Unresolved Scales Research Articles

Synchronizing spatio-temporal chaos with imperfect models: a stochastic surface growth picture.

We study the synchronization of two spatially extended dynamical systems where the models have imperfections. We show that the synchronization error across space can be visualized as a rough surface governed by the Kardar-Parisi-Zhang equation with both upper and lower bounding walls corresponding to nonlinearities and model discrepancies, respectively. Two types of model imperfections are considered: parameter mismatch and unresolved fast scales, finding in both cases the same qualitative results. The consistency between different setups and systems indicates that the results are generic for a wide family of spatially extended systems.

Subgrid-scale modeling for implicit large eddy simulation of compressible flows and shock-turbulence interaction

We derive and analyze a model for implicit Large Eddy Simulation (LES) of compressible flows that is applicable to a broad range of Mach numbers and particularly efficient for LES of shock-turbulence interaction. Following a holistic modeling philosophy, physically sound turbulence modeling and numerical modeling of unresolved subgrid scales (SGS) are fully merged, in a manner quite different from that of traditional implicit LES approaches. The implicit subgrid model is designed in such a way that asymptotic consistency with incompressible turbulence theory is maintained in the low Mach number limit. Compressibility effects are properly accounted for by a novel numerical flux function, which can capture strong shock waves in supersonic flows and also ensures an accurate representation of smooth waves and turbulence without excessive numerical dissipation. Simulations of shock-tube problems, Noh's three-dimensional implosion problem, large-scale forced and decaying three-dimensional homogeneous isotropic turbulence, supersonic turbulent boundary layer flows, and a Mach = 2.88 compression-expansion ramp flow demonstrate the good performance of the SGS model; across this range of flows, predictions are in excellent agreement with theory, direct numerical simulations, and experimental reference data. Results for implicit LES of canonical shock-turbulence interaction are compared with results of explicit LES using the dynamic Smagorinsky model. The analysis shows that details of the numerical method used for shock capturing clearly outweigh the effect of different turbulence modeling strategies in explicit and implicit LES. The implicit LES model recovers the ideal 2nd-order grid convergence of shock-capturing errors that has been predicted using Rapid Distortion Theory. The dynamic Smagorinsky model in conjunction with a hybrid method that combines sixth-order central differences with a seventh-order weighted essentially non-oscillatory scheme yields turbulence statistics that are very similar to the implicit LES results. However, while the explicit LES requires a tailored high-order low-dissipative numerical method that applies numerical dissipation only in shock normal direction, no such ad hoc adjustments are necessary with the proposed implicit LES method.

Tandem cylinder flow and noise predictions using a hybrid RANS/LES approach

The performance of a novel hybrid RANS/LES methodology for accurate flow and noise predictions of the NASA Tandem Cylinder Experiment is investigated. The proposed approach, the modified Flow Simulation Methodology (FSM), is based on scaling the turbulence viscosity and the turbulence kinetic energy dissipation rate with a damping function. This damping function consists of three individual components, a function based on the Kolmogorov length-scale ensuring correct behaviour in the direct numerical simulation (DNS) limit, a function ensuring that FSM provides the correct damping in large-eddy simulation (LES) mode, and a shielding function that forces the switch from Reynolds-Averaged Navier–Stokes (RANS) to LES to occur outside the boundary layer. The FSM is proposed for the kω-SST two-equation model (FSM-SST) and for an Explicit-Algebraic-Stress-Model (FSM-EASM), which is better suited to resolve anisotropy and non-equilibrium of the unresolved scales and the strain and rotation-rate dependent coefficients introduce a dynamic response of the model to the resolved flow field. Simulations are performed on a relatively coarse grid and the FSM data are compared with results obtained from the Scale-Adaptive-Simulation (SAS) and IDDES approaches. Acoustic predictions are obtained using an acoustic analogy approach based on Curle’s theory. The FSM-SST approach was found to predict the hydrodynamic field in very good agreement with reference data, whereas the FSM-EASM did not improve the predictions. The acoustic spectra predicted show good agreement with experimental results at various microphone positions, with some deficiencies in capturing the broadband noise levels at high Strouhal numbers.

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.

Large eddy simulations in 2030 and beyond.

