Shock interaction computations on unstructured,two-dimensional grids using a shock-fitting technique

A new shock-fitting technique has been recently proposed and implemented by the authors in conjunction with an unstructured shock-capturing solver. In the present paper, the attention is addressed towards the computation of shock–shock and shock–wall interactions by means of this novel computational technique.     

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie–Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.     

A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids

A balanced force refined level set grid method for two-phase flows on structured and unstructured flow solver grids is presented. To accurately track the phase interface location, an auxiliary, high-resolution equidistant Cartesian grid is introduced. In conjunction with a dual-layer narrow band approach, this refined level set grid method allows for parallel, efficient grid convergence and error estimation studies of the interface tracking method. The Navier–Stokes equations are solved on an unstructured flow solver grid with a novel balanced force algorithm for level set methods based on the recently proposed method by Francois et al. [M.M. Francois, S.J. Cummins, E.D. Dendy, D.B. Kothe, J.M. Sicilian, M.W. Williams, A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys. 213 (2006) 141–173] for volume of fluid methods on structured grids. To minimize spurious currents, a second order converging curvature evaluation technique for level set methods is presented. The results of several different test cases demonstrate the effectiveness of the proposed method, showing good mass conservation properties and second order converging spurious current magnitudes.     

Hierarchical reconstruction for spectral volume method on unstructured grids

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442–2467; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194–2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

A compatible and conservative spectral element method on unstructured grids

We derive a formulation of the spectral element method which is compatible on very general unstructured three-dimensional grids. Here compatible means that the method retains discrete analogs of several key properties of the divergence, gradient and curl operators: the divergence and gradient are anti-adjoints (the negative transpose) of each other, the curl is self-adjoint and annihilates the gradient operator, and the divergence annihilates the curl. The adjoint relations hold globally, and at the element level with the inclusion of a natural discrete element boundary flux term.     

L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.     

Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

The present paper deals with an efficient and accurate limiting strategy for the multi-dimensional hyperbolic conservation laws on unstructured grids. The multi-dimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The basic idea of the proposed limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured meshes are obtained.Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the newly developed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Numerous test cases are presented to validate the basic features of the proposed approach.     

Parallel re-initialization of level set functions on distributed unstructured tetrahedral grids

Level set functions are employed to track interfaces in various application areas including simulation of two-phase flows and image segmentation. Often, a re-initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function. In this note, we present a parallel algorithm for re-initializing level set functions on unstructured, three-dimensional tetrahedral grids. The main idea behind this new domain decomposition approach is to combine a parallel brute-force re-initializing algorithm with an efficient way to compute distances between the interface and grid points. Time complexity and error analysis of the algorithm are investigated. Detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes.     

A three-dimensional electrostatic particle-in-cell methodology on unstructured Delaunay–Voronoi grids

The mathematical formulation and computational implementation of a three-dimensional particle-in-cell methodology on unstructured Delaunay–Voronoi tetrahedral grids is presented. The method allows simulation of plasmas in complex domains and incorporates the duality of the Delaunay–Voronoi in all aspects of the particle-in-cell cycle. Charge assignment and field interpolation weighting schemes of zero- and first-order are formulated based on the theory of long-range constraints. Electric potential and fields are derived from a finite-volume formulation of Gauss’ law using the Voronoi–Delaunay dual. Boundary conditions and the algorithms for injection, particle loading, particle motion, and particle tracking are implemented for unstructured Delaunay grids. Error and sensitivity analysis examines the effects of particles/cell, grid scaling, and timestep on the numerical heating, the slowing-down time, and the deflection times. The problem of current collection by cylindrical Langmuir probes in collisionless plasmas is used for validation. Numerical results compare favorably with previous numerical and analytical solutions for a wide range of probe radius to Debye length ratios, probe potentials, and electron to ion temperature ratios. The versatility of the methodology is demonstrated with the simulation of a complex plasma microsensor, a directional micro-retarding potential analyzer that includes a low transparency micro-grid.     

