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Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes equations with variable density and artificial compressibility
Abstract
We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier–Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split–form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi–Rebay 1 (BR1) scheme at the inter–element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge–Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving a manufactured solution, the Kovasznay flow, a lid driven cavity, the inviscid Taylor–Green vortex, and the Rayleigh–Taylor instability.
... However, in most of the literature, the problems of two-dimensional incompressible fluid flow are governed by the form of primitive variables or vorticity-stream function formulation. In the form of primitive variables, several techniques are usually employed for processing the coupling of velocity and pressure in the momentum conversation equations, such as the SIMPLE method [8,9], the fractional step method [10,11], the artificial compressibility methods [12,13], the penalty method [14,15], the full coupled solution method [16], etc. The processed equations of primitive variables method and vorticity-stream function equations can be discreted by various mesh-based and meshless methods, such as the finite-element method (FEM) [17], the finitedifference method (FEM) [18,19], the finite volume method (FVM) [20,21], the lattice Boltzmann method (LBM) [22,23], the radial basis function (RBF)-based method [24][25][26], the smoothed particle hydrodynamics method (SPH) [27,28], and so on. ...
... The numerical solution of stream function ̂ is approximated by Meanwhile, the hybrid kernel function H for the correction function has to fulfill the homogeneous boundary condition to ensure that the numerical solution satisfies the boundary conditions. The parameter in Eq. (12) can be calculated with the following derivation: ...
... , the boundary approximation p and hybrid kernel function H can be evaluated by Eq. (19) and Eq. (12). Then, the correction function in Eq. (19) is substituted in the linearized governing equation (17a) as ...
In this study, a novel meshless collocation method based on the Gaussian–cubic hybrid kernel function in conjunction with the ghost-points method and the general Newton–Raphson method is proposed for solving the four-order stream function formulation of Navier–Stokes equations. The decrease of variables and equations results in better computational efficiency of stream function formulation than primitive variable formulations of 2D incompressible viscous flow. The original Naiver–Stokes equations can be transformed into a four-order stream function partial differential equation by introducing the vorticity and stream function. The new proposed Gaussian–cubic backward substitution method is used to solve the corresponding system of equations where the nonlinear stream function formulation is linearized by the general Newton–Raphson method. The ghost-points method is applied as the layout scheme of center points, which can effectively improve the accuracy without requiring more computational resources. Several examples are provided and the results demonstrate the feasibility of the proposed novel approach.
... Following this, many developments on fluxes satisfying entropy conservation/dissipation condition for various hyperbolic systems were made. These include developments specific for shallow water equations [16,43,29], Euler's equations [2,20,32,8,35,36,17,11,10,45], Navier-Stokes equations [44,27,33] and magneto hydro-dynamics equations [9]. Recently, several interesting studies such as, entropy stability for conservation laws with non-convex flux functions [24], and characterisation of stability [15] and robustness (for under-resolved flows) [7] of high order entropy stable schemes were carried out. ...
... m satisfying the moment constraints in eqs. (26) and (27) and rendering the eigenvalues of ∂ U F m to be positive, then ...
... (25) satisfy the moment constraints in eqs. (26) and (27). We also know that, if the convex entropy function for vector-kinetic model (eq. ...
The moment of entropy equation for vector-BGK model results in the entropy equation for macroscopic model. However, this is usually not the case in numerical methods because the current literature consists only of entropy conserving/stable schemes for macroscopic model (to the best of our knowledge). In this paper, we attempt to fill this gap by developing an entropy conserving scheme for vector-kinetic model, and we show that the moment of this results in an entropy conserving scheme for macroscopic model. With the numerical viscosity of entropy conserving scheme as reference, the entropy stable scheme for vector-kinetic model is developed in the spirit of [33]. We show that the moment of this scheme results in an entropy stable scheme for macroscopic model. The schemes are validated on several benchmark test problems for scalar and shallow water equations, and conservation/stability of both kinetic and macroscopic entropies are presented.
... The three-phase model adopted in this work is a Cahn-Hilliard diffuse interface model, which was derived by Boyer and Lapuerta [1]. The Cahn-Hilliard model is coupled to the entropy-stable incompressible Navier-Stokes equations model derived by Manzanero et al. [2]. The spatial discretization uses a high-order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries, while an implicit-explicit (IMEX) method is adopted as temporal discretization. ...
... The three-phase model is numerically approximated in space with a high-order Discontinuous Galerkin Spectral Element Method (DGSEM) [35] that uses the Symmetric Interior Penalty (SIP) method [36,37,38,39,40]. The DGSEM has been used in the past to discretize multiphase (two phase) flows [41,42,43,44,45], and it is popular for its arbitrary order of accuracy [46,35], low dissipative and dispersive errors [47,48,49,50], the representation of arbitrary three-dimensional complex geometries through the use of unstructured meshes with curvilinear elements [51], efficient mesh adaptation techniques [52,53,54] and the design of provably stable schemes [55,56,57,58,59,2,44]. Previously, three component Cahn-Hilliard models have been discretized by means of the finite element method [20], local discontinuous Galerkin method [60] or spectral element method [61]. ...
... In this work, we couple the three-phase Cahn-Hilliard model of Boyer et al. [1,62] to the incompressible Navier-Stokes with artificial compressibility [2]. We define the concentration of Phase j as the relative volume occupied by that phase. ...
In this work we introduce the development of a three--phase incompressible Navier--Stokes/Cahn--Hilliard numerical method to simulate three--phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three--phase model adopted in this work is a Cahn--Hilliard diffuse interface model, which was derived by Boyer and Lapuerta et al. 2006. The Cahn--Hilliard model is coupled to the entropy--stable incompressible Navier--Stokes equations model derived by Manzanero et al. 2019. The spatial discretization uses a high--order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries, while an implicit--explicit (IMEX) method is adopted as temporal discretization. The developed numerical tool is tested for two and three dimensional problems, including a convergence study, a two--dimensional jet, a three--dimensional annular flow, and realistic geometries like T--shaped pipe intersections.
... This kind of approach has been adopted for incompressible flows with variable density, see e.g. 35,36 , and we aim here to consider an artificial compressibility formulation for immiscible, isothermal two-phase flows with gravity. The model equations can be therefore rewritten as follows: ...
... It is important to notice that Γ does not change during the reinizialization procedure, but is computed using the initial value of the level set function. The relation (36) has been originally introduced as an intermediate step between the level set advection and the Navier-Stokes equations to keep the shape of the profile 7 and to stabilize the advection 8 . Two fluxes are considered: a compression flux which acts where 0 < < 1 and in normal direction to the interface, represented by (1 − ) Γ , and a diffusion flux, represented by ...
We propose an implicit discontinuous Galerkin (DG) discretization for incompressible two-phase flows using an artificial compressibility formulation. The conservative level set (CLS) method is employed in combination with a reinitialization procedure to capture the moving interface. A projection method based on the L-stable TR-BDF2 method is adopted for the time discretization of the Navier-Stokes equations and of the level set method. Adaptive mesh refinement (AMR) is employed to enhance the resolution in correspondence of the interface between the two fluids. The effectiveness of the proposed approach is shown in a number of classical benchmarks. A specific analysis on the influence of different choices of the mixture viscosity is also carried out.
... The discretization of incompressible Navier-Stokes equations poses several major computational issues. In particular, the velocity u and the pressure p are coupled by the incompressibility constraint ∇· u = 0. We adopt here the so-called artificial compressibility formulation, originally introduced in [7] and employed in [3], [4], [18], [22], [24] among many others. The incompressibility constraint is relaxed and a time evolution equation for the pressure is introduced. ...
