Under-resolved Direct Numerical Simulations Research Articles

Characterization of the transitional boundary layer flow on a marine propeller blade through Under-resolved Direct Numerical Simulation

In this paper, the full boundary layer flow on a C-series marine propeller is investigated through Under-resolved Direct Numerical Simulation (U-DNS). The simulated Reynolds number of 600,000 corresponds to experimental conditions for which a lab-scale propeller was designed and tested at MARIN’s towing tank in the late 1970s. The operating conditions of the simulations are set by matching the experimental advance ratio of 0.73. The simulation uses the high-order spectral element code Nek5000 to solve the boundary layer flow accurately. A specific computational domain is set to simulate the flow in the near-wall region of one of the propeller blades. The results are validated using experimental data on the propeller’s performances and the boundary layer regimes obtained from previous paint test measurements. The detailed boundary layer flow analysis provides new insights into understanding the transition mechanisms on both sides of the blade. Results suggest that centrifugal instabilities occur at the blade’s surface, inducing laminar-to-turbulent transition. A new flow map of the boundary layer regime is then discussed. The laminar cross-flow vortice inception and breakdown are studied in detail to characterize the transition dynamics.

Numerical Investigation of the Inviscid Taylor-Green Vortex Using an Adaptive Filtering Method for a Modal Discontinuous Galerkin Method

Implicit Large Eddy Simulation and under-resolved Direct Numerical Simulation bypass the complexity and uncertainty of turbulence modelling by using the numerical dissipation of the scheme as a subgrid scale model. High-order methods allow for more accurate capturing of smaller scale structures but suffer from energy pile-up in the higher modes which leads to instability in under-resolved applications. This work presents a filtered modal Discontinuous Galerkin method which adaptively determines the filter strength, avoiding unnecessary degradation of accuracy while maintaining stability. The method is applied to the inviscid Taylor-Green Vortex, a challenging test case which exhibits under-resolved turbulence for which few published results exist. This goal of this work is to present the adaptive filtering method which achieves robustness and accuracy despite a low number of degrees of freedom, as well as to publish a quantity of relevant data for the inviscid TGV problem.

Design of a Direct numerical Simulation of flow and heat transfer in a T-junction

Several investigations have been undertaken to study the velocity and temperature fields associated with the thermal mixing between fluids, and resulting thermal striping in a T-junction. However, the available experimental databases are not sufficient to describe the involved physics in adequate detail, and, due to experimental limitations, accurate data on velocity and temperature fluctuations in regions close to the wall are not available. Computational Fluid Dynamics (CFD) can play an important role in predicting such complex flow features. However, predicting complex thermal fatigue phenomena is a challenge for the available momentum and heat flux turbulence models. Furthermore, such models need to be extensively validated.The aim of the present work is to design a reference numerical experiment for Direct Numerical Simulation (DNS) of a thermal fatigue scenario using Reynolds-Averaged Navier-Stokes (RANS) simulations. First, the feasibility of scaling down the Reynolds number from experimental cases to a computationally-feasible range is investigated. The junction corner shape is also modified to a slightly rounded corner, ensuring that the underlying fundamental physical phenomena of turbulence and thermal mixing flow features are preserved. Finally, the pipe lengths of the model were calibrated to ensure there would be no interference of the upstream developing region and the outlet boundary conditions on the thermal mixing at the junction. A sample under-resolved DNS case, with unity and low-Prandtl number passive temperature scalars, with iso-temperature, iso-flux and mixed (Robin) wall boundary conditions, are presented. This proof-of-concept simulation contributes to the finalization of the set-up for fully-resolved DNS with respect to the computational grid size selection and transient characteristics.

