Inhomogeneous Turbulent Flows Research Articles

Self-Energy Closure for Inhomogeneous Turbulent Flows and Subgrid Modeling

A new statistical dynamical closure theory for general inhomogeneous turbulent flows and subgrid modeling is presented. This Self-Energy (SE) closure represents all eddy interactions through nonlinear dissipation or forcing ‘self-energy’ terms in the mean-field, covariance and response function equations. This makes the renormalization of the bare dissipation and forcing, and the subgrid modeling problem, transparent. The SE closure generalizes the quasi-diagonal direct interaction closure to allow for more complex interactions. The SE closure is applicable to flows in different geometries, is exact near maximum entropy states corresponding to canonical equilibrium, and provides a framework for deriving simpler realizable closures.

HYBRID LARGE EDDY SIMULATION/LAGRANGIAN STOCHASTIC MODEL FOR TURBULENT PASSIVE AND REACTIVE SCALAR DISPERSION IN A PLANE JET

A hybrid large eddy simulation (LES)-Lagrangian model is tested by computing passive and reactive scalar dispersion in inhomogeneous turbulent flows. Fluid particles are tracked in a Lagrangian way. In order to obtain the subgrid scale velocity component of fluid particles, a Lagrangian stochastic subgrid model is coupled at the subgrid scale level with the Eulerian LES. An additional pairing exchange model is used to take into account molecular diffusion. A second-order irreversible chemical reaction is considered to take place within each fluid particle. Validation is achieved by comparing the mixture fraction that behaves as a passive scalar to a passive scalar simulation. The results of the LES/Lagrangian modeling are compared with experimental results and with pure Eulerian LES where a passive scalar transport equation is solved. The objective of the work is to test the model and its ability to reproduce inhomogeneous reactive cases.

Modeling molecular mixing in a spatially inhomogeneous turbulent flow

Simulations of spatially inhomogeneous turbulent mixing in decaying grid turbulence with a joint velocity–concentration probability density function (PDF) method were conducted. The inert mixing scenario involves three streams with different compositions. The mixing model of Meyer [“A new particle interaction mixing model for turbulent dispersion and turbulent reactive flows,” Phys. Fluids 22(3), 035103 (2010)], the interaction by exchange with the mean (IEM) model and its velocity-conditional variant, i.e., the IECM model, were applied. For reference, the direct numerical simulation data provided by Sawford and de Bruyn Kops [“Direct numerical simulation and lagrangian modeling of joint scalar statistics in ternary mixing,” Phys. Fluids 20(9), 095106 (2008)] was used. It was found that velocity conditioning is essential to obtain accurate concentration PDF predictions. Moreover, the model of Meyer provides significantly better results compared to the IECM model at comparable computational expense.

Modeling and simulation of inertial drop break-up in a turbulent pipe flow downstream of a restriction

This work deals with the modeling of drop break-up in an inhomogeneous turbulent flow that develops downstream of a concentric restriction in a pipe. The proposed approach consists in coupling Euler–Lagrange simulations of the drop motion to an interface deformation model. First the turbulent flow downstream of the restriction is solved by means of direct numerical simulation. Single drop trajectories are then calculated from the instantaneous force balance acting on the drop within the turbulent field (one-way coupling). Concurrently, the interface deformation is computed assuming the drop to behave as a Rayleigh–Lamb type oscillator forced by the turbulent stress along its trajectory. Criterion for break-up is based upon a critical value of drop deformation.This model has been tested against experimental data. The flow conditions and fluids properties have been chosen to match those experimental investigations. Both turbulent flow statistics and break-up probability calculations are in good agreement with experimental data, strengthening the relevance of this approach for modeling break-up in complex unsteady flow.

