Body-fitted Mesh Research Articles

A Maximum Entropy-Inspired Interpolative Closure for the Prediction of Radiative Heat Transfer in Laminar Co-Flow Diffusion Flames

ABSTRACT An interpolative-based first-order maximum entropy (M1) moment closure for providing approximate solutions to the equation of radiative transfer in non-gray participating media is proposed and described. This newly developed non-gray moment closure technique results in significant computational savings compared to an approach that makes use of the direct numerical solution of the optimization problem for entropy maximization. Its predictive capabilities are also assessed by comparing its solutions to those of the more commonly adopted first-order spherical harmonics (P1) moment closure technique, as well as the popular discrete ordinates method (DOM), which is used as a benchmark for the model comparisons. The evaluations are performed for sooting co-flow laminar diffusion flames for blends of ethanol and methane fuels at atmospheric as well as at elevated pressures and include comparisons to available experimental data for soot volume fraction and flame temperature. The strong spectral dependence exhibited by the absorption coefficient of real gases is treated herein using the statistical narrow-band correlated-k method, and the chemical kinetics of the underlying species are modeled using a reduced mechanism for methane and ethanol fuels. Theoretical details of the proposed interpolative M1, along with a description of the proposed Godunov-type finite-volume scheme developed for the numerical solution of the resulting system of hyperbolic moment equations are briefly discussed. The finite-volume method makes use of limited second-order solution reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton–Krylov–Schwarz (NKS) iterative method for solution of the resulting nonlinear algebraic equations arising from the finite-volume discretization procedure. The numerical results for laminar co-flow flames show that the flame solutions and predictions of soot formation of the non-gray M1 maximum-entropy moment closure are very promising and in very good agreement with those of both the DOM and P1 spherical harmonics model, while offering a substantial reduction in the number of dependent solution variables, compared to the DOM.

Axisymmetric compact finite-difference lattice Boltzmann method for blood flow simulations.

An axisymmetric compact finite-difference lattice Boltzmann method is proposed to simulate both Newtonian and non-Newtonian flow of blood through a lumen. The curvature of the arteries could be accurately resolved using body-fitted mesh owing to the proposed finite-difference formulation. The axisymmetric nature of the flow, as well as the non-Newtonian nature of blood, are incorporated into the lattice Boltzmann equation using separate source terms. Using Chapman-Enskog expansion it is shown that the resulting lattice Boltzmann equation with these additional source terms recovers the macroscopic axisymmetric hydrodynamic equations. The solver is verified for (1) steady inflow of a Newtonian fluid through a stenosed lumen, (2) temporally developing pulsatile flow (Womersley flow) through a straight lumen with Newtonian fluid, and (3) steady inflow of a non-Newtonian fluid through a straight lumen. The solver is then applied to simulate the steady flow of a non-Newtonian fluid through a stenosed lumen, and it was found that a smaller recirculation zone and lower WSS values are obtained when compared with the flow of a Newtonian fluid. The capability of the solver to simulate spatially developing (velocity-driven) pulsatile flow is then demonstrated by simulating physiological pulsatile flow through an axisymmetric abdominal aortic aneurysm. From this simulation, the cycle-averaged wall shear stress is observed to have a steep gradient going from a minimum (negative) to a maximum (positive) value towards the distal end of the aneurysm, which is prone to the risk of rupture. An iterative procedure to select the geometric and flow parameters for unsteady inflow condition in the lattice Boltzmann method framework is demonstrated that accurately resolves all the timescales to achieve incompressibility. Overall, the present solver seems to be promising to simulate axisymmetric flow of blood with steady and pulsatile inflows while considering the blood rheology.

Moving meshes in complex configurations using the composite sliding grid method

The study is an extension of the sliding method to composite grids whereby the grid generation complexity is diminished and one can control grid features during the motion. In this approach, a fixed mesh is attached to the moving body, and together they slide through a fixed connectivity reference mesh. The connectivities of both, the reference and the attached mesh remain fixed during the motion. Only the connections between the reference mesh and the attached body-fitted mesh are modified by a stitching operation. This allows to retain the mesh features (e.g. concentration, orthogonality, etc.) inside the moving block. This also allows the moving block to combine translation and rotation. Since the attached mesh is always fixed, one can handle grid generation process for complex combined motion. The overall methodology is applied to complex geometric configurations with large and complex motion, representative of engineering applications. These procedures were applied to solve simple flows around moving bodies with varying geometric features (circular, square and wavy boundaries) to test the capability of the global algorithm and to verify the satisfaction of geometric conservation law (GCL).

