Tangent Of Hyperbola For Interface Capturing Research Articles

A high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for compressible multi-material flows

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor’s expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows.

Shock wave induced by the collapse of a bubble cluster with each bubble distributed randomly

The phenomenon of bubble collapse is ubiquitous in both natural and industrial settings, often manifesting in the form of bubble clusters. Despite its prevalence, there has been limited research on the dynamics of bubble clusters. In order to gain a better understanding of shock wave emission, liquid jet propagation, and the resulting pressure loads during bubble cluster collapse, we have developed a high-fidelity numerical approach aimed at accurately capturing shock waves and phase interfaces. The interface compression technique is introduced into the compressible two-phase model to improve the sharpness of the phase interface. We solve the physical model using the BVD (Boundary Variation Diminishing) principle. Both the 5th order WENO (Weighted Essentially Non-Oscillatory) scheme and the interface capturing function THINC (Tangent of Hyperbola for INterface Capturing) are utilized to reconstruct primary variables at cell boundaries, effectively minimizing numerical dissipation. For further resolution improvement, we implement these numerical methods on the AMR (Adaptive Mesh Refinement) mesh. Our numerical results demonstrate that the distance between the wall and the bubble cluster, as well as the nearest bubble to the wall, significantly influence the wall peak pressure. However, they have a minimal impact on the overall evolution of the bubble cluster and the peak pressure induced in the flow field. Remarkably, the spatial distribution of bubbles and the total number of bubbles play a crucial role in shaping the dynamic behavior of the bubble cluster and the pressure loads exerted both on the solid wall and in the flow field. The crucial findings offer profound insights into a range of applications, including cavitation erosion, underwater explosions, and drug delivery.

A hybrid WENO5IS-THINC reconstruction scheme for compressible multiphase flows

We present a hybrid reconstruction scheme for compressible multiphase flows governed by the five-equation model with surface tension and molecular viscosity. The Godunov-type method utilizes a fifth-order incremental stencil weighted essential non-oscillatory (WENO5IS) scheme to resolve flow structures in each component, while the Tangent of Hyperbola for INterface Capturing (THINC) maintains sharp fluid interfaces. Supplemented with positivity- and boundedness-preserving limiting procedures, the scheme exhibits a high level of robustness for multiphase flow problems with strong compressibility and large acoustic impedance mismatch. One- and two-dimensional benchmark cases demonstrate the efficiency and accuracy of interface-sharpening, and the high resolution capability for each fluid component. Numerical tests under extreme conditions corroborate the excellent robustness, and simulations with surface tension and viscosity also exhibit the versatility of the proposed approach.

Numerical study of liquid jet and shock wave induced by two-bubble collapse in open field

The shock waves and high-speed jets induced by the violent collapse of bubbles can cause serious damage to nearby structures. To fully understand the shock wave and jet characteristics induced by bubble-bubble interaction, we developed a diffuse-interface method for the simulation of compressible two-phase flow, which combined interface compression technique with interface sharpening technique to keep the sharpness of the shock wave and phase interface. The thermodynamically consistent five-equation model was improved using interface compression technique to maintain a constant thickness of the phase interface. To further suppress numerical dissipation, the high-order scheme WENO (Weighted Essentially Non-Oscillatory) and interface sharpening function THINC (Tangent of Hyperbola for INterface Capturing) were used for the flux reconstruction at the grid boundary to ensure that the total variation at the cell boundary was minimized (Boundary Variation Diminishing, BVD), thus the spatial reconstruction scheme, named WENO-THINC-BVD, was developed. Moreover, we extended the present method to a block-structured adaptive mesh refinement framework to improve grid resolution and save computing resources. Numerical results of several benchmark tests fully demonstrated the advantages of the present method in simulating complex compressible flows containing shock-shock and shock-interface interactions. Based on the present high-fidelity numerical methods, we investigated the shock wave and liquid jet induced by the two-bubble interaction in an open space. A phase diagram of the liquid jet propagation direction as a function of the initial bubble pressure and the inter-bubble distance was summarized. Finally, according to the bubble-bubble interaction process, an empirical formula for the collapse strength and another for the attenuation of shock waves were developed.