Since its introduction, in the early 1970s, large eddy simulations (LES) have advanced considerably, and their application is transitioning from the academic environment to industry. Several landmark developments can be identified over the past 40 years, such as the wall-resolved simulations of wall-bounded flows, the development of advanced models for the unresolved scales that adapt to the local flow conditions and the hybridization of LES with the solution of the Reynolds-averaged Navier-Stokes equations. Thanks to these advancements, LES is now in widespread use in the academic community and is an option available in most commercial flow-solvers. This paper will try to predict what algorithmic and modelling advancements are needed to make it even more robust and inexpensive, and which areas show the most promise.

Characterizing velocity fluctuations in partially resolved turbulence simulations

Many current turbulence model simulations follow the accuracy-on-demand paradigm to partially resolve the flow field for achieving optimal compromise between accuracy and computational effort. In many such approaches, the sub-grid viscosity is modeled using kinetic energy and dissipation of unresolved scales. It is of scientific value and practical utility to characterize the resolved velocity fluctuations of such flow fields to: (i) establish if the computed fields exhibit the self-similarity and scaling properties consistent with physical turbulence and (ii) demonstrate if the flow field tends to the limit of fully-resolved turbulence in a prescribed and controlled manner. Toward this end, we begin by characterizing partially-resolved flow computations as direct numerical simulations of non-Newtonian fluids. This paradigm permits the extension of the Kolmogorov and finite Reynolds number scaling to the computed fields. The statistical behavior of the intermediate and smallest computed scales is then postulated as a function of the degree of resolution. Then it is demonstrated that the flow field in partially-averaged Navier-Stokes simulations behaves in accordance with the adapted Kolmogorov hypotheses in isotropic turbulence computations. Further, the statistics of sub-Kolmogorov fluctuations are demonstrated to be self-similar over a range of resolutions.

Simulation of laminar–turbulent transition in compressible Taylor–Green flow basing on quasi-gas dynamic equations

We report the results of numerical simulation of laminar–turbulent transition in the Taylor–Green vortex for viscous compressible gas flow basing on quasi-gas-dynamic (QGD) equations. Here the QGD system is obtained by a temporal averaging of the Navier–Stokes equations. The additional dissipative terms in QGD system serve to model the effects of the unresolved subgrid scales. Comparison with direct numerical simulation and large eddy simulation reference data demonstrates that QGD numerical algorithm provides a uniform and adequate simulation of both laminar and turbulent evolution of the vortex for Reynolds numbers from 100 up to 5000, including transition.

A Fast Inverse Algorithm Based on the Multigrid Technique for Cloud Tomography

Abstract A fast inverse algorithm based on the half-V cycle scheme (HV) of the multigrid technique is developed for cloud tomography. Fourier analysis shows that the slow convergence problem caused by the smoothing property of the iterative algorithm can be effectively alleviated in HV by performing iterations from the coarsest to the finest grid. In this way, the resolvable scales of information contained in observations can be retrieved efficiently on the coarser grid level and the unresolvable scales are left as errors on the finer grid level. Numerical simulations indicate that, compared with the previous algorithm without HV (NHV), HV can significantly reduce the runtime by 89%–96.9% while retaining a similar level of retrieval accuracy. For the currently feasible two-level flight scheme for a 20-km-wide target area, convergence can be accelerated from 407 s in NHV to 13 s in HV. This reduction in time would be multiplied several times if the target area were much wider; but then segmental retrieval would be required to avoid exceeding the time limit of cloud tomography. This improvement represents an important saving in terms of computing resources and ensures the real-time application of cloud tomography in a much wider range of fields.

Numerical Prediction of Jet Noise from Nozzle Using Seventh-Order Dissipative Compact Scheme Satisfying Geometric Conservation Law

Jet noise from nozzle is predicted by seventh-order hybrid cell-edge and cell-node dissipative compact scheme (HDCS-E8T7). Turbulent flow, which has the main responsibility for the noise radiation, is simulated by a new high-order implicit large eddy simulation (HILES) based on the HDCS-E8T7. The conception of HILES is similar to that of monotone integrated LES (MILES), i.e., truncation error of the discretization scheme itself is employed to model the effects of unresolved scales. The Mach number in this case is 0.75 and the jet Reynolds number is . Sound pressure levels (SPL) in far-field observer locations are evaluated by Ffowcs Williams-Hawkings (FW-H) integral method. Predicted near-field flow statistics and far-field SPL are both in good agreement with the corresponding experimental data.

