An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

Symmetric Gauss-Seidel multigrid solution of the Euler equations on structured and unstructured grids

The efficient symmetric Gauss-Seidel (SGS) algorithm for solving the Euler equations of inviscid, compressible flow on structured grids, developed in collaboration with Jameson of Stanford University, is extended to unstructured grids. The algorithm uses a nonlinear formulation of an SGS solver, implemented within the framework of multigrid. The earlier form of the algorithm used the natural (lexicographic) ordering of the mesh cells available on structured grids for the SGS sweeps, but a number of features of the method that are believed to contribute to its success can also be implemented for computations on unstructured grids. The present paper reviews, the features of the SGS multigrid solver for structured gr0ids, including its nonlinear implementation, its use of “absolute” Jacobian matrix preconditioning, and its incorporation of multigrid, and then describes the incorporation of these features into an algorithm suitable for computations on unstructured grids. The implementation on unstructured grids is based on the agglomerated multigrid method developed by Sørensen, which uses an explicit Runge-Kutta smoothing algorithm. Results of computations for steady, transonic flows past two-dimensional airfoils are presented, and the efficiency of the method is evaluated for computations on both structured and unstructured meshes.     

Mono‐block and non‐matching multi‐block structured mesh adaptation based on aerodynamic functional total derivatives for RANS flow

An enhanced goal‐oriented mesh adaptation method is presented based on aerodynamic functional total derivatives with respect to mesh nodes in a Reynolds‐Averaged Navier‐Stokes (RANS) finite‐volume mono‐block and non‐matching multi‐block‐structured grid framework. This method falls under the category of methods involving the adjoint vector of the function of interest. The contribution of a Spalart–Allmaras turbulence model is taken into account through its linearization. Meshes are adapted accordingly to the proposed indicator. Applications to 2D RANS flow about a RAE2822 airfoil in transonic, and detached subsonic conditions are presented for the drag coefficient estimation. The asset of the proposed method is patent. The obtained 2D anisotropic mono‐block mesh well captures flow features as well as global aerodynamic functionals. Interestingly, the constraints imposed by structured grids may be relaxed by the use of non‐matching multi‐block approach that limits the outward propagation of local mesh refinement through all of the computational domain. The proposed method also leads to accurate results for these multi‐block meshes but at a fraction of the cost. Finally, the method is also successfully applied to a more complex geometry, namely, a mono‐block mesh in a 3D RANS transonic flow about an M6 wing. Copyright © 2016 John Wiley & Sons, Ltd.     

Developing a unified FVE‐ALE approach to solve unsteady fluid flow with moving boundaries

In this study, an arbitrary Lagrangian–Eulerian (ALE) approach is incorporated with a mixed finite‐volume–element (FVE) method to establish a novel moving boundary method for simulating unsteady incompressible flow on non‐stationary meshes. The method collects the advantages of both finite‐volume and finite‐element (FE) methods as well as the ALE approach in a unified algorithm. In this regard, the convection terms are treated at the cell faces using a physical‐influence upwinding scheme, while the diffusion terms are treated using bilinear FE shape functions. On the other hand, the performance of ALE approach is improved by using the Laplace method to improve the hybrid grids, involving triangular and quadrilateral elements, either partially or entirely. The use of hybrid FE grids facilitates this achievement. To show the robustness of the unified algorithm, we examine both the first‐ and the second‐order temporal stencils. The accuracy and performance of the extended method are evaluated via simulating the unsteady flow fields around a fixed cylinder, a transversely oscillating cylinder, and in a channel with an indented wall. The numerical results presented demonstrate significant accuracy benefits for the new hybrid method on coarse meshes and where large time steps are taken. Of importance, the current method yields the second‐order temporal accuracy when the second‐order stencil is used to discretize the unsteady terms. Copyright © 2009 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

Large eddy simulations of wall‐bounded flows using a simplified immersed boundary method and high‐order compact schemes

This paper presents a solution algorithm based on an immersed boundary (IB) method that can be easily implemented in high‐order codes for incompressible flows. The time integration is performed using a predictor‐corrector approach, and the projection method is used for pressure‐velocity coupling. Spatial discretization is based on compact difference schemes and is performed on half‐staggered meshes. A basic algorithm for body‐fitted meshes using the aforementioned solution method was developed by A. Tyliszczak (see article “A high‐order compact difference algorithm for half‐staggered grids for laminar and turbulent incompressible flows” in Journal of Computational Physics) and proved to be very accurate. In this paper, the formulated algorithm is adapted for use with the IB method in the framework of large eddy simulations. The IB method is implemented using its simplified variant without the interpolation (stepwise approach). The computations are performed for a laminar flow around a 2D cylinder, a turbulent flow in a channel with a wavy wall, and around a sphere. Comparisons with literature data confirm that the proposed method can be successfully applied for complex flow problems. The results are verified using the classical approach with body‐fitted meshes and show very good agreement both in laminar and turbulent regimes. The mean (velocity and turbulent kinetic energy profiles and drag coefficients) and time‐dependent (Strouhal number based on the drag coefficient) quantities are analyzed, and they agree well with reference solutions. Two subfilter models are compared, ie, the model of Vreman (see article “An eddy‐viscosity subgrid‐scale model for turbulent shear flow: algebraic theory and applications” in Physics and Fluids) and σ model (Nicoud et al, see article “Using singular values to build a subgrid‐scale model for large eddy simulations” in Physics and Fluids). The tests did not reveal evident advantages of any of these models, and from the point of view of solution accuracy, the quality of the computational meshes turned out to be much more important than the subfilter modeling.     

