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ÿþShock Waves (1998) 8: 23 28 Shock wave trapping B.W. Skews, M.A. Draxl, L. Felthun, M.W. Seitz School of Mechanical Engineering, University of the Witwatersrand, Johannesburg, South Africa Received 4 August 1995 / Accepted 12 December 1995 Abstract. This paper presents a two-dimensional investiga- tion into the effectiveness of trapping shock and blast waves in a duct in order to enhance attenuation, by placing an ar- ray of opposing wedges in the channel. The concept of the wedge arrangement in the trap is to allow easy shock wave entry, with weak reflected shocks, into the trap, but stronger internal reflected shocks if a wave is re-emering. The inter- nal reflections, including those of vortices shed from earlier Fig. 1. Flow around a triangular bar shock passage, result in strong shock attenuation. Differ- ent wedge placements, wedge angles, and area blockages are investigated numerically, as well as experimentally for a particular case, using pressure measurement and schlieren photography. Key words: Shock wave attenuation, Reflection, Safety Fig. 2. Maze of triangular rods in the path of a shock wave with a weak reflected wave) but would then see greater re- sistance in emerging in both directions, and the greater part of the shock would be reflected back into the space between 1 Introduction the barriers. Even with non-directional barriers, Frolov et al. (1989) have shown that quenching is increased if two In order to ameliorate the effects of blast waves from an systems of barriers are separated by an air gap. The barrier external explosion, entering buildings or other structures geometry used is based on the fact that flow resistance, and through air conditioning ducts or other apertures, or to min- reflected shock strength, are less for a triangular body with imise the effects of accidental blasts in industrial pipe work the body positioned with its apex facing upstream than when and galleries, it is necessary to take steps to attenuate the the base faces the flow (Fig. 1). strength of the wave. The performance of various permeable Shock tube tests using the grid arrangement shown in screens such as perforated plates, gauze, and beds of granu- Fig. 2, with 50 percent blockage and using equilateral tri- lar material have been used in the past to attenuate the shock angles as the elements, showed that for an incident shock wave (Mori 1975, Tong 1980, Frolov 1989). Grid arrange- Mach number of 1.27, the pressure rise across both the wave ments investigated in the past generally consist of a series reflected back from the above arrangement, as well as the of barriers or screens in a regular arrangement having both transmitted wave were less than 50 percent of that across the lateral and axial symmetry. incident wave. Schlieren photographs also showed that the Skews and Broido (1992) recently proposed an arrange- column of vortices shed behind each row of elements also ment of barriers which have a greater resistance to flow, and acted as a further wave reflection site. shock passage in one direction than the other. If a number of such arrangements are placed back-to-back then it was suggested that the incident shock could enter the space be- 1.1 Geometrical factors tween the two arrangements without too much difficulty (i.e. This paper considers a variety of geometrical influences on An abridged version of this paper was presented at the 15th Int. Colloquium the primary concept discussed above. Thus, for example, on the Dynamics of Explosions and Reactive Systems at Boulder, Colorado, from July 30 to August 4, 1995. Bird (1952) has shown that for a shock wave moving into 24 a converging channel, the shock wave energy transmittance increases with decreasing incidence angle of the shock front. Thus the strength of the reflected wave for the geometry indicated in Fig. 2 would be expected to reduce as the leading edge angle of the triangular elements is reduced. However this would be expected to have little, if any, effect on the wave strength change for waves propagating through the grid Fig. 3a e. Different wedge arrangements tested with Euler code. a Cho- from the opposite direction. Some experimental work with sen standard wedge arrangement, b narrow wedge arrangement, c compact an array of wedges with apex angles of 10, 15, 20, and 30 wedge arrangement, d opposing wedges, e assembled maze arrangement degrees has shown that this is indeed true, with the weakest reflected wave being found for the smallest wedge angle, with the same percentage of duct blockage and shock Mach number. Fig. 4. Arrangement for physical test The second factor which will have an influence is the degree of duct blockage, i.e. the size of the base of the triangular elements together with the number of elements, ing all three different angles. Pressure traces from the start as as a fraction of total duct area. This would also have an well as the end to the wedge arrangement were used to com- influence on the possibility of the flow choking between pare the reflected and transmitted shocks with the incident adjacent elements. A third factor is the distance between shock. successive rows of elements. This can in turn influence both For the first set of simulations the half base of the wedge the column of vortices shed, as well as the development of was taken as 0.2 of the pitch width (arbitrarily chosen as the wave back into a planar front, and thus on the reflection 1 m), and were laterally separated by two wedge lengths be- behaviour in the next column