A Modified Gas-Kinetic Scheme for Turbulent Flow

AbstractThe implementation of a turbulent gas-kinetic scheme into a finite-volume RANS solver is put forward, with two turbulent quantities, kinetic energy and dissipation, supplied by an allied turbulence model. This paper shows a number of numerical simulations of flow cases including an interaction between a shock wave and a turbulent boundary layer, where the shock-turbulent boundary layer is captured in a much more convincing way than it normally is by conventional schemes based on the Navier-Stokes equations. In the gas-kinetic scheme, the modeling of turbulence is part of the numerical scheme, which adjusts as a function of the ratio of resolved to unresolved scales of motion. In so doing, the turbulent stress tensor is not constrained into a linear relation with the strain rate. Instead it is modeled on the basis of the analogy between particles and eddies, without any assumptions on the type of turbulence or flow class. Conventional schemes lack multiscale mechanisms: the ratio of unresolved to resolved scales – very much like a degree of rarefaction – is not taken into account even if it may grow to non-negligible values in flow regions such as shocklayers. It is precisely in these flow regions, that the turbulent gas-kinetic scheme seems to provide more accurate predictions than conventional schemes.

Swirling flow in a tube with variably-shaped outlet orifices: An LES and VLES study

The swirling flow in a tube with the outlet designed in the form of an orifice nozzle with centered and eccentrical openings, investigated experimentally by Grundmann et al. (2012), was studied computationally by employing Large Eddy Simulation (LES) method and a Hybrid LES/RANS (Reynolds-Averaged Navier–Stokes) method. The latter method, denoted by VLES (Very Large Eddy Simulation) according to Speziale (1998), represents a variable resolution computational scheme enabling a seamless transition from RANS to the direct numerical solution of the Navier–Stokes equations (DNS) depending on the ratio of the turbulent viscosities associated with the unresolved scales corresponding to the LES cut-off and the ‘unsteady’ scales pertinent to the turbulent properties of the VLES residual motion, which varies within the flow domain. Before computing the swirling pipe configuration, the VLES model is interactively validated in the process of the model derivation in some generic flows featured by natural decay of the homogeneous isotropic turbulence and separation from a curved continuous surface. The background RANS model representing the basis of the VLES method is the eddy-viscosity-based ζ-f model proposed by Hanjalic et al. (2004). The inflowing swirl generated by two tangential inlets has the same intensity in all cases considered. However, the abrupt outlet cross-section contraction created by variably-shaped orifices causes strong modification of the flow within the tube resembling a three-layered structure characterized by an alternating axial velocity directions. Both LES and VLES methods, unlike the RANS method employing the same turbulence model, returned such a behavior in good agreement with experimental data.

Ensemble Kalman filters for dynamical systems with unresolved turbulence

Ensemble Kalman filters are developed for turbulent dynamical systems where the forecast model does not resolve all the active scales of motion. Coarse-resolution models are intended to predict the large-scale part of the true dynamics, but observations invariably include contributions from both the resolved large scales and the unresolved small scales. The error due to the contribution of unresolved scales to the observations, called ‘representation’ or ‘representativeness’ error, is often included as part of the observation error, in addition to the raw measurement error, when estimating the large-scale part of the system. It is here shown how stochastic superparameterization (a multiscale method for subgridscale parameterization) can be used to provide estimates of the statistics of the unresolved scales. In addition, a new framework is developed wherein small-scale statistics can be used to estimate both the resolved and unresolved components of the solution.The one-dimensional test problem from dispersive wave turbulence used here is computationally tractable yet is particularly difficult for filtering because of the non-Gaussian extreme event statistics and substantial small scale turbulence: a shallow energy spectrum proportional to k−5/6 (where k is the wavenumber) results in two-thirds of the climatological variance being carried by the unresolved small scales. Because the unresolved scales contain so much energy, filters that ignore the representation error fail utterly to provide meaningful estimates of the system state. Inclusion of a time-independent climatological estimate of the representation error in a standard framework leads to inaccurate estimates of the large-scale part of the signal; accurate estimates of the large scales are only achieved by using stochastic superparameterization to provide evolving, large-scale dependent predictions of the small-scale statistics. Again, because the unresolved scales contain so much energy, even an accurate estimate of the large-scale part of the system does not provide an accurate estimate of the true state. By providing simultaneous estimates of both the large- and small-scale parts of the solution, the new framework is able to provide accurate estimates of the true system state.