Statistical Dynamical Closures and Subgrid Modeling for Inhomogeneous QG and 3D Turbulence

Statistical dynamical closures for inhomogeneous turbulence described by multi‑field equations are derived based on renormalized perturbation theory. Generalizations of the computationally tractable quasi-diagonal direct interaction approximation for inhomogeneous barotropic turbulent flows over topography are developed. Statistical closures are also formulated for large eddy simulations including subgrid models that ensure the same large scale statistical behavior as higher resolution closures. The focus is on baroclinic quasigeostrophic and three-dimensional inhomogeneous turbulence although the framework is generally applicable to classical field theories with quadratic nonlinearity.

A statistical theory for inhomogeneous turbulent flow formulated in the mean-Lagrangian coordinates

An analytical way for investigating the inhomogeneous turbulent flow has been proposed by introducing Lagrangian-like view. The present method is developed based on TSDIA (Two-Scale Direct Interaction Approximation) which is one of the few theoretical works for the inhomogeneous turbulent flow formulated in the Eulerian frame work and has fundamental inconsistency with the general covariance under the coordinate transformation. The present work reached certain success in the following two aspects. First, the algebraic representation for the Reynolds stress derived by the present method has been proved to be generally covariant. Second the algebraic Reynolds stress is in much better agreement with the result of a DNS of the channel flow.

RANS modelling of the turbulent boundary layer under a solitary wave

The flow field in the boundary layer at the bottom of a solitary wave is computed by means of the Reynolds averaged Navier–Stokes equations. Reynolds stresses are determined by means of the two-equation turbulence model by Saffman (Saffman, P.G., 1970. A model for inhomogeneous turbulent flow. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 317 (1530), 417–433.) and Saffman and Wilcox (Saffman, P.G., Wilcox, D.C., 1974. Turbulence-model predictions for turbulent boundary layers. AIAA J. 12, 541–546.) which allows both the laminar and turbulent regime to be investigated along with the transition process. The results obtained for the smooth wall case show a fair agreement with the experimental measurements of Sumer et al. (Sumer, B.M., Jensen, P.M., Søerensen, L.B., Fredsøe, J., 2010. Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech. 646, 207–231.) and with the direct numerical simulations of Vittori and Blondeaux (Vittori, G., Blondeaux, P., 2008. Turbulent boundary layer under a solitary wave. J. Fluid Mech. 615, 433–443.). The model allows to investigate the rough wall case, too. Attention is focused on the bottom shear stress and on its maximum positive (in the direction of wave propagation) and negative (in the direction opposite to the direction of wave propagation) values.

On the probability distribution function of the velocity field and its derivative in multi-scale turbulence

Using a combination of active and passive grids, we produce an inhomogeneous turbulent flow with strongly non-Gaussian velocity probability distribution (density) functions (pdfs). This new, multi-scale flow shares features with the turbulent jet in freestream turbulence, the near field of turbulence-generating grids, and atmospheric flows. In particular, we study unskewed velocity distributions with both super- and sub-Gaussian tails and demonstrate the linear dependence of the normalized odd and even moments of the distributions on the first and second spatial derivatives of the turbulence r.m.s. fields, respectively. We also note bimodal behavior of the pdfs. The experiments demonstrate a clear effect of the flow organization and the large-scale intermittency on the velocity derivative pdfs and the small scales of the turbulence. The work is motivated by the complex wind fields, and associated, intermittent high stresses, encountered by wind turbines. We also draw comparisons to recent studies of multi-scale turbulence produced by fractal grids.