of elements. Finally the effect tween rows (leading edge to leading edge). For each of the of staggering the elements and not having them in distinct three wedge angles and two shock Mach numbers, two sim- rows needs to be considered. ulations were run, one with the shock entering from the left (the low resistance case), and one from the right, (the high resistance case) in order to obtain information on the effects 1.2 Computational parameters of the inlet and outlet legs. Target points were positioned at various points on the centre line of the model and pressure- In view of the large number of combinations of geometrical time traces generated. In the second set of simulations the factors the initial optimisation of the geometry was under- pitch width was reduced to 0.6 m (Fig. 3b), all of the other taken using an adaptive finite element flux-corrected trans- factors mimicking those of the first set. The third set was port Euler equation solver originally developed by Löhner similar to the first except that the lateral spacing between et al. (1985, 1987a, 1987b). A shock wave was simulated the wedges was reduced to one wedge length (Fig. 3c). In to travel through the wedge maze. This is not the same as a the fourth simulation an inlet and outlet leg were coupled blast wave as there is no theoretical pressure drop after the together, and were tested for a Mach 1.5 shock, and the initial shock. The main difference here is that the reflected three wedge angles (Fig. 3d). The final test consists of a shock has to move back into a region of sustained high pres- maze made up of all the different wedges. The shock first sure which is not the case with a blast wave. This correlation strikes two staggered 10 degree wedges, followed by two results in a conservative results as the blast wave could be staggered 20 degree wedges, in turn followed by three 30 broken up inside the maze and its transmission reduced as degree wedges. It has then reached the centre of the maze opposed to the shock wave which is driven by a sustained and encounters a mirror image of the geometry on leaving air flow behind it. This is particularly evident with the blast the maze (Fig. 3e). wave simulated (see Fig. 9c) where no significant blast was To verify the numerical results a physical wedge set with reflected and only a small portion transmitted. a 10 degree incident angle was tested and pressure traces as Geometric factors such as the wedge angle, their spacing well as schieren photographs obtained. In this case the pitch as well as the channel constriction play a major role in the width was taken as 28 mm (half the channel width of 56 mm behaviour of the blast or shock wave inside the channel. as a complete wedge was used) with no spacing between All these were considered in the optimisation of the blast the rows. The spacing between the inlet and outlet leg was barrier. This optimisation was done using a finite element chosen to be 50 mm. Reasons for these parameters were due Euler code by considering two different shock waves. A to the constraints of the physical shock tube used (Fig. 4). 1.5 Mach shock wave with subsonic flow behind it and a 3.0 Mach shock wave with supersonic flow behind it. Due to the axial symmetry of both wedges and the shock wave 2 Results and discussion reflections only two halves of the wedges were used in the numerical simulations to conserve computing power. 2.1 Comparing the effect of Mach numbers In order to obtain some understanding of the influences of the various factors five sets of simulations were run. These In the case of the incident shock striking the rising slope investigated the effect of different geometric arrangements, of the wedge the pressure traces could not be used for a spacing between the rows and a narrow channel, with the dif- meaningful comparison as the shock intensity was increased ferent wedge incident angles as well as a simulation combin- towards the centre by the gradual narrowing of the channel. 25 Fig. 5a,b. Pressure contour plots for Mach 3 shock wave in the compact 10æ% wedge arrangement. Delay time between plots 0.4 ms Fig. 6a d. Contour plots for a Mach 3 shock in the narrow 10æ% wedge arrangement. Delay time between plots: 0.5 ms The actual shape of the transmitted and reflected shock waves are very similar for a shock with a Mach number of 3 and 1.5. The Mach 3 shock is more strongly defined than Fig. 7. Optimum blast barrier design (half only) the Mach 1.5 shock and this is especially true for the shock wave reflected off the base of a wedge. The Mach 3 bow Table 1. Percentage of shock wave transmitted on striking the base of the shock is stronger and due to the supersonic air flow behind it wedges moves more slowly upstream than the Mach 1.5 shock. The speed of the Mach 3 bow shock is decelerating and it will Mach No 1.5 3.0 1.5 3.0 1.5 3.0 First battery Second battery Third battery eventually form a stable stationary shock further upstream Angle as opposed to the shock travelling into subsonic flow. The latter will continue to move upstream. 