... In this Section, we briefly outline the numerical method employed for the discretization of system (18). We refer to [22] for a detailed description of the numerical scheme. ...
We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.
... This kind of approach has been adopted for incompressible flows with variable density, see e.g. [7,36], and we aim here to consider an artificial compressibility formulation for immiscible, isothermal two-phase flows with gravity. The model equations can be therefore rewritten as follows: ...
... It is important to notice that n Γ does not change during the reinizialization procedure, but is computed using the initial value of the level set function. The relation (36) has been originally introduced as an intermediate step between the level set advection and the Navier-Stokes equations to keep the shape of the profile [38] and to stabilize the advection [39]. Two fluxes are considered: a compression flux which acts where 0 < ϕ < 1 and in normal direction to the interface, represented by u c ϕ (1 − ϕ) n Γ , and a diffusion flux, represented by βεu c (∇ ϕ · n Γ ) n Γ . ...
We propose an implicit Discontinuous Galerkin (DG) discretization for incompressible two-phase flows using an artificial compressibility formulation. Conservative level set (CLS) method is employed in combination with a reinitialization procedure to capture the moving interface. A projection method based on the L-stable TR-BDF2 method is adopted for the time discretization of the Navier-Stokes equations and of the level set method. Adaptive Mesh Refinement (AMR) is employed to enhance the resolution in correspondence of the interface between the two fluids. The effectiveness of the proposed approach is shown in a number of classical benchmarks, such as the Rayleigh-Taylor instability and the rising bubble test case, for which a specific analysis on the influence of different choices of the mixture viscosity is carried out.
... In this work, we investigated some numerical challenges involved in implicit highorder accurate DG discretizations of the VDI model introduced in [19] for the simulation of multi-component incompressible flow problems. This discretization follows the idea of coupling the artificial compressibility method to the VDI flow model [19][20][21][22][23][24] in order to introduce suitable Godunov fluxes at the inter-element boundaries. In particular, the formulation relies on the exact solution of local Riemann problems based on an artificial compressibility perturbation of the incompressible Navier-Stokes equations. ...
... where ∆t is computed by either Equation (23) or Equation (24) in agreement with the condition (22) and ∆t is the new limited value for the time step size. The smooth limiter bounds are defined through the user-defined parameter ξ. ...
Multi-component flow problems are typical of many technological and engineering applications. In this work, we propose an implicit high-order discontinuous Galerkin discretization of the variable density incompressible (VDI) flow model for the simulation of multi-component problems. Indeed, the peculiarity of the VDI model is that the density is treated as an advected property, which can be used to possibly track multiple (more than two) components. The interface between fluids is described by a smooth, but sharp, variation in the density field, thus not requiring any geometrical reconstruction. Godunov numerical fluxes, density positivity, mass conservation, and Gibbs-type phenomena at material interfaces are challenges that are considered during the numerical approach development. To avoid Courant-related time step restrictions, high-order single-step multi-stage implicit schemes are applied for the temporal integration. Several test cases with known analytical solutions are used to assess the current approach in terms of space, time, and mass conservation accuracy. As a challenging application, the simulation of a 2D droplet impinging on a thin liquid film is performed and shows the capabilities of the proposed DG approach when dealing with high-density (water–air) multi-component problems.
... To achieve this, we have utilized the artificial compressibility method proposed by [9], and further developed their method to fit in the variable density context. We emphasize that this extension is not straightforward and there are many different approaches to tackle this problem, see the dissertation by [22] and recent works by [23,24] in the context of discontinuous Galerkin methods. More specifically, [23,24] presented a high-order method where the time-stepping was performed explicitly which could lead to stiffness problems if the artificial compressibility penalty parameter is chosen too greedily. ...
... We emphasize that this extension is not straightforward and there are many different approaches to tackle this problem, see the dissertation by [22] and recent works by [23,24] in the context of discontinuous Galerkin methods. More specifically, [23,24] presented a high-order method where the time-stepping was performed explicitly which could lead to stiffness problems if the artificial compressibility penalty parameter is chosen too greedily. This is in contrast to [9], who used the Taylor series method as an implicit time-stepping method. ...
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.
... The scheme developed by Qian et al. [17] has been tested further [20-26], used to model wave overtopping [27][28][29], and extended to 3D [30]. The scheme developed by Shin et al. [18] has also been tested further [31], and discontinuous Galerkin schemes using the exact Riemann solver put forward by Bassi et al. [19] have been tested as well [32,33]. This is good, but more is needed. ...
... The exact Riemann solver determines the location of the waves with respect to the τ -axis, and then substitutes the appropriate values of density, velocity, and pressure into the flux function F x,inv . As this exact solution is so simple, it has been used to calculate numerical fluxes in some discontinuous Galerkin models [19,32,33]. However, Bassi et al. [19, Appendix B4] did acknowledge that it is possible for the contact wave to overtake the acoustic wave, meaning that the assumptions break down and the Riemann solver is no longer exact. ...
Variable density incompressible flows are governed by parabolic equations. The artificial compressibility method makes these equations hyperbolic-type, which means that they can be solved using techniques developed for compressible flows, such as Godunov-type schemes. While the artificial compressibility method is well-established, its application to variable density flows has been largely neglected in the literature. This paper harnesses recent advances in the wider field by applying a more robust Riemann solver and a more easily parallelisable time discretisation to the variable density equations than previously. We also develop a new method for calculating the pressure gradient as part of the second-order reconstruction step. Based on a rearrangement of the momentum equation and an exploitation of the other gradients and source terms, the new pressure gradient calculation automatically captures the pressure gradient discontinuity at the free surface. Benchmark tests demonstrate the improvements gained by this robust Riemann solver and new pressure gradient calculation.
... The scheme developed by Qian et al. [17] has been tested further [20-26], used to model wave overtopping [27][28][29], and extended to 3D [30]. The scheme developed by Shin et al. [18] has also been tested further [31], and discontinuous Galerkin schemes using the exact Riemann solver put forward by Bassi et al. [19] have been tested as well [32,33]. This is good, but more is needed. ...
... The exact Riemann solver determines the location of the waves with respect to the τ -axis, and then substitutes the appropriate values of density, velocity, and pressure into the flux function F x,inv . As this exact solution is so simple, it has been used to calculate numerical fluxes in some discontinuous Galerkin models [19,32,33]. However, Bassi et al. [19, Appendix B4] did acknowledge that it is possible for the contact wave to overtake the acoustic wave, meaning that the assumptions break down and the Riemann solver is no longer exact. ...
Variable density incompressible flows are governed by parabolic equations. The artificial compressibility method makes these equations hyperbolic-type, which means that they can be solved using techniques developed for compressible flows, such as Godunov-type schemes. While the artificial compressibility method is well-established, its application to variable density flows has been largely neglected in the literature. This paper harnesses recent advances in the wider field by applying a more robust Riemann solver and a more easily parallelisable time discretisation to the variable density equations than previously. We also develop a new method for calculating the pressure gradient as part of the second-order reconstruction step. Based on a rearrangement of the momentum equation and an exploitation of the other gradients and source terms, the new pressure gradient calculation automatically captures the pressure gradient discontinuity at the free surface. Benchmark tests demonstrate the improvements gained by this robust Riemann solver and new pressure gradient calculation.
