Rayleigh-Taylor Instability Problem Research Articles

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

Comparative study of WCSPH, EISPH and explicit incompressible-compressible SPH (EICSPH) for multi-phase flow with high density difference

This study presents three Smoothed Particle Hydrodynamics (SPH) methods capable of handling high-density differences in violent incompressible multiphase flows. The conventional Weakly Compressible SPH (WCSPH) is reformulated into a quasi-Lagrangian framework based on Arbitrary Lagrangian–Eulerian (ALE) context. The Explicit Incompressible SPH (EISPH) method is extended to handle multiphase flows and reformulated in the ALE framework. The Explicit Incompressible-Compressible SPH (EICSPH) method, which can handle both compressible and incompressible phases, is developed by combining these two approaches in a fully explicit algorithm. The proposed methods are validated and compared through four benchmark problems including Rayleigh-Taylor instability, hydrostatic, liquid sloshing and dam break problems. Firstly, the stability and accuracy of interface modeling of the three methods were verified through Rayleigh-Taylor instability with small density differences. In the hydrostatic problem with a large density difference, the results from all three methods exhibit good agreement in the pressure distribution. Particularly, EICSPH demonstrates the faster convergence when compared to WCSPH, with EISPH showing fastest convergence overall. In the transient sloshing problem, all three methods quantitatively converged to the experimental results and exhibited good agreement, although slight pressure noise was observed in WCSPH initially. Finally, in the dam break problem, all three methods successfully simulated sharp interfaces without non-physical voids. The temporal variations of pressure and height were well predicted by all three methods, with EISPH showing better conformity to the water-air interface morphology obtained from other incompressible numerical methods compared to WCSPH and EICSPH. Additionally, the impact of air speed of sound was examined in EICSPH, where while the difference in pressure variation at the probe was minimal, differences were observed in energy dissipation and the progression of the free surface due to the air cushion effect.

An adaptive-correction algorithm for suppressing interface smearing in incompressible multiphase flows with complex interfacial behavior

In this paper, a numerical framework for modelling multiphase flows with interface correction is proposed. In original Volume-of-Fluid method, the interface smearing may appear since the interface profile deviates from its equilibrium state, especially in the simulations of complex interface deformation. To solve this problem, a novel correction step is introduced into the procedure after solving the volume fraction transport equation, and this step just works on the region of phase interface. The difference between the current method and the previous work is that the correction coefficient can be adjusted adaptively with the gradient of normal velocity at the interface, which is strongly correlated to the intensity of interface smearing and changes with the computational time and interface position. The validation computations are performed for bubble deformation (initial shapes of two-dimensional four-lobed-star and three-dimensional (3D) box with holes), bubble rising in a channel and Rayleigh–Taylor instability problems (Reynolds number of 100 and 1000). The obtained results show that the unphysical phenomenon of interface smearing is suppressed effectively, and the interface sharpness is improved greatly by the present method. In addition, the mass of bubble deformation by the original method will decrease by 13.3%. While these results obtained by the present method are in good agreement with the analytical solutions or published data.

A three-dimensional fully threaded tree adaptive mesh phase-field lattice Boltzmann method for gas–liquid phase change problems

A fully threaded tree adaptive mesh lattice Boltzmann method based on the phase-field model with the conservative Allen–Cahn equation is presented for the simulation of multiphase flows and heat transfer, especially the gas–liquid phase change problems in three dimensions. The presented model incorporates the conservative Allen–Cahn equation for interface tracking and employs hydrodynamics and temperature evolution D3Q19 lattice Boltzmann equations to recover the corresponding Navier–Stokes equations and energy equations. The gas–liquid phase change at the phase interface can be reflected with introducing the mass production rate in the lattice Boltzmann evolution equations. With the fully threaded tree adaptive mesh implemented to capture the phase interface, the computational efficiency can obviously be raised while ensuring the accurate capture of gas–liquid interface. The present method is used to reproduce several classical benchmarks, namely, the droplet evaporation in superheated gas, the buoyancy-driven bubble rising in viscous liquid, the 3-dimensional Rayleigh Taylor instability problem, the nucleate boiling on a wall with constant temperature, and the film boiling on superheated bottom.

A preconditioning for the spectral solution of incompressible variable-density flows

In the present study, the efficiency of preconditioners for solving linear systems associated with the discretized variable-density incompressible Navier–Stokes equations with semi-implicit second-order accuracy in time and spectral accuracy in space is investigated. The method, in which the inverse operator for the constant-density flow system acts as preconditioner, is implemented for three iterative solvers: the General Minimal Residual, the Conjugate Gradient and the Richardson Minimal Residual. We discuss the method, first, in the context of the one-dimensional flow case where a top-hat like profile for the density is used. Numerical evidence shows that the convergence is significantly improved due to the notable decrease in the condition number of the operators. Most importantly, we then validate the robustness and convergence properties of the method on two more realistic problems: the two-dimensional Rayleigh–Taylor instability problem and the three-dimensional variable-density swirling jet.

