An anti-singularity Mie-Grüneisen mixture model based on isentropic and hugoniot curves
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摘要: Mie-Grüneisen多介质混合模型可以很方便地应用于Mie-Grüneisen状态方程下的多介质问题中。在Mie-Grüneisen状态方程中，等熵曲线与Hugoniot绝热曲线是两种典型的参考状态曲线。然而，这两类曲线存在奇点，利用体积分数进行界面处理时则会造成困难，而传统混合模型却习惯将体积分数作为色函数来使用。其中的难点在于体积分数模型会因界面的扩散形态而产生零碎的流体体积。这些零碎的体积使得部分等熵曲线中会出现声速在界面附近不合理的偏高，并在收敛性条件下需要耗费更多的时间步来计算。另一方面，奇点会使得Hugoniot参考状态曲线下声速出现负值，阻断计算的进行。为了避免产生零碎的体积，这里将质量分数代替体积分数，并将流场中占一定比例的介质密度的倒数定义为比容。重定义后的比容，可以使得相对体积不小于流场整体的相对体积。这样，声速在界面附近形成一个低谷形状并避免出现峰值。另外，Mie-Grüneisen混合模型中部分方程含有参考状态参数的导数项，这些导数项在界面附近被定义成加权平均，但如果用质量分数直接做加权平均容易出现负值。为了保证界面附近不出现负值，对界面处的参考状态进行了优化。数值算例显示，质量分数虽然对计算结果的影响十分微弱，但是可以在等熵状态曲线下保证声速稳定，从而使用比体积分数更少的时间步。同时，质量分数还能在Hugoniot曲线下很好的处理声速的负值。这样可以保证计算既平稳又准确。
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关键词:
- Mie-Grüneisen混合模型 /
- Mie-Grüneisen状态方程 /
- 防奇点 /
- 参考状态
Abstract: The Mie-Grüneisen mixture model is conveniently used in the multi-component problem with Mie-Grüneisen EOS (equation of states). In the Mie-Grüneisen EOS, the isentropic and Hugoniot curves are two typical reference states curves. However, the curves of these two reference states contain singularity points and cause difficulty when the interface is treated by volume fraction, which is accustomed used as a color function in traditional model. The difficulty lies in that the volume fraction model produces fragments of fluid volumes near the interface due to its diffused style, these volume fragments may encounter the singularity points and make the sound velocity abnormally high at the interface in some isentropic reference curves. On the other side, the singularity points may cause the sound velocity negative for some Hugoniot reference states and interrupt the calculation. To avoid volumes fragments near the interface area, the volume fraction is replaced by mass fraction, and the relative volume is defined by the reciprocal of proportional density of fluid component. This definition makes the relative volume no less than which of fluids mixture. Thanks to the reconstructed relative volume, the sound velocity forms a trough shape at the interface and does not cause high peak value. Moreover, some equations in Mie-Grüneisen mixture model contains the derivatives items of reference states parameters, when these items are defined as weighted average mixture at the interface, they often become negative if weighted average of mass fraction are directly used. To prevent the negative value at the interface, the reference states are optimized at the interface. Numerical examples show that the mass fraction has tiny improvement on the accuracy of results, it makes the sound velocity steady on the isentropic reference states of medium and spend less time steps than volume fraction model. And the mass fraction can be used to correct the negative sound velocity in Hugoniot reference states. Then the calculation is kept smooth and accurate. -
图 4 73 μs时刻，铜-惰性炸药冲击问题中，密度、质点速度、色函数和声速计算结果
Figure 4. The density, particle velocity, color function and sound velocity results of 73 μs for the impact problem between copper and inert explosive.
图 5 铜与惰性炸药CC方程在比容V非常小时的pref曲线
Figure 5. The pref curves of CC equation of copper and inert explosive for small values of relative volume V
图 6 体积分数模型中，两种介质声速平方项随x轴分布情况
Figure 6. The distribution of square items of sound velocity along x-axis for volume fraction model
图 7 质量分数模型中，两种介质声速平方项随x轴分布情况
Figure 7. The distribution of square items of sound velocity along x-axis for mass fraction model.
图 8 CJ态爆轰气体冲击问题中,介质声速平方项随x轴分布情况
Figure 8. The distribution of square items of sound velocity along x-axis for impact problem of CJ states detonation product.
图 9 45 μs时刻，CJ态爆轰气体冲击问题中，密度、压力、质点速度和质量分数的计算结果
Figure 9. The density, pressure, particle velocity and mass fraction results of 45 μs for impact problem of CJ states detonation product.
图 10 两种Mie-Grüneisen状态方程下，Cp'与Cref变化曲线
Figure 10. The curves of Cp' and Cref for different Mie-Grüneisen equation of states
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