... This paper aims to shed some light on the robustness and cost of high-order accurate entropy stable collocated discontinuous Galerkin methods for compressible viscous flows. The literature that reports robustness studies for standard and entropy stable DC/DG formulations in CFD is scarce and focuses exclusively on inviscid flows [34,68,93] or low-speed viscous flows [29,48,54]. Hence, a detailed analysis of the numerical simulation of compressible viscous flows with under-resolved physical features or discontinuities is needed, given that the additional dissipation introduced by the viscous terms, despite obvious expectations, may not help in resolving robustness issues. ...
... This robustness study, although simple, has never been reported in literature. In fact, published work on robustness of standard and entropy stable DC/DG formulations for the TGV test case is scarce and focuses exclusively on the inviscid version of the problem [34,54,68,93]. ...
In computational fluid dynamics, the demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability at exascale and beyond. However, equally important (if not more so) is algorithmic robustness with minimal user intervention, which becomes progressively more challenging to achieve as problem size and physics complexity increase. We numerically show that low and high order entropy stable collocated discontinuous Galerkin discretizations based on summation-by-part operators and simultaneous-approximation-terms technique provide an essential step toward a truly enabling technology in terms of reliability and robustness for both under-resolved turbulent flow simulations and flows with discontinuities.
... This paper aims to shed some light on the robustness and cost of high-order accurate entropy stable discontinuous collocation discretizations for compressible viscous flows. The literature that reports robustness studies for standard and entropy stable DC/DG formulations in CFD is scarce and focuses exclusively on inviscid flows [34,68,93] or low-speed viscous flows [29,48,54]. ...
... This robustness study, although simple, has never been reported in literature. In fact, published work on robustness of standard and entropy stable DC/DG formulations for the TGV test case is scarce and focuses exclusively on the inviscid version of the problem [34,54,68,93]. Furthermore, we report the results for a very wide spectrum of solution polynomial orders, ranging from p = 1 to p = 15. ...
This work reports on the performances of a fully-discrete hp-adaptive entropy stable discontinuous collocated Galerkin method for the compressible Naiver–Stokes equations. The resulting code framework is denoted by SSDC, the first S for entropy, the second for stable, and DC for discontinuous collocated. The method is endowed with the summation-by-parts property, allows for arbitrary spatial and temporal order, and is implemented in an unstructured high performance solver. The considered class of fully-discrete algorithms are systematically designed with mimetic and structure preserving properties that allow the transfer of continuous proofs to the fully discrete setting. Our goal is to provide numerical evidence of the adequacy and maturity of these high-order methods as potential base schemes for the next generation of unstructured computational fluid dynamics tools. We provide a series of test cases of increased difficulty, ranging from non-smooth to turbulent flows, in order to evaluate the numerical performance of the algorithms. Results on weak and strong scaling of the distributed memory implementation demonstrate that the parallel SSDC solver can scale efficiently over 100,000 processes.
... The most popular methods of approximation use the velocity as primary unknown and projection-correction methods [2,24,10,15]. Recent works also focus on the development of artificial compressibility methods that have introduced a high order method in time [39], an entropy stable method for discontinuous finite element [40], and an unconditional stable method based on SAV technology [54]. One of the main drawback of the above methods is that the resulting linear algebra requires to reassemble the mass matrix, that takes the form ρ∂ t u, and the diffusive operator at each time iteration which is either computationally expensive for high order finite element discretization or cannot be made implicit for spectral and pseudo-spectral methods. ...
We introduce a novel artificial compressibility technique to approximate the incompressible Navier-Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed scheme used the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a scheme with time independent stiffness matrices that is suitable for pseudo-spectral methods. The stability and temporal convergence of the semi-implicit version of the scheme is established under the hypothesis that the density is approximated with a method that conserves the minimum-maximum principle. Numerical illustrations confirm that both the semi-implicit and explicit scheme are stable and converge with order one under classic CFL condition. Moreover, the proposed scheme is shown to perform better than a momentum based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi-fluids in a cylinder.
... DGSEM, in particular, allows the construction of provably stable discretization schemes, enabling kinetic energy-preserving and entropystable formulations for a range of equations, including the Euler equations [22], compressible Navier-Stokes [23], the Spalart-Allmaras turbulence model for compressible Reynolds-Averaged Navier-Stokes equations [24], the compressible multiphase Baer-Nunziato equations [25][26][27], and magnetohydrodynamics [28,29]. We have also found success in applying provably entropystable DGSEM schemes to the Cahn-Hilliard equation [30], incompressible Navier-Stokes with artificial compressibility [31], and the multiphase incompressible Navier-Stokes/Cahn-Hilliard iNS/CH [1], with p-adaptivity [32]. ...
We present a novel approach for simulating acoustic (pressure) wave propagation across different media separated by a diffuse interface through the use of a weak compressibility formulation. Our method builds on our previous work on an entropy-stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes/Cahn-Hilliard system \cite{manzanero2020entropyNSCH}, and incorporates a modified weak compressibility formulation that allows different sound speeds in each phase. We validate our method through numerical experiments, demonstrating spectral convergence for acoustic transmission and reflection coefficients in one dimension and for the angle defined by Snell's law in two dimensions. Special attention is given to quantifying the modeling errors introduced by the width of the diffuse interface. Our results show that the method successfully captures the behavior of acoustic waves across interfaces, allowing exponential convergence in transmitted waves. The transmitted angles in two dimensions are accurately captured for air-water conditions, up to the critical angle of . This work represents a step forward in modeling acoustic propagation in incompressible multiphase systems, with potential applications to marine aeroacoustics.
... Lately, by deriving the viscosity coefficients through a residual-based shock-capturing approach, Lundgren et al. [24] presented a novel symmetric and tensor-based viscosity method, which can ensure the conservation of angular momentum and the dissipation of kinetic energy. For the variable density incompressible flows, an entropy-stable scheme was explored in [27] by combining the discontinuous Galerkin method with an artificial compressible approximation. Recognizing the significance of density bounds in numerical simulations, a bound-preserving discontinuous Galerkin method was introduced in [17]. ...
In this paper, we consider a mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density. Utilizing the square transformation, we first ensure the positivity of the numerical fluid density, which is form-invariant and regardless of the discrete scheme. Then, by proposing a new recovery technique to eliminate the numerical dissipation of the energy and to balance the loss of the mass when approximating the reformation form, we preserve the original energy identical-relation and mass conservation of the proposed scheme. To the best of our knowledge, this is the first work that can preserve the original energy identical-relation for the Navier-Stokes equations with variable density. Moreover, the error estimates of the considered scheme are derived. Finally, we show some numerical examples to verify the correctness and efficiency.
... Even when such solutions exist, they are often not convenient from a computational perspective. Therefore, various numerical methods are commonly used to obtain approximate solutions of Eq. (1.1), including the boundary element method (BEM) [2,9], finite element method (FEM) [13][14][15], discontinuous Galerkin method (DGM) [16][17][18][19], finite difference method (FDM) [20][21][22][23], finite volume method (FVM) [24,25], spectral method (SM) [26,27], Bernoulli matrix method [11,[28][29][30][31], and others. When approximating time-dependent PDEs with initial-boundary value conditions (1.1), FEM, DGM, FDM, and FVM always produce time-stepping schemes which require the implementation of sparse matrix-vector products (for explicit schemes) or the solution of a sequence of sparse linear systems (for implicit schemes), but their convergence accuracy is often severely constrained by suitable mesh sizes in space and time; this phenomenon may become more apparent when high accuracy numerical solutions of evolutionary PDEs (1.1) are demanded in some real applications. ...