Lateral dispersion from a concentrated line source in turbulent channel flow

The dispersion of a passive scalar (temperature) from a concentrated line source in fully developed, high-aspect-ratio turbulent channel flow is studied herein. The line source is oriented in the direction of the inhomogeneity of the velocity field, resulting in a thermal plume that is statistically three-dimensional. This configuration is selected to investigate the lateral dispersion of a passive scalar in an inhomogeneous turbulent flow (i.e. dispersion in planes parallel to the channel walls). Measurements are recorded at six wall-normal distances (y/h= 0.10, 0.17, 0.33, 0.50, 0.67 and 1.0), six downstream positions (x/h= 4.0, 7.4, 10.8, 15.2, 18.6 and 22.0) and a Reynolds number ofRe≡ 〈U〉y=hh/v= 10200 (Reτ≡u∗h/v= 502). The lateral mean temperature excess profiles were found to be well represented by Gaussian distributions. The root-mean-square (r.m.s.) profiles, on the other hand, were symmetric, but non-Gaussian. Consistent with homogeneous flows (and in contrast to the work of Lavertu & Mydlarski (J. Fluid Mech., vol. 528, 2005, p. 135) studying transverse dispersion in the same flow), (i) the downstream growth rate of the centreline mean temperature excess, centreline r.m.s. temperature fluctuation and half-width of the mean and r.m.s. temperature profiles followed a power law evolution in the downstream direction, and (ii) the r.m.s. profiles evolved from single-peaked to double-peaked profiles far downstream. By comparing the measured ratios of the centreline r.m.s. temperature fluctuation to the mean temperature excess to the ratios measured in other flows, it was hypothesized that the mean-flow shear, as well as the turbulence intensity, played an important, cooperative role in increasing the mixedness of the flow. The probability density functions (PDFs) were quasi-Gaussian near the wall as well as for large-enough downstream distances. Closer to both the source and the channel centreline, the PDFs were better approximated by exponential distributions, with a sharp peak corresponding to the free-stream temperature. For intermediate downstream distances, the PDFs of the lateral dispersion were better mixed than analogous PDFs of the transverse dispersion, consistent with the mixedness measurements.

Simple models of turbulent flows

Stochastic Lagrangian models provide a simple and direct way to model turbulent flows and the processes that occur within them. This paper provides an introduction to this approach, aimed at the nonspecialist, and providing some historical perspective. Basic models for the Lagrangian velocity (i.e., the Langevin equation) and composition are described and applied to the simple but revealing case of dispersion from a line source in grid turbulence. With simple extensions, these models are applied to inhomogeneous turbulent reactive flows, where they form the core of probability density function (PDF) methods. The use of PDF methods is illustrated for the case of a lifted turbulent jet flame. Lagrangian time series are now accessible both from experiments and from direct numerical simulations, and this information is used to scrutinize and improve stochastic Lagrangian models. In particular, we describe refinements to account for the observed strong Reynolds-number effects including intermittency. It is emphasized that all models of turbulence are necessarily approximate and incomplete, and that simple models are valuable in many applications in spite of their limitations.

Application of statistical dynamical turbulence closures to data assimilation

We describe the development of an accurate yet computationally tractable statistical dynamical closure theory for general inhomogeneous turbulent flows, coined the quasi-diagonal direct interaction approximation closure (QDIA), and its application to problems in data assimilation. The QDIA provides prognostic equations for evolving mean fields, covariances and higher-order non-Gaussian terms, all of which are also required in the formulation of data assimilation schemes for nonlinear geophysical flows. The QDIA is a generalization of the class of direct interaction approximation theories, initially developed by Kraichnan (1959 J. Fluid Mech. 5 497) for isotropic turbulence, to fully inhomogeneous flows and has been further generalized to allow for both inhomogeneous and non-Gaussian initial conditions and long integrations. A regularization procedure or empirical vertex renormalization that ensures correct inertial range spectra is also described. The aim of this paper is to provide a coherent mathematical description of the QDIA turbulence closure and closure-based data assimilation scheme we have labeled the statistical dynamical Kalman filter. The mathematical formalism presented has been synthesized from recent works of the authors with some additional material and is presented in sufficient detail that the paper is of a pedagogical nature.

