Anisotropic Cartesian grid method for steady inviscid shocked flow computation

The anisotropic Cartesian grid method, initially developed by Z.N. Wu (ICNMFD 15, 1996; CFD Review 1998, pp. 93–113) several years ago for efficiently capturing the anisotropic nature of a viscous boundary layer, is applied here to steady shocked flow computation. A finite‐difference method is proposed for treating the slip wall conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations is established by using the specific mass flux as a dependent variable. Thus, the dimensionality of the problem to be solved is reduced by one and the computational mesh to be generated is needed only on the boundary of the domain. It is shown that the boundary integral formulation developed in this paper is equivalent of the results of distributions of the fundamental solutions of the Laplacian operator equation with a different order along the boundaries of the domain. Hence, we have succeeded in establishing the fundamental-solution method for compressible inviscid flows governed by non-linear equations.     

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

This paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L² = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L² = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0–4.5.     

A symmetry-preserving Cartesian grid method for computing a viscous flow past a circular cylinder

This article deals with a numerical method for solving the unsteady, incompressible Navier–Stokes equations in domains with arbitrarily-shaped boundaries, where the boundary is represented using the Cartesian grid approach. We introduce a novel cut-cell discretization which preserves the spectral properties of convection and diffusion. Here, convection is discretized by a skew-symmetric operator and diffusion is approximated by a symmetric, positive-definite coefficient matrix. Such a symmetry-preserving discretization conserves the kinetic energy (if the dissipation is turned off) and is stable on any grid. The method is successfully tested for an incompressible, unsteady flow around a circular cylinder at $\mathit{Re}=100$. To cite this article: R. Verstappen, M. Dröge, C. R. Mecanique 333 (2005).     

A numerical comparison of two different approximations of the error in a meshless method

Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. In the Element Free Galerkin (EFG) method, it is obviously important that the error of approximation should be estimated, as it is in the Finite Element Method (FEM).In this paper we compare two different procedures to approximate the a posteriori error for the EFG method, both procedures are recovery based errors. The performance of the two different approximations of the error is illustrated by analysing different examples for 2-D potential and elasticity problems with known analytical solutions, using regular and irregular clouds of points. For irregular clouds of points, it is recommended to use smooth transition of nodes, thus creating areas of decreasing nodal densities.     

A numerical solution of 3D inviscid rotational flow in turbines and ducts

The numerical solutions of inviscid rotational (Euler) flows were obtained using an explicit hexahedral unstructured cell vertex finite volume method. A second-order-accurate, one-step Lax–Wendroff scheme was used to solve the unsteady governing equations discretized in conservative form. The transonic circular bump, in which the location and the strength of the captured shock are well predicted, was used as the first test case. The nozzle guide vanes of the VKI low-speed turbine facility were used to validate the Euler code in highly 3D environment. Despite the high turning and the secondary flows which develop, close agreements have been obtained with experimental and numerical results associated with these test cases. © 1998 John Wiley & Sons, Ltd.     

Simple treatment of non‐aligned boundaries in a Cartesian grid shallow flow model

A simple method is proposed for treating curved or irregular boundaries in Cartesian grid shallow flow models. It directly evaluates fictional values in ‘ghost’ cells adjacent to boundary cells and requires no interpolation or generation of cut cells. The boundary treatment is implemented in a dynamically adaptive quadtree grid‐based solver of the hyperbolic shallow water equations and validated against several test cases with analytical or alternative numerical solutions. The method is easy to code, accurate, and demonstrably effective in dealing with irregular computational domains in shallow flow simulations. Results are presented for still water in a basin of complicated geometry, steady hydraulic jump in an open channel with a converging sidewall, wind‐induced circulation in a circular shallow lake, and shock wave diffraction in a channel containing a contraction and expansion. Copyright © 2007 John Wiley & Sons, Ltd.     

Application of a vortex lattice numerical model in the calculation of inviscid incompressible flow around delta wings

The flow over a flat plate delta wing at incidence and in sideslip is studied using vortex lattice models based on streamwise penelling. For the attached flow problem the effect of sideslip is simulated by modifying the standard vortex lattice model for zero sideslip by aligning the trailing vortices aft of the wing along the resultant flow direction. For the separated flow problem a non-linear vortex lattice model is developed for both zero and non-zero sideslip angles in which the shape and position of the leading edge separation vortices are calculated by an iterative procedure starting from an assumed initial shape. The theoretical values are compared with available theoretical and experimental results.     

