Approximate Deconvolution Model Research Articles

A comprehensive study of large eddy simulation models in capturing the dynamics of gas-particle turbulent suspensions

The particle phase attenuates the fluid fluctuations with an increase in volume fraction, and a sudden collapse in the turbulence is observed at a particular particle volume fraction, called critical particle volume loading (CPVL) [P. Muramulla et al. J. Fluid Mech. 889, A28 (2020)]. The present study reports the capability of two different classes of large eddy simulation (LES), viz. anisotropic and eddy viscosity-based, models to capture the turbulence modulation and the sudden disruption of the fluid fluctuations in the particle-laden vertical channel flows. The simulations are performed at two bulk Reynolds numbers of 3300 and 5600 based on the channel width and the bulk averaged fluid velocity. Our study on different LES models shows that approximate deconvolution (ADM) and scale similarity (SS) models accurately predict the critical loading for the Reynolds number of 3300. However, these models predict the critical loading qualitatively only for the Reynolds number of 5600 in the sense that they fail to predict the discontinuity as shown by the direct numerical simulation (DNS) study. The coherent structure model (CSM) predicts the critical loading with an 80% accuracy at both Reynolds numbers. The energy spectral density, production, and particle-induced dissipation spectra are plotted to analyze the distribution across wavenumbers. For all the LES models, a decrease in more than one order of magnitude is observed in the energy spectrum density at the critical loading compared to the unladen flow. The energy density decreases more in the channel center than in the near-wall region for the same particle volume loading. The mean component of particle-induced dissipation is almost two orders of magnitude larger than the particle dissipation spectra of fluctuating energy. The magnitude of streamwise and spanwise dissipation spectra of fluctuating components is higher in the near-wall region than the channel center. However, the magnitude of wall-normal dissipation spectra is higher in the channel center than near the wall region.

Validation of LES-C turbulence models

A new family of turbulence models, Large Eddy Simulation with Correction (LES-C) has been proposed in Labovsky (2020), that reduces the modeling error by treating an LES model as a defect solution and then correcting it on the same spatial mesh. Herein, we investigate numerically several LES-C models, that stem from popular LES approaches: Approximate Deconvolution Model (ADM), Leray-α, NS-α, and NS-ω. The resulting LES-C models ADC, Leray-α-C, NS-α-C and NS-ω-C are tested on the two-dimensional problems (flow past a circular object, flow past a step) and on the three-dimensional benchmark problem of turbulent channel flow. In all the numerical tests all LES-C models are shown to outperform their LES counterparts on coarse spatial meshes.

Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence

An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence. The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows. The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation using the gradient-based optimization. The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer. The VOMM model outperforms the dynamic Smagorinsky model, dynamic mixed model (DMM), and approximate deconvolution model in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times. In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios. These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.

Large eddy simulation on unstructured grids using explicit differential filtering: A case study of Taylor-Green vortex

The application of an approximate deconvolution model was extended for large eddy simulation (LES) on unstructured grids by further developing a three dimensional explicit differential filter. Decaying isotropic homogeneous turbulence at a Reynolds number 3,400 was modeled on both structured and unstructured grids. Results were compared to a reference direct numerical simulation and other results reported in the literature. A Taylor-Green vortex at a Reynolds number of 1,600 was also studied as it is a canonical problem for laminar to turbulent transitional flow. Investigative studies on the ADM order and under-relaxation coefficient, the degree of anisotropy of the grid, and the grid resolution were performed to benchmark the filter performance in conjunction with ADM. Finally, a large eddy simulation on a fully unstructured grid was performed to demonstrate the range of applications of the proposed filter. All numerical results showed strong LES numerical stability when the explicit differential filter was used.

Dynamic iterative approximate deconvolution model for large-eddy simulation of dense gas compressible turbulence

We study large-eddy simulation of compressible decaying isotropic turbulence of dense gas at initial turbulent Mach numbers of 0.4 and 0.8. The unclosed subgrid-scale (SGS) terms are approximated by the dynamic iterative approximate deconvolution (DIAD) model proposed by Yuan et al. [“Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence,” Phys. Fluids 33, 085125 (2021)], and compared with the dynamic Smagorinsky (DSM) model. In an a priori test, the correlation coefficients of the DIAD model for most SGS terms are larger than 0.98, and the relative errors are smaller than 0.2, except for the SGS internal energy flux. In an a posteriori test, the DIAD model can well predict the probability density functions (PDFs) of SGS terms involving thermodynamic variables. Moreover, the DIAD model shows greater advantages than the DSM model in predicting various statistics and structures of compressible turbulence of dense gas, including spectra of velocity and thermodynamic variables, PDFs of SGS kinetic energy flux, deviatoric SGS stress and normalized strain-rate tensor, and the instantaneous spatial structures of vorticity.

