaller, the rogram brn reaction zone stays fied and the nmber of comtational cells within it increases. In articlar, if L is the rogram brn reaction zone length, then we select a vale of k, the nmber of comtational cells within the zone, sch that k = L. For the condensed hase eamle, the hysical reaction zone length is mm when no crvatre is resent, and slightly less when crvatre is resent. We fi L =mm for simlicity. We refer to this modification of the brn-times sing DSD theory and distribted rogram brn reaction zone thickness as the modified ressre-based rogram brn model II, or MPB-I. We show in Figres 9 and reslts for two different grid resoltions. In the first figre the grid resoltion is mm and so we chose k =, giving two gird oints within the reaction zone. Note that the shock strctre is qite different from that of Model I, MPB- (comare Figre ). When the stream vale of e deends on crvatre, the strctre looks like a weak detonation; when e is fied, the strctre looks like a strong detonation. Decreasing the grid size to.mm, and hence increasing k to 8, not only catres the overall flow strctre better than the coarse resoltion, bt also better catres the detonation strctre. The same is tre at the longer times, where we show a blow of the strctre for a grid resoltion of mm with k = (Figre ) and a grid resoltion of.mm with k = 8 (Figre )... Model III or TPB- In this section we resent another class of models aimed at imroving the TPB models when crvatre is resent. The eqations governing this new class of models are given by where U t + F = G + QR δ( s(t)), () U =[,, E] T, () F = [, +, (E + ) ] T, (6) G = j [,, (E + ) ] T, Q =[Q,Q,Q ] T, (7)

26 6 BDZIL STEWART AND JACKSON FIG. 9. Plot of the strctre for cylindrical geometry with = mm at s(t) = mm, with k =. Circles corresond to the MPB- (t =.µs), and the solid crve to DNS (t =.µs) FIG.. Plot of the strctre for cylindrical geometry with =.mm at s(t) = mm, with k = 8. Circles corresond to the MPB- (t =.µs), and the solid crve to DNS (t =.µs).

27 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS FIG.. Plot of the strctre for cylindrical geometry with = mm at s(t) = 9mm, with k =. Circles corresond to the MPB- (t =.µs), and the solid crve to DNS (t = 7.6µs) FIG.. Plot of the strctre for cylindrical geometry with =.mm at s(t) = 9mm, with k = 8. Circles corresond to the MPB-I (t =.µs), and the solid crve to DNS (t = 7.6µs).

28 8 BDZIL STEWART AND JACKSON and again E is the total energy defined by E = (e + ), (8) and e is the internal energy. Secific choices for e will be given below. Note that R δ( s(t)) is a delta fnction centered on the rogram shock locs = s (t) and is assmed to be known. The geometric sorce term G is identical in its first three comonents to its DNS conterart. The choice of Q and R δ will be made later in the corse of the analysis. We will consider two models for the internal energy. The first model mimics the TPB model and is written, for an ideal gas, as e = /Y (γ ), (9) where γ is the ratio of secific heats and Y is the brn fraction with Y. The stream vale of the internal energy is given by (). We will refer to this as Model IIIa or TPB-a. The second model is the standard eqation of state withot the brn fraction, and for an ideal gas is written as e = (γ ). () The stream vale of the internal energy is given by e =, consistent with the strong shock aroimation. We will refer to this as Model IIIboe TPB-b. To comletely secify the rogram brn PDEs one mst identify the sorce term strength Q. Varios secifications are made and analyzed below. One way to determine vales of Q is to make the qasi-steady assmtion and neglect the elicit deendence of crvatre in the rogram brn eqations. The lead shock is taken to be at = s (t) with seed D n, which can deend on crvatre and is given by the D n, relation from DSD theory. Across the shock we allow for doses to the mass, momentm and energy, and the jm conditions across the rogram-brn shock are given by [( n D n )] = Q [Y ], () [ n ( n D n )+] =Q [Y ], () [E( n D n )+ n ]=Q [Y ], () where [φ] =φ o φ, and the star states are determined sing DSD theory. Note that in writing down these normal jm conditions we assmed that R δ = dy/dn, where n is the normal coordinate. Since the shock location is known, and both the stream states and the star states are known, the jm relations become formlas for elicit evalation of the doses Q. Evalating the jms leads to the following secifications for the comonents of Q

29 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS 9 6 Model IIIa Model IIIb Q 6 Q FIG.. Plot of Q (solid), Q (dotted) and Q (dashed) as a fnction of for models TPB-a and TPB-b, resectively. Q = o D + ( D n ), Q = + ( D n ), Q = o e o D n + E ( D n )+. () A lot of the star states was given reviosly in Figre for the condensed hase elosive eamle given in []. The vales of Q as comted from these formlas is shown in Figre for both TPB-a and TPB-b, resectively. Reslts for the two models are given in Figres throgh 7 for TPB-a and in Figres 8 throgh for TPB-b. These reslts are in qalatative agreements with TPB (Model I). 6. CONCLUSIONS We have resented a comrehensive review of the traditional rogram brn algorithm, and have comared soltions to those of a direct nmerical simlation. It was shown that if crvatre is resent, the traditional rogram brn alogrithm overredicts the shock seed. A slight modification to the brn times, based on detonation shock dynamic theory, can correct the shock seed difficlty. Varios models are resented and comared to DNS; overall, the reslts of Model II (constant stream vale for the internal energy, fiing the rogram brn reaction zone length) give reslts which srrisingly catre the DNS strctre, even with a grid resoltion of abot times larger than that of the DNS. We are crrently investigating these models in two-dimensional geometries and etension to real rodct eqations of state. ACKNOWLEDGMENT The work of D. S. Stewart was sorted by a contract with the Deartment of Energy, DOE- LANL nder the resent contract I9-9 and revios contracts with LANL. T. L. Jacson was sorted by a revios LANL contract and the DOE contract that sorts the UIUC ASCI center, Center for Simlation of Advanced Rockets. J. B. Bdzil is sorted at Los Alamos by the Deartment of Energy.

30 BDZIL STEWART AND JACKSON FIG.. Plot of the strctre for cylindrical geometry with =mm at s(t) = mm. Circles corresond to the model TPB-a (t =.µs), and the solid crve to DNS (t =.µs) FIG.. Plot of the strctre for cylindrical geometry with =mm at s(t) = 9mm. Circles corresond to the model TPB-a (t = 6.µs), and the solid crve to DNS (t = 7.µs).

31 PROGRAM BURN BASED ON DETONATION SHOCKDYNAMICS 9. D n 7 *... *. * FIG. 6. Plot of D n and the star states as a fnction of for DSD (solid) and model TPB-a (circles) with = mm. 9. D n 7 *... *. * FIG. 7. Plot of D n and the star states as a fnction of for DSD (solid) and model TPB-a (circles) with =.mm.

32 BDZIL STEWART AND JACKSON FIG. 8. Plot of the strctre for cylindrical geometry with =mm at s(t) = mm. Circles corresond to the model TPB-b (t =.µs), and the solid crve to DNS (t =.µs) FIG. 9. Plot of the strctre for cylindrical geometry with =mm at s(t) = 9mm. Circles corresond to the model TPB-b (t = 6.µs), and the solid crve to DNS (t = 7.µs).