A level set approach for computing solutions to inviscid compressible flow with moving solid boundary using fixed Cartesian grids

A level set approach for computing solutions to inviscid compressible flow with moving solid surface is presented. The solid surface is considered to be sharp and is described as the zero level set of a smooth explicit function of space and time. The finite volume TVD–MacCormack's two‐step procedure is used. The boundary conditions on the solid surface are easily implemented by defining the smooth level set function. The present treatment of the level set method allows the handling of fluid flows in the presence of irregularly shaped solid boundaries, escaping from the bookkeeping complexity in the so‐called ‘surface‐tracking’ method. Using the proposed numerical techniques, a two‐dimensional numerical simulation is made to investigate the aerodynamic phenomena induced by two high‐speed trains passing by each other in a tunnel. Copyright © 2001 John Wiley & Sons, Ltd.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

A survey of exact solutions of inviscid field equations in the theory of shear flow instabilty

This paper presents a survey of undisturbed flows that take one or another of the field equations of inviscid shear flow instability theory (e.g. theRayleigh equation,Taylor-Goldstein-Haurwitz equation or theKuo equation) to a differential equation satisfied by aknown transcendental function forarbitrary complex values of the parameters. Some mean velocity profiles having this feature are well known. Thus, piecewise linear mean velocity profiles take theRayleigh equations to a constant-coefficient differential equation and the exponential mean velocity profile takes theRayleigh equation to theGauss hypergeometric equation. Less well known is the fact that a variety of mean velocity profiles take theRayleigh equation to a differential equation due toKarl Heun. These profiles include: (i) the sinusoidal profile; (ii) the hyperbolic tangent profile (an example pointed out byMiles (1963)); (iii) the profile in the form of the square of a hyperbolic secant (theBickley jet); and, (iv) a skewed velocity profile in which each component has the form of a quadratic function of the variable exp(–z/l) (in whichz is the cross-stream coordinate andl is a length scale). In all of these cases, one or another author has previously identified aregular neutral mode solution of theRayleigh equation and has expressed that solution in the form of elementary functions. Such regular neutral modes apparently represent cases in which the solution ofHeun's equation (which is normally an infinite series) truncates to a single term. The survey concludes by noting that the parabolic mean velocity profile takes theRayleigh equation to thedifferential equation of the spheroidal wave function.Dedicated to Mårten T. Landahl on the occasion of his sixty-fifth birthday by a former apprentice as a token of his respect and gratitude.     

A numerical study on a Cartesian-based body-fitted adaptive grid method

ABSTRACT

A hybrid Cartesian-based body-fitted adaptive grid method for compressible Navier–Stokes equations is implemented and investigated. In this method, the body-fitted structured grids are generated around the geometries, and the left regions are filled with Cartesian grids. To transfer the data between the different grids, the donor cell searching technique is adopted. An unstructured data-based finite volume update procedure is used, and least squares method is suggested to retain the second order in the overlap region. The moving shock waves with different speeds and vortex passing through the interfaces of the hybrid Cartesian grid are used to explore the accuracy and conservation. A new technique is presented to deal with the non-physical stagnation of slowly moving shock wave around the interface of grid. Numerical examples are presented to demonstrate the results. The three-dimensional extension has also been shown by a benchmark problem.     

Failures of the Burnett and super-Burnett equations in steady state processes

Linearized Burnett and super-Burnett equations are considered for steady state Couette flow. It is shown that the linear super-Burnett equations lead to periodic velocity and temperature curves, i.e. unphysical solutions. The problem is discussed as well for the so-called augmented Burnett equations by Zhong et al. (AIAA Journal 31, 1036-1043 (1993)), and for the recently introduced regularized 13 moment equations (R13) of Struchtrup and Torrilhon (Phys. Fluids 15(9), 2668-2680 (2003) ). It is shown that both theories exhibit proper Knudsen boundary layers for velocity and temperature. However, the heat flux parallel to the wall has different signs for the Burnett and the R13 equations, and a comparison with DSMC results shows that only the R13 equations predict the proper sign.Received: 30 March 2004, Accepted: 22 April 2004, Published online: 22 February 2005PACS: 51.10.-y, 47.45.-n     

A composite grid approach for the Euler equations

A zonal grid methodology has been developed for the calculation of compressible fluid flows. The domain subdivision is based on patched grid systems composed of zones or blocks within which a distinct curvilinear grid is generated. The flow simulation is then carried out with a modified scheme based on the Euler finite volume solver of Ni. This scheme uses a distribution procedure that provides an easy and accurate way for the transfer of information from one block to another. This method results in a naturally conservative computation at the interfaces. It is analysed and developed for the treatment of embedded grids with a grid point common to more than four blocks.     

A non-horizontal multi-layer numerical model for the flow in natural rivers

A non-horizontal multi-layer element model is developed for the simulation for the flow in natural rivers. Either Cartesian coordinates or sigma coordinates will experience difficulties in dealing with the water surface and irregular bed topography of natural rivers. To create the surface-fitting and non-deformed cells, the newly developed model divides the water column into several layers with non-horizontal interfaces which are nearly parallel to the water surface. The irregular bed topography is also represented by the layered integration between non-horizontal interfaces. Two case studies for the flow in a straight channel and the flow in an S-shaped meander channel are conducted with good agreement between the numerical predictions and the analytical or experimental results. The model is further applied for the investigation of the flow in a 12-km-long and 3.46-m-drop reach of the Yangtze River with the water surface evaluation and the stream-wise velocity satisfactory accordance with the observed data.     

On numerical solutions of the time-dependent Euler equations for incompressible flow

This paper presents finite element methods for the non-stationary Euler equations of a two dimensional inviscid and incompressible flow. For the time discretization, we compare numerical results obtained by the use of a leap-frog scheme and a semi-implicit scheme of order two.