The effect of sub-filter scale dynamics in large eddy simulation of turbulence

We study the effect of sub-filter scale (SFS) dynamics on the accuracy of direct deconvolution models (DDM) in large eddy simulation of isotropic turbulence at different filter-to-grid ratios (FGR), by using several types of invertible filters including the Gaussian, Helmholtz I and II, Butterworth, Chebyshev I and II, Cauchy, Pao, and rapidly decaying filters. We show that the FGR is crucial in controlling errors to ensure an accurate prediction of SFS stresses. In the case of FGR of 1, the DDM models cannot accurately reconstruct SFS stress, since the effect of SFS dynamics on SFS stress is not properly resolved by the coarse grid. The prediction abilities of most DDM models are significantly improved at FGR of 2, giving rise to quite an accurate reconstruction of SFS stresses, except for the situation of Helmholtz I and II filters. All the DDM models give very accurate results at FGR of 4. Moreover, the DDM models are comprehensively compared against various traditional SFS models, including the velocity gradient model, dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and the approximate deconvolution model. In the a priori study, the correlation coefficients of SFS stress for the DDM are much larger than those of the traditional models. In the a posteriori study, DDM outperforms DSM and DMM models in the prediction of various velocity statistics and instantaneous flow structures. These results indicate that the DDM framework with an appropriate FGR has much potential in developing high-fidelity SFS models in the LES of turbulence.

Density-unweighted subgrid-scale models for large-eddy simulations of compressible turbulence

Density-unweighted methods in large-eddy simulations (LES) of turbulence have received little attention, and the modeling of unclosed terms using density-unweighted methods even less. We investigate the density-unweighted subgrid-scale (SGS) closure problem for LES of decaying compressible isotropic turbulence at initial turbulent Mach numbers 0.4 and 0.8. Compared to the LES with Favre (density-weighted) filtering, there are more unclosed SGS terms for density-unweighted LES, which can be reconstructed using different SGS models, including the gradient model (GM), approximate deconvolution model (ADM), dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and the dynamic iterative approximate deconvolution (DIAD) models proposed by Yuan et al. “Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence,” Phys. Fluids 33, 085125 (2021). We derive GM models suitable for density-unweighted methods. We also, for the first time, apply the DIAD model to investigate compressible turbulence. In the a priori tests, the correlation coefficients of the GM, ADM, and DIAD models are larger than 0.9. Particularly, the correlation coefficients of DIAD models exceed 0.98 and the relative errors are below 0.2, which is superior to that in other SGS models. In the a posteriori tests of the density-unweighted LES, the DIAD model shows great advantages over other SGS models (including GM, ADM, DSM, and DMM models) in predicting the various statistics and structures of compressible turbulence, including the velocity spectrum, probability density functions (PDFs) of SGS fluxes and the instantaneous spatial structures of SGS heat flux, SGS kinetic energy flux, and vorticity.

Approximate deconvolution models for a fluid-fluid interaction problem with high Reynolds numbers

We propose and investigate three approaches to a fluid-fluid interaction problem with a nonlinear rigid lid condition on the joint boundary when one or both flows are at a high Reynolds number. To do so, we try to combine the Approximate Deconvolution Model of turbulence with a partitioned method of [1], called the Geometric Averaging. The nonlinear interface condition poses an extra difficulty, as there are different approaches to modeling the Reynolds stresses on the joint boundary of the two flows. We investigate three such approaches; all three models are shown to be quantitatively similar when tested on a model with a manufactured solution. Two of them are proven to be unconditionally stable, and yet it is the third model that clearly outperforms the others in the most computationally appealing case when high Reynolds number flows are modeled on a coarse mesh with large filtering width.

Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction

Abstract We propose and investigate two regularization models for fluid flows at higher Reynolds numbers. Both models are based on the reduced ADM regularization (RADM). One model, which we call DC-RADM (deferred correction for reduced approximate deconvolution model), aims to improve the temporal accuracy of the RADM. The second model, denoted by RADC (reduced approximate deconvolution with correction), is created with a more systemic approach. We treat the RADM regularization as a defect in approximating the true solution of the Navier–Stokes equations (NSE) and then correct for this defect, using the defect correction algorithm. Thus, the resulting RADC model can be viewed as a first member of the class that we call “LESC-reduced”, where one starts with a regularization that resembles a Large Eddy Simulation turbulence model and then improves it with a defect correction technique. Both models are investigated theoretically and numerically, and the RADC is shown to outperform the DC-RADM model both in terms of convergence rates and in terms of the quality of the produced solution.

Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence

Dynamic iterative approximate deconvolution (DIAD) models with Galilean invariance are developed for subgrid-scale (SGS) stress in the large-eddy simulation (LES) of turbulence. The DIAD models recover the unfiltered variables using the filtered variables at neighboring points and iteratively update model coefficients without any a priori knowledge of direct numerical simulation (DNS) data. The a priori analysis indicates that the DIAD models reconstruct the unclosed SGS stress much better than the classical velocity gradient model and approximate deconvolution model with different filter scales ranging from viscous to inertial regions. We also propose a small-scale eddy viscosity (SSEV) model as an artificial dissipation to suppress the numerical instability based on a scale-similarity-based dynamic method without affecting large-scale flow structures. The SSEV model can predict a velocity spectrum very close to that of DNS data, similar to the traditional implicit large-eddy simulation. In the a posteriori testing, the SSEV-enhanced DIAD model is superior to the SSEV model, dynamic Smagorinsky model, and dynamic mixed model, which predicts a variety of statistics and instantaneous spatial structures of turbulence much closer to those of filtered DNS data without significantly increasing the computational cost. The types of explicit filters, local spatial averaging methods, and initial conditions do not significantly affect the accuracy of DIAD models. We further successfully apply DIAD models to the homogeneous shear turbulence. These results illustrate that the current SSEV-enhanced DIAD approach is promising in the development of advanced SGS models in the LES of turbulence.

Crank-Nicholson Scheme of the Zeroth-Order Approximate Deconvolution Model of Turbulence Based On a Mixed Formulation

This report presents a method with high spatial and temporal accuracy for estimating solutions of Navier-Stokes equations at high Reynolds number. It employs Crank-Nicolson time discretization along with the zeroth-order ap-proximate deconvolution model of turbulence to regularize the flow prob-lem; solves a deviation of the Navier Stokes equation instead. Both theoreti-cal and computational findings of this report illustrate that the model pro-duces a high order of accuracy and stability. Furthermore, measurements of the drag and lift coefficients of a benchmark problem verify the potential of the model in this kind of computations.

A structural subgrid-scale model for the collision-related statistics of inertial particles in large-eddy simulations of isotropic turbulent flows

In large-eddy simulations of particle-laden isotropic turbulent flows, the collision of inertial particles is strongly influenced by missing small-scale turbulence. In this paper, we apply the Kinematic Simulation with Approximate Deconvolution (KSAD) model to determine the contribution of small-scale turbulence to the motion of inertial particles and improve the prediction accuracy of the radial distribution function (RDF) and radial relative velocity (RRV), which are closely related to particle collisions. Different values of Stokes numbers (St), which are defined as the ratio of the particle response time to the Kolmogorov time scale, are considered. The KSAD model significantly improves the prediction accuracy of the RRV for all considered St. For the prediction of RDF, good agreement between the KSAD model and direct numerical simulations is only observed for large St, i.e., St ≥ 2.0. To explore the reason for the poor prediction of the KSAD model for small St, we compare the Eulerian statistics of the flow fields and the Lagrangian properties of the particles from different simulations and find the key reason is that the Gaussian turbulence generated in the kinematic simulation model is inadequate in recovering the vortex centrifugal effect of small-scale turbulence on the inertial particle clustering at small St.

Temporal large-eddy simulation based on direct deconvolution

In this work, we propose an approach for Temporal Large-Eddy Simulation (TLES) with direct deconvolution. In contrast to previous approaches such as the Temporal Approximate Deconvolution Model (TADM) by Pruett et al. [“A temporal approximate deconvolution model for large-eddy simulation,” Phys. Fluids 18, 028104 (2006)], the non-filtered velocity field is recovered using the differential form of the filter operation rather than from a truncated series expansion of the inverse filter operator. This direct deconvolution is used to obtain formal closure of an analytic evolution equation of the temporal residual-stress tensor. Thus, the Temporal Direct Deconvolution Model (TDDM) has advantages relative to the TADM in being both more accurate and requiring less computational effort. As for the TADM, a secondary regularization term based on selective frequency damping is employed. The TDDM was implemented in the spectral element code Nek5000 (Argonne National Laboratory, NEK5000 Version 17.0, 2019, https://nek5000.mcs.anl.gov) to simulate two canonical incompressible flows as three test cases: an a priori test case of Homogeneous Isotropic Turbulence (HIT) at Reλ = 50, a greatly coarsened a posteriori HIT case at Reλ = 190, and an a posteriori highly anisotropic turbulent channel flow at Reτ = 180. Analyses of the energy spectrum, the mean flow, the root-mean-square of the velocity fluctuations, and the Reynolds stresses are presented. The results demonstrate a significant improvement compared to no-model solutions regarding the mean flow in the turbulent channel and the energy spectrum in the HIT case, while the computational cost is reduced dramatically compared to the direct numerical simulation.

