An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows

This paper presents the latest developments of a discontinuous Galerkin (DG) method for incompressible flows introduced in [Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J Comput Phys 2006;218(2):794–815] for the steady Navier–Stokes equations and extended in [Bassi F, Crivellini A. A high-order discontinuous Galerkin method for natural convection problems. In: Wesseling P, Oñate E, Periaux J, editors. Electronic proceedings of the ECCOMAS CFD 2006 conference, Egmond aan Zee, The Netherlands, September 5–8; 2006. TU Delft] to the coupled Navier–Stokes and energy equations governing natural convection flows.

The method is fully implicit and applies to the governing equations in primitive variable form. Its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the Euler equations. The tight coupling between pressure and velocity so introduced stabilizes the method and allows using equal-order approximation spaces for both pressure and velocity. Since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, the resulting method is strongly consistent.

In this paper, we present a review of the method together with two recently developed issues: (i) the high-order DG discretization of the incompressible Euler equations; (ii) the high-order implicit time integration of unsteady flows. The accuracy and versatility of the method are demonstrated by a suite of computations of steady and unsteady, inviscid and viscous incompressible flows.     

A zonal grid algorithm for DNS of turbulent boundary layers

A zonal grid algorithm for direct numerical simulation (DNS) of incompressible turbulent flows within a Finite-Volume framework is presented. The algorithm uses fully coupled embedded grids and a conservative treatment of the grid-interface variables. A family of conservative prolongation operators is tested in a 2D vortex dipole and a 3D turbulent boundary layer flow. These tests show that both, first- and second-order interpolation conserves the overall second-order spatial accuracy of the scheme. The first-order conservative interpolation has a smaller damping effect on the solution but the second-order conservative interpolation has better spectral properties. The application of this algorithm in boundary layer flow separating and reattaching due to the presence of a streamwise pressure gradient reveals the power and usefulness of the presented algorithm. This simulation has been made possible by the zonal grid algorithm by reducing the required number of grid points from about 500 × 10⁶ to 130 × 10⁶ grid cells.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower-upper symmetric Gauss-Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams-Bashforth scheme is employed for the advection terms and the Crank-Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.     

Interactive time-dependent particle tracing using tetrahedraldecomposition

Streak lines and particle traces are effective visualization techniques for studying unsteady fluid flows. For real time applications, accuracy is often sacrificed to achieve interactive frame rates. Physical space particle tracing algorithms produce the most accurate results although they are usually too expensive for interactive applications. An efficient physical space algorithm is presented which was developed for interactive investigation and visualization of large, unsteady, aeronautical simulations. Performance has been increased by applying tetrahedral decomposition to speed up point location and velocity interpolation in curvilinear grids. Preliminary results from batch computations showed that this approach was up to six times faster than the most common algorithm which uses the Newton-Raphson method and trilinear interpolation. Results presented show that the tetrahedral approach also permits interactive computation and visualization of unsteady particle traces. Statistics are given for frame rates and computation times on single and multiprocessors. The benefits of interactive feature detection in unsteady flows are also demonstrated     

Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).     

Simple and accurate scheme for fluid velocity interpolation for Eulerian-Lagrangian computation of dispersed flows in 3D curvilinear grids

Particle tracking in turbulent flows in complex domains requires accurate interpolation of the fluid velocity field. If grids are non-orthogonal and curvilinear, the most accurate available interpolation methods fail. We propose an accurate interpolation scheme based on Taylor series expansion of the local fluid velocity about the grid point nearest to the desired location. The scheme is best suited for curvilinear grids with non-orthogonal computational cells. We present the scheme with second-order accuracy, yet the order of accuracy of the method can be adapted to that of the Navier-Stokes solver.An application to particle dispersion in a turbulent wavy channel is presented, for which the scheme is tested against standard linear interpolation. Results show that significant discrepancies can arise in the particle displacement produced by the two schemes, particularly in the near-wall region which is often discretized with highly-distorted computational cells.     

An adaptive phase field method for the mixture of two incompressible fluids

This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a half-staggered, moving quadrilateral mesh to keep the velocity field divergence-free, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and mesh-redistribution. In the first part, the incompressible Navier-Stokes equations are solved on a fixed half-staggered mesh by the rotational incremental pressure-correction scheme, and the Allen-Cahn type of phase equation is approximated by a conservative, second-order accurate central difference scheme, where the Lagrangian multiplier is used to preserve the mass-conservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the high-resolution conservative interpolation, while the non-conservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergence-free velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.     