A revised WENO‐THINC scheme for the general structured mesh and applications in the direct numerical simulation of compressible turbulent flows

AbstractThe Tangent of Hyperbola for INterface Capturing (THINC) scheme allows a jump‐like reconstruction and brings about a significant improvement in resolving the discontinuous part of the numerical solutions. However, it is found that the original THINC scheme loses accuracy when working on the stretched, curvilinear or highly‐skewed grid. In this study, we propose a simple strategy to determine the jump thickness parameter in the THINC function, so as to effectively suppress the unphysical oscillation. A Boundary Variation Diminishing (BVD) guideline is introduced to make options between the Weighted Essentially Non‐Oscillatory (WENO) scheme and the modified THINC scheme, thus both the smooth and discontinuous solutions can be reconstructed properly avoiding distortion of grids. Numerical validations indicate that the improved WENO‐THINC‐BVD approach maintains high resolution and is more robust on various types of non‐uniform meshes. The present method is further extended to validate the low‐dissipation property in resolving higher wave numbers portions by simulating an isotropic turbulence decay problem. Finally, we perform the direct numerical simulation (DNS) of a spatially evolving adiabatic flat plate boundary‐layer flow problem at a supersonic Mach number (). Numerical results show the predicted mean flow variables and the normalized shear stress agree well with the experimental data, significant improvements are found in the resolution of the small‐scale vortices, especially in the transition process.

A high-robustness hybrid scheme of finite-difference WENO-THINC for compressible multicomponent flow scheme on general curvilinear grids

In this study, a high-robustness hybrid scheme of weighted essentially non-oscillatory (WENO) scheme with a modified tangent of hyperbola for interface capturing (THINC) algorithm is developed for compressible multicomponent flow on general curvilinear grids. Numerical errors induced by mesh deformation bring loss of numerical accuracy and simulation instability, resulting in inaccurate results such as interface distortion, numerical oscillations or even simulation failure. To address this issue, the WENO scheme combined with an improved THINC strategy is designed to alleviate these errors and maintain the high resolution of interfaces. A modified THINC algorithm is developed for the non-uniform grids, in which the steepness parameter is scaled adaptively according to varying grid spacings. This approach is capable of reducing numerical dissipations for interface reconstruction. The overestimated quasi-conservative WENO formulation are employed to hold the equilibriums of velocity, pressure, and temperature at the material interface. Numerical validations are tested on non-uniform grids with various randomness amplitudes to verify the effectiveness in one- and two-dimensional benchmark problems showing the better performances in shock- and interface-capturing capabilities.

A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order

Discretization of the convection term in a three-cell compact stencil has been widely used in Computational Fluid Dynamics (CFD) of engineering because the second-order interpolation scheme is considered to be more robust and algorithmically simple than very high-order schemes. However, according to Godnunov's theorem, it has never been trivial to design an accurate and oscillation-free discretization scheme in a compact stencil. Over decades, various kinds of schemes have been proposed to increase accuracy beyond second-order without producing non-physical numerical oscillations. Representative schemes are polynomial-based non-linear interpolations such as TVD (Total Variation Diminishing), NVD (Normalized Variable Diagram) and third-order WENO (Weighted Essentially Non-oscillatory); and non-polynomial-based interpolations including RBF (Radial Basis Functions), LDLR (Local Double Logarithmic Reconstruction) and Tangent of Hyperbola for INterface Capturing (THINC). Within the semi-discretization finite volume method, existing reconstruction schemes suffer from different issues to some extent. For example, existing TVD schemes tend to cause smearing, clipping and squaring effects. Most WENO schemes and inconsistent RBF schemes are not scale-invariant. Especially, the improved WENO-Z scheme is neither scale-invariant nor mesh-size-independent. Computationally expensive non-polynomial-based interpolations usually have singularity and are dependent on tunable parameters. Thus, this work reviews reconstruction schemes by reformulating or calculating the reconstructed boundary value in the normalised-variable space where the characteristics of reconstruction schemes can be read directly. Through the accuracy analysis and the induction of existing schemes, this work proposes a Unified Normalised-variable Diagram (UND) where the following properties of a reconstruction scheme can be given diagrammatically: 1) the second-order and the third-order accuracy condition, 2) the TVD and CBC (Convection Boundedness Criterion) constrain, 3) the ENO condition, 4) the High Resolution (HR) property, 5) and the numerical dissipation and anti-dissipation error.The proposed UND framework is firstly demonstrated by the improvement of the existing schemes with a shape-preserving TVD limiter and a scale-invariant WENO scheme. Then, by defining the normalised reconstruction operator, a brand-new scheme named ROUND (Reconstruction Operator on Unified Normalised-variable Diagram) is proposed. The ROUND scheme is devised by directly configuring the normalised reconstruction operator of favourable properties on UND. This work presents a dissipative ROUND scheme in which normalised reconstruction operators are fully in the dissipative region of UND, and a low-dissipative ROUND scheme where rigorously adjusted anti-dissipation errors are introduced. Different formulations of ROUND schemes are presented to show the flexibility of designing schemes within the UND framework. The proposed dissipative and low-dissipative ROUND schemes are evaluated via benchmark tests which show the superior performance of ROUND on accuracy and resolution. Especially, in some cases, the low-dissipative ROUND schemes obtain comparable and even better resolution than classic fifth-order WENO. The significance of this work is further demonstrated by the preliminary results of applying ROUND on the FDM-IBM (Finite Difference Method and Immersed Boundary Method) framework.

