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 parallel unstructured dynamic mesh adaptation algorithm for 3‐D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three‐dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell‐centred finite‐volume method and the Roe's flux‐difference splitting. Second‐order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time‐dependent solutions was enhanced by adopting an h‐refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed‐up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock‐tube problem. The flows around oscillating NACA0012 wing and F‐5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows

High‐speed compressible turbulent flows typically contain discontinuities and have been widely modeled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1.5 and Re_τ = 222 is conducted using OpenSBLI, a high‐order finite difference computational fluid dynamics framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against the WALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The sixth‐order adaptive TENO scheme is found to produce comparable results to those obtained with nondissipative fourth‐ and sixth‐order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.     

Multiscale coupling of molecular dynamics and peridynamics

We propose a multiscale computational model to couple molecular dynamics and peridynamics. The multiscale coupling model is based on a previously developed multiscale micromorphic molecular dynamics (MMMD) theory, which has three dynamics equations at three different scales, namely, microscale, mesoscale, and macroscale. In the proposed multiscale coupling approach, we divide the simulation domain into atomistic region and macroscale region. Molecular dynamics is used to simulate atom motions in atomistic region, and peridynamics is used to simulate macroscale material point motions in macroscale region, and both methods are nonlocal particle methods. A transition zone is introduced as a messenger to pass the information between the two regions or scales. We employ the “supercell” developed in the MMMD theory as the transition element, which is named as the adaptive multiscale element due to its ability of passing information from different scales, because the adaptive multiscale element can realize both top-down and bottom-up communications. We introduce the Cauchy–Born rule based stress evaluation into state-based peridynamics formulation to formulate atomistic-enriched constitutive relations. To mitigate the issue of wave reflection on the interface, a filter is constructed by switching on and off the MMMD dynamic equations at different scales. Benchmark tests of one-dimensional (1-D) and two-dimensional (2-D) wave propagations from atomistic region to macro region are presented. The mechanical wave can transit through the interface smoothly without spurious wave deflections, and the filtering process is proven to be efficient.     

Mesh adaptivity for the transport equation led by variational multiscale error estimators

Recently, we developed an explicit a posteriori error estimator especially suited for fluid dynamics problems solved with a stabilized method. The technology is based upon the theory that inspired stabilized methods, namely, the variational multiscale theory. The salient features of the formulation are that it can be readily implemented in existing codes, it is a very economical procedure, and it yields very accurate local error estimates uniformly from the diffusive to the advective regime. In this work, the variational multiscale error estimator is applied to develop adaptive strategies for the advection–diffusion‐reaction equation. The performance of L₁ and L₂ local error norms combined with three strategies to adapt the mesh is investigated. Emphasis is placed on flows with sharp boundary and interior layers but also attention is given to diffusion‐dominated flows. Computational results show that the method generates meshes with a smooth transition of the element size, which capture all the flow features. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of central differencing and low‐dissipation weights in a weighted compact nonlinear scheme

This paper proposes WCNS‐CU‐Z, a weighted compact nonlinear scheme, that incorporates adapted central difference and low‐dissipative weights together with concepts of the adaptive central‐upwind sixth‐order weighted essentially non‐oscillatory scheme (WENO‐CU) and WENO‐Z schemes. The newly developed WCNS‐CU‐Z is a high‐resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth‐order WCNS‐CU‐Z exhibits sufficient accuracy in the smooth region through use of low‐dissipative weights. The sixth‐order WCNS‐CU‐Zs are implemented with a robust linear difference formulation (R‐WCNS‐CU6‐Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that R‐WCNS‐CU6‐Z is capable of achieving a higher resolution than the seventh‐order classical robust weighted compact nonlinear scheme and can provide a crisp result in terms of discontinuity. Among the schemes tested, R‐WCNS‐CU6‐Z has been shown to be robust, and variable interpolation type R‐WCNS‐CU6‐Z (R‐WCNS‐CU6‐Z‐V) provides a stable computation by modifying the first‐order interpolation when negative density or negative pressure arises after nonlinear interpolation. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.     

High resolution flux‐difference‐splitting scheme on adaptive grid for open‐channel flows

The effectiveness and usefulness of further enhancing the shock resolution of a second‐order accurate scheme for open‐channel flows by using an adaptive grid is investigated. The flux‐difference‐splitting (FDS) scheme based on the Lax–Wendroff numerical flux is implemented on a fixed as well as on a self‐adjusting grid for this purpose. The grid‐adjusting procedure, developed by Harten and Hyman, adjusts the grid by averaging the local characteristic velocities with respect to the signal amplitude in such a way that a shock always lies on a mesh point. This enables a scheme capable of perfectly resolving a stationary shock to capture a shock that moves from mesh point to mesh point. The Roe's approximate Jacobian is used for conservation and consistency, while theoretically sound treatment for satisfying entropy inequality conditions ensures physically realistic solutions. Details about inclusion of source terms, often left out of analyses for the homogeneous part of governing equations, are also explained. The numerical results for some exacting problems are compared with analytical as well as experimental results for examining improvements in resolution of discontinuities by the adaptive grid. Copyright © 2001 John Wiley & Sons, Ltd.     