A dynamic finite volume scheme for large-eddy simulation on unstructured grids

In recent years there has been considerable progress in the application of large-eddy simulation (LES) to increasingly complex flow configurations. Nevertheless a lot of fundamental problems still need to be solved in order to apply LES in a reliable way to real engineering problems, where typically finite-volume codes on unstructured meshes are used. A self-adaptive discretisation scheme, in the context of an unstructured finite-volume flow solver, is investigated in the case of isotropic turbulence at infinite Reynolds number. The Smagorinsky and dynamic Smagorinsky sub-grid scale models are considered. A discrete interpolation filter is used for the dynamic model. It is one of the first applications of a filter based on the approach presented by Marsden et al. In this work, an original procedure to impose the filter shape through a specific selection process of the basic filters is also proposed. Satisfactory results are obtained using the self-adaptive scheme for implicit LES. When the scheme is coupled with the sub-grid scale models, the numerical dissipation is shown to be dominant over the sub-grid scale component. Nevertheless the effect of the sub-grid scale models appears to be important and beneficial, improving in particular the energy spectra. A test on fully developed channel flow at Re_τ = 395 is also performed, comparing the non-limited scheme with the self-adaptive scheme for implicit LES. Once again the introduction of the limiter proves to be beneficial.     

The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids

Novel limiters based on the weighted average procedure are developed for finite volume methods solving multi-dimensional hyperbolic conservation laws on unstructured grids. The development of these limiters is inspired by the biased averaging procedure of Choi and Liu [10]. The remarkable features of the present limiters are the new biased functions and the weighted average procedure, which enable the present limiter to capture strong shock waves and achieve excellent convergence for steady state computations. The mechanism of the developed limiters for eliminating spurious oscillations in the vicinity of discontinuities is revealed by studying the asymptotic behavior of the limiters. Numerical experiments for a variety of test cases are presented to demonstrate the superior performance of the proposed limiters.     

Spectral difference method for compressible flow on unstructured grids with mixed elements

This paper presents the development of a 2D solver for inviscid and viscous compressible flows using the spectral difference (SD) method for unstructured grids with mixed elements. A mixed quadrilateral and triangular grid is first refined using one-level h-refinement to generate a quadrilateral grid while keeping the curvature of boundary edges. The SD method designed for quadrilateral meshes can subsequently be applied for the refined unstructured grid. Results obtained with the SD method for both inviscid and viscous compressible flows compare well with analytical solutions and other published results.     

Helicity generation in turbulent MHD flows

We study turbulent flow of a conducting liquid in a uniform external magnetic field. It is shown that intense helicity generation is possible in the presence of a mean shear flow. It is noted that even though the mean helicity of the initial flow can be zero, the presence of internal topological structure of the flow, for example the presence of helicity of different signs at different scales, is nevertheless necessary for helicity generation. Zh. éksp. Teor. Fiz. 114, 946–955 (September 1998)     

Evaporating droplets in turbulent reacting flows

Three-dimensional direct numerical simulations are carried out to determine the effects of turbulence on the preferential segregation of an evaporating spray and then to study the evolution of the resulting mixture fraction topology and propagating flame. First, the mixing between an initially randomly dispersed phase and the turbulent gaseous carrier phase is studied with non-evaporating particles. According to their inertia and the turbulence properties, the formation of clusters of particles is analyzed (formation delay, cluster characteristic size and density). Once the particles are in dynamical equilibrium with the surrounding turbulent flow, evaporation is considered through the analysis of the mixture fraction evolution. Finally, to mimic ignition, a kernel of burnt gases is generated at the center of the domain and the turbulent flame evolution is described.     

Localized turbulent flows on scouring granular beds

In many applications a sustained, localized turbulent flow scours a cohesionless granular bed to form a pothole. Here we use similarity methods to derive a theoretical formula for the equilibrium depth of the pothole. Whereas the empirical formulas customarily used in applications contain numerous free exponents, the theoretical formula contains a single one, which we show can be determined via the phenomenological theory of turbulence. Our derivation affords insight into how a state of dynamic equilibrium is attained between a granular bed and a localized turbulent flow.     