Asymmetries in an obstructed turbulent channel flow

The asymmetric flow pattern caused by a single thin-plate obstruction in a plane channel has been explored by means of direct numerical simulations. The blockage ratio was 1:2 and the bulk Reynolds number about 5700. In order to mimic an infinitely long channel section upstream of the obstruction, realistic dynamic inflow conditions were provided by a promising technique proposed by Barri et al. [“Inflow conditions for inhomogeneous turbulent flows,” Int. J. Numer. Methods Fluids 60, 227 (2009)]. The fluid downstream of the symmetric obstruction was sucked toward one side where a modestly long region of rather strong recirculating flow was observed. The weaker recirculation bubble formed at the opposite side was 17 times longer than the obstruction height and almost four times the size of the shorter bubble. The overall flow pattern turned out to be rather different from that observed in a similar study of channel flow subjected to periodically repeating obstructions by Makino et al. [“Turbulent structures and statistics in turbulent channel flow with two-dimensional slits,” Int. J. Heat Fluid Flow 29, 602 (2008)]. An anomalous variation of the pressure coefficient was observed with an excessively low pressure below the shorter of the bubbles. A locally high pressure occurred where the deflected jet flow impinges on the wall, whereas another pressure minimum could be associated with the flow acceleration caused by the severe blockage due to the major recirculation bubble. The turbulent fluctuations were suppressed due to the acceleration through the obstruction and high levels of streamwise velocity persisted far downstream. Exceptionally high turbulence levels were observed in the mixing-layers emanating from the two sides of the obstruction. The turbulence in these mixing-layers turned out to be qualitatively and quantitatively different on the two sides and exhibited distinctly different anisotropies.

High-order incompressible large-eddy simulation of fully inhomogeneous turbulent flows

The subgrid-scale (SGS) eddy-viscosity model developed by Vreman [Phys. Fluids 16 (2004) 3670] and its dynamic version [Phys. Fluids 19 (2007) 065110] are tested in large-eddy simulations (LES) of the turbulent flow in an Re = 12,000 lid-driven cubical cavity by comparison to the direct numerical simulation (DNS) data of Leriche and Gavrilakis [Phys. Fluids 12 (2000) 1363]. This appears to be the first test of this class of model to flows without any homogeneous flow directions, which is typical of flows in complex geometries. Additional LES predictions at Re = 18,000 and Re = 22,000 are compared to the DNS data of Leriche [J. Sci. Comp. 27 (2006)]. The new LES framework yielded excellent agreement for both the mean velocity and Reynolds stress profiles and matches DNS data better than the more traditional Smagorinsky-based SGS models.

A general inflow turbulence generator for large eddy simulation

This paper presents a general inflow turbulence generator for numerical simulation of a spatially correlated turbulent flow field. The novel inflow turbulence generator is developed based on the discretizing and synthesizing random flow generation (DSRFG) technique that is proved to be able to generate a fluctuating turbulent flow field satisfying any given spectrum. Then, the techniques of aligning and remapping are incorporated in the inflow turbulence generator for generation of an inhomogeneous and anisotropic turbulent flow field following arbitrary target spectra in three orthogonal directions. The performance of the present inflow turbulence generator is compared with that of Smirnov’s random flow generation (RFG) method. Detailed numerical studies show that the proposed inflow turbulence generator can strictly guarantee divergence-free condition without any additional improving step and synthetically generate inflows satisfying prescribed spatial anisotropy and correlation conditions. It is demonstrated through numerical examples that the inflow turbulence generator developed in this study is an effective tool for generation of a spatially correlated turbulent flow field for large eddy simulation (LES).

Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method

In this paper, we discuss the incorporation of dynamic subgrid scale (SGS) models in the lattice-Boltzmann method (LBM) for large-eddy simulation (LES) of turbulent flows. The use of a dynamic procedure, which involves sampling or test-filtering of super-grid turbulence dynamics and subsequent use of scale-invariance for two levels, circumvents the need for empiricism in determining the magnitude of the model coefficient of the SGS models. We employ the multiple relaxation times (MRT) formulation of LBM with a forcing term, which has improved physical fidelity and numerical stability achieved by proper separation of relaxation time scales of hydrodynamic and non-hydrodynamic modes, for simulation of the grid-filtered dynamics of large-eddies. The dynamic procedure is illustrated for use with the common Smagorinsky eddy-viscosity SGS model, and incorporated in the LBM kinetic approach through effective relaxation time scales. The strain rate tensor in the SGS model is locally computed by means of non-equilibrium moments of the MRT-LBM. We also discuss proper sampling techniques or test-filters that facilitate implementation of dynamic models in the LBM. For accommodating variable resolutions, we employ conservative, locally refined grids in this framework. As examples, we consider the canonical anisotropic and inhomogeneous turbulent flow problem, i.e. fully-developed turbulent channel flow at two different shear Reynolds numbers R e ∗ of 180 and 395. The approach is able to automatically and self-consistently compute the values of the Smagorinsky coefficient, C S . In particular, the computed value in the outer or bulk flow region, where turbulence is generally more isotropic, is about 0.155 (or the model coefficient C = C S 2 = 0.024 ) which is in good agreement with prior data. It is also shown that the model coefficient becomes smaller and approaches towards zero near walls, reflecting the dampening of turbulent length scales near walls. The computed turbulence statistics at these Reynolds numbers are also in good agreement with prior data. The paper also discusses a procedure for incorporation of more general scale-similarity based SGS stress models.

Anisotropic Turbulence and Coherent Structures in Eccentric Annular Channels

Eccentric annular pipe flows represent an ideal model for investigating inhomogeneous turbulent shear flows, where conditions of turbulence production and transport vary significantly within the cross-section. Moreover recent works have proven that in geometries characterized by the presence of a narrow gap, large-scale coherent structures are present. The eccentric annular channel represents, in the opinion of the present authors, the prototype of these geometries. The aim of the present work is to verify the capability of a numerical methodology to fully reproduce the main features of the flow field in this geometry, to verify and characterize the presence of large-scale coherent structures, to examine their behavior at different Reynolds numbers and eccentricities and to analyze the anisotropy associated to these structures. The numerical approach is based upon LES, boundary fitted coordinates and a fractional step algorithm. A dynamic Sub Grid Scale (SGS) model suited for this numerical environment has been implemented and tested. An additional interest of this work is therefore in the approach employed itself, considering it as a step into the development of an effective LES methodology for flows in complex channel geometries. Agreement with previous experimental and DNS results has been found good overall for the streamwise velocity, shear stress and the rms of the velocity components. The instantaneous flow field presented large-scale coherent structures in the streamwise direction at low Reynolds numbers, while these are absent or less dominant at higher Reynolds and low eccentricity. After Reynolds averaging is performed over a long integration time the existence of secondary flows in the cross session is proven. Their shape is found to be constant over the Reynolds range surveyed, and dependent on the geometric parameters. The effect of secondary flows on anisotropy is studied over an extensive Reynolds range through invariant analysis. Additional insight on the mechanics of turbulence in this geometry is obtained.

Inflow conditions for inhomogeneous turbulent flows

AbstractA cost‐effective method to generate inflow conditions for direct numerical simulations of wall‐bounded flows is presented. The method recycles a finite‐length time series of instantaneous velocity planes extracted from a precursor simulation and has earlier proved efficient for free shear layers. Now a spatially developing plane channel flow is considered. Different durations ts of the time series are tested and compared. Excellent agreement with fully developed channel flow statistics is observed when ts equals or exceeds the large‐eddy turnover time scale. The present results are more realistic than those obtained with synthetic turbulence generation and at the same time substantially cheaper than running an auxiliary simulation in parallel. Copyright © 2008 John Wiley & Sons, Ltd.

Analysing sludge balance in activated sludge systems with a novel mass transport model

In activated sludge systems the mechanically treated wastewater is biologically cleaned by biomass (activated sludge). The basic requirement of an efficient biological wastewater treatment is to have as a high biomass concentration in the biological reactor (BR) as possible. The activated sludge balance in activated sludge systems is controlled by the settling, thickening, scraper mechanism in the secondary settling tank (SST) and sludge returning. These processes aim at keeping maximum sludge mass in the BR and minimum sludge mass in the SST even in peak flow events (storm water flow). It can be, however, only reached by a high SST performance. The main physical processes and boundary conditions such as inhomogeneous turbulent flow, geometrical features of the SST, wastewater treatment plant (WWTP) load, return sludge flow, sludge volume index etc. all influence settling thickening and sludge returning. In the paper a novel mass transport model of an activated sludge system is presented which involves a 2-dimensional SST model coupled with a mixed reactor model of the biological reactor. It makes possible to investigate different sludge returning strategies and their influence on the sludge balance of the investigated activated sludge system, furthermore, the processes determining the flow and concentration patterns in the SST. The paper gives an overview on the first promising model results of a prevailing peak flow event investigation at the WWTP of Graz.