Assessment of adaptive and heuristic time stepping for variably saturated flow

The performance of improved initial estimates and ‘heuristic’ and ‘adaptive’ techniques for time step control in the iterative solution of Richards equation is evaluated. The so‐called heuristic technique uses the convergence behaviour of the iterative scheme to estimate the next time step whereas the adaptive technique regulates the time step on the basis of an approximation of the local time truncation error. The sample problems used to assess these various schemes are characterized by nonuniform (in time) boundary conditions, sharp gradients in the infiltration fronts, and discontinuous derivatives in the soil hydraulic properties. It is found that higher order initial solution estimates improve the convergence of the iterative scheme for both the heuristic and adaptive techniques, with greater overall performance gains for the heuristic scheme, as could be expected. It is also found that the heuristic technique outperforms the adaptive method under strongly nonlinear conditions. Previously reported observations suggesting that adaptive techniques perform best when accuracy requirements on the numerical solution are very stringent are confirmed. Overall both heuristic and adaptive techniques have their limitations, and a more general or mixed time stepping strategy combining truncation error and convergence criteria is recommended for complex problems. Copyright © 2006 John Wiley & Sons, Ltd.     

A hybrid time stepping scheme combining explicit Runge–Kutta with implicit LU‐SGS for Navier–Stokes equations

A hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU‐SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU‐SGS. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite volume scheme preserving extremum principle for convection–diffusion equations on polygonal meshes

We propose a nonlinear finite volume scheme for convection–diffusion equation on polygonal meshes and prove that the discrete solution of the scheme satisfies the discrete extremum principle. The approximation of diffusive flux is based on an adaptive approach of choosing stencil in the construction of discrete normal flux, and the approximation of convection flux is based on the second‐order upwind method with proper slope limiter. Our scheme is locally conservative and has only cell‐centered unknowns. Numerical results show that our scheme can preserve discrete extremum principle and has almost second‐order accuracy. Copyright © 2017 John Wiley & Sons, Ltd.     

Parallel adaptive refinement for unsteady flow calculations on 3D unstructured grids

A parallel adaptive refinement algorithm for three‐dimensional unstructured grids is presented. The algorithm is based on an hierarchical h‐refinement/derefinement scheme for tetrahedral elements.The algorithm has been fully parallelized for shared‐memory platforms via a domain decomposition of the mesh at the algebraic level. The effectiveness of the procedure is demonstrated with applications which involve unsteady compressible fluid flow. A parallel speedup study of the algorithm also is included. Published in 2004 by John Wiley & Sons, Ltd.     

The direct simulation Monte Carlo method using unstructured adaptive mesh and its application

The implementation of an adaptive mesh‐embedding (h‐refinement) scheme using unstructured grid in two‐dimensional direct simulation Monte Carlo (DSMC) method is reported. In this technique, local isotropic refinement is used to introduce new mesh where the local cell Knudsen number is less than some preset value. This simple scheme, however, has several severe consequences affecting the performance of the DSMC method. Thus, we have applied a technique to remove the hanging node, by introducing the an‐isotropic refinement in the interfacial cells between refined and non‐refined cells. Not only does this remedy increase a negligible amount of work, but it also removes all the difficulties presented in the originals scheme. We have tested the proposed scheme for argon gas in a high‐speed driven cavity flow. The results show an improved flow resolution as compared with that of un‐adaptive mesh. Finally, we have used triangular adaptive mesh to compute a near‐continuum gas flow, a hypersonic flow over a cylinder. The results show fairly good agreement with previous studies. In summary, the proposed simple mesh adaptation is very useful in computing rarefied gas flows, which involve both complicated geometry and highly non‐uniform density variations throughout the flow field. Copyright © 2002 John Wiley & Sons, Ltd.     

A conservative local interface sharpening scheme for the constrained interpolation profile method

A conservative local interface sharpening scheme has been developed for the constrained interpolation profile method with the conservative semi‐Lagrangian scheme, because the conservative semi‐Lagrangian scheme does not feature a mechanism to control the interface thickness, thus causing an increase of numerical error with the advance of the time step. The proposed sharpening scheme is based on the conservative level set method proposed by Olsson and Kreiss. However, because their method can cause excessive deformation of the free‐surface in certain circumstances, we propose an improvement of the method by developing a local sharpening technique. Several advection tests are presented to assess the correctness of the advection and the improved interface sharpening scheme. This is followed by the validations of dam‐breaking flow and the rising bubble flows. The mass of the fluid is exactly conserved and the computed terminal velocity of the rising bubble agrees well with the experiments compared with other numerical methods such as the volume of fluid method (VOF), the front tracking method, and the level set method. Copyright © 2011 John Wiley & Sons, Ltd.