变参数状态方程下多介质Riemann问题的质量分数方法及应用
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摘要
近些年来,多介质的Riemann问题已成为一个热点课题。这个问题主要的难点是由间断的界面所造成的。由于界面两边,流体的状态和性质都不一样,因此很难模拟界面附近的流体运动,尤其是带有复杂状态方程的介质所构成的界面。针对间断界面的问题,虽然目前已发展出了多种数值模型,但它们中的有很多是用固定参数的状态方程来描述流体介质。而适应变化参数形式的状态方程的数值算法仍然发展得很慢,原因是很难维持介质之间的压力平衡。目前,这个压力平衡条件主要是通过在计算中添加额外的程序来实现。但是,这些额外的程序使计算变得复杂,特别是包含三种以上介质的流场。为此,我们做了许多工作,为的是寻求一种简便而又有效的方法来解决多介质界面问题。
本文提出了一种改进形式的基于质量分数的多介质Mie-Gruneisen混合体计算模型。这种计算模型将一般形式的状态方程改写成包含变化参数的Mie-Griineisen形式。Mie-Gruneisen状态方程使用了随密度变化的热力学参数,使得它应用更加灵活,但它的缺点是表达式稍显复杂。为了简化计算步骤,Mie-Griineisen状态方程中的可变参数作为独立的变量引入计算中,这些参数的求解将由新添加的辅助方程来完成。这样,Riemann问题可以由原始的欧拉方程和新的辅助方程联立求解。对于多介质的流场,一种比较有效的做法是将流体的混合物看成是一个整体,然后用色函数来区分流体中的不同介质。这里,我们选用质量分数作为色函数,且仍通过欧拉方程和相应的辅助性方程来完成多介质问题的求解。考虑到流体中含有多种不同的介质,辅助性方程的构建来源于一种扩散式的平衡。这种扩散式的平衡基于一种近似的混合流体模型,它和流体中的各种成分通过质量分数联系起来。质量分数起到的作用是用光滑的过渡来代替数值的间断,确保数值解无发散。为完成整个求解系统,我们添加了关于计算质量分数的输送方程。这样,整个求解系统包括三个部分：原始的欧拉方程,关于新构造变量的辅助性方程和输送方程。这种Mie-Griineisen质量分数模型形式简单,且易于扩展到包含三种以上介质的Riemann问题中。
利用Mie-Griineisen质量分数模型,本文研究了气水剧烈作用下的Riemann问题。对于水下爆炸中气水的强冲击问题,这里考虑水在不同情况下的参考状态。当水被压缩时,参考状态选用Hugoniot冲击曲线；当水膨胀时,参考状态用Murnagham等熵曲线代替。这样,水的状态方程就表达成一种关于密度的分段形式的状态方程。这样一种分段形式的状态方程,很重要的一点就是必须利用水的密度来判断其状态。利用基于质量分数的混合体模型,可以直接导出水的密度,从而不需要添加其他程序。随后我们给出了一些数值计算实例。通过计算值和实验值的对比,证实了Mie-Gruneisen质量分数模型在气水强冲击问题中能得到比较准确的数值解。
本文还利用Mie-Gruneisen质量分数模型研究了水下爆炸中的流体—固体相互作用问题。爆炸冲击波对刚性体和弹性体的冲击,本文都进行了研究。对于刚性体,将物体的表面考虑成边界,并设定了全反射边界条件。这一条件是通过在刚性体内部设定镜像质点来实现的。而弹性体则用Mie-Griineisen状态方程来描述,参考状态为Hugoniot曲线。对于弹性体的物理模型,本文通过爆轰激波管实验证实了其可靠性。然后,利用这个模型模拟了一些二维的水下爆炸问题。在模拟过程中,本文还对结构体的变形,二次冲击波等情况作了特别的研究。
本文对水下爆炸问题中,物体表面防护层的保护作用进行了研究。将炸药,水,防护层和结构体用统一形式的Mie-Griineisen状态方程来描述以后,再用Mie-Griineisen质量分数模型可以很方便地对各种物质之间的相互作用进行模拟。随后,我们对防护层的效果进行了研究。研究发现,当结构体表面被防护层覆盖以后,爆炸的冲击波冲击防护层时会对其后的结构体形成二次激波或稀疏波。二次波性质(激波或稀疏波)由防护层的冲击阻抗所决定,阻抗高于水,则出现激波；阻抗低于水,则出现稀疏波。而防护层只有当板层材料的冲击阻抗小于水的时候才能发挥对结构体的保护作用。另外,当防护层满足条件时,可压缩的结构体也能受到防护层的保护作用。这时,在使用了防护层以后,结构体的变形将得到缓和。对于起保护作用的防护层,本文还研究了防护层厚度和炸药距离的影响,结果表明,它们主要影响第二次冲击波的防护效果,但对主冲击波的防护效果影响不大。
Multi-media Riemann problem becomes a hot issue in recently years. The main difficulty of this problem lies in the discontinuous interfaces. As the fluids show differences in state and property at two sides of interface, it is hard to simulate of the fluids motion near the interface, especially for the interface of medias with complex equation of states (EOS). Although a lot of numerical models are developed for such a discontinuous interface problem, many of them describe the media with EOS in which constant-parameter are used. The development of numerical solver for variable-parameter EOS is still very slow because it is hard to maintain the pressure equilibrium among different medias. At present, the equilibrium condition of pressure are always carried up by using additional procedures in calculation. The additional procedures, however, make the computation complex, especially for fluids which contain three or more medias. For these reasons, we made a lot of works to seek a simple and effective to solve the multi-media interface problem.