Retracted and replaced: A flow‐condition‐based interpolation finite element procedure for triangular grids

A flow‐condition‐based interpolation finite element scheme is presented for use of triangular grids in the solution of the incompressible Navier–Stokes equations. The method provides spatially isotropic discretizations for low and high Reynolds number flows. Various example solutions are given to illustrate the capabilities of the procedure. This article and been retracted and replaced. See retraction and replacement notice DOI: 10.1002/fld.1247 . Copyright © 2005 John Wiley & Sons, Ltd.     

Semi‐structured meshes for axial turbomachinery blades

This paper describes the development and application of a novel mesh generator for the flow analysis of turbomachinery blades. The proposed method uses a combination of structured and unstructured meshes, the former in the radial direction and the latter in the axial and tangential directions, in order to exploit the fact that blade‐like structures are not strongly three‐dimensional since the radial variation is usually small. The proposed semi‐structured mesh formulation was found to have a number of advantages over its structured counterparts. There is a significant improvement in the smoothness of the grid spacing and also in capturing particular aspects of the blade passage geometry. It was also found that the leading‐ and trailing‐edge regions could be discretized without generating superfluous points in the far field, and that further refinements of the mesh to capture wake and shock effects were relatively easy to implement. The capability of the method is demonstrated in the case of a transonic fan blade for which the steady state flow is predicted using both structured and semi‐structured meshes. A totally unstructured mesh is also generated for the same geometry to illustrate the disadvantages of using such an approach for turbomachinery blades. Copyright © 2000 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

Comparison of body‐fitted,embedded and immersed solutions of low Reynolds‐number 3‐D incompressible flows

The solutions obtained for low Reynolds‐number incompressible flows using the same flow solver and solution technique on body‐fitted, embedded surface and immersed body grids of similar size are compared. The cases considered are a sphere at Re = 100 and an idealized stented aneurysm. It is found that the solutions using all these techniques converge to the same grid‐independent solution. On coarser grids, the effect of higher‐order boundary conditions is noticeable. Therefore, if the manual labor required to set up a body‐fitted domain is excessive (as is often the case for patient‐specific geometries with medical devices), and/or computing resources are plentiful, the embedded surface and immersed body approaches become very attractive options. Copyright © 2007 John Wiley & Sons, Ltd.     

Acoustic upwinding for sub‐ and super‐sonic turbulent channel flow at low Reynolds number

A recently developed asymmetric implicit fifth‐order scheme with acoustic upwinding for the spatial discretization for the characteristic waves is applied to the fully compressible, viscous and non‐stationary Navier–Stokes equations for sub‐ and super‐sonic, mildly turbulent, channel flow (Re_τ=360). For a Mach number of 0.1, results are presented for uniform (32³, 64³ and 128³) and non‐uniform (expanding wall‐normal, 32³ and 64³) grids and compared to the (incompressible) reference solution found in (J. Fluid. Mech. 1987; 177 :133–166). The results for uniform grids on 128³ and 64³ nodes show high resemblance with the reference solution. Expanding grids are applied on 64³‐ and 32³‐node grids. The capability of the proposed technique to solve compressible flow is first demonstrated by increasing the Mach number to 0.3, 0.6 and 0.9 for isentropic flow on the uniform 64³‐grid. Next, the flow speed is increased to Ma=2. The results for the isothermal‐wall supersonic flows give very good agreement with known literature results. The velocity field, the temperature and their fluctuations are well resolved. This means that in all presented (sub‐ and super‐sonic) cases, the combination of acoustic upwinding and the asymmetric high‐order scheme provides sufficient high wave‐number damping and low wave‐number accuracy to give numerically stable and accurate results. Copyright © 2007 John Wiley & Sons, Ltd.     