Numerical simulations and optimizations for turbine-related configurations

In order to accelerate the numerical simulation and optimization of gas turbine-related configurations, a source based computational fluid dynamics (SCFD) approach is developed for flow and heat transfer simulations. Different sources de-pending on the fluid porosity at each grid node in the computational domain are introduced to the continuity, momentum, energy and turbulence model equations, so that both the fluid and solid regions can be solved as one region. In the present paper, test cases including a ribbed channel and a winglet shrouded turbine cascade with tip injection are investigated using the SCFD and CFD with body-fitted meshes. Impacts of grid clustering and turbulence model equation sources on the SCFD precision are examined. Numerical results show that the SCFD predicts consistent aero-thermal performance with the fluid dynamics with body-fitted meshes and experiments. The validated SCFD scheme is then employed in a response surface optimization of tip jet holes on the winglet shroud tip. A jet arrangement with the minimum energy loss and injection mass-flow rate is obtained, indicating that source based predictions can be applied to the preliminary aero-thermal design of turbine blades.

Topology optimization of compliant mechanisms with stress constraints and manufacturing error robustness

This work proposes a robust formulation to address the compliant mechanism design problem subject to both stress constraints and manufacturing uncertainty. The proposed formulation is an extension of the robust approach for compliant mechanism design based on eroded, intermediate and dilated projections. The novelty in this proposal comes from inclusion of a stress failure criterion in each projected field, in order to ensure compliant mechanisms that satisfy the stress failure criterion even in the presence of uniform manufacturing variations. The objective of the optimization problem is the minimization of the maximum displacement at the output port of the mechanism, given eroded, intermediate and dilated designs, subjected to upper and lower volume constraints and one stress constraint per finite element on each of the three projected fields. The objective function is weighted by the volume of the dilated topology, in order to avoid possible numerical instabilities that may occur when the upper volume constraint is not active. Several examples are solved and the optimized results are post-processed with body-fitted finite element meshes. Numerical results demonstrate that: 1) the proposed stress-constrained robust approach provides results in which both maximum stress and output displacements are robust with respect to uniform boundary variations; however, while the maximum stress is almost insensitive to manufacturing variations, the output displacement does show some degradation when compared with the traditional robust approach; 2) the traditional robust approach, i.e., without the stress considerations, provides results in which the maximum stress has unpredictable and non-smooth behavior after uniform boundary variation; and 3) the stress-constrained deterministic approach, i.e., without considering the manufacturing uncertainty, provides results in which both maximum stress and output displacements are non-robust with respect to uniform boundary variations.

An implicit parallel UGKS solver for flows covering various regimes

This paper presents an engineering-oriented UGKS solver package developed in China Aerodynamics Research and Development Center (CARDC). The solver is programmed in Fortran language and uses structured body-fitted mesh, aiming for predicting aerodynamic and aerothermodynamics characteristics in flows covering various regimes on complex three-dimensional configurations. The conservative discrete ordinate method and implicit implementation are incorporated. Meanwhile, a local mesh refinement technique in the velocity space is developed. The parallel strategies include MPI and OpenMP. Test cases include a wedge, a cylinder, a 2D blunt cone, a sphere, and a X38-like vehicle. Good agreements with experimental or DSMC results have been achieved.

Numerical study of flow past oscillatory square cylinders at low Reynolds number

Abstract The present work reports a numerical investigation of the flow past an oscillatory square cylinder at R e = 100 . A fixed finite element immersed technique is proposed to solve the incompressible fluid–rigid body interaction problem. The study encompasses the analyses of the flow over a fixed cylinder and transverse to the flow and streamwise oscillating cylinders. The dimensionless frequency f r , i.e., the ratio between cylinder’s oscillation and Strouhal frequencies, and motion amplitude are used to characterize the flow pattern evolution and its action on the cylinder. First, to compute the flow over the cylinder, a mesh size is determined from a mesh sensitivity analysis where the drag and lift coefficients computed using the immersed method are compared with those obtained from a body-fitted mesh. A reasonable numerical convergence is found from such analyses. Second, the oscillating square cylinder cases are studied to describe the synchronization regions. To this end, an oscillating body motion is prescribed varying the frequency of the imposed displacement with a constant amplitude. The synchronization zones are determined when the aerodynamic coefficients exhibit the most significant variations. In particular, the lock-in phenomenon is observed at f r ≈ 1 for transversely oscillating square cylinders, confirming some aspects previously reported in the literature using other numerical techniques. In the present work, lock-in is determined at f r ≈ 2 for streamwise oscillating square cylinders, results is not longer found in the literature.

