Toward a Process for Quantifying Uncertainty in a CFD Solution: Analysis of Mesh Refinement

The Complex Cepstrum is shown to remove spurious reflections from artificial boundaries in computational fluid dynamic (CFD) solutions. First, the Complex Cepstrum theory is presented. A model time sequence consisting of a direct signal and reflections is analyzed theoretically with the Complex Cepstrum, and it is shown that the direct signal uncontaminated by reflections may be recovered in the time domain. Next, the Complex Cepstrum is applied to one- and three-dimensional CFD solutions, and spurious reflections from the boundary conditions are removed. By eliminating spurious reflections introduced by artificial boundary conditions, the applicability of CFD methods to aeroacoustic problems is greatly enhanced.

ABSTRACTA framework is proposed to bridge coarse and fine meshes in a single simulation within the phase field method for fracture. Fine meshes are used in the vicinity of localized defects to accurately capture crack initiation, while coarse meshes are used away from initial defects and include only crack propagation paths. This reduces the prohibitive computational times associated with uniformly fine meshes over the entire domain, or with the use of complex adaptive meshes in the phase field method. The coupling between the two overlapping meshes is achieved using a variational formulation in which the energies of the models associated with the fine and coarse meshes are weighted in the superposition zone. Two situations are considered. The first includes the resolution of the phase field problem only in the fine mesh, while the coarse mesh is limited to the undamaged elastic problem. In the second situation, cracks can propagate in both the fine and coarse mesh. Variational formulations and associated finite element implementations are detailed. Numerical examples are presented, showing the potential of this approach to significantly reduce computational costs in the phase field method for cracking without affecting the accuracy.

This Paper presents a coupled vortex particle method–computational fluid dynamics solver. The vortex particle method is used to prevent dissipation of the vortex structure on coarse computational fluid dynamics meshes. Implementation of the approach uses the split velocity method that specifies the fluid velocity as the sum of the induced vortex particle velocity and a remaining velocity. Dissipation of the vortex velocities on coarse meshes is removed, and the computational fluid dynamics equations solved for the remaining velocity have a form identical to those for a moving mesh, but with additional source terms. The coupled solver is demonstrated on a selection of two-dimensional test cases, and the results are compared to the solutions of the computational fluid dynamics solver on its own using a coarse mesh and a fine mesh. It is shown that the coupled solver preserves the vortices on a coarse mesh and is computationally more efficient than using the fine mesh.

