# An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension
###### Abstract  
This paper presents an all-Mach method for two-phase inviscid flow in the presence of surface tension. A modified version of the Hartens-Lax-van Leer Contact (HLLC) solver is developed and combined for the first time with a widely used volume-of-fluid (VoF) method: the compressive interface capturing scheme for arbitrary meshes (CICSAM). This novel combination yields a scheme with both HLLC shock capturing as well as accurate liquid-gas interface tracking characteristics. It is achieved by reconstructing non-conservative (primitive) variables in a consistent manner to yield both robustness and accuracy. Liquid-gas interface curvature is computed via height functions and the convolution method. We emphasize the use of VoF in the interest of interface accuracy when modelling surface tension effects. The method is validated using a range of test-cases available in the literature. The results show flow features that are in sensible agreement with previous experimental and numerical work. In particular, the use of the HLLC-VoF combination leads to a sharp volume fraction and energy field with improved accuracy. VoF; compressible; HLLC; surface tension; CSF; height functions Article  
An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension  
M. Y.  
modelling [12; 27; 28] of liquid-gas systems. For this article, we shall consider the inviscid modelling of a liquid-gas flow via a homogeneous one-fluid method. To capture the compressible characteristics in the flow, different solvers are outlined in the literature [1; 12; 20; 27; 28; 29]. These comprise the popular Riemann- or Godunov-type schemes [29; 30; 31] and the semi-implicit projection solvers [32]. In this article, we shall employ the Hartens-Lax-van Leer Contact (HLLC) approximate Riemann solver [30] for its contact-preserving properties and its ability to maintain the positivity of variables. However, as will be demonstrated, when applied to track the liquid-gas interface, this solver results in smearing leading to the loss of liquid-gas interface resolution. Due to the latter, curvature cannot be computed accurately and hence surface tension is often not accounted for. In recent years, different interface handling methods have been proposed to address this smearing. For instance, [PERSON] et al. [20] extended the HLLC solver to an arbitrary Lagrangian-Eulerian (ALE) formulation to track the liquid-gas interface. However, their approach does not include any surface tension effects. Moreover, ALE methods are known to be limited by mesh distortions [33] and therefore require re-gridding procedures that are computationally expensive. Subsequent work on HLLC includes the use of a compression technique to remove numerical diffusion present at the interface. This approach was first proposed by [PERSON] et al. [1], where HLLC was re-formulated to allow for surface tension effects. However, their method does not take into account the liquid availability criteria [34] when advecting the interface. This is further discussed in the next paragraph. The final class of interface-handling methods is that of the sharp interface techniques [35]. Notably, the volume-of-fluid (VoF) method is well-known [36; 37; 34; 38] for its volume conservative properties. For example, [PERSON] et al. [12] successfully extended the unified semi-implicit compressible solver proposed by [PERSON] et al. [32] and combined their formulation with a geometric VoF method [36] to allow for the inclusion of surface tension effects. Here, the superior accuracy is due to accounting for liquid availability in mixed cells when computing phase fluxes. This ensures boundedness of the volume fraction field. However, historically, such VoF methods have been widely used for incompressible flow and have therefore mostly been used in conjunction with semi-implicit projection solvers [24; 26; 39; 40]. Only recently has the use of VoF methods gained interest in the context of Riemann-type schemes [22; 28].