Efficient unsteady high Reynolds number flow computations on unstructured grids

Despite the advances in computer power and numerical algorithms over the last decades, solutions to unsteady flow problems remain computing time intensive. Especially for high Reynolds number flows, nonlinear multigrid, which is commonly used to solve the nonlinear systems of equations, converges slowly. The stiffness induced by the high-aspect ratio cells and turbulence is not tackled well by this solution method.In this paper, it is investigated if a Jacobian-free Newton-Krylov (jfnk) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix-free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A^-1.In this paper, it is shown that a jfnk solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5-12 over nonlinear multigrid for two-dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three-dimensional laminar problems. A modest memory consumption is achieved with partly lumping the Jacobian before constructing the approximate factorization preconditioner, whereas robustness is ensured with enhanced diagonal dominance.     

High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients

An improved high resolution finite volume method based on linear and quadratic variable reconstructions using solution dependent weighted least squares (SDWLS) gradients has been presented here. An extended stencil consisting of vertex-based neighbours of a cell is used in the higher order reconstructions for inviscid flux computations. A QR algorithm with Householder transformation is used to solve the weighted least squares problem. In case of Navier–Stokes equations, viscous fluxes are discretized in a central differencing manner based on the Coirier’s diamond path. A few inviscid and viscous test cases are solved in order to demonstrate the efficacy of the present method. Progressive improvements in solution accuracy are observed with the increase in the order of variable reconstructions. In most cases, results of quadratic reconstruction show significant improvements over that of linear reconstruction.     

Developing an ordering-based renumbering approach for triangular unstructured grids

The unstructured grid generation and employment have become very common in computational fluid dynamics applications since a few decades ago. Comparing with the structured grid data, the unstructured grid has a random data structure known as unstructured data structure (USDS). In this work, we develop a new method to convert the USDS of a triangular unstructured grid to a quasi-structured data structure (QSDS) using an ordering-based renumbering approach. In this method, the unstructured grid data is re-ordered in a manner to represent several bands in the original unstructured grid domain. Each band presents one element layer and two node lines. Then, the indices of elements and nodes are renumbered in a unified direction for the entire constructed element layers and node lines, respectively. These numbers eventually present ascending sets of element and node indices in each element layer and every node line of the resulting QSDS. This method alleviates the random USDS drawbacks because the scattered node and element numbers in the original USDS are reordered and renumbered properly. To show the robustness of the current method, we construct a few arbitrary unstructured grid distributions and convert their ordinary USDS to our innovated QSDS without requiring additional data storage.     

The hierarchial preconditioning on unstructured grids

We present the implementation of two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as thebpx-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.     

A front tracking method on unstructured grids

A numerical method is developed for tracking discontinuities which is integrated in a generalized finite-volume solution framework for systems of conservation laws on unstructured grids of arbitrary element type. The location, geometry and the movement of the discontinuities are described by a local level set method on a restricted, dynamic definition range. Special algorithms based on least square methods are developed for handling the transport and renormalization of the level set function within the restricted range. An additional error correction is employed to minimize topological errors of the tracked front geometry. The jump conditions at the front are updated by one-sided extrapolation which define the local front velocity and the Riemann problem. A flux separation concept enables the treatment of the discontinuity within the finite-volume concept. The front tracking method is demonstrated by a number of computational examples for shock wave problems.     

Parallel multilevel methods with adaptivity on unstructured grids

We present two parallel multilevel methods for solving large-scale discretized partial differential equations on unstructured 2D/3D grids. The presented methods combine three powerful numerical algorithms: overlapping domain decomposition, multigrid method and adaptivity. As the foundation of the methods we propose an algorithm for generating and partitioning a hierarchy of adaptively refined unstructured grids, so that adaptivity can be incorporated up to a certain grid level. We ensure that the resulting subgrid hierarchies are well balanced and no inter-processor communication is needed across different grid levels, thus obtaining high parallel efficiency. Numerical experiments show that the parallel multilevel methods offer almost equally fast convergence as their sequential multigrid counterpart. And the resulting implementation has reasonably good scalability. Received: 4 December 1998 / Accepted: 12 January 2000     

Implementation of a transient adaptive sub-cell module for the parallel-DSMC code using unstructured grids

A method of transient adaptive sub-cells (TAS) suitable for unstructured grids that is modified from the existing one for the structured grids of DSMC is introduced. The TAS algorithm is implemented within the framework of a parallelized DSMC code (PDSC). Benchmarking tests are conducted for steady driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, steady hypersonic flow over a cylinder/flare and the unsteady vortex shedding behind a two-dimensional cylinder. The use of TAS enables a reduction in the computational expense of the simulation since larger sampling cells and less simulation particles can be employed. Furthermore, the collision quality of the simulation is maintained or improved and the preservation of property gradients and vorticity at the scale of the sub-cells enables correct unsteady vortex shedding frequencies to be predicted. The use of TAS in a parallel-DSMC code allows simulations of unsteady processes at a level to be carried out efficiently, accurately and with acceptable computational time.     