The exergy concept and compressible turbulence

Turbulence models facilitated by Kolmogorov’s theory play an important role for compressible flows. Typically the basis of these models is the power spectrum of the velocity $${\mathbf {u}}$$ or of the density-weighted velocity $${\mathbf {w}}\equiv \rho ^{1/3}{\mathbf {u}}$$ . While for incompressible flow the quantity turbulent kinetic energy characterises turbulent motions, from the thermodynamic point of view, due to fluctuations of the density and the temperature other kinds of energies play a role at the different scales in compressible turbulence. We generalise the power spectrum of the velocity $${\mathbf {u}}$$ from incompressible flows to compressible flows by introducing the exergy spectrum as an application of the exergy concept. Furthermore, we discuss the application of the concept of turbulent exergy to turbulence modelling and demonstrate this approach with a direct numerical simulation and a Large-Eddy-Simulation of homogeneous isotropic turbulence. The advantage of turbulence modelling based on turbulent exergy is shown on the example of the Approximate Deconvolution Model (ADM) where, at smallest scales for its newly introduced entropy formulation, more available energy is extracted from the flow, and this occurs in a more physical way than for the classical equation set of the model using the total energy.

A discontinuous Galerkin method for the Stolz–Adams approximate deconvolution model for turbulent flows

We consider the zeroth order model of the family of approximate deconvolution models of Stolz and Adams. We propose and analyze fully discrete schemes using discontinuous finite elements. Optimal error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is O(ReeRe), which is an improvement with respect to the classical continuous finite element method where the dependence is O(ReeRe3), Layton [1].

Two approaches to creating a turbulence model with increased temporal accuracy

When modeling a turbulent fluid flow, an Approximate Deconvolution Model (ADM) is sometimes chosen - in particular, due to the high order spatial accuracy. A method has been presented in [1], that demonstrates an approach to increase the temporal accuracy of the ADM, by combining it with the Deferred Correction method (DCM). The resulting model, DC-ADM, is at least second order accurate in both space and time, and theoretically an arbitrarily high order of accuracy is achievable, provided that enough computational resources are available. However, in some applications, especially when random input data are present, the efficiency of the ADM becomes at least as important, as its accuracy. In order to keep the second order accuracy, but reduce the computational time, we propose a modification of the DC-ADM: in the first, defect step, we replace the turbulence model with the computationally cheaper artificial viscosity approximation. Obviously, this defect step approximation loses accuracy, compared with that obtained in the DC-ADM method, but we will show that the accuracy is recovered in the correction step. The numerical tests demonstrate that the new approach is more efficient, if the compared algorithms are run sequentially: the results from the proposed method are obtained in 35% less time, than those obtained by DC-ADM. The tests also demonstrate that there is no loss of quality of the solutions, when switching from the DC-ADM to the new model. When parallelization is introduced, the methods are comparable in both accuracy and computational time.

Numerical simulation of turbulent gas–solid flow using an approximate deconvolution model

In our prior study (Schneiderbauer and Saeedipour, Phys. Fluids, 2018; 30:023301), an a-priori analysis on the spatially-filtered two-fluid model (TFM) was presented for turbulent gas–solid flows, where the unresolved terms were modeled by an approximate deconvolution model (ADM). With such an approach, an approximation of the unfiltered solution is obtained by repeated filtering allowing the determination of the RSFS (resolved sub-filter scales) contribution of unclosed terms of the filtered equations directly. In the present study, this ADM-TFM approach is implemented in an a-posteriori manner for the coarse grid simulation of a fluidized bed of Geldart type B particles and is verified against highly-resolved TFM simulations. Furthermore, different regularization methods, which account for the sub-filter scale (SFS) contribution, are presented and discussed with respect to their predictiveness and stability. It is shown that employing an appropriate rheological model for the solid phase considerably improves stability without removing too much energy from the resolved scales. Finally, the ADM-TFM predictions are in fairly good agreement with the fine grid reference case and do not show a notable grid-dependency. Compared to TFM simulations using the same grid resolution, the ADM-TFM approach does only require marginally more computational resources.

