A Truncation Error-Based Approach to Understanding and Improving Mesh Quality in CFD

Freeform surface models are conventionally used to model sheet metal components, such as automobile body parts. Finite element meshes generated automatically for such models have poor quality around small detailed features. Manual correction of the mesh is extremely tedious. An approach presently receiving attention in industry aims to alleviate this problem by automatically simplifying these features in the surface model such that an acceptable mesh is automatically generated. Simplification involves recognition of the feature and modification of its geometry or complete suppression of the feature. Since features such as holes, notches, etc. are punched after the basic shape has been formed, such a simplification will also help in the modelling of forming dies and molds. The ability to detect features will also allow part comparison and classification in surface models. This paper proposes techniques to directly query the CAD data structure to recognize and suppress two basic features, viz. holes and fillets in freeform surface models. It further demonstrates how these techniques can be extended to suppress compound features that are composed of a combination of basic features. Results of a software implementation for the same are discussed with suitable examples and the improvement in mesh quality is demonstrated.

A uniform finite element mesh rarely provides the best discretization of a domain to accommodate a solution with both optimal efficiency and minimal error. Mesh adaptation can approach a more optimal solution by accommodating regions of the mesh with higher or lower element density. Extensive attention has been given to mesh adaptation in both computational mechanics and computer graphics to provide or improve methods for increasing the model resolution or solution accuracy. The algorithm developed in this paper, entitled Automated Quadrilateral Coarsening by Ring Collapse (AQCRC), provides a unique solution to allow mesh coarsening of both structured and unstructured quadrilateral meshes. The algorithm is based on modification and removal operations utilizing the dual description of the quadrilateral mesh. The AQCRC algorithm iterates on five steps: 1) input of a coarsening region and a coarsening factor, 2) selection of coarsening rings, 3) mesh quality improvement, 4) removal of coarsening rings, and 5) mesh clean-up. Examples are presented showing the application of the algorithm.

We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.

Multiresolution Representation of Triangular Mesh Model with Surface Detail -Improvement of Mesh Quality and Application to FEM -

A priori mesh quality metric error analysis applied to a high-order finite element method

formula omitted]MST: A robust unified algorithm for quadrilateral mesh adaptation in 2D and 3D

A survey of diffusion operators for compressible CFD solvers on unstructured grids is conducted to understand different formulation procedures for viscous fluxes. A patch-wise version of the method of manufactured solutions is used to test the robustness and accuracy of the diffusion operators. These diffusion operators are tested and compared on different mesh topologies in 1D and 2D to study the effect of mesh quality, mesh resolution, and the solution behavior. In 1D the mesh quality is determined by stretching the mesh by a constant factor, whereas in 2D the mesh quality includes stretching, aspect ratio, skewness, and curvature. Insight into the role of mesh quality on the diffusion terms is gained by examining the truncation error for diffusion terms evaluated with central differences in transformed coordinates. Quantities examined include the numerical approximation errors and order of accuracy associated with face gradient reconstruction and diffusion term integration. In addition, the positivity of the diffusion term is examined which provides a measure of the numerical stability of the gradient operators. In the present work, the testing of diffusion operators is limited to cell-centered finite volume methods that are formally second order accurate.

Direct numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to $3\times 10^{5}$, corresponding to Taylor numbers of $\mathit{Ta}=4.6\times 10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

The flow of a viscous incompressible fluid in the gap between two concentric spheres was investigated for the case where only the inner sphere rotates and the outer one is stationary. Flow visualization studies were carried out for a wide range of Reynolds numbers (Re≦2·105) and aspect ratios (0.08≦β≦0.5) to determine the instabilities during the laminar-turbulent transition and the corresponding critical Reynolds numbers as a function of the aspect ratio. It was found that the laminar basic flow loses its stability at the stability threshold in different ways. The instabilities occurring depend strongly on the aspect ratio and the initial conditions. For small and medium aspect ratios (0.08≦β≦0.25), experiments were carried out as a function of the Reynolds number to determine the regions of existence for basic flow, Taylor vortex flow, supercritical basic flow and furthermore, to give the best fit for the maximum number of pairs of Taylor vortices as a function of aspect ratio. For wide gaps (0.33≦β≦0.5), however, Taylor vortices could not be detected. The first instability manifests itself as a break of the spatial symmetry and non-axisymmetric secondary waves with spiral arms appear depending on the Reynolds number. For β=0.33, secondary waves with an azimuthal wave numbern=six, five and four were found, while in the gap with an aspect ratio of β=0.5 secondary waves withn=five, four and three spiral arms exist. Frequencies of these secondary waves were measured, the corresponding critical Reynolds numbers and the transition Reynolds numbers during the transition to turbulence were found. The flow modes occurring at the poles look very similar to those found in the flow between two rotating disks. Effects of non-uniqueness and hysteresis were determined as a function of the acceleration rate.

