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.     

Vorticity,backscatter and counter-gradient transport predictions using two-level simulation of turbulent flows

The two-level simulation (TLS) method evolves both the large-and the small-scale fields in a two-scale approach and has shown good predictive capabilities in both isotropic and wall-bounded high Reynolds number (Re) turbulent flows in the past. Sensitivity and ability of this modelling approach to predict fundamental features (such as backscatter, counter-gradient turbulent transport, small-scale vorticity, etc.) seen in high Re turbulent flows is assessed here by using two direct numerical simulation (DNS) datasets corresponding to a forced isotropic turbulence at Taylor’s microscale-based Reynolds number Re_λ ≈ 433 and a fully developed turbulent flow in a periodic channel at friction Reynolds number Re_τ ≈ 1000. It is shown that TLS captures the dynamics of local co-/counter-gradient transport and backscatter at the requisite scales of interest. These observations are further confirmed through a posteriori investigation of the flow in a periodic channel at Re_τ = 2000. The results reveal that the TLS method can capture both the large- and the small-scale flow physics in a consistent manner, and at a reduced overall cost when compared to the estimated DNS or wall-resolved LES cost.     

Large-eddy simulation of plane channel flow with Vreman's model

In this paper, large-eddy simulations of Vreman's model (VM) have been carried out to investigate its performances in a temporal transitional channel flow and in high Reynolds number turbulent channel flows. As a preliminary work, it is found that cubic root of the cell volume is the best choice of filter width for both VM and dynamic VM based on Germano identity (DVM), according to comparative studies and a-posteriori analyses at Re_τ = 590. VM and DVM are then used to simulate the temporal laminar–turbulent transitional channel flow, and the results turn out that VM and DVM are capable to simulate this temporal transient flow. In simulating high Reynolds number turbulent channel flows with a relatively coarse grid resolution, DVM itself shares the same weakness as the dynamic Smagorinsky model, while it can successfully predict the mean velocity profile and skin friction coefficient when it is coupled with the constrained large eddy simulation methodology. The coupling highly promotes the capability of Vreman's model, offering a new promising approach to simulate high Reynolds number wall-bounded turbulent flows.     

Large-eddy simulations of temporally accelerating turbulent channel flow

The unsteady turbulent channel flow subject to the temporal acceleration is considered in this study. Large-eddy simulations were performed to study the response of the turbulent flow to the temporal acceleration. The simulations were started with the fully developed turbulent channel flow at an initial Reynolds number of Re₀ = 3500 (based on the channel half-height and the bulk-mean velocity), and then a constant temporal acceleration was applied. During the acceleration, the Reynolds number of the channel flow increased linearly from the initial Reynolds number to the final Reynolds number of Re₁ = 22,600. The effect of grid resolution, domain size, time step size on the simulation results was assessed in a preliminary study using simulations of the accelerating turbulent flow as well as simulations of the steady turbulent channel flow at various Reynolds numbers. Simulation parameters were carefully chosen from the preliminary study to ascertain the accuracy of the simulation. From the accelerating turbulent flow simulations, the delays in the response of various flow properties to the temporal acceleration were measured. The distinctive features of the delays responsible for turbulence production, energy redistribution, and radial propagation were identified. Detailed turbulence statistics including the wall shear stress response during the acceleration were examined. The results reveal the changes in the near-wall structures during the acceleration. A self-sustaining mechanism of turbulence is proposed to explain the response of the turbulent flow to the temporal acceleration. Although the overall flow characteristics are similar between the channel and pipe flows, some differences were observed between the two flows.     

A one-equation turbulence model for recirculating flows

A one-equation turbulence model which relies on the turbulent kinetic energy transport equation has been developed to predict the flow properties of the recirculating flows. The turbulent eddy-viscosity coefficient is computed from a recalibrated Bradshaw’s assumption that the constant a₁ = 0.31 is recalibrated to a function based on a set of direct numerical simulation (DNS) data. The values of dissipation of turbulent kinetic energy consist of the near-wall part and isotropic part, and the isotropic part involves the von Karman length scale as the turbulent length scale. The performance of the new model is evaluated by the results from DNS for fully developed turbulence channel flow with a wide range of Reynolds numbers. However, the computed result of the recirculating flow at the separated bubble of NACA4412 demonstrates that an increase is needed on the turbulent dissipation, and this leads to an advanced tuning on the self-adjusted function. The improved model predicts better results in both the non-equilibrium and equilibrium flows, e.g. channel flows, backward-facing step flow and hump in a channel.     