Introduction A very efficient formulation for the inverse design of two-dimensional turbomachinery blades in turbulent transonic flow conditions is presented. The proposed optimization technique employs a highly accurate space discretization scheme for the accurate computation of the flow solution and of the objective function. The sensitivity derivatives are evaluated very efficiently by using a discrete adjoint formulation. The use of an auxiliary dissipative flow solver in the adjoint equations, rather than the accurate basic flow solver, allows a robust evaluation of the sensitivity derivatives, even in presence of shocks and of numerical noise. Moreover, if the variable prescribed on the blade surface is the pressure, as usual, a simplified inviscid formulation of the adjoint problem can be used, even in the real case of turbulent flows. The entire procedure is termed progressive optimization, because both the flow computation and the iterative solution of the adjoint problem, based on the addition of an artificial time-dependent term in the adjoint equations, are brought to convergence while approaching the optimum. The idea of the simultaneous convergence of all iterative solutions suggests also the use of progressively finer grids for the progressive computation of the flow field; the cost of the adjoint problem is strongly reduced by solving the adjoint equations always at the coarsest level. The proposed approach has been tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline. The numerical results demonstrate that the methodology is robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work needed to perform from 0.7 to 1.3 converged flow analyses on the finest grid. 'Associate Professor. ^Professor and Department Head, AIAA senior member. Copyright © 2000 by L. A. Catalano and A. Dadone. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Until few years ago, the optimization of the aerodynamic efficiency of turbomachinery bladings has been mostly approached by subsequent improvements based on the experience of the designers, each one of them followed by expensive and timeconsuming wind tunnel tests. For such a reason, the recent development of automatic tools based on the numerical solution of the flow equations and on an optimization process for the shape changes has greatly attracted the interest of turbomachinery designers. Due to the complex flow patterns about turbomachinery bladings, these numerical methods have been first developed and applied to the inverse and the direct design of airfoils; some of them have been later extended to the optimization of turbine and compressor blades. Nevertheless, many of the inverse design methods for turbomachinery bladings currently used are based on the use of a transpiration model', which cannot be extended to the direct design of the blade. For such a reason, the authors find very attractive to extend more general metodologies, already developed and tested for the optimization of airfoils or wings, to the design of turbomachinery bladings. In the application of these methods to transonic flows, typical of high-load bladings, a major difficulty commonly arises: the objective function is generally noisy or non-smooth, so that the convergence of standard optimization techniques may be compromised or even prevented, as shown in Ref. 3 for the case of a transonic nozzle. In such situations a standard finite-difference approach for design sensitivities or an alternative route based on the solution of a set of sensitivity equations' may result in an optimization process hung up at one of the many local minima artificially created by the shock shifting. The adjoint procedure proposed by Jameson et al overcomes this problem by effectively smoothing the objective function, which however may change the result of the optimization process, as outlined in Ref. 7. An alternative smoothing procedure which operates only on the design sensitivities and not on the objective function itself is proposed in Ref. 8, where it is (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. shown to be effective and efficient in designing transonic diffusers and airfoils by using two-dimensional inviscid flow equations. A second, very important challenge is the development of an optimization technique able to converge to the optimum in a computational work corresponding to few flow analyses. Indeed, most existing methodologies are based on a sequence of global iterations, each one of them requiring a computation of the converged flow field, a solution for the sensitivity derivatives and an appropriate update of the design variables. The repeated solution of the flow equations typical of such a serial approach results in a high computational cost of the entire design procedure. The efficiency of the smoothing procedure developed in Ref. 8 has been recently improved by introducing a progressive optimization strategy''', whereby the optimization process is based on partially converged flow solutions with the aim to converge the flow solution while converging the design problems. In particular, in Ref. 11 this efficient approach is applied to the inverse design of turbine blades in inviscid transonic flow conditions. In Refs. 9 — 1 2 , the adjoint equations are solved using a direct solver, appropriate for two-dimensional flow problems, but not effective for three-dimensional flow problems. The extension to three dimensions has been considered in Ref. 13, where the progressive procedure suggested in Ref. 9 is modified by adding a time derivative in the adjoint equations, and by solving them iteratively. in a progressive manner quite similar to the one employed for the flow equations. The smoothing procedure and the use of partially converged flow solutions as well as of partially converged adjoint equation solutions in the progressive optimization implies that the optimization process is based on approximate values of the gradient of the objective function. Another example of approximate design sensitivity has been discussed by Matsuzawa and Hafez, where adjoint equations based on an inviscid flow formulation have been used successfully to determine approximate gradients of the objective function for airfoil inverse design in laminar flow conditions. The aim of the present paper is to apply all of the aforementioned strategies that contribute to improve the robustness and the efficiency of the optimization process to the inverse design of two-dimensional cascades in turbulent flow conditions. The final goal is to validate the robustness and efficiency of the outlined procedure also for this important class of design problems and, more importantly, to perform a first important step towards the development of an automatic tool for the direct optimization of turbomachinery bladings in real flow conditions. The adjoint equations will be derived using an inviscid solver involving a traditional centered scheme with added numerical viscosity. The inviscid formulation will be used in order to preserve the simplicity of the adjoint equations. Moreover, an additional time derivative will be introduced in order to solve the time dependent adjoint equations progressively in time. Finally, the flow equations will be solved on a progressively finer mesh, while the adjoint equations will be always solved on the coarsest mesh in order to reduce the cost of the computation of the gradient of the objective function to a few percents of the total cost of the optimization procedure. The following sections will be devoted to the description of the main ingredients of the proposed approach, that will be tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline.