An efficient finite difference IFWENO-THINC hybrid scheme for capturing discontinuities

The weighted essentially non-oscillatory (WENO) scheme is a famous shock-capturing scheme, but for high-precision numerical simulation, the classical WENO scheme is always considered to have excessive dissipation and insufficient resolution near discontinuities. Improving the accuracy order or reducing the dissipation of the WENO by optimizing the linear weights can effectively improve the resolution of this scheme in the smooth regions. However, under the same method, the improvement of shock-capturing abilities is generally limited and may lead to numerical oscillations. In order to simultaneously improve the resolution in discontinuous regions and ensure the performance of the smooth regions, this study uses the shock detector to combine the tangent of hyperbola for interface capturing (THINC) scheme, which is a better discontinuity-capturing scheme, with a given WENO discretization, such as the improved fast WENO (IFWENO) scheme that equips with high economy and does not loss accuracy at any critical point. The hybrid method in this study is an apriori algorithm, which is more efficient and universal than the method in (Takagi et al. J. Comput. Phys. 452(2022)110899). Additionally, the performance of the THINC scheme in different regions near the discontinuity is analyzed in detail through numerical experiments, and a shock detector that can fully exploit the advantages of the THINC scheme is provided. By using this shock detector, several one-dimensional and two-dimensional numerical cases are employed to verify that the hybrid scheme can greatly improve the performance in discontinuities without affecting the flow structure in the smooth regions, which proves the effectiveness of the proposed shock detector and the excellence of the hybrid IFWENO-THINC scheme.

A fifth-order low-dissipation discontinuity-resolving TENO scheme for compressible flow simulation

For compressible flows characterized by a wide range of flow length scales and discontinuities, it is still an open challenge to design the optimal schemes, which resolve the small-scale flow structures with low numerical dissipation and capture the shock waves without artificial oscillations. In Takagi et al. [1], a novel TENO5-THINC scheme with the combination of classical TENO5 (fifth-order Targeted Essentially Non-Oscillatory) scheme and the non-polynomial THINC (Tangent of Hyperbola for INterface Capturing) reconstruction has been proposed. Building upon the strategy of isolating discontinuities from smooth and high-wavenumber regions, in the present work, a new very low-dissipation TENO scheme with discontinuity-resolving property is proposed for compressible flow simulations based on three new concepts: (1) an improved discontinuity-detecting criterion is devised based on the TENO weighting strategy, which significantly enhances the discontinuity-detecting accuracy compared to that in TENO5-THINC; (2) A local interpolation-like strategy is proposed to represent the detected discontinuity with subcell resolutions, and this strategy can minimize the numerical dissipation even when compared to the THINC reconstruction scheme; (3) According to the varying sharpness of the discontinuities separated by the discontinuity-detecting indicator, the local interpolation-like strategy is extended with a two-steepness approximation. Specifically, the discontinuities will be classified as genuinely sharp discontinuities and general ones. For the genuinely sharp discontinuities, the interface flux will be estimated by a steeper step-like function with even less numerical dissipation. The resulting scheme maintains the high-order and low-dissipation properties of the TENO scheme for smooth flow scales, while further improving the discontinuity-resolving capability and suppressing the numerical oscillations in the vicinity of discontinuities. A variety of benchmark cases with broadband length scales as well as discontinuities is presented to demonstrate the high wave-resolution property and the sharp shock-capturing capability of the proposed scheme.