Editorial: Celebrating the contributions of Professor Guieppe Marrucci to rheology and non-Newtonian fluid mechanics

Direct numerical simulations (DNS) of generalized Newtonian fluids in a rectangular channel at a friction velocity of 180 are performed using OpenFOAM. The simulations are performed for a Newtonian fluid, a power-law fluid, and a Herschel-Bulkley (HB) fluid. Two-point correlations of the streamwise velocity reveal that a significant increase in the computational domain size is required to avoid statistical correlation due to the shear-thinning and yield stress. The analogous Kolmogorov scale shows that the dissipative structures may be orders of magnitude larger with these two non-Newtonian behaviors. This is demonstrated to be linked to the significant increase in the viscosity. The mean streamwise velocity field integrates to a larger bulk velocity with increased shear-thinning as compared with the Newtonian fluid flow. The addition of the yield stress does the same with respect to the power-law fluid flow. The difference in the velocity profiles is more pronounced in the outer layer. It, however, is observed that the rheology does not play a significant role in the outer layer for the mean streamwise velocity. It is shown that an HB fluid flow does not necessarily show an increase in the fluctuating streamwise velocity with respect to a power-law fluid flow. It is also shown that low power index and high Hedstrom number lead to strong turbulence anisotropy due to the weakening of turbulence. The turbulent kinetic energy (TKE) budget reveals that the studied non-Newtonian behavior mainly affects the inner layer of the flow. The yield stress is also seen to show small variations in the TKE budget in the outer layer. This is attributed to the weakening turbulence as a result of the increased viscosity near the center. It is also demonstrated that the studied HB fluid flow results can be successfully reproduced by under-resolved DNS.

Status, perspectives, and added value of high fidelity simulations for safety and design

With the increase in computational power, High Fidelity (HiFi) simulations become increasingly important to nuclear thermal hydraulics for fundamental understanding of turbulent flow and heat transfer phenomena and for further development and validation of engineering CFD models. We consider Direct Numerical Simulations (DNS) and Large Eddy Simulation (LES) as HiFi analyses. In this paper, we present an overview of single-phase and two-phase flow DNS simulations relevant for nuclear reactor safety and design purposes. In order to limit the scope, this overview has a focus on DNS simulations of fully turbulent non-reacting incompressible flows. LES approaches are briefly mentioned in certain areas, where LES-to-DNS upgrade is expected to happen in the near future.We first explain the concept of DNS and related concepts like Under-resolved DNS (UDNS) and quasi-DNS (q-DNS). Subsequently, we present the numerical methods which are generally used for HiFi simulations. A wider span of numerical methods can be observed in single-phase flow simulations than in two-phase flow simulations, where the later require additional algorithms for tracking of the interfaces between the two phases. Consequently, two-phase DNS simulations are not immune to modelling errors, and are much more complex than their single-phase counterparts. Next, an overview is given of single-phase flow DNS simulations. We start with basic cases like turbulent channel and pipe flows, which are mainly useful for development and validation of less detailed engineering CFD models. We continue with more recent and more complex cases like, e.g., flows in a rod bundle and a pebble bed, which are directly applicable in nuclear engineering. Next, an overview of two-phase flow HiFi simulations is presented in a similar fashion. The importance of the availability of the presented HiFi simulations is explained by addressing their role for further development and validation of engineering CFD models which are used for the eventual industrial applications. The importance of HiFi simulation data, complementary to experimental data, for uncertainty quantification and machine learning is explained. Finally, future challenges for HiFi simulations, especially for two-phase flow, are elucidated.

Large-eddy simulation of turbulent pipe flow of Herschel-Bulkley fluids - Assessing subgrid-scale models