Analytical Equilibrium Swirling Inflow Conditions for Computational Fluid Dynamics

T HE specification of swirling inflow conditions is an integral and important part of computational fluid dynamics (CFD) of swirling flows, which are complex flows with many practical applications [1], for instance, in turbulent combustion [2]. To specify a swirling velocity profile with a desired swirl number in a traditional CFDanalysis based on aReynolds-averagedNavier–Stokes (RANS) modeling approach is not an easy task, which becomes even more difficult in advanced CFD approaches such as direct numerical simulation (DNS) or large-eddy simulation (LES). DNS or LES of spatially inhomogeneous turbulent flows requires turbulent inflow boundary conditions [3], which may be generated either from an auxiliary simulation, from a proper orthogonal decomposition and linear stochastic estimation [4], from a digital filter reproducing specified statistical data [5], or simply taken as the mean velocity profile with superimposed fluctuations [6], which is essentially a perturbed inflow. With the presence of swirling, inflow conditions are difficult to specify. For RANS-based CFD, an appropriate mean velocity profile is needed. For DNS and LES, apart from the mean velocity profile, information on an appropriate “turbulent” component is needed as well. Pierce and Moin [7] developed a method for generating equilibrium swirling inflow conditions, which represents a logical, idealized starting point for generating swirling velocity profiles. In their method, the swirling inflow is obtained numerically by solving for the flow driven by fictitious axial and azimuthal body forces, where the axial body force represents themean pressure gradient that drives the physical flow. The azimuthal body force is not physically producible and may be thought of as existing only to overcome drag from the walls. This technique was used in LES of a coaxial jet combustor and good agreement with the experimental data was observed [7]. The method can be regarded as an effective and simple means of generating realistic swirling inflow conditions. In this study, efforts have beenmade to derive an analytical formof the equilibrium swirling inflow conditions. The analytical swirling inflow is easy to implement, as it does not require solving for the flow numerically [7]. The particular forms of the analytical equilibrium swirling velocity profiles have been obtained for swirling annular and round jets, which can be directly applied in the specification of swirling inflow conditions for RANS approaches and further explored for the specification of swirling inflow conditions for DNS and LES, regardless of the turbulent component needed in the timedependent simulations. A desired swirl number at the inflow can also be conveniently achieved by adjusting a constant.

Dynamics of drop breakup in inhomogeneous turbulence at various volume fractions

We report experimental and numerical determinations of the breakup probability of a drop travelling through inhomogeneous turbulent flow generated in a pipe downstream of a restriction. The model couples the Rayleigh–Lamb theory of drop oscillations with the Kolmogorov–Hinze theory of turbulent breakup. The interface deformation is modelled by a linear oscillator forced by the Lagrangian turbulent Weber number measured in experiments. The interface is assumed to rupture when either (i) the instantaneous Weber number exceeds a critical value or (ii) the predicted deformation exceeds a given threshold. Seven flow configurations have been tested, corresponding to various Reynolds numbers, damping coefficients and drop volume fractions. The history of the drop deformation proves to play an important role, and simulations assuming a critical Weber number fail to reproduce the experiments. Simulations assuming a critical deformation predict well the main features observed in the experiments. The linear oscillator appears able to describe the main feature of the dynamics of the drop deformation in inhomogeneous turbulence. Provided the oscillation frequency and the damping rate are known, the model can be used to compute the breakup probability in concentrated dispersed two-phase flows.