We propose a Bernoulli‐barycentric rational matrix collocation method for two‐dimensional evolutionary partial differential equations (PDEs) with variable coefficients. This method absorbs Bernoulli polynomials and barycentric rational interpolations as the basis functions in time and space, respectively. The theoretical accuracy of the proposed numerical scheme is proven to be , where is the number of basis functions in time, and and are the grid sizes in the and directions, respectively. Additionally, . To efficiently solve the linear systems arising from the discretizations, we introduce a class of dimension‐expanded preconditioners that leverage the structural properties of the coefficient matrices. A theoretical analysis of the eigenvalue distributions of the preconditioned matrices is provided. The effectiveness of the proposed method and preconditioners is demonstrated through numerical experiments on real‐world examples, including the heat conduction equation, the advection‐diffusion equation, the wave equation, and telegraph equations.
... In [38][39][40], the convex splitting method or the stabilized explicit method was used to compute the Cahn-Hilliard equations of two-phase incompressible flows, and the energy for the proposed scheme was unconditionally stable. For the Navier-Stokes equations, there are also a series of successful techniques to deal with the divergence free condition, for instance the projection method [41][42][43][44][45], the artificial compressibility method [46][47][48][49][50][51][52][53][54][55] and so on. Furthermore, the artificial compression method is used to solve the multiphysics coupling model [56][57][58]. ...
... Unlike constant density flow, variable density flow has additional conserved properties and no fully conservative formulation is currently known in the literature. The current state-of-the-art formulations for variable density flow are based on skew-symmetric formulations that can be shown to conserve squared density and kinetic energy [13] or a formulation that can be shown to conserve mass, kinetic energy, momentum and angular momentum [30,22]. In this work, we extend the modified pressure technique [5] to variable density flow which leads to a new formulation that, when discretized by a Galerkin method, is shift-invariant and also conserves mass, squared density, momentum and angular momentum. ...
This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to favorable discrete conservation properties without the divergence-free constraint being strongly enforced. In addition, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.
... Artificial compressibility methods involve a penalty parameter that determines the strength of the imposed divergence-free constraint, leading to a trade-off between accuracy and computational effort. For explicit artificial compressibility methods [61,45,39], this limitation arises from time-step restrictions. On the other hand, implicit artificial compressibility methods [18,19,37] are affected by the resulting condition number of the linear systems. ...
In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios.
... Some researchers combined artificial compressibility with other methods. For instance, Manzanero et al. [9] combined the artificial compressibility with the discontinuous Galerkin spectral element method to solve incompressible Navier-Stokes equations. A mathematical entropy function was defined to consider the energy terms. ...
The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies. Therefore, this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow. To improve the stability and convergence speed of time discretizations, the fifth-order Runge-Kutta method is applied. A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids. The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation. The optimum artificial compressibility is found to be between 1 and 5. These findings can be utilized to enhance the performance of commercial numerical simulation tools, including ANSYS and COMSOL.
... We select Gauss-Lobatto (GL) points, as they are becoming very popular in newly energy-stable and entropy-conserving schemes ( [39,62]). Using GL points, the nodal (grid point) values become u k h,j (t) = u k h (ξ j , t), and l j (ξ) is the N -th order Lagrange interpolating polynomial, ...
The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection-diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immerse boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection-diffusion equations.
... Artificial compressibility methods involve a penalty parameter that determines the strength of the imposed divergence-free constraint, leading to a trade-off between accuracy and computational effort. For explicit artificial compressibility methods [61,45,39], this limitation arises from time-step restrictions. On the other hand, implicit artificial compressibility methods [18,19,37] are affected by the resulting condition number of the linear systems. ...
... Many works were focused on finding an entropy stable semidiscretization (in space), isolating the entropy conservative flux. These techniques have been applied to many applications, inter alia for shallow water equations [35,71,50], for Euler's equation [51,34,13,56,55,19], Navier Stokes problems [33,72,20,47,46], magneto hydro-dynamics problems [14], multiphase or multicomponent problems [18,58,48] and generally for hyperbolic problems [2,4,42,43,15,10,27,28,41,38,57], also in the Lagrangian framework [11,22,12]. More recently also fully discrete entropy stable or entropy conservative schemes have been introduced, e.g. ...
In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.
... Despite the apparent numerical and computational beneficial properties of SEM, they present important temporal stability constraints, and they lack robustness in under-resolved turbulent flows simulations due to aliasing errors and the inherent low numerical dissipation of these methods. To alleviate the latter issues, several techniques exist: the spectral vanishing viscosity (SVV) [36], dealiasing techniques [48], modal filtering [19], application of skew-symmetric formulations [2], use of entropy-stable schemes, [37], domain-invariant methods [43], etc.. Nevertheless, such methods are usually accompanied by an increased computational cost and/or the need for appropriate tune-in of a certain set of parameters. ...
The purpose of this work is to describe in detail the development of the spectral difference Raviart–Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the flux reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows building a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough analysis of the numerical dissipation and dispersion of both aforementioned schemes and the nodal discontinuous Galerkin FR (FR-DG) method with two and three-dimensional elements is proposed through the use of the combined-mode Fourier approach. SDRT is shown to possess an enhanced temporal linear stability in comparison to FR-DG. On the contrary, SDRT displays larger dissipation and dispersion errors with respect to FR-DG. Finally, the study is concluded with a set of numerical experiments, the linear advection-diffusion problem, the Isentropic Euler Vortex, and the Taylor-Green Vortex (TGV). The latter test case shows that SDRT schemes present a non-linear unstable behavior with simplex elements and certain polynomial degrees. For the sake of completeness, the matrix form of the SDRT method is developed and the computational performance of SDRT with respect to FR schemes is evaluated using GPU architectures.
... )(Hesthaven and Warburton, 2007;Manzanero et al., 2020;Kirby and Sherwin, 2006) on = [−0.5, 1] × [−0.5, 1.5] (m 2 ). ...
A high-order hybrid continuous-Galerkin numerical method, designed for the simulation of non-linear, non-hydrostatic internal waves and turbulence in long computational domains with complex bathymetry, is presented. The spatial discretization in the non-periodic wave-propagating directions, utilizes the nodal spectral element method. Such a high-order element-based discretization allows the highly accurate representation of complex domain geometry along with the flexibility of concentrating resolution in areas of interest. Under the assumption of the normal-to-isobath propagation of non-linear internal waves, a third periodic direction is incorporated via a Fourier-Galerkin discretization. The distinct non-hydrostatic nature of non-linear internal waves and, any instabilities and turbulence therein, necessitates the numerically challenging solution of the pressure Poisson problem. A defining feature of this work is the application of a domain decomposition approach, combined with block-Jacobi/deflation-based preconditioning to the pressure Poisson problem. Such a combined approach is particularly suitable for the long high aspect-ratio complex domains of interest and enables the efficient high-accuracy reproduction of the non-hydrostatic dynamics of non-linear internal waves. Implementation details are also described in the context of the stability of the solver and its parallelization strategy. A series of benchmarks of increasing complexity demonstrate the robustness of the flow solver. The benchmarks culminate with the three-dimensional simulation of a convectively breaking mode-one non-linear internal wave over a realistic South-China-Sea bathymetric transect and background current/stratification profiles.
... hp-Versions and handling of meshes with small faces have been considered in [28,29]; see also the recent monograph [30]. More recently, Tavelli and Dumbser [31,32] and Dumbser et al. [33] proposed to use staggered meshes, while Manzanero et al. [34] devised an entropy-stable nodal DG spectral element method. ...
We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on h-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.