An improved Mie-Gruneisen mxiture model based on mass fraction is presented here. The model expresses the general EOS in a Mie-Gruneisen form which contains variable parameters. The Mie-Gruneisen EOS use thermodynamical parameters which vary with density, that make the EOS flexible in application, but the drawback of Mie-Gruneisen EOS is that the expression is a bit complex. To simplify the computation process, the variable parameters in Mie-Gruneisen EOS are taken as independent variables and introduced in calculation, and new auxiliary equations are added to compute these variables. So the Riemann problem can be solved by a combination of the original Euler equations and new auxiliary equations. For multi-media fluids, it is very effective to consider the fluids mixture as an entire fluid, and use color functions to make a distinction of the fluid components. In our works, mass fraction is chosen as color function and the multi-media problem is still solved by the Euler equations and other corresponding auxiliary equations. Considering that the fluids mixture contains different medias, the auxiliary equations are constructed by establishing a diffused balance among different medias here. The diffused balance is based on an approximation of fluid mixture model which connects its medias by mass fraction. The role of mass fraction is replacing the discountinous jumping by smooth transition and guaranteeing a non-oscillation solution near the interface. To make the system completed, transport equations, in terms of mass fraction, are added in the equations system. The whole equations system of the Mie-Griineisen mass fraction model includes three parts:original Euler equations for the whole fluids, auxiliary equations of new created variables, and transport equations. The Mie-Griineisen mass fraction model is in a simple style and can be easily extend to Riemann problem with three or more medias.
The strong gas-water Riemann problem is studied with the help of Mie-Griineisen mass fraction model. For strong shock problem of gas-water interaction in underwater explosion, the different reference states are considered here. For compressed water, shock-Hugoniot curve is taken as reference state; while for expanding water, it is replaced by Murnagham isentropic curve. Therefore, the EOS for water is in a piecewise form with respect to density. For such a piecewise EOS of water, there's a crucial point that the state estimation relies on density of water. As the mixture model based on mass fraction is used here, the density of water can be directly deduced and there's no need to use other additional condition. And some numerical tests about gas-water problem are present here. The comparisons of calculated and experimental data show that the Mie-Griineisen mass fraction model gives accuracy solutions for strong shock gas-water flow.
The Mie-Griineisen mass fraction model is applied to the fluid-structure interaction in underwater explosion. The impact effects of explosion shock wave on rigid and elastic body are both studied here. For rigid body, the edge of body are taken as boundary, the boundary conidition is set as total reflection condition. The condition is satisfied by setting illusion points at inner region the rigid body. While for elastic body, it is modeled by a Mie-Griineisen EOS, which taking Hugoniot curve as reference state. The feasibility of elastic body model is tested and verified by detonation shock tube tests. Then the model is applied to the simulation of some2D underwater explosion problems. In the simulation, deformation of structure under explosion loads, as well as second shock wave, are specially studied here.
The protection effects of mitigation layer against shock wave in underwater explosion are investigated here. By using uniform Mie-Griineisen EOS to model explosive charge, water and mitigation layer, as well as other elastic structure, it is easy and effective for Mie-Gruneisen mass fraction model to simulate the interaction among different medias. Then an investigation about the effects of mitigation layer is given here. In this investigation, it is found that when structure is covered by mitigation layer, the impact of explosion wave produces second shock wave or rarefaction wave on structure. The property of the second wave depends on the shock impedance of layer, if this shock impedance higher than water, it generates shock wave; or else, it generates rarefaction wave. And the protection effects only occur when the impedance of layer is smaller than water. In addition, the compressible structure can also be protected by mitigation layer. The deformation will be reduced when mitigation layer is used. For structure mitigated by low impedance layer, the affection of layer thickness and explosive-structure distance is studied here. It is concluded that they mainly affect the protection effects on second shock while almost don't affect the protection effects on main shock.