三维非定常不可压涡量—速度Navier—Stokes方程组的有限差分法

提出了一种数值求解三维非定常涡量—速度形式的不可压Navier-Stokes方程组的有限差分方法,该方法在空间方向上具有二阶精度,并且系数矩阵具有对角占优性,因此适合高雷诺数问题的数值求解．同时,给出了适合的二阶涡量边界条件．通过对有精确解的狄利克雷边值问题和典型的驱动方腔流问题的数值实验,验证了本文格式的精确性、稳定性和有效性．     

Numerical simulation of the unsteady three-dimensional flow in confined domains crossed by moving bodies

A numerical method devoted to the prediction of unsteady flows in complex domains with moving boundaries is presented. Based on the unsteady Euler equations with source terms to take diffusive effects into account as well as additional mass, momentum or enthalpy sources, it has been specially developed to model the thermal and dynamic behavior of the ambient air inside underground stations in the presence of moving trains. The numerical solution method is a unstructured finite-volume cell-centered scheme using the SIMPLE algorithm coupled with a second-order intermediate time stepping scheme. The spatial discretization is realized with an automatic Cartesian grid generator, complemented by a technique of sliding grids to handle straight moving bodies inside the domain.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

A numerical method for the simulation of free surface flows with surface tension

A numerical model for the simulation of three-dimensional liquid–gas flows with free surfaces and surface tension is presented. The incompressible Navier–Stokes equations are assumed to hold in the liquid domain, while the gas pressure is assumed to be constant in each connected component of the gas domain and to follow the ideal gas law. The surface tension effects are imposed as a normal force on the interface.

An implicit splitting scheme is used to decouple the physical phenomena. Given the curvature of the liquid–gas interface, the method described in [Caboussat A, Picasso M, Rappaz J. Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas. J Comput Phys 2005;203(2):626–49] is used to track the liquid domain and compute the velocity and pressure in the liquid and the pressure in the gas domain. Then the surface tension effects are added. A variational method for the computation of the curvature is presented by smoothing the characteristic function of the liquid domain and using a finite element unstructured mesh.

The model is validated and numerical results in two and three space dimensions are presented for bubbles and/or droplets flows.     

Computation of unsteady flows with moving boundaries using body fitted curvilinear moving grids

An algorithm is proposed for the computation of unsteady incompressible laminar flows involving moving boundaries using body-fitted coordinates. It is shown analytically and numerically that if the non-conservative form of equations (only suitable for incompressible flows) is utilized, a uniform flow field when substituted in the transport equations does not lead to any spurious terms. Therefore, computation of incompressible flows on moving or deforming grids with this methodology would eliminate any consideration like the geometrical conservation law. The results of the bench mark cases show that the procedure is quite robust and yields accurate results.     

A direct numerical simulation of axisymmetric cryogenic two-phase flows in a pipe with phase change

An accurate finite-volume based numerical method for the simulation of a cryogenic two-phase flow with phase change heat transfer, consisting of a saturated liquid slug translating in its own superheated vapor in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed-grid with a cut-cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and vapor phases are solved separately. The mass continuity, momentum flux conditions and conservation of energy are explicitly matched at the true boundary between the two phases to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures.It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity, pressure and temperature fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, Jakob number, Ja and large property jumps at the interface.     

Comparison of different solution algorithms for collocated method of MCIM to calculate steady and unsteady incompressible flows on unstructured grids

In this paper, the collocated method of MCIM (Mass Corrected Interpolation Method) is employed for solving two-dimensional incompressible flows on unstructured grid systems. A control-volume-based finite element (CVFEM) approach has been taken in which conservation equations are formed for each control volume throughout the solution domain. In this study, three different algorithms are proposed and investigated. In the first algorithm a fully coupled formulation is used, however in the second algorithm a segregated approach without pressure correction is employed. In the third solution algorithm, the scheme of MCIM is implemented to SIMPLE-like segregated algorithm. The linear system of equations obtained in these algorithms is solved using a direct sparse solver. The accuracy and performance of the proposed algorithms are demonstrated by solving different steady and unsteady benchmark problems.