... Many works were focused on finding an entropy stable semidiscretization (in space), isolating the entropy conservative flux. These techniques have been applied to many applications, inter alia for shallow water equations [33,63,45], for Euler's equation [46,32,12,50,49,17], Navier Stokes problems [31,64,18,43,42], magneto hydrodynamics problems [13], multiphase or multicomponent problems [16,52,44] and generally for hyperbolic problems [1,3,39,40,14,9,25,26,38,35,51], also in the Lagrangian framework [10,20,11]. More recently also fully discrete entropy stable or entropy conservative schemes have been introduced, e.g. ...
In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.
... Until the present paper, numerical methods to solve the variable-density artificialcompressibility equations have not been implemented on 3D unstructured meshes. They have been implemented on 2D structured meshes [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36], 3D structured meshes [17,37,38,39], and 2D unstructured meshes [40]. While most of these papers use the Cartesian cut-cell method to model simple obstacles, an implementation on 3D unstructured meshes is needed for more complex geometries. ...
Free-surface flows and other variable density incompressible flows have numerous important applications in engineering.
One way such flows can be modelled is to extend established numerical methods for compressible flows to incompressible flows using the method of artificial compressibility. Artificial compressibility introduces a pseudo-time derivative for pressure and, in each real-time step, the solution advances in pseudo-time until convergence to an incompressible limit - a fundamentally different approach than SIMPLE, PISO, and PIMPLE, the standard methods used in OpenFOAM. Although the artificial compressibility method is widespread in the literature, its application to free-surface flows is not. In this paper, we apply the method to variable density flows on 3D unstructured meshes for the first time, implementing a Godunov-type scheme with MUSCL reconstruction and Riemann solvers, where the free surface gets captured automatically by the contact wave in the Riemann solver. The critical problem in this implementation lies in the slope limiters used in the MUSCL reconstruction step. It is well-known that slope limiters can inhibit convergence to steady state on unstructured meshes; the problem is exacerbated here as convergence in pseudo-time is required not just once, but at every real-time step. We compare the limited gradient schemes included in OpenFOAM with an improved limiter from the literature, testing the solver against dam-break and hydrostatic pressure benchmarks. This work opens OpenFOAM up to the method of artificial compressibility, breaking the mould of PIMPLE and harnessing high-resolution shock-capturing schemes that are easier to parallelise.
... ℎ -Versions and handling of meshes with small faces have been considered in [3,4]; see also the recent monograph [22]. More recently, Tavelli and Dumbser [58,59] and Dumbser et al. [41] proposed to use staggered meshes, while Manzanero et al. [52] devised an entropy-stable nodal dG spectral element method. ...
We propose two Hybrid High-Order (HHO) methods for the incompressible Navier-Stokes equations and investigate their robustness with respect to the Reynolds number. While both methods rely on a HHO formulation of the viscous term, the pressure-velocity coupling is fundamentally different, up to the point that the two approaches can be considered antithetical. The first method is kinetic energy preserving, meaning that the skew-symmetric discretization of the convective term is guaranteed not to alter the kinetic energy balance. The approximated velocity fields exactly satisfy the divergence free constraint and continuity of the normal component of the velocity is weakly enforced on the mesh skeleton, leading to H-div conformity. The second scheme relies on Godunov fluxes for pressure-velocity coupling: a Harten, Lax and van Leer (HLL) approximated Riemann Solver designed for cell centered formulations is adapted to hybrid face centered formulations. The resulting numerical scheme is robust up to the inviscid limit, meaning that it can be applied for seeking approximate solutions of the incompressible Euler equations. The schemes are numerically validated performing steady and unsteady two dimensional test cases and evaluating the convergence rates on h-refined mesh sequences. In addition to standard benchmark flow problems, specifically conceived test cases are conducted for studying the error behaviour when approaching the inviscid limit.
... The choice [31] is based on the formula L 2 r /ν where L r taken as 1/2π. From the tabulated results of the velocity components, shown in Tables 2 and 3, it can be observed equal order of accuracy for the velocity and its [55][56][57] where the pressure convergence order is consistently smaller than that of the velocity. ...
A high-order Flux reconstruction implementation of the hyperbolic formulation for the incompressible Navier-Stokes equation is presented. The governing equations employ Chorin's classical artificial compressibility (AC) formulation cast in hyperbolic form. Instead of splitting the second-order conservation law into two equations, one for the solution and another for the gradient, the Navier-Stokes equation is cast into a first-order hyperbolic system of equations. Including the gradients in the AC iterative process results in a significant improvement in accuracy for the pressure, velocity, and its gradients. Furthermore, this treatment allows for taking larger time-steps since the hyperbolic formulation eliminates the restriction due to diffusion. Tests using the method of manufactured solutions show that solving the conventional form of the Navier-Stokes equation lowers the order of accuracy for gradients, while the hyperbolic method is shown to provide equal orders of accuracy for both the velocity and its gradients which may be beneficial in several applications. Two- and three-dimensional benchmark tests demonstrate the superior accuracy and computational efficiency of the developed solver in comparison to the conventional method and other published works. This study shows that the developed high-order hyperbolic solver for incompressible flows is attractive due to its accuracy, stability and efficiency in solving diffusion dominated problems.
... This is of major importance with DTS is presented. In addition, the stability limit of the DTS approach is 31 derived using a von Neumann analysis, leading to a closed form expression for 32 the critical dual time step. Finally, an optimal choice for the dual time step is 33 proposed by imposing maximum damping over the whole range of frequencies. ...
This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a von Neumann analysis, leading to a closed form expression for the critical dual time step. An optimal choice for the dual time step, producing the maximum damping for all the frequencies, is also derived. Steady and transient convection-diffusion problems are considered to demonstrate the performance of the proposed DTS approach, with particular emphasis on convection dominated problems. Two simple approaches to accelerate the convergence of the DTS approach are also considered and three different time marching approaches for the dual time are compared.
... But using the SBP property of the DGSEM-LGL operators, it is possible to apply ideas similar to Fisher et al. and construct a novel DGSEM with LGL quadrature, that is discretely L 2 -stable for the nonlinear Burgers' equation, without the assumption on exact evaluation of the integrals [111]. These first results have been extended and compounded upon for the compressible Euler equations [114][115][116][117][118], the shallow water equations [63,[119][120][121], the compressible Navier-Stokes equations [32,122,124], non-conservative multi-phase problems [124], magnetohydrodynamics [125,126], relativistic Euler [127], relativistic magnetohydrodynamics [128], the Cahn-Hilliard equations [129], incompressible Navier-Stokes (INS) [130], and coupled Cahn-Hilliard and INS [131] among many other complex PDE models and DG discretization types e.g., [132]. ...
In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.
... There are two alternatives which avoid the nonlinear saddle point system. The first option, which is not discussed here, is to introduce an artificial compressibility as a numerical parameter, see e.g., Noventa et al. (2016), Loppi et al. (2018), Guermond and Minev (2019), or Manzanero et al. (2020). The second option are projection methods, where each time step is divided into several sub-steps with a simpler structure. ...
In this book chapter, the high-performance implementation of discontinuous Galerkin methods is reviewed, with the main focus on sum factorization algorithms. The main computational properties of the algorithms are compared to capabilities of modern computer hardware, highlighting the opportunities and limitations of discontinuous Galerkin discretizations. The chapter closes with a presentation of how to apply these algorithms to the compressible Euler equations, the acoustic wave equation, and the incompressible Navier–Stokes equations.
High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously---sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin--Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation.
Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes, as FD schemes typically have more unresolved modes than nodal DG methods.
This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.
The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection–diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immersed boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection–diffusion equations.
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.
