THREE DISTINCT STAGES OF THE UNSTEADY FLOW BEHIND THE SHOCK WAVE FORMED BY THE NORMAL REFLECTION OF A PLANAR STRONG POINT BLAST WAVE FROM A WALL

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1 ACTA MECHANCA SNCA, Vol. 2, No.l, Mar., Science Press, Beijing, China 27 THREE DSTNCT STAGES OF THE UNSTEADY FLOW BEHND THE SHOCK WAVE FORMED BY THE NORMAL REFLECTON OF A PLANAR STRONG PONT BLAST WAVE FROM A WALL Ting Aili (Department of Mathematics, Beijing University) ABSTRACT: The method of characteristics with shock fitting and a family of continuous C- characteristic lines are used for the analysis of the normal reflection of a planar strong point blast wave from a wall. Several techiques are devised in order to increase the accuracy of calculation for the problem including two complex singularities. New data for the flow field near the wall till very late time are obtained. Descriptions are given for three stages, each possessing distinct gasdynamic features such as the orientation of characteristics, paths of fluid particles, distribution of entropy, and so on. KEY WORDS: unsteady flow, reflecting shock, shock wave, local sound speed.. NTRODUCTON An unsteady reflecting shock wave occurs during the normal reflection of a planar strong point blast wave from a wall. t is almost impossible to solve the whole Eulerian equations for the reflecting shock and the flow field behind it because of the nonlinearity of the equations and the complexity of the problem. However, there were some numerical work and some asymptotic analyses for the problem in China in the past few years. The singularities occur as the reflecting shock wave has just left the wall and approached the explosive center (Refs. [1] [2]). Two different kinds of numerical methods have been used, i.e., the shock fitting method, including the characteristics method and the separating singularities method, and the shock capture method such as the van Jeer second-order monotonic upwind scheme, the MacCormack scheme, the Ru~anov scheme, the Godunov scheme, the random choice method and so on. But most of these methods can only be used to calculate the situation shortly after reflection. The difficulty lies in the singularity appearing in the flow field. As the reflecting shock wave approaches the explosive center, the flow field around there varies very strongly: density becomes extremely rare, temperature and local sound speed increase infinitely, and the speed of the reflecting shock becomes extremely high but its strength very weak. Thus, the numerical errors are large and the shock smearing very wide. n addition, the time step limited by the Courant condition becomes very small due to the extremely large density and it may even stop the calculation immediately. n 1983, with some special treatment, numerical results passing the explosive center were obtained for the first time by the separating singularities method (by Wu et al [3]). They have found a new left-propagating shock wave and calculated the approximate flow field after the reflecting shock reaching_ the Received 11 June 1984.

2 28 ACTA MECHANCA SNCA 1986 explosive center by the time t = 2.6t B (Here t 8 is the time at which reflecting shock reaches the explosive center). The results presented in this paper indicate that by using charcteristic cells together with shock fitting, a flow field can be obtained near the wall till very late time t = 31t B (Figs.1 & 4)and there is no previous report of getting similar outcome with other methods. With the characteristics method, we may give a much better physical picture of the problem since it helps to draw clearly the influence zones and dependence zones with high accuracy and less computer time. Moreover, for this complicated problem, we have designed several special skills for calculating the flow field in different stages. From the numerical results, we have found that the flow field behind the reflecting shock can be explicitly divided into three distinct stages with different features, i.e. the early, intermediate and late stages. They are separated respectively by two C-characteristics starting from reflecting shock at about 1/5 and 3/5 of the distance between the wall and the explosive center (Fig. 4). Three stages possess distinct particle paths, distributions of entropy and different shapes of influence zones and dependence zones. n the third stage, particularly, the speed of reflecting shock becomes higher and higher while its strength weaker and weaker, tending to equalize the pressures on both sides of the reflecting shock because of the limit state at the explosive center with extremely high temperature and low density. tb... B 0 N t~r-~ m- ~r reflecting shock -- t o points delected Fig.1 Fig;2 C partic~ path ~ 4 e 3 J~ 2 Fig. 3(a) cells reflecting shock Fig. 3(b) 2 3 Ms Step contralled