本文提出了一种改进形式的基于质量分数的多介质Mie-Gruneisen混合体计算模型。这种计算模型将一般形式的状态方程改写成包含变化参数的Mie-Griineisen形式。Mie-Gruneisen状态方程使用了随密度变化的热力学参数,使得它应用更加灵活,但它的缺点是表达式稍显复杂。为了简化计算步骤,Mie-Griineisen状态方程中的可变参数作为独立的变量引入计算中,这些参数的求解将由新添加的辅助方程来完成。这样,Riemann问题可以由原始的欧拉方程和新的辅助方程联立求解。对于多介质的流场,一种比较有效的做法是将流体的混合物看成是一个整体,然后用色函数来区分流体中的不同介质。这里,我们选用质量分数作为色函数,且仍通过欧拉方程和相应的辅助性方程来完成多介质问题的求解。考虑到流体中含有多种不同的介质,辅助性方程的构建来源于一种扩散式的平衡。这种扩散式的平衡基于一种近似的混合流体模型,它和流体中的各种成分通过质量分数联系起来。质量分数起到的作用是用光滑的过渡来代替数值的间断,确保数值解无发散。为完成整个求解系统,我们添加了关于计算质量分数的输送方程。这样,整个求解系统包括三个部分：原始的欧拉方程,关于新构造变量的辅助性方程和输送方程。这种Mie-Griineisen质量分数模型形式简单,且易于扩展到包含三种以上介质的Riemann问题中。
利用Mie-Griineisen质量分数模型,本文研究了气水剧烈作用下的Riemann问题。对于水下爆炸中气水的强冲击问题,这里考虑水在不同情况下的参考状态。当水被压缩时,参考状态选用Hugoniot冲击曲线；当水膨胀时,参考状态用Murnagham等熵曲线代替。这样,水的状态方程就表达成一种关于密度的分段形式的状态方程。这样一种分段形式的状态方程,很重要的一点就是必须利用水的密度来判断其状态。利用基于质量分数的混合体模型,可以直接导出水的密度,从而不需要添加其他程序。随后我们给出了一些数值计算实例。通过计算值和实验值的对比,证实了Mie-Gruneisen质量分数模型在气水强冲击问题中能得到比较准确的数值解。
本文还利用Mie-Gruneisen质量分数模型研究了水下爆炸中的流体—固体相互作用问题。爆炸冲击波对刚性体和弹性体的冲击,本文都进行了研究。对于刚性体,将物体的表面考虑成边界,并设定了全反射边界条件。这一条件是通过在刚性体内部设定镜像质点来实现的。而弹性体则用Mie-Griineisen状态方程来描述,参考状态为Hugoniot曲线。对于弹性体的物理模型,本文通过爆轰激波管实验证实了其可靠性。然后,利用这个模型模拟了一些二维的水下爆炸问题。在模拟过程中,本文还对结构体的变形,二次冲击波等情况作了特别的研究。
本文对水下爆炸问题中,物体表面防护层的保护作用进行了研究。将炸药,水,防护层和结构体用统一形式的Mie-Griineisen状态方程来描述以后,再用Mie-Griineisen质量分数模型可以很方便地对各种物质之间的相互作用进行模拟。随后,我们对防护层的效果进行了研究。研究发现,当结构体表面被防护层覆盖以后,爆炸的冲击波冲击防护层时会对其后的结构体形成二次激波或稀疏波。二次波性质(激波或稀疏波)由防护层的冲击阻抗所决定,阻抗高于水,则出现激波；阻抗低于水,则出现稀疏波。而防护层只有当板层材料的冲击阻抗小于水的时候才能发挥对结构体的保护作用。另外,当防护层满足条件时,可压缩的结构体也能受到防护层的保护作用。这时,在使用了防护层以后,结构体的变形将得到缓和。对于起保护作用的防护层,本文还研究了防护层厚度和炸药距离的影响,结果表明,它们主要影响第二次冲击波的防护效果,但对主冲击波的防护效果影响不大。
Multi-media Riemann problem becomes a hot issue in recently years. The main difficulty of this problem lies in the discontinuous interfaces. As the fluids show differences in state and property at two sides of interface, it is hard to simulate of the fluids motion near the interface, especially for the interface of medias with complex equation of states (EOS). Although a lot of numerical models are developed for such a discontinuous interface problem, many of them describe the media with EOS in which constant-parameter are used. The development of numerical solver for variable-parameter EOS is still very slow because it is hard to maintain the pressure equilibrium among different medias. At present, the equilibrium condition of pressure are always carried up by using additional procedures in calculation. The additional procedures, however, make the computation complex, especially for fluids which contain three or more medias. For these reasons, we made a lot of works to seek a simple and effective to solve the multi-media interface problem.