On the supremum of the steepness parameter in self-adjusting discontinuity-preserving schemes

Self-adjusting steepness (SAS)-based schemes preserve various structures in the compressible flows. These schemes provide a range of desired behaviors depending on the steepness-adjustable limiters with the steepness measured by a steepness parameter. These properties include either second-order accuracy with exact steepness infima that are theoretically given or having anti-diffusive/compression properties with a larger steepness parameter. Nevertheless, the supremum of the steepness parameter has not been determined theoretically yet. In this study, we demonstrate that any anti-diffusive limiter should be limited by Ultra-bee limiter according to Sweby's total variation diminishing (TVD) condition. Two typical steepness-adjustable limiters are analyzed in detail including the tangent of hyperbola for interface capturing (THINC) limiter and the steepness-adjustable harmonic (SAH) limiter. Applying this constraint, we derive for the first time two inequalities which the steepness parameters much satisfy. Furthermore, we obtain the analytical expression of the Courant-Friedrichs-Lewy (CFL) number-dependent supremum of the steepness parameter. Using this solution, we then propose supremum-determined SAS schemes. These schemes are further extended to solve the compressible Euler equations. The results of typical numerical tests confirm our theoretical conclusions and show that the final schemes are capable of sharply capturing contact discontinuities and minimizing numerical oscillations.

Breaking wave simulations for a high-speed surface vessel with hybrid THINC and HRIC schemes

To simulate the ship-induced breaking waves with high-accuracy VOF (volume of fluid) schemes, the unstructured THINC (tangent of hyperbola for interface capturing) type scheme was selected as the main interface capturing method.Particularly, an extension version of THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature), namely THINC/QQ-SF (THINC/QQ extended for split-face unstructured cells) was used. The computational domain was discretized by the trimmed meshes of STAR-CCM+. The mesh cells were categorized into three groups, including regular unstructured (i.e. hexahedral, tetrahedral, prismatic and pyramidal) cells, split-face unstructured cells and polyhedral cells. THINC/QQ-SF was applied for the first two cell groups, which take up most of the space. The algebraic VOF scheme of HRIC (high-resolution interface capturing) which has the advantage of algorithm simplicity, was implemented for the third cell group. To validate the performance of THINC/QQ-SF and HRIC, a series of testing meshes were designed for the pure VOF advection cases. Additionally, the geometric VOF scheme of isoAdvector was also involved for comparisons. In terms of the accuracy, THINC/QQ-SF is similar with isoAdvector, both of which are much better than HRIC. For the efficiency, both THINC/QQ-SF and HRIC have obvious advantages over isoAdvector under the present conditions. Then, the hull form of DTMB (David Taylor model basin) 5415 was chosen for testing, under the Froude number of 0.35. By using the combination of THINC/QQ-SF and HRIC (i.e. THINC/QQ-SF & HRIC), the compactness of the interface can be maintained. Besides, the wave elevations are well agreed with the published experiment. For the averaged calm-water resistance, the present strategy can obtain more accurate results with fewer (or much fewer) cells, compared with the previous CFD studies. To better show the advantages of the hybrid method, the simulations with pure HRIC were also conducted. Compared with pure HRIC, THINC/QQ-SF & HRIC can better capture the wave breaking details with less numerical diffusion, and the reliability of the resistance prediction is higher.

Numerical simulation for liquid sloshing with baffle by the CLSVOF/IB method

The liquid sloshing problem in a partially filled rectangular tank is numerically investigated by a Coupled Level Set and Volume Of Fluid (CLSVOF) method. In the present CLSVOF method, the Level Set (LS) equation needs to be solved to advect the LS function for calculating interface curvature; the re-distancing equation is also taken into account for the LS function being maintained as a distance function. For capturing the interface, the Tangent of Hyperbola for INterface Capturing (THINC) scheme with Weighted Linear Interface Calculation (WLIC) is implemented to advect the Volume of Fluid (VOF) function. To preserve mass conservation, the interface reconstruction procedure needs to be carried out on the basis of the connection between LS and VOF functions. Rectangular and prismatic tanks are used to validate the accuracy and mass conservation of the present CLSVOF method. Then, further simulations of the problem of partially filled liquid sloshing with baffle considered are conducted by combining the CLSVOF method and the Immersed Boundary (IB) method. Finally, the influence of different baffle lengths on the slapping pressure and free-surface is discussed.