In this work we introduce the development of a three–phase incompressible Navier–Stokes/Cahn–Hilliard numerical method to simulate three–phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three–phase model adopted in this work is a Cahn–Hilliard diffuse interface model, which was derived by Boyer and Lapuerta (2006). The Cahn–Hilliard model is coupled to the kinetic–energy stable incompressible Navier–Stokes equations model derived by Manzanero et al. (2020). The spatial discretization uses a high-order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries. An implicit–explicit (IMEX) method is adopted as temporal scheme for the Cahn–Hilliard equation, while Runge–Kutta 3 (RK3) is used for the Navier–Stokes equations. The developed numerical tool is validated with a manufactured solution test case and used to simulate multiphase flows in pipes, including and a three–phase T–shaped pipe intersection.
We develop a novel entropy–stable discontinuous Galerkin approximation of the incompressible Navier–Stokes/Cahn–Hilliard system for p–non–conforming elements. This work constitutes an evolution of the work presented by Manzanero et al. ((2020) [10]), as it extends the discrete analysis into supporting p–adaptation (p–refinement/coarsening). The scheme is based on the summation–by–parts simultaneous–approximation term property along with Gauss–Lobatto points and suitable numerical fluxes. The p–non–conforming elements are connected through the classic mortar method, the use of central fluxes for the inviscid terms, and the BR1 scheme with additional dissipation for the viscous fluxes. The scheme is proven to retain its properties of the original conforming scheme when transitioning to p–non–conforming elements and to mimic the continuous entropy analysis of the model. We focus on dynamic polynomial adaptation as the applications of interest are unsteady multiphase flows. In this work, we introduce a heuristic adaptation criterion that depends on the location of the interface between the different phases and utilises the convection velocity to predict the movement of the interface. The scheme is verified to be total phase conserving, entropy–stable and freestream preserving for curvilinear p–non–conforming meshes. We also present the results for a rising bubble simulation and we show that for the same accuracy we get a ×2 to ×6 reduction in the degrees of freedom and a 41% reduction in the computational time. We compare our results for the three–dimensional dam break test case against experimental and numerical data and we show that a ×4.3 to ×9.5 reduction of the degrees of freedom and a 51% reduction in the computational time can be achieved compared to the p–uniform solution.
We present an entropy–stable formulation for the compressible Reynolds Averaged Navier–Stokes (RANS) Discontinuous Galerkin (DG) equations and the Spalart–Allmaras one–equation closure. The model is designed to satisfy an entropy law, which includes free– and no–slip wall boundary conditions. We then construct a high–order DG approximation of the model that satisfies the summation–by–parts simultaneous–approximation–term (SBP-SAT) property. With the help of a discrete stability analysis, we construct two approximations: a kinetic energy preserving scheme based on Pirozzoli's two–point flux and a thermodynamic entropy conserving one based on Chandrashekar's split–form. The schemes are applicable to, and the stability proofs hold for, three–dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution for increasing polynomial orders and mesh refinement levels, to then assess their numerical stability by propagating a flow from a random initial condition, and finally solve the flow around a two–dimensional flat plate and a NACA 0012 airfoil, comparing numerical results with those available in the literature. The proposed schemes are entropy–stable, and provide accurate solutions for the selected test cases.
We present a high–order discontinuous Galerkin (DG) discretization for the three–phase Cahn–Hilliard model of [Boyer, F., & Lapuerta, C. (2006). Study of a three component Cahn–Hilliard flow model]. In this model, consistency is ensured with an additional term in the chemical free–energy. The model considered in this work includes a wall boundary condition that allows for an arbitrary equilibrium contact angle in three–phase flows. The model is discretized with a high–order discontinuous Galerkin spectral element method that uses the symmetric interior penalty to compute the interface fluxes, and allows for unstructured meshes with curvilinear hexahedral elements. The integration in time uses a first order IMplicit–EXplicit (IMEX) method, such that the associated linear systems are decoupled for the two Cahn–Hilliard equations. Additionally, the Jacobian matrix is constant, and identical for both equations. This allows us to solve the two systems by performing only one LU factorization, with the size of the two–phase system, followed by two Gauss substitutions. Finally, we test numerically the accuracy of the scheme providing convergence analyses for two and three–dimensional cases, including the captive bubble test, the study of two bubbles in contact with a wall and the spinodal decomposition in a cube and in a curved pipe with a “T” junction.
We present a Computational Fluid Dynamics (CFD)–based methodology for the modeling of erosion and corrosion in hydrocarbon pipes. The novelty of this work is the use of a high–order Discontinuous Galerkin Spectral Element Method (DGSEM) approximation of the incompressible Navier–Stokes/Cahn–Hilliard model for the CFD simulation. This technique permits a very detailed three dimensional representation of the flow regime, phases distribution and contact surfaces that conform the pipe, which results in accurate computations of erosion and corrosion rates and distribution over the pipeline surface.
The developed methodology is validated with experiments relevant for oil and gas industry. In particular, we simulate the erosion in a one–phase ascending pipe with two elbows and the corrosion in a two–phase pipe under several flow regimes.
Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems. This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier–Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.
We present an implicit Large Eddy Simulation (iLES) h/p high order (≥2) unstructured Discontinuous Galerkin–Fourier solver with sliding meshes. The solver extends the laminar version of Ferrer and Willden, 2012 [34], to enable the simulation of turbulent flows at moderately high Reynolds numbers in the incompressible regime. This solver allows accurate flow solutions of the laminar and turbulent 3D incompressible Navier–Stokes equations on moving and static regions coupled through a high order sliding interface. The spatial discretisation is provided by the Symmetric Interior Penalty Discontinuous Galerkin (IP-DG) method in the x–y plane coupled with a purely spectral method that uses Fourier series and allows efficient computation of spanwise periodic three-dimensional flows.
Since high order methods (e.g. discontinuous Galerkin and Fourier) are unable to provide enough numerical dissipation to enable under-resolved high Reynolds computations (i.e. as necessary in the iLES approach), we adapt the laminar version of the solver to increase (controllably) the dissipation and enhance the stability in under-resolved simulations. The novel stabilisation relies on increasing the penalty parameter included in the DG interior penalty (IP) formulation. The latter penalty term is included when discretising the linear viscous terms in the incompressible Navier–Stokes equations. These viscous penalty fluxes substitute the stabilising effect of non-linear fluxes, which has been the main trend in implicit LES discontinuous Galerkin approaches.
The IP-DG penalty term provides energy dissipation, which is controlled by the numerical jumps at element interfaces (e.g. large in under-resolved regions) such as to stabilise under-resolved high Reynolds number flows. This dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps, thus providing an efficient stabilization mechanism for iLES.
The IP-DG stabilisation is complemented with a Spectral Vanishing Viscosity (SVV) method, in the z-direction, to enhance stability in the continuous Fourier space. The coupling between the numerical viscosity in the DG plane and the SVV damping, provides an efficient approach to stabilise high order methods at moderately high Reynolds numbers.
We validate the formulation for three turbulent flow cases: a circular cylinder at Re=3900, a static and pitch oscillating NACA 0012 airfoil at Re=10000 and finally a rotating vertical-axis turbine at Re=40000, with Reynolds based on the circular diameter, airfoil chord and turbine diameter, respectively. All our results compare favourably with published direct numerical simulations, large eddy simulations or experimental data.
We conclude that the DG-Fourier high order solver, with IP-SVV stabilisation, proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery.