Aerosol nucleation in a turbulent jet using Large Eddy Simulations

This work presents a model for particle nucleation and transport in the context of Large Eddy Simulation. A turbulent Dibutyl-Phthalate-laden Nitrogen jet diffusing in atmospheric air is used for validation. The proposed nucleation model treats the process in a probabilistic manner where the frequency of events is determined from local equilibrium conditions. Two methodologies for the sub-grid influence on nucleation rates are implemented: the presumed β-PDF and the source expansion method. Good agreement is found with respect to the experimental results for particle concentrations. The differences between using mean and instantaneous values for the evaluation of the nucleation rate are shown. For the grid spacing used, the unresolved scales seem to have little influence on the calculated particle concentrations. It is concluded that the use of instantaneous nucleation rates is advantageous and therefore it is important to consider a particle coupling that allows for the full use of instantaneous values.

Impact of dynamic subgrid-scale modeling in variational multiscale large-eddy simulation of bluff-body flows

The effects of dynamic subgrid-scale (SGS) modeling are investigated in variational multiscale (VMS) large-eddy simulation (LES) simulations of bluff-body flows. The spatial discretization is based on a mixed finite-element/finite-volume formulation on unstructured grids. In the VMS approach used in this work, the separation between the largest and the smallest resolved scales is obtained through a variational projection operator and a finite-volume cell agglomeration. The dynamic and non-dynamic versions of Smagorinsky and wall-adapted local eddy-viscosity SGS models are used to account for the effects of the unresolved scales. In the VMS approach, these effects are only modeled in the smallest resolved scales. The VMS-LES and classical LES approaches, combined with the considered dynamic and non-dynamic SGS models, are applied to the simulation of the flow around a circular cylinder at Reynolds numbers 3,900 and 20,000 and to the flow around a square cylinder at Reynolds numbers 22,000 and 175,000.

Large eddy simulation requirements for the Richtmyer-Meshkov instability

The shock induced mixing of two gases separated by a perturbed interface is investigated through Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). In a simulation, physical dissipation of the velocity field and species mass fraction often compete with numerical dissipation arising from the errors of the numerical method. In a DNS, the computational mesh resolves all physical gradients of the flow and the relative effect of numerical dissipation is small. In LES, unresolved scales are present and numerical dissipation can have a large impact on the flow, depending on the computational mesh. A suite of simulations explores the space between these two extremes by studying the effects of grid resolution, Reynolds number, and numerical method on the mixing process. Results from a DNS are shown using two different codes that use a high- and low-order numerical method and show convergence in the temporal and spectral dependent quantities associated with mixing. Data from an unresolved, high Reynolds number LES are also presented and include a grid convergence study. A model for an effective viscosity is proposed which allows for an a posteriori analysis of the simulation database that is agnostic to the LES model, numerics, and the physical Reynolds number of the simulation. An analogous approximation for an effective species diffusivity is also presented. This framework is then used to estimate the effective Reynolds number and Schmidt number of future simulations, elucidate the impact of numerical dissipation on the mixing process for an arbitrary numerical method, and provide guidance for resolution requirements of future calculations in this flow regime.

Bayesian inversion for the filtered flow at the Earth's core‐mantle boundary

AbstractThe inverse problem of determining the flow at the Earth's core‐mantle boundary according to an outer core magnetic field and secular variation model has been investigated through a Bayesian formalism. To circumvent the issue arising from the truncated nature of the available fields, we combined two modeling methods. In the first step, we applied a filter on the magnetic field to isolate its large scales by reducing the energy contained in its small scales, we then derived the dynamical equation, referred as filtered frozen flux equation, describing the spatiotemporal evolution of the filtered part of the field. In the second step, we proposed a statistical parametrization of the filtered magnetic field in order to account for both its remaining unresolved scales and its large‐scale uncertainties. These two modeling techniques were then included in the Bayesian formulation of the inverse problem. To explore the complex posterior distribution of the velocity field resulting from this development, we numerically implemented an algorithm based on Markov chain Monte Carlo methods. After evaluating our approach on synthetic data and comparing it to previously introduced methods, we applied it to a magnetic field model derived from satellite data for the single epoch 2005.0. We could confirm the existence of specific features already observed in previous studies. In particular, we retrieved the planetary scale eccentric gyre characteristic of flow evaluated under the compressible quasi‐geostrophy assumption although this hypothesis was not considered in our study. In addition, through the sampling of the velocity field posterior distribution, we could evaluate the reliability, at any spatial location and at any scale, of the flow we calculated. The flow uncertainties we determined are nevertheless conditioned by the choice of the prior constraints we applied to the velocity field.