High-order adaptive quadrature-free spectral volume method on unstructured grids

The high-order quadrature-free spectral volume (SV) method is extended to handle local adaptive hp-refinement (grid and order refinement). Efficient edge-based adaptation utilizing a binary tree search algorithm is employed. An adaptation criteria is selected which focuses computational effort near discontinuities, and effectively reduces the physical area of the domain necessitating data limiting for stability. This makes the method very well suited for capturing and preserving discontinuities with high resolution. Both h- and p-refinement are presented in a general framework where it is possible to perform either or both on any grid cell at any time. Several well-known inviscid flow test cases, subjected to various levels of adaptation, are utilized to demonstrate the effectiveness of the method.     

Flux-based level-set method for two-phase flows on unstructured grids

In this paper we deal with the application of the flux-based level set method to moving interface computations on unstructured grids. The focus lies on the overcoming of the known difficulties of level set methods, e.g. accurate computations of important geometric properties, reliable and precise reinitialization of the level set function and the adaption of standard discretization methods to the moving boundary case. The basic building block of our approach is the high-resolution flux-based level set method for general advection equation (Frolkovi? and Mikula in SIAM J Sci Comput 29(2):579–597, 2007, Frolkovi? and Wehner in Comput Vis Sci 12(6):626–650, 2009). We extend this method for the problem of reinitialization of the level set function on unstructured grids by using quadratic interpolation to compute distances for nodes close to the interface. To realize numerical simulation for some applications with moving boundaries, we adapt the approach of ghost fluid method (Gibou and Fedkiw in J Comput Phys 202:577–601, 2005) for unstructured grids. The idea is to describe the development of the moving boundary with a level set formulation while the computational grid remains fixed and the boundary conditions are enforced using some extrapolation. Our main motivation is the numerical solution of two-phase incompressible flow problems. Additionally to previously mentioned steps, we introduce further numerical schemes in the framework of finite volume discretization for the flow. Possible jumps of the pressure and the directional derivative of velocity at the interface are modeled directly within the method using the approach of extended approximation spaces. Besides that, an algorithm for the computations of curvature is considered that exhibits the second order accuracy for some examples. Numerical experiments are provided for the presented methods.     

A residual-based scheme for computing compressible flows on unstructured grids

A second-order residual-based compact scheme, initially developed for computing flows on Cartesian and curvilinear grids, is extended to general unstructured grids in a finite-volume framework. The scheme is applied to the computation of several inviscid and viscous compressible flows governed by the Euler and Navier-Stokes equations. Its efficiency and accuracy properties are compared with those of conventional second-order upwind schemes based on variable reconstruction.     

A MIMD implementation of a parallel Euler solver for unstructured grids

A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on a MIMD (multiple instruction/multiple data stream) parallel computer. Three partitioning strategies for distributing the work load onto the processors are discussed. Issues pertaining to the communication costs are also addressed. We find that the spectral bisection strategy yields the best performance. The performance of this unstructured computation on the Intel iPSC/860 compares very favorably with that on a one-processor CRAY Y-MP/1 and an earlier implementation on the Connection Machine.The authors are employees of Computer Sciences Corporation. This work was funded under contract NAS 2-12961     

One- and two-equation turbulence models for the prediction of complex cascade flows using unstructured grids

One- and two-equation, low-Reynolds eddy-viscosity turbulence models are employed in the context of a primitive variable, finite volume, Navier-Stokes solver for unstructured grids. Through the study of the complex flow in a controlled-diffusion compressor cascade at off-design conditions, the ability of the models under consideration to predict the laminar separation bubble close to the leading edge and the boundary layer development is investigated. In order to control the unphysical growth of turbulent kinetic energy near the leading edge stagnation point, appropriate modifications to the conventional models are employed and tested. All of them improve the leading edge flow patterns and significantly affect the size of the predicted laminar separation bubble. The use of an adequately refined mesh around the airfoil, that is formed by triangles placed in a quasi-structured way, allows for the generation of grid elements of moderate aspect ratios. This helps to readily overcome any relevant problems of accuracy; a second-order upwind scheme without flux limiters or least squares approximations is successfully employed for the gradients. The test case includes quasi-3D effects by considering the streamtube thickness variation in the governing equations.     