A new paradigm of dissipation-adjustable, multi-scale resolving schemes for compressible flows

The scale-resolving simulation of high speed compressible flow through direct numerical simulation (DNS) or large eddy simulation (LES) requires shock-capturing schemes to be more accurate for resolving broadband turbulence and robust for capturing strong shock waves. In this work, we develop a new paradigm of dissipation-adjustable, shock capturing scheme to resolve multi-scale flow structures in high speed compressible flow. The new scheme employs a polynomial of n-degree and non-polynomial THINC (Tangent of Hyperbola for INterface Capturing) functions of m-level steepness as reconstruction candidates. These reconstruction candidates are denoted as PnTm. From these candidates, the piecewise reconstruction function is selected through the boundary variation diminishing (BVD) algorithm. Unlike other shock-capturing techniques, the BVD algorithm effectively suppresses numerical oscillations without introducing excess numerical dissipation. Then, an adjustable dissipation (AD) algorithm is designed for scale-resolving simulations. This novel paradigm of shock-capturing scheme is named as PnTm−BVD−AD. The proposed PnTm−BVD−AD scheme has following desirable properties. First, it can capture large-scale discontinuous structures such as strong shock waves without obvious non-physical oscillations while resolving sharp contact, material interface and shear layer. Secondly, the numerical dissipation property of PnTm−BVD−AD can be effectively adjusted between n+1 order upwind-biased scheme and non-dissipative n+2 order central scheme through a simple tunable parameter λ. Thirdly, with λ=0.5 the scheme can recover to n+2 order non-dissipative central interpolation for smooth solution over all wavenumber, which is preferable for solving small-scale structures in DNS as well as resolvable-scale in explicit LES. Finally, the under-resolved small-scale can be solved with the dissipation adjustable algorithm through the so-called implicit LES (ILES) approach. Through simulating benchmark tests involving multi-scale flow structures and comparing with other central-upwind schemes, the superiority of the proposed scheme is evident. For instance, the simulation results of the supersonic planar jet show that PnTm−BVD−AD schemes can achieve competitive results as Pn+2Tm−BVD schemes which utilize a higher degree of reconstruction polynomial. Thus, in comparison with the previous work, the proposed PnTm−BVD−AD schemes have the benefit of a more compact stencil and lower cost. In summary, this work provides an alternative scheme for solving multi-scale problems in high speed compressible flows.

A vertex-centered finite volume method with interface sharpening technique for compressible two-phase flows

A robust and efficient finite volume method with interface sharpening technique has been developed to solve the six-equation multi-fluid single-pressure model for compressible two-phase flows. The numerical method is implemented in a three-dimensional vertex-centered code. A least-squares reconstruction with Kuzmin's vertex-based (VB) limiter is implemented for the volume fraction and a set of primitive variables in the presented finite volume framework. In regions where two different fluid components are present within a cell, a sharpening technique based on THINC (Tangent of Hyperbola for Interface Capturing) is adopted to provide a sharp resolution for the transitioning interface. These reconstructed values are then used as the initial data for Riemann problems. The enhanced AUSM+ -up scheme is applied to both liquid and gas flows. The multi-stage Runge-Kutta method is used for time marching. A number of benchmark test cases are presented to assess the performance of the present method. These include: an air-water interface moving at a constant velocity, Ransom's faucet problem, air-water/water-air shock tube problems with high pressure ratios, a shock in air impacting a water column case, an underwater explosion case and an air bubble blast case. In all of these cases, the shock and rarefaction waves are captured accurately, especially with the THINC interface sharpening technique.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

A novel high-order low-dissipation TENO-THINC scheme for hyperbolic conservation laws