On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

Understanding of the structure of turbulent flows at extreme Reynolds numbers (Re) is relevant because of several reasons: almost all turbulence theories are only valid in the high Re limit, and most turbulent flows of practical relevance are characterized by very high Re. Specific questions about wall-bounded turbulent flows at extreme Re concern the asymptotic laws of the mean velocity and turbulence statistics, their universality, the convergence of statistics towards their asymptotic profiles, and the overall physical flow organization. In extension of recent studies focusing on the mean flow at moderate and relatively high Re, the latter questions are addressed with respect to three canonical wall-bounded flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Main results reported here are the asymptotic logarithmic law for the mean velocity and corresponding scale-separation laws for bulk flow properties, the Reynolds shear stress, the turbulence production and turbulent viscosity. A scaling analysis indicates that the establishment of a self-similar turbulence state is the condition for the development of a strict logarithmic velocity profile. The resulting overall physical flow structure at extreme Re is discussed.     

On minimum aspect ratio for duct flow facilities and the role of side walls in generating secondary flows

To the surprise of some of our colleagues, we recently recommended aspect ratios of at least 24 (instead of accepted values over last few decades ranging from 5 to 12) to minimise effects of side walls in turbulent duct flow experiments, in order to approximate the two-dimensional channel flow. Here we compile available results from hydraulics and civil engineering literature, where this was already documented in the 1980s. This is of great importance due to the large amount of computational studies (mainly direct numerical simulations, DNSs) for spanwise-periodic turbulent channel flows, and the extreme complexity of constructing a fully developed duct flow facility with aspect ratio of 24 for high Reynolds numbers with adequate probe resolution. Results from this non-traditional literature for the turbulence community are compared to our recent database of DNS of turbulent duct flows with aspect ratios ranging from 1 to 18 at Re_{τ, c} values of 180 and 330, leading to very good agreement between their experimental and our computational results at these low Reynolds numbers. The DNS results also reveal the complexity of a multitude of streamwise vortical structures in addition to the secondary corner flows (which extend up to z ? 5h). These time-dependent and meandering streamwise structures are located at the core of the duct and scale with its half-height. Comparisons of these structures with the vortical motions found in spanwise-periodic channels reveal similitudes in their time-averages and the same rate of decay of their mean kinetic energy ～ T^{? 1}_A, with T_A being the averaging time. However, differences between the two flows are identified and ideas for their future analysis are proposed.     

Aspect ratio effects in turbulent duct flows studied through direct numerical simulation

Three-dimensional effects in turbulent duct flows, i.e., sidewall boundary layers and secondary motions, are studied by means of direct numerical simulation (DNS). The spectral element code Nek5000 is used to compute turbulent duct flows with aspect ratios 1–7 (at Re_{b, c} = 2800, Re_{τ, c} ? 180) and aspect ratio 1 (at Re_{b, c} = 5600, Re_{τ, c} ? 330), in streamwise-periodic boxes of length 25h. The total number of grid points ranges from 28 to 145 million, and the pressure gradient is adjusted iteratively in order to keep the same bulk Reynolds number in the centreplane with changing aspect ratio. Turbulence is initiated via a trip forcing active during the initial stages of the simulation, and the statistical convergence of the data is discussed both in terms of transient approach and averaging period. Spanwise variations in wall shear, mean-flow profiles, and turbulence statistics are analysed as a function of aspect ratio, and also compared with the spanwise-periodic channel (as idealisation of an infinite aspect ratio duct). The computations show good agreement with experimental measurements carried out in parallel at the Illinois Institute of Technology (IIT) in Chicago, and highlight the relevance of sidewall boundary layers and secondary vortices in the physics of the duct flow. The rich array of secondary vortices extending throughout the upper and lower walls of the duct, and their dependence on Reynolds number and aspect ratio, had not been reported in the literature before.     