Investigation on turbulence in the vicinity of liquid-liquid interfaces – Large eddy simulation and PIV experiment

In this work, the turbulence in the vicinity of the liquid-liquid interfaces in an incompressible interfacial flow is investigated numerically and experimentally. A new large eddy simulation formulation is employed where all the subgrid scale (SGS) terms appearing in the filtered governing equations are closed by a volume of fluid-based Approximate Deconvolution Model (ADM-VOF) (Saeedipour et al., 2018). In this structural turbulence modelling approach, the SGS terms of stress tensor and surface tension in the Navier-Stokes equations, as well as the interface dynamics in the VOF transport equation, are mathematically reconstructed from the resolved scales. An oil-water benchmark experiment was designed for validation purposes where an oil layer is poured on the top of the water into which a submerged turbulent jet is injected and interacts with the oil–water interface. The interactions of the turbulence with the liquid-liquid interfaces are then studied by shadowgraphy visualization and time-resolved PIV measurement. Two sets of LES were performed using the ADM-VOF and a conventional LES model based on eddy-viscosity approach. The statistics of interfacial turbulence including velocity fluctuations and interface dynamics are analyzed and compared with the ones obtained from the experiment. The results reveal that in the interface vicinity the ADM-VOF approach, which reconstructs more turbulent scales and includes the SGS terms of turbulence-interface interactions, is of superior performance in predicting the energy spectra and statistics of turbulence. Far from the interface no particular difference between the LES models was observed. In addition, the temporal average of the interface motion as well as the evolution of the interfacial area is studied. While both models predict the averaged temporal evolution of the interface similarly, the ADM-VOF revealed its potentials for high fluctuating regions as it accounts for the effect of SGS term interface dynamics. This study provides a perspective for connecting the interfacial turbulence to metallurgical aspects of the continuous casting process.

Large eddy simulation of turbulent interfacial flows using Approximate Deconvolution Model

Large eddy simulation (LES) is currently widespread in turbulence modeling research. Although LES has reached a mature level in modelling single-phase turbulent flows even in industrial scales, the challenges with LES of turbulent interfacial flows are still remaining. The main issue arises in subgrid closure modelling where the small-scale physics of the flow, as well as the small-scale interfacial topological changes, must be accounted for in filtered governing equations. In this paper, the Approximate Deconvolution Model (ADM) (Stolz and Adams, 1999) is extended to the two-phase LES problems to model all the subgrid terms appearing in the filtered governing equations. Then, the ADM-VOF approach is developed comprehensively and implemented in the frame of C++ libraries of OpenFOAM. The ADM-VOF is then employed for an a posteriori LES on the phase inversion benchmark which represents a buoyancy-driven turbulent interfacial flow with several interfacial events such as coalescence and rupture. To investigate the performance of this approach, a series of quantitative comparisons with quasi-DNS data and conventional LES (i.e. with single-phase assumption) is conducted based on macroscopic characteristics of the flow including enstrophy and interfacial length. The results reveal that the structural ADM-VOF approach improves the prediction of macroscopic flow characteristics associated with unresolved contributions compared to the conventional LES. The dependency of ADM-VOF performance on the grid resolution is also investigated. It shows that ADM-VOF exhibits more potentials in predicting the interfacial scales on a finer grid. Additionally, our study unveils that the choice of functional methods for modeling relaxation term may not be extendable to the two-phase ADM.

A structural subgrid-scale model for relative dispersion in large-eddy simulation of isotropic turbulent flows by coupling kinematic simulation with approximate deconvolution method

A kinematic simulation with an approximate deconvolution (KSAD) hybrid model is proposed to predict the Lagrangian relative dispersion of fluid particles in a large eddy simulation (LES) of isotropic turbulent flows. In the model, a physical connection between the resolved and subgrid scales is established through the energy flux rate at the filter width scale. Due to the lack of subgrid-scale (SGS) turbulent structures and SGS model errors, the LES cannot accurately predict the two- and multi-point Lagrangian statistics of the fluid particles. To improve the predictive capability of the LES, we use an approximate deconvolution model to improve the resolved scales near the filter width and a kinematic simulation to recover the missing velocity fluctuations beneath the subgrid scales. To validate the proposed hybrid model, we compare the Lagrangian statistics of two- and four-particle dispersion with the corresponding results from the direct numerical simulation and the conventional LES. It is found that a significant improvement in the prediction of the Lagrangian statistics of fluid particles is achieved through the KSAD hybrid model. Furthermore, a parametric study regarding the wavenumbers and orientation wavevectors is conducted to reduce the computational cost. Good results can be obtained using a small number of wavenumber modes and orientation wavevectors. Thus, we can improve the prediction of the Lagrangian dispersion of fluid particles in the LES by applying the KSAD hybrid model at an acceptable computational cost.