The high-order low-dissipation shock-capturing scheme is important for compressible fluid simulations, in particular for cases where both shock and turbulence present. However, the competing requirements for resolving the small-scale turbulence structures with low dissipation and capturing the discontinuities sharply render it non-trivial for numerical development. In this work, based on a novel parameter-free discontinuity-detection criterion, a new shock-capturing framework is proposed by combining the standard TENO (targeted essentially non-oscillatory) scheme for smooth regions with the non-polynomial based THINC (tangent of hyperbola for INterface capturing) reconstruction for non-smooth discontinuities. The resulting hybrid scheme retains the low-dissipation property of TENO scheme for smooth flow scales and the discontinuity-resolving capability of THINC reconstruction for shock and contact waves. A set of benchmark simulations involving a wide range of length scales demonstrates that the new TENO-THINC scheme performs significantly better than the standard TENO scheme without the necessity of parameter tuning case by case, and thus is promising for more complex compressible flow predictions where the excessive numerical dissipation of existing schemes prevents the targeted flow structures from being properly resolved. Moreover, the present results serve as the state-of-the-art references for future numerical method development.

A pressure‐based numerical scheme for compressible–incompressible two‐phase flows

AbstractThis article presents a numerical scheme for two‐phase flows consisting of compressible gas and incompressible liquid. Assuming that the gas density is determined by pressure only and that the liquid density is constant, we develop a new set of governing equations for compressible–incompressible two‐phase flows, which can be seen as a simplification of the previous unified governing equations. The volume fraction of gas, velocity, and pressure are the primary variables. Given the liquid's usual leading role in the flow motion, a pressure‐based method is employed to solve the equations. A fifth‐order accurate weighted essentially nonoscillatory (WENO) scheme is applied to discretize the advection terms, and the tangent of hyperbola for interface capturing (THINC) scheme is utilized to capture the interface. The numerical scheme proposed is validated against the generalized Bagnold model and the free drop of a water patch in a closed tank. The results show that the scheme can tackle low Mach number compressible–incompressible two‐phase flows. For the one‐dimensional Bagnold model, the present numerical model achieves first‐order convergence and gives better results than that based on the previous unified governing equations with the same numerical methods. In the simulation of the free drop of a water patch in a closed tank, a similar pressure curve with other reported work is given.

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

Hybrid Discontinuous Galerkin/Finite Volume Method with Subcell Resolution for Shocked Flows

A hybrid discontinuous Galerkin (DG)/finite volume (FV) method under the multistage boundary variation diminishing (BVD) reconstruction paradigm is developed for shocked inviscid and viscous flows. The methodology uses high-order unlimited polynomials and the THINC (which stands for tangent of hyperbola for interface capturing) function as the candidate reconstruction functions to perform FV computation on embedded subcells. The reconstruction scheme on the FV subcells can be designed in a way to keep the resulting approach as compact as the underlying DG method. The reconstruction information based on the modified BVD selection algorithm has been applied as an indicating strategy so that no additional discontinuity indicator is required for the hybrid DG/FV computation. The designed approach has been investigated through the coupling with the explicit and implicit time discretization scheme for both inviscid and viscous flows. Numerical results demonstrate the high-fidelity characteristics of the method by using a much coarser grid to capture both the discontinuity and the smooth flow structure with high resolution.

A Hybrid Targeted Eno-Thinc Scheme With a Modified BVD Algorithm and its Application in Stiff Detonation

This paper presents a new hybrid scheme which combines the fifth-order Targeted Essentially Non-Oscillatory (TENO) reconstruction and the tangent of hyperbola for interface capturing (THINC) reconstruction by using a modified boundary variation diminishing (MBVD) algorithm to select one of the two candidate reconstructions. The THINC scheme has a jump-like distribution can approximate a discontinuity well within a mesh cell with high fidelity. The original BVD algorithm is based on minimisation of jumps of reconstructed variables at cell interfaces in order to reduce the dissipation in numerical flux. The present MBVD algorithm minimises the dissipation terms of the HLL flux which includes jumps of both variables and fluxes at cell interfaces. A number of benchmark examples demonstrate that the MBVD algorithm is more robust than the original BVD algorithm. Applications in stiff detonation problems show that the present hybrid scheme can capture detonation fronts better than the single TENO scheme.