A multi-scale simulation method for high Reynolds number wall-bounded turbulent flows

We present an assessment and enhancement of the hybrid two-level large-eddy simulation method (A.G. Gungor and S. Menon, A new two-scale model for large eddy simulation of wall-bounded flows, Prog. Aerosp. Sci. 46 (2010), pp. 28–45), a multi-scale formulation for simulation of high Reynolds number wall-bounded turbulent flows. The assessment of the method is performed by examining role of static and dynamic blending functions used to perform hybridisation of two-level simulation (K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows, J. Comput. Phys. 220 (2006), pp. 290–311; K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows. Part 2: Non-homogeneous flows, J. Comput. Phys. 222 (2007), pp. 673–701) and large-eddy simulation methods. The sensitivity of first- and second-order turbulence statistics to the type of blending functions is investigated by simulating a fully developed turbulent flow in a channel at a friction Reynolds number Re_τ = 395 and comparing the results with those obtained using a direct numerical simulation. The first-order statistics do not show any significant differences for different blending functions, but the second-order statistics show some minor differences. The dynamic evaluation of the hybrid region and the blending function is necessary for non-equilibrium and complex flows where use of a static blending function can lead to inaccurate results. We propose two criteria for the dynamic evaluation; first evaluates extent of the hybrid region based on the subgrid turbulent kinetic energy and the second estimates the blending function based on a characteristic length scale. The computational efficiency of the method is enhanced by incorporating a hybrid programming paradigm where a standard domain decomposition by the message-passing-interface library is combined with the open multi-processing based parallelisation. A further enhancement of the method is achieved by incorporating a closure model for the unclosed hybrid terms in the governing equations, which appear due to hybridisation of two-level- and large-eddy-simulation methods. The model is based on an order of magnitude approximation and a preliminary assessment of the model shows improvement of turbulence statistics when used to simulate turbulent flow in a periodic channel. The assessment and improvements to the multi-scale method make it more suitable for simulation of practical wall-bounded turbulent flows at higher Reynolds number than a conventional large-eddy simulation. This is demonstrated by simulating two representative cases; turbulent flow at high Reynolds number in a periodic channel and flow over a bump placed on the lower surface of a channel, where a relatively coarser computational grid is found to be sufficient for reasonably accurate results.     

On mean flow universality of turbulent wall flows. I. High Reynolds number flow analysis

ABSTRACT

The universality and mathematical physical structure of wall-bounded turbulent flows is a topic of discussions over many decades. There is no agreement about questions like what is the physical mean flow structure, how universal is it, and how universal are theoretical concepts for local and global flow variations. These questions are addressed by using latest direct numerical simulation (DNS) data at moderate Reynolds numbers Re and experimental data up to extreme Re. The mean flow structure is explained by analytical models for three canonical wall-bounded turbulent flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Thorough comparisons with DNS and experimental data provide support for the validity of models. Criteria for veritable physics derived from observations are suggested. It is shown that the models presented satisfy these criteria. A probabilistic interpretation of the mean flow structure shows that the physical constraints of equal entropies and equally likely mean velocity values in a region unaffected by boundary effects impose a universal log-law structure. The structure of wall-bounded turbulent flows is much more universal than previously expected. There is no discrepancy between local logarithmic velocity variations and global friction law and bulk velocity variations. Flow effects are limited to the minimum: the difference of having a bounded or unbounded domain, and the variation range of mean velocity values allowed by the geometry.     

Large eddy simulation of turbulent horizontal buoyant jets

Large eddy simulations (LESs) of turbulent horizontal buoyant jets are carried out using a high-order numerical method and Sigma subgrid-scale (SGS) eddy-viscosity model, for a number of different Reynolds (Re) and Richardson (Ri) numbers. Simulations at previous experimental flow conditions (Re = 3200, 24, 000 and Ri = 0, 0.01) are carried out first, and the results are found to be qualitatively and quantitatively similar to the experimental results, thus validating the numerical methodology. The effect of varying Ri (values 2×10^?4, 0.001, 0.005, and 0.01) and Re (3200 and 24, 000) is studied next. The presence of stable stratification on one side and unstable stratification on the other side of the jet centreline leads to an asymmetric development of horizontal buoyant jets. It is found that this asymmetry, the total radial spread and the vertical deflection are significantly affected by Ri, while Re affects only the radial asymmetry. The need for developing improved integral models, accounting for this asymmetry, is pointed out. Turbulent production and dissipation rates are investigated, and are found to be symmetric in the horizontal plane, but asymmetric in the mid-vertical plane. A previously proposed model, for correlation between the vertical component of the fluctuating scalar flux vector and the vertical cross-correlation component of the Reynolds tensor, is modified based on the current LES results. Instantaneous scalar and velocity fields are analysed to reveal the structure of horizontal buoyant jets. Similar to the developed turbulent jet, the flow close to the nozzle too is found to be markedly different in the stable and unstable stratification regions. Persistent coherent vortex rings are found in the stable stratification region, while intermittent breakdown of vortex rings into small-scale structures is observed in the unstable stratification region. Similarities and differences between the flow structures in the horizontal buoyant jet configuration and those in the jet in crossflow configuration are discussed. Finally, a dynamic mode decomposition analysis is carried out, which indicates that the flow in the unstable stratification region is more energetic and prone to instabilities, as compared to the flow in the stable stratification region.     