High Order DG methods with Riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behaviour: dispersion errors are very low for a broad range of scales, while dissipation errors are very low for well resolved scales and are very high for scales close to the Nyquist cutoff. This observation motivates the trend that DG methods with Riemann solvers are used without an explicit LES model added. Due to under-resolution of vortical dominated structures typical for LES type setups, element based high order methods suffer from stability issues caused by aliasing errors of the non-linear flux terms. A very common strategy to fight these aliasing issues (and instabilities) is so-called polynomial de-aliasing, where interpolation is exchanged with projection based on an increased number of quadrature points. In this paper, we start with this common no-model or implicit LES (iLES) DG approach with polynomial de-aliasing and Riemann solver dissipation and review its capabilities and limitations. We find that the strategy gives excellent results, but only when the resolution is such, that about 40\% of the dissipation is resolved. For more realistic, coarser resolutions used in classical LES e.g. of industrial applications, the iLES DG strategy becomes quite in-accurate. The core of this work is a novel LES strategy based on split form DG methods that are kinetic energy preserving. Such discretisations offer excellent stability with full control over the amount and shape of the added artificial dissipation. This premise is the main idea of the work and we will assess the LES capabilities of the novel split form DG approach. We will demonstrate that the novel DG LES strategy offers similar accuracy as the iLES methodology for well resolved cases, but strongly increases fidelity in case of more realistic coarse resolutions.
Over the past few years, high-order discontinuous Galerkin (DG) methods for Large-Eddy Simulation (LES) have emerged as a promising approach to solve complex turbulent flows. However, despite the significant research investment, the relation between the discretization scheme, the subgrid-scale (SGS) model and the resulting LES solver remains unclear. This paper aims to shed some light on this matter. To that end, we investigate the role of the Riemann solver, the SGS model, the time resolution, and the accuracy order in the ability to predict a variety of flow regimes, including transition to turbulence, wall-free turbulence, wall-bounded turbulence, and turbulence decay. The transitional flow over the Eppler 387 wing, the Taylor-Green vortex problem and the turbulent channel flow are considered to this end. The focus is placed on post-processing the LES results and providing with a rationale for the performance of the various approaches.
In this work we prove that the original Bassi and Rebay [F. Bassi and S. Rebay, A High Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics, 131:267--279, 1997] scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto (GL) nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier-Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. [M. Carpenter, T. Fisher, E. Nielsen, and S. Frankel, Entropy Stable Spectral Collocation Schemes for the Navier--Stokes Equations: Discontinuous Interfaces, SIAM Journal on Scientific Computing, 36: B835-B867, 2014] for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.
Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier-Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy.
This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.
Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.
We present the development of a sliding mesh capability for an unsteady high order (order ⩾ 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.
In the discontinuous Galerkin (DG) community, several formulations have been proposed to solve PDEs involving second-order spatial derivatives (e.g. elliptic problems). In this paper, we show that, when the discretisation is restricted to the usage of Gauss–Lobatto points, there are important similarities between two common choices: the Bassi-Rebay 1 (BR1) method, and the Symmetric Interior Penalty (SIP) formulation. This equivalence enables the extrapolation of properties from one scheme to the other: a sharper estimation of the minimum penalty parameter for the SIP stability (compared to the more general estimate proposed by Shahbazi [1]), more efficient implementations of the BR1 scheme, and the compactness of the BR1 method for straight quadrilateral and hexahedral meshes.
In this work we present and compare three Riemann solvers for the artificial compressibility perturbation of the 1D variable density incompressible Euler equations. The goal is to devise an artificial compressibility flux formulation to be used in Finite Volume or discontinuous Galerkin discretizations of the variable density incompressible Navier-Stokes equations. Starting from the constant density case, two Riemann solvers taking into account density jumps at fluid interfaces are first proposed. By enforcing the divergence free constraint in the continuity equation, these approximate Riemann solvers deal with density as a purely advected quantity. Secondly, by retaining the conservative form of the continuity equation, the exact Riemann solver is derived. The variable density solution is fully coupled with velocity and pressure unknowns. The Riemann solvers are compared and analysed in terms of robustness on harsh 1D Riemann problems. The extension to multidimensional problems is described. The effectiveness of the exact Riemann solver is demonstrated in the context of an high-order accurate discontinuous Galerkin discretization of variable density incompressible flow problems. We numerically validate the implementation considering the Kovasznay test case and the Rayleigh-Taylor instability problem.
Fisher and Carpenter (High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518--557, 2013) found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes. We show that it is possible to exploit the equivalent subcell formulation even further. With the specific choice of subcell finite volume fluxes we recover split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as e.g. the Ducros splitting and the Kennedy and Gruber splitting. %This equivalence holds in general for diagonal norm summation-by-parts operators. In this work, we recast the discontinuous Galerkin spectral element method with Gauss-Lobatto nodes into the subcell finite volume form. Thus, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor Green vortex and show that the new split forms enhance the stability of computations in comparison to the standard scheme when solving turbulent vortical dominated flows.
We report development of a high-order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. This compact high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids.
In engineering applications critical complex unsteady flows often are, at least in certain flow areas, only marginally resolved. Within these areas, the truncation error of the underlying difference schemes strongly affects the solution. Therefore, a significant gain in computational efficiency is possible if the truncation error functions as physically consistent, i.e. reproducing the correct evolution of resolved scales, subgrid-scale (SGS) model. The truncation error of high-order WENO-based schemes can be exploited to function as an implicit subgrid-scale (SGS) model. A recently developed sixth-order adaptive central-upwind weighted essentially non-oscillatory scheme with implicit scale-separation has been demonstrated to incorporate a physically consistent implicit SGS model for compressible turbulent flows. We consider the implicit SGS modeling capabilities of an improved version of this scheme simultaneously for underresolved turbulent and non-turbulent incompressible flows, thus extending previous works on this subject to a more general scope. With this model we are able to reach very long integration times for the incompressible Taylor–Green vortex at infinite Reynolds number, and recover in particular a low-mode transition to isotropy. Inviscid shear-layer instabilities are resolved to highly nonlinear stages, which is shown by considering the doubly periodic two-dimensional shear layer as test configuration. Proper resolved-scale prediction is also obtained for viscous–inviscid interactions and fully confined viscous flows. These properties are demonstrated by applying the model to a vortex–wall interaction problem and lid-driven cavity flow.
A level set method for capturing the interface between two fluids is
combined with a variable density projection method to allow for
computation of two-phase flow where the interface can merge/break and
the flow can have a high Reynolds number. A distance function
formulation of the level set method enables one to compute flows with
large density ratios (1000/1) and flows that are surface tension driven;
with no emotional involvement. Recent work has improved the accuracy of
the distance function formulation and the accuracy of the advection
scheme. We compute flows involving air bubbles and water drops, to name
a few. We validate our code against experiments and theory.
Developing stable and robust high-order finite-difference schemes requires mathematical
formalism and appropriate methods of analysis. In this work, nonlinear entropy
stability is used to derive provably stable high-order finite-difference methods
with formal boundary closures for conservation laws. Particular emphasis is placed
on the entropy stability of the compressible Navier-Stokes equations. A newly derived
entropy stable weighted essentially non-oscillatory finite-difference method is
used to simulate problems with shocks and a conservative, entropy stable, narrowstencil
finite-difference approach is used to approximate viscous terms.
Several methods have been previously used to approximate free boundaries in finite-difference numerical simulations. A simple, but powerful, method is described that is based on the concept of a fractional volume of fluid (VOF). This method is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations. To illustrate the method, a description is given for an incompressible hydrodynamics code, SOLA-VOF, that uses the VOF technique to track free fluid surfaces.