10æ% 85 88 85 80 86 78 20æ% 80 82 81 87.5 81 71 30æ% 93 97 86 92 62 60 2.2 Comparing the effect of total channel area reduction This resulted in a very high localised pressure rise to be Increasing the change in area from 5:4 to 2:3 significantly indicated by the traces which was often larger than the initial reduces the amount of shock transmitted. This holds for both pressure rise. the incoming and outgoing shock. Here again the position As a result only the simulation representing a shock wave of the pressure traces have to be considered. The narrower travelling out of the maze were considered to analyze the channel allows the shock to reach a uniform strength more effect of shock wave Mach number. This is because the quickly than the wider one. As a result the pressure trace of pressure trace was taken at the end of the wedge where the narrower channel is not as affected by local shock wave width of the channel has been gradually restored. Here the behaviour as is the pressure trace of the wider channel. shock wave has reached a more uniform magnitude across The entire improvement of the reduction of shock wave its width as it had more time to form into a planar front. transmittance cannot however be attributed solely to the dif- Table 1 indicates the percentage of shock wave strength ference of localised effects. This is because a shock wave out of the reverse simulation. The percentages were obtained profile is considerably more complex in the narrower chan- by measuring the pressure rise from the trace plots for the nel with an area ratio of 3:2. incident wave and the transmitted wave. This fraction is then The narrower geometry causes the bow shock to quickly given as a percentage. affect the entire width of the channel and develop into a The first and second battery of tests indicate an increased planar reflected shock. Furthermore, part of the reflected bow transmission of shock strength for a higher Mach number. shock is reflected off the walls of the channel and develops The percentages of the 10æ% wedges in the second battery of into a horizontal shock being reflected back and forth. In a simulations and the percentage of shock transmitted in the real situation this reflected shock would be quickly dissipated third battery of simulations indicate a greater transmission of and lose much of its energy. The narrower channel thus the Mach 1.5 shock. These two differences in values seem to dissipates the shock more efficiently (Fig. 6a d). correspond to the formation of a secondary reflected shock The narrower channel, however also restricts the amount front across the entire width of the channel in the Mach 3 of shock wave to be transmitted into the maze causing strong case (Fig. 5a,b). reflected shocks to develop. 26 Table 2. Reflected pressure as percentage of initial shock Table 4. Transmitted and reflected shock strength for complete maze section Mach No. 1.5 1.5 Angle Transmitted Reflected Wide Narrow Angle channel channel 10æ% 78 % 43 % 20æ% 74 % 58 % 10æ% 21 100 30æ% 63 % 77 % 20æ% 29 32 30æ% 39 48 Theoretical 36.4 % > 43 % combination < 77 % Table 3. Percent transmitted on reverse run Simulated 42 % 32 % combination 75 % Mach no. 3 3 1.5 1.5 incident spaced compact spaced compact angle geometry geometry geometry geometry are obtained. The shock wave in the compact geometry did 10æ% 88 80 85 85 not have as much time to expand and strengthen at its edges 20æ% 82 87.5 80 81 as when a spacing is provided between the rows of wedges. 30æ% 92 97 86 93 2.4 Comparing the effect of incident angles The Mach 3 shocks are followed by a supersonic air flow. This results in virtually stationary reflected shocks to As the pressure traces depend considerably on the placement develop. These are often not indicated on the pressure trace of the pressure trace point a fourth battery of tests was run. as they often did not reach the location of the pressure trans- This time the traces were placed at a constant distance further ducer. The flow behind the Mach 1.5 shock is subsonic. As away from the wedges. This means the shock waves have a result pressure rises can be transmitted upstream. had the same time to stabilise. The values obtained can thus The stronger reflected pressure rise of the narrow channel be compared. A complete maze section was simulated with arrangement indicate clearly that more of the shock wave is two wedges pointing away from the centre on each side as reflected in a narrower channel with the same size of wedge. seen in Fig. 3d. The percentages in Table 2 are the ratios of the highest Table 4 represents the ratios in percent of the reflected pressure rise registered due to reflection of the shock and and transmitted shock to the initial shock. The highest stable the pressure rise of the incident shock. As the running time peak value of the reflected shock was chosen to represent and the dimensional lengths are equal at a particular angle. the degree of reflection. This ratio was used as a indicator Table 2 can be used to evaluate the effect of the area change of the reflected shock strength. for each wedge angle. The values in Table 4 are representative of a Mach 1.5 The reflected shock wave with the 10æ% incident angle shock. The values for a Mach 3 shock were not simulated for the narrow channel seems excessively large compared to as they are very similar. Table 4 shows a decreasing shock the others. It should be below 32% to keep with the trend wave transmission and an increasing reflection of the shock indicated by the wide channel. with increasing incident angle as was expected. Furthermore a theoretical value for the combination of all three types of wedges was calculated by simply multiplying 2.3 Comparing the effect of compact design the percentages of previously simulated maze sections. This resulted in a prediction with an error of only 13% for the Table 3 represents the ratios of the pressure rise of the shock transmitted shock. transmitted to that of the incident shock as a percentage. Predicting the reflected shock is more difficult as it de- From the table