3 Vol. 2, No.1 Ting Ailh Three Distinct Stages of the Unsteady Flow ~ 1 late (C_)ll particle path ,5 t 8 2 early 1 0 intermediate J 1/5 R o 3/5R o,, 1 ~ J Ro Fig.4 Three stages in zone H. PROBLEM POSED The problem to be solved is a one-dimensional, unsteady, adiabatic flow of inviscid, perfect gas with a constant specific heat. The governing equations are omitted here. The coordinates xlsed are shown in Fig. 1, Assume that planar strong point explosion sets out on the plane x = Ro at t = - tl(tl > 0) with the releasing energy Eo per unit area and the blast wave impinges on the wall at x = 0 and reflects from there after t seconds. n Fig.l, the incident blast wave is AO, the reflecting shock wave 013; there are three regions on the right half of x-t plane. H is a still zone before the strong blast wave, in which flow velocities and pressures are set to be u = p -- 0, density p = po > 0, specific inner energy e = 0 with the opposed pressure neglected; -- --the zone located behind the strong blast wave, in which the selfsimilarity solution of the planar point blast wave can be used; the zone located behind the reflecting shock, in which is an unsteady flow with neither uniform entropy nor self-similarity. To determine the shape of the reflecting shock 013 and to find out the whole flow field in are the objects of our numerical work, but the top right boundary of H is still awaiting to be fixed.. NUMERCAL METHOD The calculation starts from point 0. 0 is a singular point. The pressure, density and velocity tend to three different limit values respectively as the point 0 is approached

4 30 ACTA MECHANCA SNCA 1986 along three different directions of,,. n this paper, the asymptotic solution of the reflecting shock at the point 0 given in Ref. [1] is used as the initial value. Firstly, a C- characteristics MN very close to 0 is taken, on which, the point N is the initial point at the reflecting shock and other two points on MN are again chosen as the initial points, while the point M is that on the wall. Then from MN the characteristic cell and the reflecting shock path OB can be constructed step by step. The features of numerical method with some special skills are as follows: 1. High resolution, large time steps and wide range. By using the method we can save computer time with the given accuracy compared with other numerical methods. f it is compared with the characteristics method of computing at every same time level, it is not restricted by the most serious Courant condition of that at the same time level. f it is compared with the finite difference method (or various shock capture techniques), then it has the advantage of determining the most accurate shock location with at least 4--6 times larger time steps. n Fig.l, from 0 to B, more than 400 C-characteristics have been used. The points on each C- is increasing initially from three points and gradually to about 400 mesh points. The method can cover the large area near the wall until very late time as the cells are constructed in the physical influence zone. We have calculated up to about t = 31tB. Beyond this, more careful work needs to be done but there is not enough time for doing it and figuring out the upper boundary of. 2. Point deleting. The C+ characteristics behind the reflecting shock always go faster than the reflecting shock. As they get very close, a new intersecting point of C+ and C- is very ~se to a new reflecting shock point. n this case, it is hard to determine the new shock point series any more, so that we delete the intersecting point. Actually, that means deleting the whole C + line passing through the intersecting point to avoid using other special complicated skill. 3. The variable steps on the reflecting shock. The calculation of the reflecting shock point is of great importance. Just before it, the Sedov solution in varies violently, and the errors at that point will propagate to the place behind the shock. Thus the steps on the reflecting shock are chosen so that the characteristic cell next to the reflecting shock form a proper shape to make the particle path passing the point C as close as possible to the diagonal CB of the cell (Fig. 3(a), cell ABCJt, Jt new shock point, J old shock point), With this arrangement, well-defined velocity and entropy distributions near reflecting shock can be obtained. At the same time, large errors due to the slopes of C_ approaching to - and the slopes of C+ approaching to + ~ near the explosive center can be avoided. After some tests, a step control expression of reflecting shock points varying with the reflecting shock Mach number Ms is derived (Fig.3(b)). 4. The iteration for calculating reflecting shock points has the advantage that every new shock point has come out with no more than two iterations. Within each iteration, the calculation of the location and speed of the new reflecting shock point embeds the calculation of reflecting shock Mach number. 5. The Sedov solution in is hard to calculate accurately, so we use its asymptotic expansion at singularity point 0 and B, developed by Professor Huang Dun. V. NUMERCAL RESULTS --THE FEATURES OF FLOW FELD N ZONE The data used are as follows:

5 Vol. 2, No.1 Ting Aili: Three Distinct Stages of the Unsteady Flow 31 E 0 = x K 10~J/m 2, t = x 10-Sks, R0 = 0.1km, P0 = 1.29kg/m, ~ = 1.4 This set of data is probably equivalent to the case of the explosion of 0.7kg per square metre TNT explosive. K is an adjustable parameter. By the above data, the pressure on the wall as the blast wave attacks is x 108N/m 2 ~ 100atm, and the pressure on the wall after the reflecting shock has just emitted is x 108N/m 2 ~ 800atm. Without loss of generality, we only have to examine the case of K-1. The initial values of reflecting shock used are: x = 0' m, t = K 10-5s. The new results obtained are: the flow field near the wall till very late time is secured, which is ten times later than that of other methods. Bisides, we have found that the evolution in the zone can be divided into three stages: early, intermediate and late ones, according to various directions of C_ characteristics and other features. n Fig. 4, lines (C_)~, (C_)u, (C-)m are the boundaries of different stages. 1. The early stage. t is a curved triangular zone whose boundaries consist of the wall, the reflecting shock and a straight line (C_) emitting from the reflecting shock at about 0.2R 0, and one of its vertex is the point 0. At this stage, the reflecting shock Mach number Ms decreases from 2.6 to 2.0, while the shock speed increases from 1167 m/s to 3195 m/s. n the early stage, all of the paths of gas particles are straight lines. t indicates that every particle is moving at a constant speed and no forces is applied on it. And the scattered particle paths are shaped into a fan-like form. t demonstrates that the gas is expanding over and over and the isentropic lines are radiating-like ones. We apply the x direction momentum equation of gas on the wall. Due to u(0, t) = 0 and ~u/~t = 0 on the wall, by substituting these two conditions into the equation, ~p/~x = 0 is obtained. t shows that the pressure does not vary with x at time t near the wall, which agrees with our numerical results. However, the region with no acceleration of particles is rather wide. 2. The intermediate stage. t is the zone following the early stage and preceding (C_)tl line emitting from the place that the reflecting shock reaches about 0.6R o. n this stage the reflecting shock Mach number Ms decreases continuously from 2.0 to 1.48 while the shock speed increases from 319~ m/s to 8270 m/s. Every physical quantity varies rapidly, not only the speed of reflecting shock in~2c~sing greatey, but the particles also moving faster and faster so that the particle 'p~aths and isentropic lines gradually bend to the right and are no longer straight lines. n addition, the local sound speed and temperature of the gas both enhance rapidly, but its density declines very fast, and its pressure also decreases along the x direction. n summary, the gas becomes rarer and rarer and its temperature higher and higher. From Fig.4, it can be seen that the slopes of the (C_hl characteristics at both edges, i.e., at the wall and the reflecting shock, are negative and their absolute values are very large. However, in the middle parts of (C_), there is a point with zero slope, which indicates that the velocities of a few particles are accelerated to the local sound speed and that the late stage begins. 3. The late stage. t is' the zone following the intermediate stage and preceding the (C-)u line emitting from B. At this stage, Ms decreases continuously from 1.48 to nearly 1, the limit value, and the shock speed increases from 8270m/s to infinity. The singularity of

