Explicit Filtering Research Articles

Accuracy, stability, and performance comparison between the spectral difference and flux reconstruction schemes

We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.

Large-Eddy Simulation of Compressible Flow in Asymmetric and Symmetric Planar Nozzles

Large-eddy simulations of supersonic, turbulent flow in asymmetric and symmetric planar nozzles are carried out using high-order finite difference schemes and a large-eddy simulation approach based on explicit filtering. The inflow to the nozzles is from supersonic, fully developed turbulent channel flows at and 360 with centerline Mach number 1.2. The combined effects of expansion and acceleration of the flow in this geometry result in reductions of mean pressure, density, and temperature as well as a reduction of Reynolds stresses. The asymmetry of the nozzle leads to notable differences in the aforementioned effects near the straight and the curved walls. It is shown that the decay in Reynolds stresses is more near the curved wall than near the straight wall of asymmetric nozzle, and it is nearly the same near both walls of the symmetric nozzle. Effects of longitudinal streamline curvature are found to be localized near the curved wall of the asymmetric nozzle and are quantified. The aforementioned effects of acceleration, expansion, and streamline curvature are found to be similar in the flow cases with two different Reynolds numbers, although with increasing Reynolds number, subtle differences in these effects are found.

The LMARS Based Shallow‐Water Dynamical Core on Generic Gnomonic Cubed‐Sphere Geometry

AbstractThe rapidly increasing computing powers allow global atmospheric simulations with aggressively high resolutions, challenging traditional model design principles. This study presents a Low Mach number Approximate Riemann Solver (LMARS) based unstaggered finite‐volume model for solving the shallow‐water equations on arbitrary gnomonic cubed‐sphere grids. Using a novel reference line‐based grid‐generation process, it unifies the representation of arbitrary gnomonic cubed‐sphere grid projections and permits high‐efficiency 1D reconstruction in the halo regions. The numerical discretization also extends a widely used pressure gradient algorithm with the LMARS viscous term, thus improves the model's stability for various numerical applications. The solver demonstrates a broad range of organic diffusion control without any explicit filters, validated by a comprehensive set of test cases. Lastly, a newly introduced splash on the sphere test verifies the solver's desirable dispersion properties and consistent performance among different grid types. This study paves a solid foundation for a new generation of global circulation models with kilometer horizontal scales.

Hybrid Large Eddy Simulation for low-order Discontinuous Galerkin methods using an explicit filter

In this paper we present a simple, easily implemented and effective approach for explicitly-filtered Large Eddy Simulation with a Discontinuous Galerkin (DG) discretisation for velocity. DG formulations are often desirable due to their stability and increased accuracy, however this can come at greater computational expense due to the additional degrees of freedom in the velocity field. Additionally, data output can also be an issue, due to the increased storage requirements. Here we present a hybrid approach, based upon the construction of an approximation of the velocity shear tensor using information from a projected Continuous Galerkin (CG) version of the discontinuous velocity field. The resulting turbulence algorithm is implemented within Fluidity, an open-source computational fluid dynamics solver. The model is then validated with a well known test case, and shown to agree favourably with published results. Comparisons are also made between the CG/DG hybrid LES with DG-only LES, which demonstrate the superior computational performance of the hybrid model.

Boundary-consistent B-spline filtering schemes and application to high-fidelity simulations of turbulence

A filtering operation, based on B-spline discretizations, is introduced to target weakly growing mesh-scale oscillations that can arise in high-fidelity turbulence simulations. This is a spectral regularization that can be described using the singular values of a banded matrix operator, with the filtering strength set by a scalar- or vector-valued penalty parameter. The penalty parameter can be specified though it can also be advantageously selected to minimize the generalized cross validation (GCV) measure of distance between the pre- and post-filtered solutions. Efficient algorithms are developed to compute both the scalar and vector penalty parameters. The B-spline filter has a sharper localization to high-wavenumber than compact or explicit filters of the same stencil width and is demonstrated for solutions of the Burgers' equation, decaying Burgers' turbulence, and compressible Navier–Stokes turbulent channel flow. These simulations confirm the scheme's numerical stability and ability to narrowly target the high wavenumber components of numerical solutions. An advantage over finite-difference filters is that these B-spline filters are stable on bounded domains and even preserve formal order of accuracy.

Impact of collision models on the physical properties and the stability of lattice Boltzmann methods.