This paper investigates the turbulent pipe flow of viscoplastic fluids using large-eddy simulation (LES) and the OpenFOAM package. The Herschel-Bulkley model is employed to describe the viscosity function with shear-thinning and viscoplastic behavior. Many LES studies evaluating different subgrid-scale (SGS) models for Newtonian fluids are available in the literature. However, studies regarding LES and SGS models involving non-Newtonian fluids are scarce, despite the importance of the subject. The present paper focuses on evaluating the performance of the Dynamic Smagorinsky and Wall Adapting Local Eddy-viscosity (WALE) models, and the under-resolved direct numerical simulation (UDNS) with viscoplastic fluids. The first- and second-order turbulence statistics and flow features, such as mean wall shear stress, mean wall viscosity, friction factor, mean shear rate, and mean viscosity profiles are used to assess the different models. Simulations are carried out at two different generalized Reynolds numbers (ReG) of 5000 and 7500. The numerical results computed with LES are compared with direct numerical simulation (DNS) results and experimental data for turbulent pipe flow of a Newtonian fluid to validate the numerical method. LES results for the Herschel-Bulkley fluid show a good agreement for the first- and second-order turbulence statistics compared with DNS data. Moreover, mean flow quantities obtained with LES follow the trend of DNS data.

Partially-averaged Navier–Stokes simulations of turbulence within a high-order flux reconstruction framework

High-order methods and hybrid turbulence models have independently shown promise as means of decreasing the computational cost of scale-resolving simulations. The objective of this work is to develop the combination of these methods and analyze the effects of high-order discretizations on hybrid turbulence models, particularly with respect the optimal model parameters and the relative accuracy benefits compared to approaches such as under-resolved direct numerical simulation (URDNS). We employ the Partially-averaged Navier–Stokes (PANS) approach using the flux reconstruction scheme on the flow around a periodic hill and the wake flow of a circular cylinder at a Reynolds number of 3900, the latter of which we provide direct numerical simulation results and novel statistical analysis. By increasing the order of the discretization while fixing the total degrees of freedom, it was observed that larger improvements in the prediction of the statistics and flow physics were generally seen with PANS than URDNS. Furthermore, less sensitivity to the resolution-control parameter was observed with a high-order discretization, indicating that high-order discretizations may be an effective approach for increasing the accuracy and reliability of hybrid turbulence models for scale-resolving simulations without a significant increase in computational effort.

Lattice-Boltzmann hydrodynamics of single-square-grid generated turbulence - a partial entropic stabilisation approach

Under-resolved direct numerical simulations of turbulence in the wake of a single-square-grid (SSG) are conducted using the lattice-Boltzmann method (LBM), with varying degrees of partial moment entropic stabilisation. Turbulence statistics along the grid centreline show satisfactory agreement with literature, greatly outperforming a control study employing the single-relaxation-time Smagorinsky LES model with equivalent resolution. The coarse-resolution entropic LBM simulations are able to capture the qualitative behaviour of the anisotropic and non-homogeneous​ turbulence generated by the SSG, indicating possible extension of the capability of the entropic lattice-Boltzmann method to simulate more exotic forms of turbulence beyond those reported by the majority of current studies. However, in the extreme near field of the single-square-grid, pseudo-compressibility effects, even at a low lattice Mach number, undermine LBM predictions for turbulence which is generated by action of the pressure Hessian — this is found to be an issue for all models studied, and not a numerical artefact of the entropic stabilisation. With regard to the action of the entropic stabiliser, a comparison of models employing differing degrees of stabilisation on higher-order natural moments suggest that the trace of the stress tensor should not be subject to entropic control for better accuracy in turbulence prediction.

High-Frequency Fluctuations in Post-stenotic Patient Specific Carotid Stenosis Fluid Dynamics: A Computational Fluid Dynamics Strategy Study