Stress-constrained topology optimization considering uniform manufacturing uncertainties

This paper proposes a robust design approach, based on eroded, intermediate and dilated projections, to handle uniform manufacturing uncertainties in stress-constrained topology optimization. In addition, a simple scheme is proposed to increase accuracy of stress evaluation at jagged edges, based on limiting sharpness of the projections to intentionally allow a thin layer of intermediate material between solid and void phases. A reference problem is analyzed through voxel-based finite element models, demonstrating that, in association with a proper choice of stiffness and stress interpolation functions, the proposed scheme can ensure consistent stress magnitude and smooth stress behavior for uniform boundary variation. Optimization problems are solved and post-processing with body-fitted meshes is performed over optimized solutions, demonstrating that: (1) stresses evaluated with voxel-based meshes containing thin soft transition boundaries are consistent with stresses evaluated with body-fitted meshes; and (2) optimized structures are robust with respect to uniform boundary variations.

Lattice Boltzmann simulation of melting of a phase change material confined within a cylindrical annulus with a conductive inner wall using a body-fitted non-uniform mesh

In this paper, the lattice Boltzmann method on a body-fitted non-uniform mesh is employed to simulate the outward melting of a pure phase change material (PCM) confined in a cylindrical annulus with a conducting inner wall, for Ra=104−105 and Ste=1. To perform the propagation step on the non-uniform mesh, the Taylor series least square method is used. After verification of the developed code against the experimental and numerical benchmark solutions, the effects of Reynolds number, radius ratio rr, wall to PCM conductivity ratio λ, dimensionless inner wall thickness t∗, and thermal boundary condition on the melting process are studied. It is concluded that the smaller radius ratios enhance the melting process. It is seen that the results of cases with λ=10 and λ=100 are in good agreement with each other and with the corresponding cases where the wall thickness is ignored. However, this is not the case for λ=1. Moreover, it is seen that for λ=10 and λ=100, the bigger t∗ enhances the melting process whereas, for λ=1, the reverse is true. Finally, it is figured out that the effect of applying the adiabatic conditions on the outer cylinder, instead of the constant temperature, enhances the liquid fraction in such a way that this improvement is seen to be more pronounced for the times after the melted PCM first touches the outer cylinder.

Highly accurate simplified lattice Boltzmann method

In this paper, we present a highly accurate simplified lattice Boltzmann method (HSLBM) which can achieve the third-order of accuracy in space. By introducing virtual streaming nodes and decoupling the streaming distance from the mesh spacing, HSLBM effectively combines the local second-order simplified and highly stable lattice Boltzmann method (SHSLBM) and the overall high order scheme. The correlation between the streaming distance and the mesh spacing is accomplished by a high-order Lagrange interpolation algorithm. Through a series of tests, it is found that using 5 interpolation points and setting streaming distance as 1/5 of the mesh spacing can give optimal results. In general, HSLBM improves the accuracy of SHSLBM from the second order to the third order, while maintaining its merits like low memory cost, convenient implementation of physical boundary conditions, and good numerical stability. To match the accuracy of the flow solver, a linear extrapolation scheme with the third-order of accuracy is also proposed to determine the boundary values of the non-equilibrium distribution functions. Various benchmark tests are performed to demonstrate the robustness of HSLBM in simulating two- and three-dimensional incompressible viscous flows as well as its flexibility in problems with curved boundaries and on body-fitted meshes.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime. II. Application to curved boundaries.

