Home The Institute of Space and Astronautical Science Report Experimental Studies on Characteristics of Shock Wave Propagation through Cylinder Array

3. 4 Model for Description of Overall Features

To describe the interaction process of the shock wave with the cylinder array, the unsteady one-dimensional model2) is introduced. The governing equations are the one-dimensional Euler equations with the term of the drag force of cylinders and are given in the conservative form:

 ......(2)

where represents the drag force of the cylinder array. The gas is assumed to be calorically perfect. The term of the heat flux to the cylinder surface is neglected, since the temperature difference between the flow and the cylinder surface is not large in the present experiments and the effect of the energy exchange between the flow and the cylinders is expected to be negligibly small. In fact, the numerical results considering the heat flux model given in Ref. 2 are almost the same as those without the heat flux term. After Rogg2 ), the drag force ~ is assumed to be in proportion to the local dynamic pressure:

......(3)

where B is the drag parameter and represents the drag coefficient per unit length in the flow direction. The drag parameter is assumed to be constant in the cylinder array. This assumption is valid when the cylinders are packed in the channel and the cylinder array is considered as the porous media. The present model is not suitable for coarse cylinder arrangement. The equations (2) and (3) are numerically solved by the finite difference method using Yee's symmetric TVD scheme 7) for the spatial discretization and the two-stage Runge-Kutta method for the time integration. We use the uniform grid with the spacing of 5 mm for computation. The CFL number is set as 0.1.

The drag parameter must be determined by comparison with the experimental data. We consider the cylinder arrangement of the category III(see Table 1 ) . In this category, the porosity , which is the volume fraction of void, is considered to be constant at 0.89, since there are four cylinders in each row. The length of the cylinder array is varied from 10 mm (O004-R) to 40 mm (4444-R). The variations of the pressure behind the transmitted shock wave (P 5/P 1 ) and the pressure behind the reflected shock wave (P7/P1) with the length of the cylinder array at the shock Mach number I .36 are shown in Fig. 25. The results at the shock Mach number I .47 are shown in Fig. 26. The drag parameter in the range from 30 to 50 m-1 provides good agreement with the experimental data both for the transmitted shock wave and for the reflected shock wave. The best fit value of the drag parameter is 30 m-1 and 40 m-1 for the array length 40 mm and 20-30 mm, respectively.



(L)Fig.25.Comparison of Computational Results
with Experimental Data for Pressure behide Transmitted and Reflected Shock Wave
(R)Fig.26.Comparison of Computational Results
with Experimental Data at Shock Mach Number 1.36 and 1.47
Figure 27 shows the computational and experimental results on the time history of the pressure at port # 1 and # 8. The cylinder arrangement is 4444-R. Good agreement is obtained not only on the pressure level but also on the time of arrival of the reflected wave (port # 1 ) and the transmitted wave (port # 8). The pressure rise at the shock wave surface is not so sharp in the numerical results due to the insufficient temporal and spatial resolution. It should be noted that the diffusive nature of the reflected shock wave, which is already pointed out in the experimental results in Fig. 9, is also observed in the computational result. The typical spatial distributions of the pressure, temperature and flow velocity are shown in Fig. 28. These patterns are consistent with the wave diagram in Fig. 10. The computational results indicate the presence of the contact surface in the downstream region of the cylinder array. It is expected to correspond to the location of the starting vortices which are observed in the experiments (see Fig. 12).



(L)Fig.27.Comparison of Computational Results
with Experimental Data for Pressure Records at Cylinder Arrangement 444-R
(R) Fig.28.Spatial Distributions of Pressure,Temperature and Velocity Obtained in Computation

For the flow resistance of the solid matrix of packed cylinders, some empirical relations 2 ) are available. These relations are given in terms of the porosity , the solid Reynolds number Res , and the drag coefficient F defmed as follows:
......(4)
......(5)
Figure 29 shows the comparison of the best fit values of the drag coefficient obtained from Fig. 25 with the relation by Ergun :
......(6)
and the relation by Jones and Krier:
......(7)
Though the porosity in the present experiments is quite large, the present results are close to these relations.



Fig.29.Variation of Drag Coefficient with Solid Reynolds Number

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