6 32 ACTA MECHANCA SNCA 1986 strong point explosion at explosive center can be seen clearly when the velocity and the speed of the reflecting shock approach infinity. The reflecting shock tends increasingly to be parallel to the x axis on the x-t plane. Because the gas is rarefied to near vacuum and its temperature tends to infinity, the reflecting shock Mach number Ms approaches unity. At this stage there is a rather large size of local sound speed area in which the velocities of particles is eguivalent to local sound speeds. And all of the C_ characteristics have inflection points and thus have a vertical part of zero slope. The data in the late stage are expected to be helpful in analyzing the new leftpropagating shock emitting from B. The asymptotic properties of the explosive center before the reflecting shock reaches it are; according to Ref. [2] u 1 ~ r -* 0, Pl ~ r~ (slow varying with time), Pl ~ r 2"5 ~ 0, a ~ r ~ where r is the distance to the explosive center x = Ro, al is the local sound speed. And the asymptotic properties of the explosive center after reflecting shock reaches it are, according to the same paper, u2 ~ r -~ -~ oc, P2 ~ r~ (slowly varying with time), a2 ~ r -1"25 --* ~ > u2, P2 ~ r2"5 ~ 0. The numerical results can be explained with the above analysis. The slopes of every C_ characteristics approach - ~ as they have just left the reflecting shock. The reason is that the increments of the particle velocity and local sound speed at the reflecting shock are of different orders and the order of the latter is much higher. Later, they cancel each other at the local sound area due to'the decrease in their orders. As C_ reaches the wall, the particle velocity decreases to zero and its slope turns to negative again. On the other hand, in the later period, the slope of C+ characteristics which intersects with reflecting shock approaches to + ~ at that point, and the slopes of the other C+ 's which do not intersect with the reflecting shock are smaller because both the particle velocity and the local sound speed are decreasing along a C_ line. concerning the propagation of shock in the non-unifor-m medium, approximate calculating method proposed by chester and chisnel s around 1956 gave a surprisingly accurate result. Witham had summarized it as a characteristics rule in Ref. [4], by which the relation among physical parameters on the C+ characteristics is applied to the reflecting shock. As the ratio of the pressures before and behind the reflecting.shock is nearly 1, theapproximation is rather good. An expression derived from above viewpoint in Ref. [2] is as follows: (Ms 2-1)r-~ 2 = const, as r--, 0. By using this equation to check our numerical results at the late stage, the initial constant decreases monotonically slower and slower, and finally it approaches a real constant about 5.6. The relative errors of the characteristic relation on the reflecting shock are less than 0.2% as the spacial step is shorter than Ro/200. The whole numerical results have described the evolution process of reflecting shock from emitting singularity to explosive center singularity. Three local pressures of x = 0, 0.1R o, 0.5R o have been calculated with interpolation. t can be seen that the pressure at the wall is below 1/4 atm and the density there is only 7% of that before explosion as t = 31t B. V. CONCLUSON The existing numerical results of the normal reflection of a planar strong point blast wave either require a larger amount of computing work or bring about unsatisfactory accuracy. Furthermore, it is hard to make a dynamical analysis for the flow field after reflection. n this paper, a characteristic cell technique together with some uncomplicated approaches based on physical features is presented in order to get more accurate numerical results with ease. t is for the first time that the flow field fill very late time has been obtained. After some dynamical analyses, we have found that

7 Vol. 2, No.1 Ting Aili: Three Distinct Stages of the Unsteady Flow 33 the flow field behind the reflecting shock can be divided into three different stages with 9 u distinct gasdynamic features. This analysis provides a reliable basis for calculating the flow field disturbed by the reflecting shock passing through the explosive center in the future. The author expresses her deep gratitude here to Professor Huang Dun in Peking University and Professor Li Yinfan and late Cao Yiming in the Computer Center of Academia Scinica for their valuable assistance and suggestions during discussions and computation. REFERENCES [11 Huang Dun, Explosion and Shock, 1, 1 (1981) (in Chinese). [2l Huang Dun, Rong Sheng, On the propagation of a shock wave near a high temperature singularity of singularity of strong planar explosion. Proc. of the 2nd Asian Congress of Fluid Mechanics. (Oct. 1983). [3] Wu Xionghua, Huang Dun and Zhu Youlan, Explosion and Shock, 3, 2 (1983). [4] G.B.Witham, F.R.S, Linear and Nonlinear Waves, John Wiley and Sons (1974).