In this article, a novel multiscale modeling method is proposed for transient computational fluid dynamics (CFD) simulations of the human airways. The developed method is the first attempt to incorporate spatial coupling and temporal coupling into transient human airway simulations, aiming to improve the flexibility and the efficiency of these simulations. In this method, domain decomposition was used to separate the complex airway model into different scaled domains. Each scaled domain could adopt a suitable mesh and timestep, as necessary: the coarse mesh and large timestep were employed in the macro regions to reduce the computational cost, while the fine mesh and small timestep were used in micro regions to maintain the simulation accuracy. The radial point interpolation method was used to couple data between the coarse mesh and the fine mesh. The continuous micro solution–intermittent temporal coupling method was applied to bridge different timesteps. The developed method was benchmarked using a well‐studied four‐generation symmetric airway model under realistic normal breath conditions. The accuracy and efficiency of the method were verified separately in the inhalation phase and the exhalation phase. Similar airflow behavior to previous studies was observed from the multiscale airway model. The developed multiscale method has the potential to improve the flexibility and efficiency of transient human airway simulations without sacrificing accuracy.

The influence of various aspects of Malaise trap design on efficiency in collecting various groups of Hymenoptera was compared using commercially available traps. The influence of fine and coarse mesh sizes and the importance of the incorporation of pan traps into the design were evaluated in two sampling periods in an old dune community at Pinery Provincial Park in southern Ontario. Numbers of individuals collected in Malaise trap heads and pan traps were presented for each paired comparison of mesh size (fine and coarse) and location (top and bottom of a hill) for various families of Hymenoptera. A graphical analysis of the normalized catch data for more inclusive groupings, viz. Symphyta, Aculeata, Ichneumonoidea, and microhymenoptera, was presented. The major results of these comparisons were as follows: (1) pan traps were an important component of efficient Malaise traps, especially for Aculeata and microhymenoptera; (2) coarse mesh was more effective in collecting Aculeata; (3) coarse and fine mesh were both effective in collecting Ichneumonoidea; and (4) fine mesh was more effective in collecting microhymenoptera. There was an interaction effect between the type of trap used and groups collected and it was not possible to maximize simultaneously the collection of all groups of Hymenoptera. The use of various mesh types and a trap design that incorporates pan traps was recommended.

Modelling of fluid–particle interactions is a major area of research in many fields of science and engineering. There are several techniques that allow modelling of such interactions, among which the coupling of computational fluid dynamics (CFD) and the discrete element method (DEM) is one of the most convenient solutions due to the balance between accuracy and computational costs. However, the accuracy of this method is largely dependent upon mesh size, where obtaining realistic results always comes with the necessity of using a small mesh and thereby increasing computational intensity. To compensate for the inaccuracies of using a large mesh in such modelling, and still take advantage of rapid computations, we extended the classical modelling by combining it with a machine learning model. We have conducted seven simulations where the first one is a numerical model with a fine mesh (i.e. ground truth) with a very high computational time and accuracy, the next three models are constructed on coarse meshes with considerably less accuracy and computational burden and the last three models are assisted by machine learning, where we can obtain large improvements in terms of observing fine-scale features yet based on a coarse mesh. The results of this study show that there is a great opportunity in machine learning towards improving classical fluid–particle modelling approaches by producing highly accurate models for large-scale systems in a reasonable time.