A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, det⁡∇ψ(x,t)=G∘ψ(x,t). The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map ψ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method [21] that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2D and 3D mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers [22,18], while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2D gas dynamics. Specifically, we implement the C-method [64,65] and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.

Numerical modeling of liquid spills from the damaged container and collision of two rising bubbles in partially filled enclosure using modified Volume-Of-Fluid (VOF) method

In the present work, two crucial shortcomings associated with Volume-Of-Fluid (VOF) model namely: (1) spurious interface smearing arising from false-diffusion errors, and (2) non-physical velocity fluctuation across the physical discontinuities, are systematically addressed, aiming to establish a unique methodological foundation and guidelines for the enhancement of interface-capturing techniques in handling multi-fluid flows. To accomplish this objective, first, a novel third-order bounded convection scheme is derived based on the Normalized Variable Diagram and Total Variation Diminishing concepts (NVD-TVD) and is then applied for the discretization of the volume-fraction equation. To cope with the instability issue induced by pressure fluctuation, the standard version of the implicit non-iterative PISO algorithm is first modified by incorporating the third pressure-correction step into the algorithm and is then utilized for the treatment of the pressure-velocity coupling. A feasibility and applicability of the proposed modifications in dealing with violent free-surface and multi-fluid flows are demonstrated against the five different challenging benchmark cases including two-dimensional dam-break flow over the dry bed, oil spill from the damaged container, single bubble rising, merging of two rising bubbles and two-fluid Rayleigh-Taylor Instability problems. The comparison of the obtained results with previously published literatures vividly corroborates the robustness and versatility of the modified VOF model in handling multi-fluid flows involving interface coalescence and breakup events. In the last staged, three new benchmark solutions namely (1) coalescence of two consecutive bubbles inside the partially filled enclosure, (2) two-dimensional three-fluid Rayleigh-Taylor Instability, and (3) oil/water spilling from the damaged tank are analyzed using the verified VOF method, aiming to provide a high-quality validation data for CFD simulations.

Leidenfrost instability in a circular container and its suppression using a rod

Leidenfrost drops have demonstrated promising applications in, for example, drag reduction. However, large Leidenfrost drops may be unstable when their diameters exceed a critical value, leading to less control of such drops in their applications. In this work, through theoretical and experimental investigations, the authors explore the instability of a Leidenfrost drop in a circular configuration, as well as the suppression of this instability using a small rod. There are four findings. First, the diameter of the largest inscribed cylinder inside a rod–container configuration is the critical dimension for determining Leidenfrost instability. Second, in the cases of water and isopropyl alcohol, the threshold value of this diameter is 8.3λ ± 0.3λ, where λ is the capillary length of the liquid. Third, due to the specific interface profile between the liquid drop and the surrounding vapor layer, the threshold diameter of a circular container for instability to occur is slightly larger than its counterpart in the corresponding Rayleigh–Taylor instability problem. Fourth and finally, placing a rod inside a circular container reduces the size of the largest inscribed cylinder in the container. If the diameter of this inscribed cylinder is below the threshold value, the instability inside the container is suppressed.

A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow

In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier–Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylor series method is unsuitable for problems where regularity is lost by solving the 2D Rayleigh–Taylor instability problem.

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Numerical simulation of three-fluid Rayleigh-Taylor instability using an enhanced Volume-Of-Fluid (VOF) model: New benchmark solutions

The main objective of the present study is to introduce two novel benchmark solutions namely: two-dimensional three-fluid Rayleigh-Taylor Instability problems, aiming to provide an up-to-date data set and a unique fundamental insight into morphology and hydrodynamic behavior of coupled Rayleigh-Taylor-Kelvin-Helmholtz instability phenomenon. To this end, the Volume-Of-Fluid (VOF) model is adopted to probe the complex configurations and kinetic processes of highly nonlinear multi-fluid flow problems with large topological changes and moving interfaces. However, to improve the performance and accuracy of the classical VOF model and preserve monotonicity for the density and viscosity, a novel high-order bounded advection scheme is first proposed in the context of the Total Variation Diminishing and Normalized Variable Diagram (TVD-NVD) constraints and then is utilized for the discretization of the convection terms in the Navier-Stokes and transport equations. To further increase the accuracy of the numerical simulations, the second-order PLIC-ELVIRA is implemented for the reconstruction of the physical discontinuity between phases and the determination of its curvature. Furthermore, to enhance the consistency and stability of the classical VOF model in handling incompressible multi-fluid flows, a novel semi-iterative pressure-velocity coupling algorithm is constructed by the combination of the standard PISO and SIMPLEC algorithms and is then applied to ensure mass conservation in each grid cell. To demonstrate the versatility and robustness of the proposed model in dealing with the multiphase flows involving large interface deformation and breaking phenomena, a series of canonical test cases such as dam-break over a dry bed with and without stationary obstacle, 2D three-fluid rising bubble, two-fluid and three-fluid Rayleigh-Taylor Instability are adopted. In the last stage, an improved VOF model is applied to solve two new three-fluid Rayleigh-Taylor Instability benchmark problems on a staggered grid system. The results of this study can provide a wide panorama on the improvements of standard VOF model and may be utilized as benchmark solutions for validation of various CFD tools or simply to understand more complex related multi-fluid flows.