Nuclear coups: dynamics of black holes in galaxy mergers

We study the dynamical evolution of supermassive black holes (BHs) in merging galaxies on scales of hundreds of kpc to 10 pc, to identify the physical processes that aid or hinder the orbital decay of BHs. We present hydrodynamical simulations of galaxy mergers with a resolution of $\leq$20 pc, chosen to accurately track the motion of the nuclei and provide a realistic environment for the evolution of the BHs. We find that, during the late stages of the merger, tidal shocks inject energy in the nuclei, causing one or both nuclei to be disrupted and leaving their BH `naked', without any bound gas or stars. In many cases, the nucleus that is ultimately disrupted is that of the larger galaxy (`nuclear coup'), as star formation grows a denser nuclear cusp in the smaller galaxy. We supplement our simulations with an analytical estimate of the orbital-decay time required for the BHs to form a binary at unresolved scales, due to dynamical friction. We find that, when a nuclear coup occurs, the time-scale is much shorter than when the secondary's nucleus is disrupted, as the infalling BH is more massive, and it also finds itself in a denser stellar environment.

Turbulence characteristics in free-surface flow over two-dimensional dunes

The turbulent structure of open-channel flows over two-dimensional dunes is investigated numerically using large-eddy simulation (LES), in order to improve our understanding of the interaction between the dune-generated turbulence and the free surface dynamics. The filtered Navier–Stokes equations in the LES model have been discretised using the finite volume method, with a dynamic sub-grid model being employed for the unresolved scales of turbulence. The partial cell treatment has been implemented in a Cartesian grid form to deal with the dune topography. Both the volume of fluid method and rigid lid approach have been employed in the numerical framework to investigate the effects of the free surface treatment on the flow characteristics. The numerical model predicted mean flow velocities, turbulence intensities and Reynolds stresses have been compared with experimental measurements published in the literature, with a detailed analysis being undertaken to assess the accuracy of the model results and the effects of the free surface treatment on the velocity and turbulence predictions. The instantaneous flow structure has been investigated, with emphasis being focused on the free surface dynamics and coherent structures.

Estimating the effective Reynolds number in implicit large-eddy simulation

In implicit large-eddy simulation (ILES), energy-containing large scales are resolved, and physics capturing numerics are used to spatially filter out unresolved scales and to implicitly model subgrid scale effects. From an applied perspective, it is highly desirable to estimate a characteristic Reynolds number (Re)-and therefore a relevant effective viscosity-so that the impact of resolution on predicted flow quantities and their macroscopic convergence can usefully be characterized. We argue in favor of obtaining robust Re estimates away from the smallest scales of the simulated flow-where numerically controlled dissipation takes place and propose a theoretical basis and framework to determine such measures. ILES examples include forced turbulence as a steady flow case, the Taylor-Green vortex to address transition and decaying turbulence, and simulations of a laser-driven reshock experiment illustrating a fairly complex turbulence problem of current practical interest.

Variational Multiscale Analysis: The Fine-Scale Green's Function for Stochastic Partial Differential Equations

We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect of the unresolved fine scales through a model term that contains a fine-scale stochastic Green's function. For a natural choice of an “optimal” coarse-scale solution and $L^2$-orthogonal stochastic basis functions, we demonstrate that the fine-scale stochastic Green's function is intimately linked to its deterministic counterpart. In particular, (i) we demonstrate that whenever the deterministic fine-scale function vanishes, the stochastic fine-scale function satisfies a weaker and discrete notion of vanishing stochastic coefficients, and (ii) we derive an explicit formula for the fine-scale stochastic Green's function that only involves quantities needed to evaluate the fine-scale deterministic Green's function. We present numerical results that support our claims about the physical support of the stochastic fine-scale function and demonstrate the benefit of using the VMS method when the fine-scale Green's function is approximated by an easier to implement element Green's function.