Effects of forcing in three-dimensional turbulent flows

We present the results of a numerical investigation of three-dimensional homogeneous and isotropic turbulence, stirred by a random forcing with a power-law spectrum, E(f)(k) approximately k(3-y). Numerical simulations are performed at different resolutions up to 512(3). We show that at varying the spectrum slope y, small-scale turbulent fluctuations change from a forcing independent to a forcing dominated statistics. We argue that the critical value separating the two behaviors, in three dimensions, is y(c)=4. When the statistics is forcing dominated, for yy(c), we find the same anomalous scaling measured in flows forced only at large scales. We connect these results with the issue of universality in turbulent flows.     

Statistics of Lagrangian velocities in turbulent flows

We present a generalized Fokker-Planck equation for the joint position-velocity probability distribution of a single fluid particle in a turbulent flow. Based on a simple estimate, the diffusion term is related to the two-point two-time Eulerian acceleration-acceleration correlation. Dimensional analysis yields a velocity increment probability distribution with normal scaling v approximately t(1/2). However, the statistics need not be Gaussian.     

Fractal modelling of turbulent combustion

In a previous paper we proposed a new model for turbulent flows, called the fractal model (FM), which is applicable both to RANS and LES formulations. Here, the model is extended to the reactive case with the goal of simulating turbulent flames, both premixed and non-premixed.

FM is a subgrid model that describes the physics of the small scales of turbulence building on the phenomenological concept of vortex cascade and on fractal theory. The physics of the small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls.

The fundamental assumption in treating combustion in this work is that chemical reactions take place only at the dissipative scales of turbulence, i.e. near the so-called ‘fine structures’ (the eddy dissipation concept). FM predicts the growth of dissipative scales due to heat release; therefore, it enables a local DNS in the hot regions of the flow where the dissipative scale may grow up to the cell dimension. FM can also estimate the volume fraction γ* occupied by the ‘fine structures’; this quantity is critical for modelling the reaction rate, and therefore the source terms in the energy and species equations. FM can also estimate the local surface of the reactive ‘fine structures’, that is, the local flame front area. It also takes into account, although in approximate manner, the formation of islands of unburnt mixture. In this paper, the model (in the isotropic formulation (IFM)) is used in conjunction with a time-dependent LES (but with the limitations of an isotropic model) approach and is validated through a three-dimensional axisymmetric diffusion flame studied experimentally by Correa and Gulati and numerically by many researchers. The time-dependent results obtained are in good agreement with the experiments. Moreover, the IFM solution offers a possible explanation for the stabilization process of this flame, which undergoes local stretching of the order of 46 000 s^?1.     

Fractal modelling of turbulent mixing

The aim of this work is to propose a new model for turbulent flows, called the fractal model (FM), applicable both in a Reynolds averaged Navier–Stokes (RANS) and a large-eddy simulation (LES) formulation, with the ultimate goal of applying it to simulate turbulent combustion irrelevant of its mode (premixed or non-premixed). The model is able to turn itself off in the laminar zones of the flow, and in particular near walls. It is based on the fractal theory. It describes the physics of the smaller spatial scales and therefore represents a small-scales model.

FM describes the physics of the small scales of turbulence based on the phenomenological concept of vortex cascade and on the self-similar behaviour of turbulence in the inertial range. Such a model is used in each cell of a numerical calculation. A characteristic length Δ is associated to each cell, and the local energy u ³ _Δ/Δ is distributed over a certain number of eddies, which depends on the local Reynolds number Re _Δ. Each vortex of the cascade generates N _c vortices; the recursive process of vortex generation terminates at the dissipative scale level, i.e. when the eddy Reynolds number is equal to one. FM is also able to estimate the volume fraction occupied by the dissipative fine structures of turbulence; this quantity is critical in reactive turbulent flows.

The physics of small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls. Assuming μ_t to be isotropic, FM is applicable in a RANS formulation (IFM, isotropic fractal model). The model can be extended to the anisotropic case (AFM, anisotropic fractal model) and therefore used to close the transport equations in an LES approach. In the present paper, the model (IFM) is used in a RANS approach and is validated through a test case studied experimentally by Johnson and Bennett, and numerically (with LES) by Akselvoll and Moin. The results obtained are in good agreement both with the experimental and the numerical ones. Other tests are being performed.