Adaptive characteristic spatial quadrature schemes for use in discrete ordinates neutral particle transport computer codes in slab geometry have been introduced recently; the extension to two-dimensional Cartesian coordinates with rectangular cells is presented here. These quadratures differ from previous schemes in using strictly non-negative finite-element-like representations of the entering fluxes and scattering source, which adapt to match both the zeroth and first spatial moments. (These input moments are obtained from the boundary conditions or as output moments of the upstream adjacent cells and from the angular quadrature of the spatial flux moments in the outer iteration, respectively.) The flux throughout the cell is found by integration of the transport equation along streaming (characteristic) lines. The moments of the flux through the cell and along the exiting boundaries are then obtained. The resulting quadratures are nonlinear and strictly non-negative, while satisfying zeroth-spatial-moment and first-spatial-moment balances within each cell and globally. For a square test problem 32 mean free paths thick, the variation in results over a range of meshes from 1 by 1 to 8 by 8 was comparable to (usually less than) the corresponding variation due to angular quadrature (S4, S8, S16). Linear adaptive quadrature performed better than step adaptive, with few exceptions. Using a mesh consisting of a single cell, both adaptive methods performed as well as or better than linear nodal and linear characteristic methods did with an 8 by 8 mesh of cells. Although they are substantially more complicated to derive and implement, the adaptive methods cost only three to nine times more per cell to compute, and require no additional storage, as compared to linear nodal and linear characteristic methods. Thus, by using these coarser meshes, step and linear adaptive were more cost-effective than linear nodal or linear characteristic (with or without negative fixups). Linear adaptive is similar to linear characteristic (but with a bilinear term) wherever linear characteristic requires no negative flux or source fixups. Thus, it should share the excellent convergence of linear characteristic. For an optically thin test problem, with a fine mesh (up to 32 by 32 cell mesh on a square 2 mfp thick), we found that linear adaptive was slightly more accurate than linear characteristic without fixup, which was substantially different than linear characteristic with fixup. This test showed that, even with a fine mesh, fixups can be required by conventional methods (linear nodal and linear characteristic), which either have negative fluxes or violate first-spatial-moment balance. Linear adaptive has no negative fluxes and conserves first spatial moments; its results were very close to those of linear characteristic (without fixup), for optically thin cells. Step adaptive converges somewhat more slowly, however; so it is less efficient than the other methods when both are used on a fine enough mesh. We conclude that linear adaptive quadrature can be an accurate and cost-effective method for problems requiring the use of optically thick cells.

Hypersonic vehicles experience different flow regimes during flight due to changes in atmospheric density. Hybrid Computational Fluid Dynamics (CFD) and direct simulation Monte Carlo (DSMC) methods are being developed to simulate the flow in different hypersonic regimes. These methods use a breakdown parameter to determine regions of the flow where the CFD physics are no longer valid. The current study investigates the effect of continuum breakdown on surface aerothermodynamic properties, such as pressure, shear stress and heat transfer rate, of a cylinder in a Mach 10 flow of argon gas for several different flow regimes, from the continuum to a rarefied gas. CFD and DSMC solutions are obtained at each flow condition. Total drag predictions are identical for a continuum flow while differing by more than 26% for a rarefied flow. Peak heat transfer rate differences range from less than 1% for a continuum flow to almost 32% for a rarefied flow. Drag depends primarily on continuum breakdown in the wake, while heat transfer rate appears to depend primarily on continuum breakdown in the shock and differences in thermal boundary layer thickness.

A reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

PurposeWind loading calculations are currently performed according to the ASCE 7 standard. Values in this standard were estimated from simplified models that do not necessarily take into account relevant flow characteristics. Thus, the standard does not have provisions to handle the majority of rooftop photovoltaic (PV) systems. Accurate solutions for this problem can be produced using a full-fledged three-dimensional computational fluid dynamics (CFD) analysis. Unfortunately, CFD requires enormous computation times, and its use would be unsuitable for this application which requires real-time solutions. To this end, a real-time response framework based on the proper orthogonal decomposition (POD) method is proposed.Design/methodology/approachA real-time response framework based on the POD method was used. This framework used beforehand and off-line CFD solutions from an extensive data set developed using a predefined design space. Solutions were organized to form the basis snapshots of a POD matrix. The interpolation network using a radial-basis function (RBF) was used to predict the solution from the POD method given a set of values of the design variables. The results presented assume varying design variables for wind speed and direction on typical PV roof installations.FindingsThe trained POD–RBF interpolation network was tested and validated by performing the fast-algebraic interpolation to obtain the pressure distribution on the PV system surface and they were compared to actual grid-converged fully turbulent 3D CFD solutions at the specified values of the design variables. The POD network was validated and proved that large-scale CFD problems can be parametrized and simplified by using this framework.Originality/valueThe solar power industry, engineering design firms and the society as a whole could realize significant savings with the availability of a real-time in situ wind-load calculator that can prove essential for plug-and-play installation of PV systems. Additionally, this technology allows for automated parametric design optimization to arrive at the best fit for a set of given operating conditions. All these tasks are currently prohibited because of the massive computational resources and time required to address large-scale CFD analysis problems, all made possible by a simple but robust technology that can yield massive savings for the solar industry.