Diffuse interface relaxation model for two-phase compressible flows with diffusion processes

The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction. Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be significant for heat conduction problems. The thermal conduction leads to instantaneous thermal relaxation so that the temperature equilibrium is always maintained in the interface region with meeting the entropy relations. A numerical method is developed to solve the model governing equations that ensures the pressure-velocity-temperature (PVT) equilibrium condition in its high-order extension. We solve the hyperbolic part of the governing equations with the Godunov method with the HLLC approximate Riemann solver. The non-linear parabolic part is solved with an efficient Chebyshev explicit iterative method without dealing with large sparse matrices. To verify the model and numerical methods proposed, we demonstrate numerical results of several numerical tests such as the multiphase shock tube problem, the multiphase impact problem, and the planar ablative Rayleigh–Taylor instability problem.

Three-dimensional direct numerical simulation of Rayleigh–Taylor instability triggered by acoustic excitation

Rayleigh–Taylor instability (RTI) occurs when the interface between two fluids of different densities is removed, with the heavier (cold) fluid resting on top of the lighter (hot) fluid in the equilibrium state. This arrangement is unstable due to buoyancy, in the absence of any other forces. RTI is noted across a range of length scales from very small in nuclear fusion to supernova explosion at astrophysical scales. RTI is viewed as a baroclinic instability if viscous actions are ignored. An accurate non-overlapping parallel algorithm is used to solve a three-dimensional RTI problem, employing more than 4 × 109 points and a refined time step (7.69×10−8s) for the direct numerical simulation. Air masses at two different temperatures are initially separated by a non-conducting partition inside a box (with a temperature difference of 200 K). The impermeable partition is removed impulsively at t = 0, and the ensuing instability is triggered by an acoustic mechanism involving infra to ultrasonic pulses that travel to either side of the interface. Present high precision petascale computations enable one to capture acoustic disturbances with unprecedented accuracy without any additional interfacial disturbances. The creation of the vorticity is studied by performing enstrophy budget for the compressible flow for RTI, which shows that the viscous terms are dominant compared to the baroclinic one.

Numerical investigation on the influence of different physical factors on the interface evolution in 3D RTI using CLSVOF method

The three-dimensional (3D) Rayleigh–Taylor instability (RTI) problem is numerically investigated using the coupled level set and volume of fluid (CLSVOF) method. The CLSVOF method combines the level set and volume of fluid methods to ensure good mass conservation performance, enabling the curvature of the interface captured by numerical simulations to be accurately calculated. A classical single-mode RTI (with only one “mushroom” generated under the initial setting) is adopted for numerical verification, and the shape of the interface and the penetration depth with respect to time are determined. Based on this initial two-dimensional single-mode RTI problem, the influence of factors such as the surface tension, density ratio, Reynolds number, and viscosity ratio on the 3D effects caused by the velocity gradient on both sides of the interface are thoroughly analyzed. This analysis allows us to obtain the basic laws governing the influence of different physical factors on the 3D effects of the RTI.

Large eddy simulation of Rayleigh–Taylor mixing based on helicity model

This paper focuses on large eddy simulation of Rayleigh–Taylor instability (RTI) based on helicity model (HM) which is derived from our previous research. Based on the RTI problem, we obtain from a priori test and theoretical analysis that the suggested model can automatically distinguish laminar flow and turbulence. Comparing the numerical simulation results with direct numerical simulation (DNS) and traditional model in RTI problem, we could find that the suggested model overcomes some defects of the traditional model, such as the traditional model cannot predict transition accurately and the suggested model can predict the turbulence mixing width and mass fraction precisely. It provides a beneficial tool for the research of RTI.