PurposeScreening of asymptomatic carotid stenoses is performed by auscultation of the carotid bruit, but the sensitivity is poor. Instead, it has been suggested to detect carotid bruit as neck’s skin vibrations. We here take a first step towards a computational fluid dynamics proof-of-concept study, and investigate the robustness of our numerical approach for capturing high-frequent fluctuations in the post-stenotic flow. The aim was to find an ideal solution strategy from a pragmatic point of view, balancing accuracy with computational cost comparing an under-resolved direct numerical simulation (DNS) approach vs. three common large eddy simulation (LES) models (static/dynamic Smagorinsky and Sigma).MethodWe found a reference solution by performing a spatial and temporal refinement study of a stenosed carotid bifurcation with constant flow rate. The reference solution left( {Delta x = 1.92 times 10^{ - 4} ;{text{m}},; Delta t = 5 times 10^{ - 5} ;{text{s}}} right) was compared against LES for both a constant and pulsatile flow.ResultsOnly the Sigma and Dynamic Smagorinsky models were able to replicate the flow field of the reference solution for a pulsatile simulation, however the computational cost of the Sigma model was lower. However, none of the sub-grid scale models were able to replicate the high-frequent flow in the peak-systolic constant flow rate simulations, which had a higher mean Reynolds number.ConclusionsThe Sigma model was the best combination between accuracy and cost for simulating the pulsatile post-stenotic flow field, whereas for the constant flow rate, the under-resolved DNS approach was better. These results can be used as a reference for future studies investigating high-frequent flow features.

A low-dissipation DG method for the under-resolved simulation of low Mach number turbulent flows

In recent years the use of high-order Discontinuous Galerkin (DG) methods for the under-resolved direct numerical simulations (uDNS) of turbulent flows has received special attention. The suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme: while the dispersion errors are small over a broad range of frequencies, a relevant dissipation error mainly acts at the smallest under-resolved scales, resembling a high frequency filter. Nevertheless, it was recognized (Flad and Gassner, 2017; Mengaldo et al., 2018) that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the uDNS approach. In this regard, the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided, in particular when dealing with low-speed flows, since this behaviour is exacerbated approaching the incompressibility limit. Fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods, see for example Weiss and Smith (1995). In this work a simple modification of the dissipation term of the low Mach preconditioned Roe scheme proposed by Weiss and Smith is considered. The aim is to reduce further the amount of numerical dissipation with the intent of improving the results of uDNS. A spatial DG discretization coupled with a linearly-implicit Rosenbrock-type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions. Results on canonical turbulent flow problems as the Taylor–Green vortex and the flow in a straight sided channel are presented. The improved accuracy of the proposed flux function is demonstrated. The new low-dissipation flux can be useful also in the context of standard, lower order, finite volume methods.

Pressure wave damping in transonic airfoil flow by means of micro vortex generators

In transonic airfoil flow, pressure waves are generated mainly at the trailing edge and in the case of a shock in the region of the shock/boundary layer interaction. Depending on the Mach number, these waves lead to oscillating shock waves and an unsteady pressure distribution. For a free steam Mach number of M=0.76 and a chord length based Reynolds number of Re=106, micro vortex generators (μVG) are applied to dampen pressure waves. This is studied experimentally in a shock tube and numerically by using a high-order finite difference scheme (under-resolved Direct Numerical Simulation). The agreement of the pressure distribution and Schlieren pictures between simulation and experiment is good. By means of numerical visualizations, instability waves are identified within the separated boundary layer above a marginal boundary layer separation bubble. The applicability of μVG for dampening the pressure waves and stabilizing the flow field is possible and is studied in this paper. By numerical Schlieren pictures and further visualizations, the flow around the VG is characterized. The spanwise oriented instability waves are partly disintegrated which is also confirmed by the analysis of the vorticity. Finally, the nonlinear wave propagation is investigated and an explanation for the typical 1 to 2 kHz pressure oscillation is given.

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

The study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods - specifically discontinuous Galerkin (DG) - via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.

Explicit and implicit LES closures for Burgers turbulence

In this work, we perform an aposteriori error analysis on implicit and explicit large eddy simulation closure models for solving the Burgers turbulence problem. Our closure modeling efforts include both functional and structural models equipped with various low-pass filters. We introduce discrete binomial smoothing filters and an enhanced version of the Van Cittert algorithm to accelerate the convergence of approximate deconvolution processes including regularization and relaxation filtering approaches. Our implicit modeling efforts consist of various high-order schemes including compact and non-compact fifth-order upwind schemes as well as weighted essential non-oscillatory (WENO) and compact reconstructed WENO (CRWENO) schemes, and the resulting schemes are shown to effectively converge to the direct numerical simulation (DNS) for increasing resolutions. Comparing with DNS and underresolved DNS computations, our numerical assessments illustrate the ability of these methods to capture the energy content near grid cut-off scale.