Gaseous flows inside microfluidic devices often fall in the slip-flow regime. According to this theoretical description, the Navier-Stokes model remains applicable in bulk, while at solid walls a slip velocity boundary model shall be considered. Physically, it is well established that, to properly account for the wall curvature, the wall slip velocity must be determined by the shear stress, rather than the normal component of the velocity derivative alone, as commonly applied to planar surfaces. It follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics (CFD) techniques. This paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice Boltzmann method (LBM). To substantiate this conclusion, this work undertakes the following three studies. First, we examine the conditions under which the generic reflection-type boundary rules used by LBM become consistent models for the slip velocity boundary condition. This effort makes use of the second-order Chapman-Enskog expansion method, where we address both planar and curved boundaries. The analysis also clarifies the capabilities and limitations behind the considered reflection-type slip schemes. Second, we revisit the family of parabolic accurate LBM slip boundary schemes, originally formulated in [Phys. Rev. E 96, 013311 (2017)2470-004510.1103/PhysRevE.96.013311] on the basis of the multireflection framework, and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate LBM slip boundary formulations, such as the linearly accurate slip schemes and the widely popular "kinetic-based" boundary schemes. In addition, we also discuss the numerical stability of the parabolic slip schemes previously developed, providing an heuristic strategy to improve their stable range of operation. Third, we evaluate the performance of the several slip boundary schemes debated in this paper. The numerical tests correspond to two classical 2D benchmark flow problems of slip over non-planar solid surfaces, namely: (i) the velocity profile of the cylindrical Couette flow, and (ii) the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles. The obtained numerical results confirm the competitiveness of the LBM when equipped with slip boundary schemes of parabolic accuracy as CFD tool to simulate slippage phenomena over arbitrarily non-planar surfaces. Indeed, although operating on a simple uniform mesh discretization, the LBM yields a similar, or even superior, level of accuracy compared to state-of-the-art FEM simulations conducted on hardworking body-fitted meshes. This conclusion establishes the LBM as a very appealing CFD technique for simulating microfluidic flows in the slip-flow regime, a result that deserves further exploration in future studies.

Steady-state and transient solutions to drop evaporation in a finite domain: Alternative benchmarks to the d2 law

The d2 law, commonly used to assess the accuracy of computational solvers, corresponds to evaporation under steady-state conditions and in an infinite gaseous domain. In the present work, we propose new alternative benchmarks that relax those constraints. The benchmarks are developed for a spherical as well as a circular drop. First, steady-state analytical relations are derived that are applicable to a finite domain. A radially-symmetric transient model is then proposed that can provide transient benchmark solutions in finite domains. Adaptive mesh-refinement and body-fitted mesh are used for numerical solution and to ensure highly accurate transient results. The proposed steady-state and transient benchmarks are examined in several evaporation problems, and the evolution of drop diameter, temperature and vapor mass fraction fields are analyzed. In one such problem, we report a secondary transient phenomenon that appears towards the end of drop lifetime. The benchmarks proposed in this work are straightforward to be implemented for systematic validation of computational solvers.

Simulation of turbulent flows with the entropic multirelaxation time lattice Boltzmann method on body-fitted meshes

We propose a body-fitted mesh approach based on a semi-Lagrangian streaming step combined with an entropy-based collision model. After determining the order of convergence of the method, we analyse the flow past a circular cylinder in the lower subcritical regime, at a Reynolds number$Re=3900$, in order to assess the numerical performances for wall-bounded turbulence. The results are compared to experimental and numerical data available in the literature. Overall, the agreement is satisfactory. By adopting an efficient local refinement strategy together with the enhanced stability features of the entropic model, this method extends the range of applicability of the lattice Boltzmann approach to the solution of realistic fluid dynamics problems, at high Reynolds numbers, involving complex geometries.

Optimization Design of Minimum Total Resistance Hull Form Based on CFD Method

In order to reduce the resistance and improve the hydrodynamic performance of a ship, two hull form design methods are proposed based on the potential flow theory and viscous flow theory. The flow fields are meshed using body-fitted mesh and structured grids. The parameters of the hull modification function are the design variables. A three-dimensional modeling method is used to alter the geometry. The Non-Linear Programming (NLP) method is utilized to optimize a David Taylor Model Basin (DTMB) model 5415 ship under the constraints, including the displacement constraint. The optimization results show an effective reduction of the resistance. The two hull form design methods developed in this study can provide technical support and theoretical basis for designing green ships.

The aggregated unfitted finite element method for elliptic problems

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

Topology optimization of a pseudo 3D thermofluid heat sink model

This paper investigates the application of density-based topology optimization to the design of air-cooled forced convection heat sinks. To reduce the computational burden that is associated with a full 3D optimization, a pseudo 3D optimization model comprising a 2D modeled conducting metal base layer and a thermally coupled 2D modeled thermofluid design layer is used. Symmetry conditions perpendicular to the flow direction are applied to generate periodic heat sink designs. The optimization objective is to minimize the heat sink heat transfer resistance for a fixed pressure drop over the heat sink and a fixed heat production rate in the base plate. Optimized designs are presented and the resulting fin geometry is discussed from a thermal engineering point of view and compared to fin shapes resulting from a pressure drop minimization objective. Parametric studies are conducted to analyze the influence of the pressure drop on the heat sink heat transfer resistance. To quantify the influence of the assumptions made in the pseudo 3D optimization model, validation simulations with a body-fitted mesh in 2D and 3D are conducted. A good agreement between optimization model and validation simulations is found, confirming the physical validity of the utilized optimization model. Two topology optimized designs are exemplarily benchmarked against a size optimized parallel fin heat sink and an up to 13.6% lower thermal resistance is found to be realized by the topology optimization.