The largest and most difficult numerical approximation error to estimate is discretization error. This study investigates the accuracy of two different residual-based discretization error estimators: discretization error transport equations and defect correction methods. Residual methods are a category of error estimators that use a discrete solution and information about the problem being solved (either the differential or the discrete equations) to calculate an error estimate using only one grid level. As few as two grid levels are needed if the reliability of the error estimate is to be assessed. This is advantageous because the reliability of all discretization error estimators require that the numerical solution, or solutions, be solved on sufficiently fine grids, which is often difficult to achieve for complex scientific computing applications. Two different linearization methods for the discretization error transport equation will also be studied. The estimated discretization error for these residual-based methods will be compared to Richardson extrapolation using exact solutions to 1D Burgers’ equation and the 2D Euler equations. The exact solutions to the Euler equations include two manufactured solutions, Ringleb's flow, and a supersonic vortex flow.

Adaptive tetrahedral meshing in free-surface flow

Optimization of twisted-tape element static mixers for industrially relevant setups

In this study, a Reynolds averaged Navier-Stokes solver is used for prediction of the propeller performance in open-water conditions at different Reynolds numbers ranging from 104 to 107. The k−ω SST turbulence model and the γ−R˜eθt correlation-based transition model are utilised and results compared for a conventional marine propeller. First, the selection of the turbulence inlet quantities for different flow regimes is discussed. Then, an analysis of the iterative and discretisation errors is made. This work is followed by an investigation of the predicted propeller flow at variable Reynolds numbers. Finally, the propeller scale-effects and the influence of the turbulence and transition models on the performance prediction are discussed. The variation of the flow regime showed an increase in thrust and decrease in torque for increasing Reynolds number. From the comparison between the turbulence model and the transition model, different flow solutions are obtained for the Reynolds numbers between 105 and 106, affecting the scale-effects prediction.

Flow around a near-bed horizontal circular cylinder mounted on a vertical wall

Predicting aerodynamic forces on bluff bodies remains to be a challenging task due to the unpredictable flow behavior, specifically at higher Reynolds numbers. Experimental approaches to determine aerodynamic coefficients could be costly and time consuming. In the meantime, use of numerical techniques could also require a considerable computational cost and time depending on complexity of the flow behavior. The research focusses on developing an effective deep learning technique to predict aerodynamic force coefficients acting on elliptical bluff bodies for a given aspect ratio and given flow condition. Collecting data for drag and lift coefficients of several aspect ratios for flow conditions starting from onset of vortex shredding to verge of subcritical region is conducted by an accurate full order model. The specified region will provide a transient flow behavior and thus lift coefficient will be represented in terms of root mean square value and drag coefficient in terms of a mean value. With variations in flow behavior and vortex shredding frequencies, it requires to select an appropriate turbulence model, optimum discretization of fluid domain and time step to obtain an accurate result. Flow simulations are conducted primarily using Unsteady Reynolds Averaged Navier-Stokes Equations (URANS) model and Detached Eddy Simulations (DES) model. Effectiveness in using different turbulence models for specified flow regimes are also explored in comparison to available experimental results. At lower Reynolds numbers, aerodynamic force coefficients for a specified body will only depend on Reynolds number. But after a certain specific Reynolds number, aerodynamic forces are dependent on the Mach number in addition to Reynolds number. Therefore, for higher Reynolds numbers, aerodynamic force coefficients are recorded for multiple Mach numbers with same Reynolds number and will be fed to the neural network. With the development of the machine learning and neural network modelling, many of the fields have nourished and created effective and efficient technologies to ease complex functions and activities. Our goal is to ease the complexity in the computational fluid dynamic field with a deep neural network tool created to predict drag and lift coefficient of elliptical bluff bodies for a given aspect ratio with an acceptable accuracy level. Researchers have developed deep neural network tools to predict various flow conditions and have succeeded with sufficient accuracy and a satisfying reduction of computational cost. In our proposed deep learning neural network, we have chosen to model the network with inputs as the geometry setup and the flow conditions with validated drag and lift coefficients. The model will extract the necessary flow features into filters with the convolution operation performed on the inputs. Our main directive is to create a deep learned neural network tool to predict the target values within an acceptable range of accuracy while minimizing the computation cost.