An improved Mie-Gruneisen mxiture model based on mass fraction is presented here. The model expresses the general EOS in a Mie-Gruneisen form which contains variable parameters. The Mie-Gruneisen EOS use thermodynamical parameters which vary with density, that make the EOS flexible in application, but the drawback of Mie-Gruneisen EOS is that the expression is a bit complex. To simplify the computation process, the variable parameters in Mie-Gruneisen EOS are taken as independent variables and introduced in calculation, and new auxiliary equations are added to compute these variables. So the Riemann problem can be solved by a combination of the original Euler equations and new auxiliary equations. For multi-media fluids, it is very effective to consider the fluids mixture as an entire fluid, and use color functions to make a distinction of the fluid components. In our works, mass fraction is chosen as color function and the multi-media problem is still solved by the Euler equations and other corresponding auxiliary equations. Considering that the fluids mixture contains different medias, the auxiliary equations are constructed by establishing a diffused balance among different medias here. The diffused balance is based on an approximation of fluid mixture model which connects its medias by mass fraction. The role of mass fraction is replacing the discountinous jumping by smooth transition and guaranteeing a non-oscillation solution near the interface. To make the system completed, transport equations, in terms of mass fraction, are added in the equations system. The whole equations system of the Mie-Griineisen mass fraction model includes three parts:original Euler equations for the whole fluids, auxiliary equations of new created variables, and transport equations. The Mie-Griineisen mass fraction model is in a simple style and can be easily extend to Riemann problem with three or more medias.
The strong gas-water Riemann problem is studied with the help of Mie-Griineisen mass fraction model. For strong shock problem of gas-water interaction in underwater explosion, the different reference states are considered here. For compressed water, shock-Hugoniot curve is taken as reference state; while for expanding water, it is replaced by Murnagham isentropic curve. Therefore, the EOS for water is in a piecewise form with respect to density. For such a piecewise EOS of water, there's a crucial point that the state estimation relies on density of water. As the mixture model based on mass fraction is used here, the density of water can be directly deduced and there's no need to use other additional condition. And some numerical tests about gas-water problem are present here. The comparisons of calculated and experimental data show that the Mie-Griineisen mass fraction model gives accuracy solutions for strong shock gas-water flow.
The Mie-Griineisen mass fraction model is applied to the fluid-structure interaction in underwater explosion. The impact effects of explosion shock wave on rigid and elastic body are both studied here. For rigid body, the edge of body are taken as boundary, the boundary conidition is set as total reflection condition. The condition is satisfied by setting illusion points at inner region the rigid body. While for elastic body, it is modeled by a Mie-Griineisen EOS, which taking Hugoniot curve as reference state. The feasibility of elastic body model is tested and verified by detonation shock tube tests. Then the model is applied to the simulation of some2D underwater explosion problems. In the simulation, deformation of structure under explosion loads, as well as second shock wave, are specially studied here.
The protection effects of mitigation layer against shock wave in underwater explosion are investigated here. By using uniform Mie-Griineisen EOS to model explosive charge, water and mitigation layer, as well as other elastic structure, it is easy and effective for Mie-Gruneisen mass fraction model to simulate the interaction among different medias. Then an investigation about the effects of mitigation layer is given here. In this investigation, it is found that when structure is covered by mitigation layer, the impact of explosion wave produces second shock wave or rarefaction wave on structure. The property of the second wave depends on the shock impedance of layer, if this shock impedance higher than water, it generates shock wave; or else, it generates rarefaction wave. And the protection effects only occur when the impedance of layer is smaller than water. In addition, the compressible structure can also be protected by mitigation layer. The deformation will be reduced when mitigation layer is used. For structure mitigated by low impedance layer, the affection of layer thickness and explosive-structure distance is studied here. It is concluded that they mainly affect the protection effects on second shock while almost don't affect the protection effects on main shock.
引文
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