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Subgrid-scale model for large-eddy simulation of transition and turbulence in compressible flows

This article focuses on a subgrid-scale (SGS) eddy viscosity model based on helicity which is derived from our previous research [Yu et al., “Subgrid-scale eddy viscosity model for helical turbulence,” Phys. Fluids 25, 095101 (2013)] for large-eddy simulation of transition and turbulence in compressible flows. Based on the character of the compressible boundary layer over a flat plate, we obtain from theoretical analysis that this model can automatically distinguish laminar flow and turbulence and can also simulate turbulence well. Meanwhile, an a priori test using direct numerical simulation (DNS) data of a spatially developing flat-plate boundary layer at Ma = 2.25 shows that the helicity model can clearly differentiate laminar, transitional, and turbulent regions. Comparing the numerical simulation results with DNS and other SGS models in the spatially developing boundary-layer over a flat plate, we find that the suggested model could precisely predict the onset of transition, transition peak, skin-friction coefficient, mean velocity profile, mean temperature profile, and turbulence intensities. In the case of a compression ramp, the model can well simulate the bypass-type transition, the separated and reattached points, and the size of the separation bubble in the corner region. Furthermore, the prominent advantage of the proposed model can predict transitional flow exactly with no explicit filtering or dynamic procedure.

Learning on the Edge: Investigating Boundary Filters in CNNs

Convolutional neural networks (CNNs) handle the case where filters extend beyond the image boundary using several heuristics, such as zero, repeat or mean padding. These schemes are applied in an ad-hoc fashion and, being weakly related to the image content and oblivious of the target task, result in low output quality at the boundary. In this paper, we propose a simple and effective improvement that learns the boundary handling itself. At training-time, the network is provided with a separate set of explicit boundary filters. At testing-time, we use these filters which have learned to extrapolate features at the boundary in an optimal way for the specific task. Our extensive evaluation, over a wide range of architectural changes (variations of layers, feature channels, or both), shows how the explicit filters result in improved boundary handling. Furthermore, we investigate the efficacy of variations of such boundary filters with respect to convergence speed and accuracy. Finally, we demonstrate an improvement of 5–20% across the board of typical CNN applications (colorization, de-Bayering, optical flow, disparity estimation, and super-resolution). Supplementary material and code can be downloaded from the project page: http://geometry.cs.ucl.ac.uk/projects/2019/investigating-edge/.

Higher-order optimized hybrid Robert-Asselin type time filters

In this paper, a class of optimized filters based on hybrid blending of implicit and explicit filters are formulated for the three time-level leapfrog time integration scheme. As the leapfrog method is the most recurrently employed in ocean and atmospheric simulations, however, it also admits a spurious mode in the numerical computations. To subdue the computational mode, the developed hybrid time filters are also optimized using constraints arising from error dynamics. The optimized filters are adept in suppressing the computational mode(s) more efficaciously without altering the physical mode. Finally, the developed hybrid filters coupled with centered and compact spatial discretization schemes are also authenticated for the numerical solutions to one-dimensional (1D) convection equation, two-dimensional (2D) dispersive rotating shallow water equation, and unsteady 2D incompressible Navier-Stokes equation at different Reynolds numbers. Present solutions are also compared with results reported in the literature.

Shock Capturing in Large Eddy Simulations by Adaptive Filtering

A method for shock capturing by adaptive filtering for use with high-resolution, high-order schemes for Large Eddy Simulations (LES) is presented. The LES method used in all the examples here employs the Explicit Filtering approach and the spatial derivatives are obtained with sixth-order, compact, finite differences. The adaptation is to drop the order of the explicit filter to two at gridpoints where a shock is detected, and to then increase the order from 2 to 10 in steps at successive gridpoints away from the shock. The method is found to be effective in a series of tests of common inviscid 1D and 2D problems of shock propagation and propagation of waves through shocks. As a prelude to LES, the 3D Taylor–Green problem for the inviscid and a finite viscosity case were simulated. An assessment of the overall performance of the method for LES was carried out by simulating an underexpanded round jet at a Reynolds number of 6.09 million, based in centerline velocity and diameter at nozzle exit plane. Very close quantitative agreement was found for the development of centerline mean pressure when compared to experiment. Simulations on several increasingly finer grids showed a monotonic extension of the computed part of the inertial range, with little change to low frequency content. Amplitudes and locations of large changes in pressure through several cells were captured accurately. A similar performance was observed for LES of an impinging jet containing normal and curved shocks.

Numerical Simulation of Isothermal Flow Across Slant Five-Tube Bundle with Spectral Element Method Code Nek5000

The helical coil steam generator (HCSG) is a specific type of shell-and-tube heat exchanger known for having a higher heat transfer coefficient than many other designs. For this reason, they are considered in small modular reactor and high temperature reactor designs. Investigation of flow behavior in HCSGs is important for better design. In this paper we present our study for modeling the coolant flow in the primary side of the HCSG. We used Nek5000, an open source, high-order spectral element computational fluid dynamics code developed in Argonne National Laboratory. Nek5000 accepts only hexahedral mesh, which makes the meshing process for the complicated HCSG geometry very challenging. A tetrahedral-to-hexahedral meshing strategy was applied to bypass the geometric complexities. In this study large eddy simulation (LES) was performed at the Reynolds number of 9000 based on the inlet velocity and the tube diameter. The employed subgrid-scale model for LES relies on explicit filtering. First- and second-order statistics were compared to available experimental data. Overall velocity and turbulent kinetic energy showed good agreement with particle image velocimetry data.