Fluid-structure interaction study of the supersonic parachute using large-eddy simulation

PurposeThe purpose of this study is to model the dynamic characteristics of an opened supersonic disk-gap-band parachute.Design/methodology/approachA fluid-structure interaction (FSI) method with body-fitted mesh is used to simulate the supersonic parachute. The compressible flow is modeled using large-eddy simulation (LES). A contact algorithm based on the penalty function with a virtual contact domain is proposed to solve the negative volume problem of the body-fitted mesh. Automatic unstructured mesh generation and automatic mesh moving schemes are used to handle complex deformations of the canopy.FindingsThe opened disk-gap-band parachute is simulated using Mach 2.0, and the simulation results fit well with the wind tunnel test data. It is found that the LES model can successfully predict large-scale turbulent vortex in the flow. This study also demonstrates the capability of the present FSI method as a tool to predict shock oscillation and breathing phenomenon of the canopy.Originality/valueThe contact algorithm based on the penalty function with a virtual contact domain is proposed for the first time. This methodology can be used to solve the negative volume problem of the dynamic mesh in the flow field.

Simulating the hydrodynamic response of a floater–net system in current and waves

We present a novel numerical model for simulating current and wave interaction with a floater–net system. The main contribution of the paper is the integration of the floater motion and the fluid–structure interaction analysis of the net structure in the same modelling framework via the computational fluid dynamic approach. The sinker and the mooring lines were not directly resolved, but their effects were partially modelled. The model couples a hydrodynamic solver, a rigid body motion solver, a mesh motion solver and a structural solver in a segregated manner. In the numerical model, the net structure was modelled as a set of dynamic porous zones. A lumped mass model was coupled with it to realize fluid–structure interaction analysis for the net structure. The floater was treated as a rigid body, which was resolved by the body-fitted computational mesh in the fluid domain. The motion equation for the floater was set up based on the principle of linear and angular momentum balance. Different motion integration schemes were implemented and tested in the numerical model. The numerical model was validated against three sets of available experimental data in the open literature. The first set of validation cases treated the floater motion in regular waves. The second set of validation cases focused on the fluid–structure interaction analysis of the net structure. The final one was related to the whole floater–net system in regular waves, and combined current and wave condition.

Fluid flow around NACA 0012 airfoil at low-Reynolds numbers with hybrid lattice Boltzmann method

We simulate the two-dimensional fluid flow around National Advisory Committee for Aeronautics (NACA) 0012 airfoil using a hybrid lattice Boltzmann method (HLBM), which combines the standard lattice Boltzmann method with an unstructured finite-volume formulation. The aim of the study is to assess the numerical performances and the robustness of the computational method. To this purpose, after providing a convergence study to estimate the overall accuracy of the method, we analyze the numerical solution for different values of the angle of attack at a Reynolds number equal to 103. Subsequently, flow fields at Reynolds numbers up to 104 are computed for a zero angle of attack configuration. A grid refinement scheme is applied to the uniformly spaced component of the overlapping grid system to further enhance the numerical efficiency of the model. The results demonstrate the capability of the HLBM to achieve high accuracy near solid curved walls, thus providing a viable alternative in the realm of off-lattice Boltzmann methods based on body-fitted mesh.

Gradient Recovery for Elliptic Interface Problem: I. Body-Fitted Mesh

In this paper, we propose a novel gradient recovery method for elliptic interface problem using body-fitted mesh in two dimension. Due to the lack of regularity of solution at interface, standard gradient recovery methods fail to give superconvergent results, and thus will lead to overrefinement when served as a posteriori error estimator. This drawback is overcome by designing an immersed gradient recovery operator in our method. We prove the superconvergence of this method for both mildly unstructured mesh and adaptive mesh, and present several numerical examples to verify the superconvergence and its robustness as a posteriori error estimator.