Abstract Numerical difficulties, notably the non-monotonic behavior of the Wood speed of sound and the volume fraction positivity, associated with the reduced five-equation two-phase flow model of Kapila et al. (2001) [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, 2001. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10), 3002–3024] have been resolved in the past through the introduction of a frozen speed of sound and an algebraic approach for mechanical relaxation afforded by a pressure non-equilibrium six-equation model proposed by [R. Saurel, F. Petitpas, R.A. Berry, 2009. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixture, J. Comput. Phys. 228, 1678–1712]. By contrast, it is explored and demonstrated in this work that these difficulties can in fact still be resolved within the numerical scheme for solving the reduced five-equation model by numerically replacing the Wood speed of sound for the estimates of wave speeds in the approximate Riemann solver HLLC with the monotonic mixture speed of sound for a transport five-equation model. For shock interface (artificial mixture separating pure or nearly pure fluids) interaction problems, with the apparent advantage of monotonic behavior of the speed of sound in the interface, the effect of the numerical replacement is also confined to the interface. Differences other than the behavior of the speed of sound within the interface in the solutions due to the replacement diminish with increasing resolution when reasonable solution can be obtained with Wood speed of sound. For cavitating/expansion problems in physical fluid mixture, it is pointed out and explained why acoustics in the numerical solutions still propagate at the Wood speed of sound (therefore consistent with the reduced five-equation model) even though in some cases a much higher speed of sound like the numerical replacement above for solving the reduced five-equation model or the frozen speed of sound for solving a six-equation model is used for the estimates of wave speeds in the HLLC scheme. A variant of the five-equation two-phase flow model by Saurel et al. (2008) [R. Saurel, F. Petitpas, R. Abgrall, 2008. Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607, 313–350] is then constructed for liquid-vapor phase transition in cavitating flows. The relaxation toward thermo-chemical equilibrium during phase transition is achieved by solving a simple system of algebraic equations for the equilibrium state variables for better efficiency, following Pelanti and Shyue (2014) [M. Pelanti, K.-M. Shyue, 2014. A mixture-energy-consistent six-equation two-phase numerical model for fluid with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259, 331–357]. Therefore, the current model retains both the simplicity afforded by the five-equation model and the efficiency of the algebraic relaxation solver. An alternative algebraic approach for handling the non-conservative term (the so-called K∇ · u term) in the reduced five-equation model for mechanical equilibrium of a liquid-vapor mixture is also explored by enforcing the thermal equilibrium at the same time. Numerical results of sample tests in both one and two dimensions in the literature as well as that in three dimensions demonstrate the effectiveness and ability of the proposed model to simulate cavitating flows. An interesting mechanism of shock generation by acoustics in water due to phase transition is then found by the numerical simulations.

中文翻译：

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空化流中液汽相变的简单有效的五方程两相数值模型

摘要 数值困难，特别是伍德声速的非单调行为和体积分数正性，与 Kapila 等人的简化五方程两相流模型相关。(2001) [AK Kapila, R. Menikoff, JB Bdzil, SF Son, DS Stewart, 2001. 颗粒材料中爆燃-爆轰转变的两相建模：简化方程，流体物理学 13(10), 3002– 3024] 过去通过引入冻结声速和代数方法来解决机械松弛问题，该方法由 [R. 3024] 提出的压力非平衡六方程模型提供。Saurel, F. Petitpas, RA Berry, 2009。用于分离可压缩流体、空化流动和多相混合物中的冲击的界面的简单而有效的松弛方法，J. Comput。物理。228, 1678–1712]。相比之下，在这项工作中探索并证明了这些困难实际上仍然可以在求解简化的五方程模型的数值方案中解决，方法是通过数值替换近似黎曼求解器 HLLC 中用于估计波速的伍德声速传输五方程模型的单调混合声速。对于激波界面（分离纯或近纯流体的人工混合物）相互作用问题，由于界面中声速单调行为的明显优势，数值置换的影响也仅限于界面。当可用伍德声速获得合理解时，由于替换而导致解中界面内声速行为的差异随着分辨率的增加而减小。对于物理流体混合物中的空化/膨胀问题，指出并解释了为什么数值解中的声学仍然以伍德声速传播（因此与简化的五方程模型一致），即使在某些情况下速度要高得多用于求解简化五方程模型的上述数值替代或用于求解六方程模型的冻结声速用于 HLLC 方案中的波速估计。Saurel 等人的五方程两相流模型的变体。(2008) [R. Saurel, F. Petitpas, R. Abgrall，2008 年。模拟亚稳态液体中的相变：应用于空化和闪蒸流，J. Fluid Mech。607, 313–350] 然后构建用于空化流中的液-气相变。遵循 Pelanti 和 Shyue (2014) [M. 佩兰蒂，K.-M. Shyue，2014 年。具有界面、空化和蒸发波的流体的混合能量一致六方程两相数值模型。J. 计算。物理。259, 331–357]。因此，当前模型保留了五方程模型提供的简单性和代数松弛求解器的效率。还通过同时实施热平衡探索了用于处理液体 - 蒸汽混合物机械平衡的简化五方程模型中的非保守项（所谓的 K∇ · u 项）的替代代数方法. 文献中的一维、二维和三维样本测试的数值结果证明了所提出的模型模拟空化流动的有效性和能力。然后通过数值模拟发现了一种有趣的水声学由于相变而产生冲击的机制。