Explicit Filtering and Reconstruction to Reduce Grid Dependence in Convective Boundary Layer Simulations Using WRF-LES

Abstract As model grid resolutions move from the mesoscale to the microscale, turbulent structures represented in atmospheric boundary layer simulations change dramatically. At intermediate resolutions, the so-called gray zone, turbulent motions are not resolved accurately, posing a challenge to numerical simulations. The representation of turbulence is also highly sensitive to the choice of closure model. Here, we examine explicit filtering and reconstruction in the gray zone as a technique to better represent atmospheric turbulence. The convective boundary layer is simulated using the Weather Research and Forecasting (WRF) Model with horizontal resolutions ranging from 25 m to 1 km. Four large-eddy simulation (LES) turbulence models are considered: the Smagorinsky model, the TKE-1.5 model, and two versions of the dynamic reconstruction model (DRM). The models are evaluated by their ability to produce consistent mean potential temperature profiles, heat and momentum fluxes, velocity fields, and turbulent kinetic energy spectra as the grids become coarser. The DRM, a mixed model that uses an explicit filtering and reconstruction technique to account for resolvable subfilter-scale (RSFS) stresses, performs very well at resolutions of 500 m and 1 km without any special tuning, whereas the Smagorinsky and TKE-1.5 models produce heavily grid-dependent results.

Large-Eddy Simulation of Sandia Flame D with Efficient Explicit Filtering

A uniform Gaussian filter has been applied explicitly to the LES conservation equations for mass, momentum and mixture to simulate a piloted non-premixed methane-air flame (Sandia Flame D). Using a basic property of exponential functions, the three dimensional Gaussian filter is decomposed into the product of three one dimensional filters, greatly reducing the cost of filtering. Seven simulations on three different grids have been performed to investigate the influence of grid refinement with a purely implicit filter, the effects of explicit filtering with increasing filter width and the effect of grid refinement at constant filter-size. Overall, consistent results have been achieved at a cost that is moderate with implicit or explicit filtering, so that explicit filtering can be applied in cases where the numerical error should be independent of the modelling error.

Grid-independent large-eddy simulation of turbulent wake flow using explicit differential filters

Large-eddy simulations (LES) using explicit filtering are performed to obtain grid-independent solutions of turbulent wake flow behind a circular cylinder at ReD = 3900 on non-Cartesian type grids. A differential elliptic equation where the filter kernel is implicitly defined is discretized in an unstructured-grid solver to enable explicit filtering on non-Cartesian grids. The separation of filtering procedure from discretization is known to produce an LES solution of which error is mainly attributed to the capability of a sub-filter scale (SFS) model. Equipped with the differential filter and the Vreman SFS model, explicitly filtered LES on unstructured grids is shown to produce nearly grid-independent solutions for flow over a circular cylinder at a critical Reynolds number.

Key Elements of Turbulence Closures for Simulating Deep Convection at Kilometer‐Scale Resolution

AbstractKilometer‐scale grid spacing is increasingly being used in regional numerical weather prediction and climate simulation. This resolution range is in the terra incognita, where energetic eddies are partially resolved and turbulence parameterization is a challenge. The Smagorinsky and turbulence kinetic energy 1.5‐order models are commonly used at this resolution range, but, as traditional eddy‐diffusivity models, they can only represent forward‐scattering turbulence (downgradient fluxes), whereas the dynamic reconstruction model (DRM), based on explicit filtering, permits countergradient fluxes. Here we perform large‐eddy simulation of deep convection with 100‐m horizontal grid spacing and use these results to evaluate the performance of turbulence schemes at 1‐km horizontal resolution. The Smagorinsky and turbulence kinetic energy 1.5 schemes produce large‐amplitude errors at 1‐km resolution, due to excessively large eddy diffusivities attributable to the formulation of the squared moist Brunt‐Väisälä frequency ( ). With this formulation in cloudy regions, eddy diffusivity can be excessively increased in “unstable” regions, which produce downward (downgradient) heat flux in a conditionally unstable environment leading to destabilization and further amplification of eddy diffusivities. A more appropriate criterion based on saturation mixing ratio helps eliminate this problem. However, shallow clouds cannot be simulated well in any case at 1‐km resolution with the traditional models, whereas DRM allows for countergradient heat flux for both shallow and deep convection and predicts the distribution of clouds and fluxes satisfactorily. This is because DRM employs an eddy diffusivity model that is dynamically adjusted and a reconstruction approach that allows countergradient fluxes.