Classical and variational multiscale LES of the flow around a circular cylinder on unstructured grids

The effects of numerical viscosity, subgrid scale (SGS) viscosity and grid resolution are investigated in LES and VMS-LES simulations of the flow around a circular cylinder at Re=3900 on unstructured grids. The separation between the largest and the smallest resolved scales in the VMS formulation is obtained through a variational projection operator and finite-volume cell agglomeration. Three different non-dynamic eddy-viscosity SGS models are used both in classical and in VMS-LES. The so-called small-small formulation is used in VMS-LES, i.e. the SGS viscosity is computed as a function of the smallest resolved scales. Two different grid resolutions are considered. It is found that, for each considered SGS model, the amount of SGS viscosity introduced in the VMS-LES formulation is significantly lower than in classical LES. This, together with the fact that in the VMS formulation the SGS viscosity only acts on the smallest resolved scales, has a strong impact on the results. However, a significant sensitivity of the results to the considered SGS model remains also in the VMS-LES formulation. Moreover, passing from classical LES to VMS-LES does not systematically lead to an improvement of the quality of the numerical predictions.     

Computation of complex turbulent flow using matrix-free implicit dual time-stepping scheme and LRN turbulence model on unstructured grids

In this study, a matrix-free implicit dual time-stepping method has been developed. It is implemented, together with a low-Reynolds-number q-ω turbulence model, in a high-order upwind finite-volume solver on unstructured grids. Semi-implicit treatment of the source terms of the q and ω equations is also introduced to further stabilize the numerical solution. It has been found that these techniques provide strong stabilization in the computation of a supersonic flow with complex shock-boundary-layer interactions in a channel with a backward-facing step. The proposed method has a low-memory overhead, similar to an explicit scheme, while it shows good stability and computational efficiency as an implicit scheme. The method developed has been validated by comparing the computed results with the corresponding experimental measurements and other calculated results, which shows good agreement. Research is being done to extend the method to calculate unsteady turbulent flows.     

An improved multislope MUSCL scheme for solving shallow water equations on unstructured grids

This paper describes an improved vector manipulation multislope monotone upstream-centred scheme for conservation laws (MUSCL) reconstruction for solving the shallow water equations on unstructured grids. This improved MUSCL reconstruction method includes a bigger stencil for the interpolation and saves time for determining the geometric relations compared to the original vector manipulation method, so it is computationally more efficient and straightforward to implement. Four examples involving an analytical solution, laboratory experiments and field-scale measurements are used to test the performance of the proposed scheme. It has been proven that the proposed scheme can provide comparable accuracy and higher efficiency compared to the original vector manipulation method. With the increasing of the number of cells, the advantage of the proposed scheme becomes more apparent.     

An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.     

A parallel,load-balanced MHD code on locally-adapted,unstructured grids in 3d

An efficient parallel code for the approximate solution of initial boundary value problems for hyperbolic balance laws is introduced. The method combines three modern numerical techniques: locally-adaptive upwind finite-volume methods on unstructured grids, parallelization based on non-overlapping domain decomposition, and dynamic load balancing. Key ingredient is a hierarchical mesh in three space dimensions.The proposed method is applied to the equations of compressible magnetohydrodynamics (MHD). Results for several testproblems with computable exact solution and for a realistic astrophysical simulation are shown.     

A new shock-capturing technique based on Moving Least Squares for higher-order numerical schemes on unstructured grids

This paper presents a shock detection technique based on Moving Least Squares reproducing kernel approximations. The multiresolution properties of these kinds of approximations allow us to define a wavelet function to act as a smoothness indicator. This MLS sensor is used to detect the shock waves. When the MLS sensor is used in a finite volume framework in combination with slope limiters, it improves the results obtained with the single application of a slope-limiter algorithm. The slope-limiter algorithm is activated only at points where the MLS sensor detects a shock. This procedure results in a decrease of the artificial dissipation introduced by the whole numerical scheme. Thus, this new MLS sensor extends the application of slope limiters to higher-order methods. Moreover, as Moving Least Squares approximations can handle scattered data accurately, the use of the proposed methodology on unstructured grids is straightforward. The results are very promising, and comparable to those of essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes. Another advantage of the proposed methodology is its multidimensional character, that results in a very accurate detection of the shock position in multidimensional flows.     

A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids

The convergence of high-order methods, such as recently developed spectral difference (SD) method, can be accelerated using both implicit temporal advancement and a p-multigrid (p = polynomial degree) approach. A p-multigrid method is investigated in this paper for solving SD formulations of the scalar wave and Euler equations on unstructured grids. A fast preconditioned lower-upper symmetric Gauss-Seidel (LU-SGS) relaxation method is implemented as an iterative smoother. Meanwhile, a Runge-Kutta explicit method is employed for comparison. The multigrid method considered here is nonlinear and utilizes full approximation storage (FAS) [Ta’asan S. Multigrid one-shot methods and design strategy, Von Karman Institute Lecture Note, 1997 [28]] scheme. For some p-multigrid calculations, blending implicit and explicit smoothers for different p-levels is also studied. The p-multigrid method is firstly validated by solving both linear and nonlinear 2D wave equations. Then the same idea is extended to 2D nonlinear Euler equations. Generally speaking, we are able to achieve speedups of up to two orders using the p-multigrid method with the implicit smoother.