Performance of subgrid-scale models in coarse large eddy simulations of a laminar separation bubble

The flow over many blades and airfoils at moderate angles of attack and Reynolds numbers ranging from 104 to 105 undergoes separation due to the adverse pressure gradient generated by surface curvature. In many cases, the separated shear layer then transitions to turbulence and reattaches, closing off a recirculation region—the laminar separation bubble. An equivalent problem is formulated by imposing suitable boundary conditions for flow over a flat plate to avoid numerical and mesh generation issues. Recent work demonstrated that accurate large eddy simulation (LES) of such a flow is possible using only O(1%) of the direct numerical simulation (DNS) resolution but the performance of different subgrid-scale models could not be properly assessed because of the effects of unquantified numerical dissipation. LES of a laminar separation bubble flow over a flat plate is performed using a pseudo-spectral Navier-Stokes solver at resolutions corresponding to 3% and 1% of the chosen DNS benchmark by Spalart and Strelets (2000). The negligible numerical dissipation of the pseudo-spectral code allows an unambiguous assessment of the performance of subgrid-scale models. Three explicit subgrid-scale models—dynamic Smagorinsky, σ, and truncated Navier-Stokes (TNS)—are compared to a no-model simulation (under-resolved DNS) and evaluated against benchmark DNS data focusing on two quantities of critical importance to airfoil and blade designers: time-averaged pressure (Cp) and skin friction (Cf) predictions used in lift and drag calculations. Results obtained with explicit subgrid-scale models confirm that accurate LES of laminar separation bubble flows is attainable with as low as 1% of DNS resolution, and the poor performance of the no-model simulation underscores the necessity of subgrid-scale modeling in coarse LES with low numerical dissipation.

Influence of the spatial resolution on fine-scale features in DNS of turbulence generated by a single square grid

We focus in this paper on the effect of the resolution of direct numerical simulations (DNS) on the spatio-temporal development of the turbulence downstream of a single square grid. The aims of this study are to validate our numerical approach by comparing experimental and numerical one-point statistics downstream of a single square grid and then investigate how the resolution is impacting the dynamics of the flow. In particular, using the Q–R diagram, we focus on the interaction between the strain-rate and rotation tensors, the symmetric and skew-symmetric parts of the velocity gradient tensor,respectively. We first show good agreement between our simulations and hot-wire experiment for one-point statistics on the centreline of the single square grid. Then, by analysing the shape of the Q–R diagram for various streamwise locations, we evaluate the ability of under-resolved DNS to capture the main features of the turbulence downstream of the single square grid.

Analysis of intermittency in under-resolved smoothed-particle-hydrodynamics direct numerical simulations of forced compressible turbulence

We perform three-dimensional under-resolved direct numerical simulations of forced compressible turbulence using the smoothed particle hydrodynamics (SPH) method and investigate the Lagrangian intermittency of the resulting hydrodynamic fields. The analysis presented here is motivated by the presence of typical stretched tails in the probability density function (PDF) of the particle accelerations previously observed in two-dimensional SPH simulations of uniform shear flow [Ellero et al., Phys. Rev. E 82, 046702 (2010)]. In order to produce a stationary isotropic compressible turbulent state, the real-space stochastic forcing method proposed by Kida and Orszag is applied, and the statistics of particle quantities are evaluated. We validate our scheme by checking the behavior of the energy spectrum in the supersonic case where the expected Burgers-like scaling is obtained. By discretizing the continuum equations along fluid particle trajectories, the SPH method allows us to extract Lagrangian statistics in a straightforward fashion without the need for extra tracer particles. In particular, Lagrangian PDF of the density, particle accelerations as well as their Lagrangian structure functions and local scaling exponents are analyzed. The results for low-order statistics of Lagrangian intermittency in compressible turbulence demonstrate the implicit subparticle-scale modeling of the SPH discretization scheme.