Joint Image Filtering with Deep Convolutional Networks.

Joint image filters leverage the guidance image as a prior and transfer the structural details from the guidance image to the target image for suppressing noise or enhancing spatial resolution. Existing methods either rely on various explicit filter constructions or hand-designed objective functions, thereby making it difficult to understand, improve, and accelerate these filters in a coherent framework. In this paper, we propose a learning-based approach for constructing joint filters based on Convolutional Neural Networks. In contrast to existing methods that consider only the guidance image, the proposed algorithm can selectively transfer salient structures that are consistent with both guidance and target images. We show that the model trained on a certain type of data, e.g., RGB and depth images, generalizes well to other modalities, e.g., flash/non-Flash and RGB/NIR images. We validate the effectiveness of the proposed joint filter through extensive experimental evaluations with state-of-the-art methods.

Evolve Filter Stabilization Reduced-Order Model for Stochastic Burgers Equation

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results are based on a stochastic Burgers equation with linear multiplicative noise. The numerical result shows that the EF-ROM is significantly better than G-ROM.

Overdamped large-eddy simulations of turbulent pipe flow up to Reτ = 1500

We present results from large-eddy simulations (LES) of turbulent pipe flow in a computational domain of 42 radii in length. Wide ranges of shear the Reynolds number and Smagorinsky model parameter are covered, 180 ≤ Reτ ≤ 1500 and 0.05 ≤ Cs ≤ 1.2, respectively. The aim is to asses the effect of Cs on the resolved flow field and turbulence statistics as well as to test whether very large scale motions (VLSM) in pipe flow can be isolated from the near-wall cycle by enhancing the dissipative character of the static Smagorinsky model with elevated Cs values. We found that the optimal Cs to achieve best agreement with reference data varies with Reτ and further depends on the wall normal location and the quantity of interest. Furthermore, for increasing Reτ, the optimal Cs for pipe flow LES seems to approach the theoretically optimal value for LES of isotropic turbulence. In agreement with previous studies, we found that for increasing Cs small-scale streaks in simple flow field visualisations are gradually quenched and replaced by much larger smooth streaks. Our analysis of low-order turbulence statistics suggests, that these structures originate from an effective reduction of the Reynolds number and thus represent modified low-Reynolds number near-wall streaks rather than VLSM. We argue that overdamped LES with the static Smagorinsky model cannot be used to unambiguously determine the origin and the dynamics of VLSM in pipe flow. The approach might be salvaged by e.g. using more sophisticated LES models accounting for energy flux towards large scales or explicit anisotropic filter kernels.

A Priori Assessment of an Iterative Deconvolution Method for LES Sub-grid Scale Variance Modelling

An alternative method is proposed as a means for providing a closure for the sub-grid-scale variance, which is a key quantity in reacting flow simulations. The method is based on deconvolution, namely a constrained iterative deconvolution method combined with explicit filtering. The assessment of the method is conducted a priori using a direct numerical simulation database, and for the conditions tested in this study the method is found to provide quantitatively good estimates of both the un-filtered progress variable and its variance.

Large-Eddy Simulation of the Stratocumulus-Capped Boundary Layer with Explicit Filtering and Reconstruction Turbulence Modeling

Abstract Large-eddy simulation (LES) has been an essential tool in the development of theory and parameterizations for clouds, but when applied to stratocumulus clouds under sharp temperature inversions, many LES models produce an unrealistically thin cloud layer and a decoupled boundary layer structure. Here, explicit filtering and reconstruction are used for simulation of stratocumulus clouds observed during the first research flight (RF01) of the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS II). The dynamic reconstruction model (DRM) is used within an explicit filtering and reconstruction framework, partitioning subfilter-scale motions into resolvable subfilter scales (RSFSs) and unresolvable subgrid scales (SGSs). The former are reconstructed, and the latter are modeled. Differing from traditional turbulence models, the reconstructed RSFS stress/fluxes can produce backscatter of turbulence kinetic energy (TKE) and, importantly, turbulence potential energy (TPE). The modeled backscatter reduces entrainment at the cloud top and, meanwhile, strengthens resolved turbulence through preserving TKE and TPE, resulting in a realistic boundary layer with an adequate amount of cloud water and vertically coupled turbulent eddies. Additional simulations are performed in the terra incognita, when the grid spacing of a simulation becomes comparable to the size of the most energetic eddies. With 20-m vertical and 1-km horizontal grid spacings, simulations using DRM provide a reasonable representation of bulk properties of the stratocumulus-capped boundary layer.