Compound wall treatment with low‐Re turbulence model

A generalized treatment for the wall boundary conditions relating to turbulent flows is developed that blends the integration to a solid wall with wall functions. The blending function ensures a smooth transition between the viscous and turbulent regions. An improved low Reynolds number k?ε model is coupled with the proposed compound wall treatment to determine the turbulence field. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. Computations with fine and coarse meshes of a few flow cases yield appreciably good agreement with the direct numerical simulation and experimental data. The method is recommended for computing the complex flows where computational grids cannot satisfy a priori the prerequisites of viscous/turbulence regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

CFD simulation of two‐ and three‐dimensional free‐surface flow

The paper describes the implementation of moving‐mesh and free‐surface capabilities within a 3‐d finite‐volume Reynolds‐averaged‐Navier–Stokes solver, using surface‐conforming multi‐block structured meshes. The free‐surface kinematic condition can be applied in two ways: enforcing zero net mass flux or solving the kinematic equation by a finite‐difference method. The free surface is best defined by intermediate control points rather than the mesh vertices. Application of the dynamic boundary condition to the piezometric pressure at these points provides a hydrostatic restoring force which helps to eliminate any unnatural free‐surface undulations. The implementation of time‐marching methods on moving grids are described in some detail and it is shown that a second‐order scheme must be applied in both scalar‐transport and free‐surface equations if flows driven by free‐surface height variations are to be computed without significant wave attenuation using a modest number of time steps. Computations of five flows of theoretical and practical interest—forced motion in a pump, linear waves in a tank, quasi‐1d flow over a ramp, solitary wave interaction with a submerged obstacle and 3‐d flow about a surface‐penetrating cylinder—are described to illustrate the capabilities of our code and methods. Copyright © 2003 John Wiley & Sons, Ltd.     

On the use of adaptive hierarchical meshes for numerical simulation of separated flows

This paper describes the use of adaptive hierarchical grids to predict incompressible separated flow at low Reynolds number. The grids consist of a quadtree system of hierarchical Cartesian meshes which are generated by recursive subdivision about seeding points. The governing equations are discretized in collocated primitive variable form using finite volumes and solved using a pressure correction scheme. The mesh is locally adapted at each time step, with panel division or removal dependent on the vorticity magnitude. The resulting grids have fine local resolution and are economical in array size. Results are presented for unidirectional, impulsively started flow past a circular and a square cylinder at various Reynolds numbers up to 5000 and 250 respectively. It is clear that hierarchical meshes may offer gains in efficiency when applied to complex flow domains or strongly sheared flows. However, as expected, the stepped approximation to curved boundaries resulting from the Cartesian quadtree representation adversely affects the accuracy of the results for flow past a circular cylinder. © 1998 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 novel non‐upwind,interconnected, multi‐grid,overlapping numerical procedure for problems involving fluid flow

A novel numerical procedure for heat, mass and momentum transfer in fluid flow is presented. The new scheme is passed on a non‐upwind, interconnected, multi‐grid, overlapping (NIMO) finite‐difference algorithm. In 2D flows, the NIMO algorithm solves finite‐difference equations for each dependent variable on four overlapping grids. The finite‐difference equations are formulated using the control‐volume approach, such that no interpolations are needed for computing the convective fluxes. For a particular dependent variable, four fields of values are produced. The NIMO numerical procedure is tested against the exact solution of two test problems. The first test problem is an oblique laminar 2D flow with a double step abrupt change in a passive scalar variable for infinite Peclet number. The second test problem is a rotating radial flow in an annular sector with a single step abrupt change in a passive scalar variable for infinite Peclet number. The NIMO scheme produced essentially the exact solution using different uniform and non‐uniform square and rectangular grids for 45 and 30° angle of inclination. All other schemes were unable to capture the exact solution, especially for the rectangular and non‐uniform grids. The NIMO scheme was also successful in predicting the exact solution for the rotating radial flow, using a uniform cylindrical‐polar coordinate grid. Copyright © 2008 John Wiley & Sons, Ltd.     

Anisotropic mesh adaptation: towards user‐independent,mesh‐independent and solver‐independent CFD. Part II. Structured grids

The present paper is the second article in a three‐part series on anisotropic mesh adaptation and its application to (2‐D) structured and unstructured meshes. In the first article, the theory was presented, the methodology detailed and brief examples given of the application of the method to both types of grids. The second part details the application of the mesh adaptation method to structured grids. The adaptation operations are restricted to mesh movement in order to avoid the creation of hanging nodes. Being based on a spring analogy with no restrictive orthogonality constraint, a wide grid motion is allowed. The adaptation process is first validated on analytical test cases and its high efficiency is shown on relevant transonic and supersonic benchmarks. These latter test cases are also solved on adapted unstructured grids to provide a reference for comparison studies. The third part of the series will demonstrate the capability of the methodology on 2‐D unstructured test cases. Copyright © 2002 John Wiley & Sons, Ltd.