Helical Turbulence Prevails over Inertial Waves in Forced Rotating Flows at High Reynolds and Low Rossby Numbers

Abstract A study of spectral laws for helical turbulence in the presence of solid body rotation up to Reynolds numbers Re ~ 1 × 105 and down to Rossby numbers Ro ~ 3 × 10−3 is presented. The forcing function is a fully helical flow that can also be viewed as mimicking the effect of atmospheric convective motions. Variants of a model developed previously by Baerenzung et al. are tested in the helical case against direct numerical simulation (DNS), using data from a run on a grid of 15363 points; its efficiency is also contrasted against a spectral large-eddy simulation (LES) by Chollet and Lesieur, as well as an underresolved DNS. The model including the contribution of helicity to the spectral eddy dissipation and eddy noise behaves best, allowing the recovery of statistical features of the flow. Even if the model is based on isotropic assumptions, the authors demonstrated in a previous study that the small scales of flows at moderate Rossby number can be considered to be isotropic in the range of parameters considered here and that therefore their model is appropriate to treat this kind of flow. An exploration of parameter space is then performed beyond what is feasible today using DNS. At a fixed Reynolds number, lowering the Rossby number leads to a regime of wave-mediated inertial helicity cascades to small scales. However, at a fixed Rossby number, increasing the Reynolds number leads the system to be dominated by turbulent energy exchanges where the role of inertial waves is to weaken the direct cascade of energy while strengthening the large scales. It is found that a useful parameter for partitioning the data is , with Urms, ν, and Ω being the rms velocity, the viscosity, and the rotation rate, respectively. The parameter that determines how much the energy cascade is direct or inverse—in which case the cascade to small scales is predominantly that of helicity—is linked to Ro.

On modelling transitional turbulent flows using under-resolved direct numerical simulations: the case of plane Couette flow

Direct numerical simulations have proven of inestimable help to our understanding of the transition to turbulence in wall-bounded flows. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable computing resources in domains of limited extent, the study of the decay of turbulence in conditions approaching those in the laboratory requires consideration of domains so wide as to exclude the recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we scrutinize the effects of a controlled lowering of the numerical resolution on the decay of turbulence in plane Couette flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds Rt and Rg where the bands appear and break down. The study is illustrated with the results of two preliminary experiments.

Temporally regularized direct numerical simulation

Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and Adams [S. Stolz, N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids II (1999) 1699–1701]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid-flow simulations. We prove existence of solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations → 0 in a precise sense. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier–Stokes equations.

Spectral modeling of rotating turbulent flows

A subgrid-scale spectral model of rotating turbulent flows is tested against direct numerical simulations (DNSs). The case of Taylor–Green forcing is considered, a configuration that mimics the flow between two counter-rotating disks as often used in the laboratory. Computations are performed for moderate rotation down to Rossby numbers of 0.03, as can be encountered in the Earth’s atmosphere. We provide several measures of the degree of anisotropy of the small scales and conclude that an isotropic model may suffice at moderate Rossby number. The model, developed previously [J. Baerenzung, H. Politano, Y. Ponty, and A. Pouquet, “Spectral modeling of turbulent flows and the role of helicity,” Phys. Rev. E 77, 046303 (2008)], incorporates eddy viscosity and eddy noise that depend dynamically on the index of the energy spectrum. We show that the model reproduces satisfactorily all large-scale properties of the DNS up to Reynolds numbers of ∼104 and for long times after the onset of the inverse cascade of energy; it is also shown to behave better than either the Chollet–Lesieur eddy viscosity model [J. P. Chollet and M. Lesieur, “Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures,” J. Atmos. Sci. 38, 2747 (1981)] or an under-resolved DNS.