It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f0(c)+κ(▿c)2]dV, where NV is the number of molecules per unit volume, ▿c the composition or density gradient, f0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (Tc−T)32.
The predicted interfacial free energy and its temperature dependence are found to be in agreement with existing experimental data. The possibility of using optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed.
We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.
To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987 b ). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.
A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.
We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.
The production of a ‘wake’ behind solid bodies has been treated by different authors. S. Goldstein (1) and S. H. Hollingdale (2) have discussed the laminar wake behind a flat plate, while the wake of a two-dimensional grid has been treated for the turbulent case alone using L. Prandtl's ‘Mischungsweg’ theory by E. Anderlik and Gran Olsson (3), (4).(Received May 20 1947)
This work describes a new finite element projection method for the computa-tion of incompressible viscous flows of nonuniform density. One original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation, to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid. It is shown that unconditional stability in the incremental ver-sion of the projection method is possible provided two projections are performed per time step. In particular, a second order accurate BDF projection method is pre-sented and its numerical performance is illustrated by test computations and comp-arisons. c 2000 Academic Press Key Words: variable density Navier–Stokes equations; incompressible flows of nonuniform density; fractional step projection method; poisson pressure equation; incremental second order projection method; BDF scheme; finite elements; mixed method.
Phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near-critical interfacial phenomena, phase separation under shear, and microstructure evolution during solidification. However, because interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations require resolution of very thin layers to capture the physics of the problems studied. This demands robust numerical methods that can efficiently achieve high resolution and accuracy, especially in three dimensions. We present here an accurate and efficient numerical method to solve the coupled Cahn–Hilliard/Navier–Stokes system, known as Model H, that constitutes a phase field model for density-matched binary fluids with variable mobility and viscosity. The numerical method is a time-split scheme that combines a novel semi-implicit discretization for the convective Cahn–Hilliard equation with an innovative application of high-resolution schemes employed for direct numerical simulations of turbulence. This new semi-implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the Cahn–Hilliard equation at the same cost as that of an explicit scheme. It is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity. Moreover, we solve the Navier–Stokes equations with a robust time-discretization of the projection method that guarantees better stability properties than those for Crank–Nicolson-based projection methods. For channel geometries, the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction. The capabilities of the method are demonstrated with several examples including phase separation with, and without, shear in two and three dimensions. The method effectively resolves interfacial layers of as few as three mesh points. The numerical examples show agreement with analytical solutions and scaling laws, where available, and the 3D simulations, in the presence of shear, reveal rich and complex structures, including strings.
This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. We extend a discontinuous finite element discretization originally considered for hyperbolic systems such as the Euler equations to the case of the Navier–Stokes equations by treating the viscous terms with a mixed formulation. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. As a consequence the method is ideally suited to compute high-order accurate solution of the Navier–Stokes equations on unstructured grids. The performance of the proposed method is illustrated by computing the compressible viscous flow on a flat plate and around a NACA0012 airfoil for several flow regimes using constant, linear, quadratic, and cubic elements.
Phase-field models provide a way to model fluid interfaces as having finite thickness. This can allow the computation of interface movement and deformation on fixed grids. This paper applies phase-field modeling to the computation of two-phase incompressible Navier–Stokes flows. The Navier–Stokes equations are modified by the addition of the continuum forcing −Cφ, where C is the composition variable and φ is C's chemical potential. The equation for interface advection is replaced by a continuum advective-diffusion equation, with diffusion driven by C's chemical potential gradients. The paper discusses how solutions to these equations approach those of the original sharp-interface Navier–Stokes equations as the interface thickness ϵ and the diffusivity both go to zero. The basic flow-physics of phase-field interfaces is discussed. Straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion. On the other hand, too large a diffusion will overly damp the flow. These two constraints result in an upper bound for the diffusivity of O(ϵ) and a lower bound of O(ϵ2). Within these two bounds, the phase-field Navier–Stokes equations appear to generate an O(ϵ) error relative to the exact sharp-interface equations. An O(h2/ϵ2) numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing. An O(h4/ϵ4) compact scheme is introduced that takes advantage of the asymptotic, comparatively smooth, behavior of the chemical potential. For O(ϵ) accurate phase-field models the optimum path to convergence for this scheme appears to be ϵ∝h4/5. The asymptotic rate of convergence corresponding to this is O(h4/5) but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear. Extensive analysis and computations show that this scheme is very effective and accurate. It allows the accurate calculation of two-phase flows with interfaces only two cells wide. Computational results are given for linear capillary waves and for Rayleigh–Taylor instabilities. The first set of computations is compared to exact solutions of the diffuse-interface equations and of the original sharp-interface equations. The Rayleigh–Taylor computations test the ability of the method to compute highly deforming flows. These flows include near-singular phenomena such as interface coalescences and breakups, contact line movement, and the formation and breakup of thin wall-films. Grid-refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed, wavelength, and the general physical characteristics of the instability and mass transfer rates.
The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations.
The effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized Navier–Stokes equations is examined in the presence of a non-trivial density spectrum. Fourier analysis shows that the existing skew-symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form. Several formulations of the convective term, including a new formulation proposed for cubically nonlinear terms, are tested in direct numerical simulation (DNS) of decaying compressible isotropic turbulence both in chemically inert (small density fluctuations) and reactive cases (large density fluctuations) and for different degrees of resolution. In the DNS of reactive turbulent flow, the new cubic skew-symmetric form gives the most accurate results, consistent with the spectral error analysis, and at the lowest cost. In marginally resolved DNS and LES (poorly resolved by definition) the new cubic skew-symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high-order dissipative filters.
All second-order, many third-order, and a few fourth-order Runge-Kutta schemes can be arranged to require only two storage locations per variable, compared with three needed by Gill's method.
We study how to approximate the metric terms that arise in the discontinuous spectral element (DSEM) approximation of hyperbolic
systems of conservation laws when the element boundaries are curved. We first show that the metric terms can be written in
three forms: the usual cross product and two curl forms. The first curl form is identical to the “conservative” form presented
by Thomas and Lombard [(1979), AIAA J. 17(10), 1030–1037]. The second is a coordinate invariant form. We prove that in two space dimensions, the typical approximation
of the cross product form does satisfy a discrete set of metric identities if the boundaries are isoparametric and the quadrature
is sufficiently precise. We show that in three dimensions, this cross product form does not satisfy the metric identities,
except in exceptional circumstances. Finally, we present approximations of the curl forms of the metric terms that satisfy
the discrete metric identities. Two examples are presented to illustrate how the evaluation of the metric terms affects the
satisfaction of the discrete metric identities, one in two space dimensions and the other in three.
We present in this paper a up-to-date review on the error analysis of a class of pseudo-compressibility methods and their time discretizations for the unsteady incompressible Navier-Stokes equations. 1.
A method is introduced to decrease the computational labor of the standard level set method for propagating interfaces. The fast approach uses only points close to the curve at every time step. We describe this new algorithm and compare its efficiency and accuracy with the standard level set approach. 1 A Fast Level Set Implementation The level set technique was introduced in [9] to track moving interfaces in a wide variety of problems. It relies on the relation between propagating interfaces and propagating shocks. The equation for a front propagating with curvature dependent speed is linked to a viscous hyperbolic conservation law for the propagating gradients of the fronts. The central idea is to follow the evolution of a function OE whose zero--level set always corresponds to the position of the propagating interface. The motion for this evolving function OE is determined from a partial differential equation in one higher dimension which permits cusps, sharp corners, and changes i...