An adaptive coupled volume-of-fluid and level set method based on unstructured grids

This paper presents an adaptive coupled volume-of-fluid and level set (VOSET) method based on unstructured grids to simulate incompressible interfacial flows. In this study, a novel high quality refinement algorithm for unstructured grid cells based on level set and surface curvature is first developed to ensure a preset bandwidth of the adaptive region around the interface. Subsequently, the adaptive VOSET algorithm is integrated into a finite volume solver with a large-scale sparse matrix preconditioned bi-stable conjugate method. Finally, the accuracy and robustness of the current method are validated by four classical test cases: the dam break, single bubble rising, a Rayleigh–Taylor instability problem, and a liquid jet problem. It is found that the adaptive coupled VOSET algorithm makes the numerical results near the interface more accurate, especially the curvature and surface tension at corners. Moreover, the adaptive VOSET is more efficient since it reduces the number of grid cells significantly. The present results show that only 4.85%–24.5% of the number of adaptive grid cells and 5.31%–15.93% of the computational time are needed for VOSET compared to that of fixed unstructured grid cells, and thus, it has outstanding properties of both time and computational cost savings.

A Novel Parallel Computing Strategy for Compact Difference Schemes with Consistent Accuracy and Dispersion

In this paper, based on the boundary approximation approach for parallelization of the compact difference schemes, a novel strategy for the sub-domain boundary approximation schemes is proposed to maintain consistent accuracy and dispersion with the compact scheme in the interior points. In this strategy, not only the order of accuracy of the sub-domain boundary scheme is the same as the interior scheme, but the coefficient of the first truncation error term is also equal to that of the internal scheme. Furthermore, to realize the consistent dispersion performance for a class of high order upwind compact schemes, which usually include two expressions, we modify the opposite expression to be the sub-domain boundary scheme. As an example of application, the present strategy is applied to a fourth-order upwind compact scheme, and its accuracy is verified by a numerical test. The resolution and efficiency of the newly proposed parallel method are examined by four numerical examples, including propagation of a wave-packet, convection of isentropic vortex, Rayleigh–Taylor instability problems, and propagation of Gauss pulse. The results obtained demonstrate that the present strategy for compact difference schemes has the feasibility to solve the flow problems with high accuracy, resolution and efficiency in parallel computation.

Rayleigh-Taylor Instability: A Status Review of Experimental Designs and Measurement Diagnostics

Abstract The focus of experiments and the sophistication of diagnostics employed in Rayleigh-Taylor instability (RTI) induced mixing studies have evolved considerably over the past seven decades. The first theoretical analysis by Taylor and the two-dimensional experimental results by Lewis on RTI in 1950 examined single-mode RTI using conventional imaging techniques. Over the next 70 years, several experimental designs have been used to creating an RTI unstable interface between two materials of different densities. These early experiments though innovative, were arduous to diagnose and provided little information on the internal, turbulent structure and initial conditions of the RT mixing layer. Coupled with the availability of high-fidelity diagnostics, the experiments designed and developed in the last three decades allow detailed measurements of various turbulence statistics that have allowed broadly to validate and verify late-time nonlinear models and mix-models for buoyancy-driven flows. Besides, they have provided valuable insights to solve several long-standing disagreements in the field. This review serves as an opportunity to discuss the understanding of the RTI problem and highlight valuable insights gained into the RTI driven mixing process with a focus on low to high Atwood number (>0.1) experiments.

A HLLC-type finite volume method for incompressible two-phase flows

• A hyperbolic system which couples LS equation and flow equations is obtained. • The HLLC riemann solver is proposed for the new hyperbolic system. • Matrix-free implicit dual-time stepping method is used to improve the efficiency. • The WENO reconstruction is used to achieve second-order accuracy. • The results agree well with the experimental data or other numerical results. A finite volume method based on the artificial compressibility scheme has been developed to simulate incompressible two-phase flows. In this method, the level set equation, used to capture the interface between two phases, is solved by coupling with the Navier-Stokes equations. The HLLC approximate Riemann solver, capable of resolving contact discontinuity accurately, is proposed to compute the intercell flux. In order to solve the interface flow accurately and efficiently, the high-order WENO reconstruction and the matrix-free implicit dual-time stepping method are applied in this study. Several classical cases, including a Rayleigh-Taylor instability problem, a broken dam problem and two bubble-rising problems are studied. The capability of the present scheme in simulating interface flows is validated by comparison with experimental data or reported results.

A divergence-free HDG scheme for the Cahn-Hilliard phase-field model for two-phase incompressible flow

We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes phase-field model. The scheme is robust in the convection-dominated regime, produces a globally divergence-free velocity approximation, and can be efficiently implemented via static condensation. Two numerical benchmark problems, namely the rising bubble problem, and the Rayleigh-Taylor instability problem are used to show the good performance of the proposed scheme.