Over the last decades, the discontinuous Galerkin (DG) method has demonstrated its excellence in accurate, higher-order numerical simulations for a wide range of applications in aerodynamics simulations. However, the development of practical, computationally accurate flow solvers for industrial applications is still in the focus of active research, and applicable boundary conditions and fluxes are also very important parts. Based on curvilinear DG method, we have developed a flow solver that can be used for solving the three-dimensional subsonic, transonic and hypersonic inviscid flows on unstructured meshes. The development covers the geometrical transformation from the real curved element to the rectilinear reference element with the hierarchical basis functions and their gradient operation in reference coordinates up to full third order. The implementation of solid wall boundary conditions is derived by the contravariant velocities, and an enhanced algorithms of Harten-Lax-van Leer with contact (HLLC) flux based on curved element is suggested. These new techniques do not require a complex geometric boundary information and are easy to implement. The simulation of subsonic, transonic and hypersonic flows shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved element overcomes this restriction. In addition, such a flow solver is stable on a reasonably coarse meshes and finer ones, and has good robustness for three-dimensional flows with various geometries and velocities. For engineering practice, a reasonable accuracy can be obtained at reasonably coarse unstructured meshes.

A method is proposed to estimate the flow-induced drag on the actuator arm inside a hard disk drive. Typically, drag forces and moments on the actuator are computed as part of a computational fluid dynamics (CFD) solution of the flow field in the entire drive. Unidirectional coupling from the flow to the structure is then imposed to determine the structural response of the arm to the flow induced forcing. The methodology proposed here aims to reduce the simulation time associated with the flow calculations by directly estimating the forcing functions. The approach involves fitting a piecewise linear model (PLM) to the forcing frequency spectrum and interpolating or extrapolating the model to provide estimates of the spectrum at different points in the parameter space. A simple guideline for the formulation of such models is the conservation of energy between the CFD and PLM spectrum. Numerical experiments show that the linear models predict the behavior of arm to within 3% accuracy of the full CFD solution. The proposed technique is applied to two parameters: the disk RPM and the radial position of the arm. Clear trends are manifested for both parameters, making it possible to use this method to estimate forcing functions for a range of disk speeds and radial positions of the arm. This technique opens up the possibility of flow related design or optimization, which was previously thought to be prohibitively expensive.

Recent advances in unsteady flow visualization

Rotor broadband noise can be predicted using various numerical and analytical methods. This paper explores the application of computational fluid dynamics (CFD) combined with Lee's wall pressure spectrum and Amiet's trailing edge noise model to predict broadband noise, comparing the results to different experiments. A simple, ideally twisted rotor blade with the NACA 0012 airfoil in hover is employed to establish the appropriate methodology for integrating CFD with other models. Two approaches are utilized: the first is a direct input approach, where flow data from the CFD solution is directly input into the models, while the second approach employs CFD to inform XFOIL inputs, which subsequently provide inputs for the wall pressure spectrum and acoustic model. The CFD/XFOIL approach demonstrates better agreement with experimental measurements and is less sensitive to CFD pre- and post-processing setups. The Straight Up Imagining (SUI) Endurance rotor is also validated for forward flight, followed by the prediction of broadband noise generated by a propeller and wing configuration. Observed trends during the validations and predictions suggest that airfoil selection and blade geometry play a more significant role in the magnitude of trailing edge noise than thrust.