it would seem that the compact design is less pends mainly on the initial wedge shape. As can be seen efficient in reducing the transmittance of the shock. This is from the table the values were inside the correct range. The only contradicted by the Mach 3 shock and the 10æ% incident two values given in the table for the reflected shock repre- angle. sent two reflection shocks separated in time. The first value It must be remembered that the shock wave has to travel a considerably longer distance for the wedges with a 10æ% being the first reflected shock on its own while the second value incorporates the second shock. incident angle before reaching the point at which a pressure trace is taken. The simulation with the 10æ% incident angle wedges is thus the only data representing the characteristic 2.5 Optimum design of reducing the longitudinal space between the wedges. The other results are inaccurate as the shock wave did not have enough time to form a uniform front and is strongest at the The accuracy of the prediction can be improved by adding centre. some space between the maze sections of different angles The Mach 3 data indicates an significant improvement to allow the transmitted shock to expand before striking the in shock strength reduction for the 10æ% wedges. The Mach next section. In other words one space every two rows of 1.5 data also indicates a improvement as the same values wedges as can be seen in Fig. 7. 27 Fig. 9a c. Pressure traces for shock and blast wave traversing the wedge arrangement of Fig. 4. Two transducers are positioned upstream of the trap and two downstream. a Simulated shock wave, b experimental shock wave, c simulated blast wave Fig. 8a f. Schlieren photographs of a Mach 1.34 shock wave in a wedge maze unit. Uneven profile of wedges is due to slight excess glue fastening them to the window. Wedge half angle is 10æ% The blast barrier can thus be tailored to meet the required reduction of the shock strength by simply adding more rows of wedges to make up a wedge maze. 2.6 Method of barrier design 1) Obtain results for single wedge maze units (wedges fac- ing in opposing directions) 2) Work out the number of maze sections needed Fig. 10a f. Simulated shock wave travelling through a wedge maze unit 3) Ensure sufficient distance between maze sections 28 4) Wedges with increasing incident angle are placed to- 4 Conclusion wards the centre of the maze. The wedges can also be of the same incident angle although the shock reflec- The optimum chosen here is one which combines the reduc- tion may be significant at large incident angles, and the tion and transmission characteristics of the parameters cho- transmission will increase with small incident angles. sen. It is however not the only optimum choice possible, it 5) The centre space between the opposing wedges has to however presents a practical solution. Decreasing the wedge be sufficiently large for the transmitted wave to expand apex angle further results in too thin wedges and decreas- and strengthen into a planar wave front before it strikes ing the area ratio extensively might, in a mine application, the opposing wedges. interfere with ventilation. A tuely optimal design would not only optimise the breaking up of the shock or blast wave but also have to 2.7 Example of optimised blast barrier geometry minimise the pressure losses generated by placing the blast barrier into the flow. Area Ratio: 10:7 (30% area reduction) Varying wedges: 10æ%, 20æ% and 30æ% Centre Distance: 1.5 times width of channel References Distance between 0.5 times width Bird GA (1958) The effect of wall shape on the reinforcemcnt of a shock maze sections: of channel wave moving into a converging channel. J. Fluid Mech. Frolov SM, Gelfand BE, Medvedev SP, Tsygonov SA (1989) Quenching of shock waves by barriers and screens. In: Kim Y (ed.) Current topics 3 Comparison of simulated shock wave in shock waves. AIP Conference Proceedings 208, 314 320 with experimental data Mori Y, Hijikata K, Shimizu T (1975) Attenuation of shock waves by multi-orifice. Proc. 10th Int. Symp. on Shock Tubes and Waves, Kyoto, pp 400 407 The schlieren photographs (Fig. 8a f) show a good correla- Skews BW, Broido PE (1992) Shock attenuation through a grid of obstacles. tion with the pressure colour plot of the simulation as shown 4th Int. Symp. on Explos. Tech. and Ballistics, Pretoria in Fig. 10. As the schlieren was set extremely sensitive the Tong KO, Knight CJ, Srivastava BN (1980) Interaction of weak shock photographs pick up considerably more detail than is present waves with screens and honeycombs. AIAA Journal 18, 1298 1305 in the simulated data. Löhner R, Morgan K, Zienkiewicz OC (1985) An adaptive finite element Similarly this is also true of the experimental pressure procedure for compressible high speed flows. Comp. Meths. Appl. Mech. Eng. 51, 441 465 traces most of which however is electronic noise in the sys- Löhner R, Morgan K, Peraire JP, Vahdati M (1987) Finite element flux- tem. Not as much of the incident shock wave is attenuated corrected transport (FEM-FTC) for the Euler and Navier-Stokes equa- as is indicated in the simulation although it is significantly tions. Int. J. Numer. Meth. Fluids 7, 1093 1109 reduced (Figs. 9a c). The reflected experimental shock has a Löhner R (1987) An adaptive finite element scheme for transient problems very much steeper pressure rise but it does however compare in CFD. Comp. Meths. Appl. Mech. Eng. 61, 323 328 to the rise indicated in the simulation.

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