Fractal modelling of turbulent mixing

The aim of this work is to propose a new model for turbulent flows, called the fractal model (FM), applicable both in a Reynolds averaged Navier–Stokes (RANS) and a large-eddy simulation (LES) formulation, with the ultimate goal of applying it to simulate turbulent combustion irrelevant of its mode (premixed or non-premixed). The model is able to turn itself off in the laminar zones of the flow, and in particular near walls. It is based on the fractal theory. It describes the physics of the smaller spatial scales and therefore represents a small-scales model.

FM describes the physics of the small scales of turbulence based on the phenomenological concept of vortex cascade and on the self-similar behaviour of turbulence in the inertial range. Such a model is used in each cell of a numerical calculation. A characteristic length Δ is associated to each cell, and the local energy u ³ _Δ/Δ is distributed over a certain number of eddies, which depends on the local Reynolds number Re _Δ. Each vortex of the cascade generates N _c vortices; the recursive process of vortex generation terminates at the dissipative scale level, i.e. when the eddy Reynolds number is equal to one. FM is also able to estimate the volume fraction occupied by the dissipative fine structures of turbulence; this quantity is critical in reactive turbulent flows.

The physics of small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls. Assuming μ_t to be isotropic, FM is applicable in a RANS formulation (IFM, isotropic fractal model). The model can be extended to the anisotropic case (AFM, anisotropic fractal model) and therefore used to close the transport equations in an LES approach. In the present paper, the model (IFM) is used in a RANS approach and is validated through a test case studied experimentally by Johnson and Bennett, and numerically (with LES) by Akselvoll and Moin. The results obtained are in good agreement both with the experimental and the numerical ones. Other tests are being performed.     

Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residual-based variational multiscale method

We present an application of the residual-based variational multiscale turbulence modeling (RBVMS) methodology to the computation of turbulent Taylor–Couette flow at high Reynolds number. We show that the RBVMS formulation globally conserves angular momentum, a feature that is felt to be important for flows dominated by rotation, and that is not shared by standard stabilized formulations of fluid flow. Weak imposition of Dirichlet boundary conditions is employed to enhance the accuracy of the RBVMS framework in the presence of thin turbulent boundary layers near solid walls. Calculation of conservative boundary forces and torques is also presented for the case of weakly enforced boundary conditions. NURBS-based isogeometric analysis is employed for the spatial discretization, and mesh refinement is performed to assess the convergence characteristics of the proposed methodology. Numerical tests show that very accurate results are obtained on relatively coarse grids. To the best of the authors’ knowledge, this paper is the first to report large eddy simulation computations of this challenging test case.     

HEAT TRANSFER TO TWO-PHASE AIR–VISCOELASTIC FLUID FLOWS OVER A HOT CYLINDER

Over a range of 70 < Re _a < 9,600, 7 < Pr _a < 130, 0 < β < 0.12, and 0.7 < n < 1, circumferential wall temperatures for air–water and air–aqueous polymer (viscoelastic) solution flows over a horizontal cylinder were measured experimentally. The 2.5-cm-diameter and 7.5-cm-long cylinder was heated by passing direct electric current through it. The peripherally averaged heat transfer coefficient for relatively dilute viscoelastic–air solutions, at any fixed flow rate of liquid phase, increases with β. Such increase is more pronounced at lower flow rates of liquid phase. For relatively more elastic solutions, the two-phase heat transfer decreases with increasing β. Such reduction is more pronounced at higher flow rates of liquid phase. A new correlation is proposed for predicting the Nusselt number for air–viscoelastic fluid flows over a heated cylinder in cross flow.     

On the effect of an anisotropy-resolving subgrid-scale model on large eddy simulation predictions of turbulent open channel flow with wall roughness

A large eddy simulation (LES) was conducted of turbulent flow in a channel with a rough wall on one side and a free surface on the other by adopting an anisotropy-resolving subgrid-scale (SGS) model. A shear Reynolds number of Re_τ = 395 was used based on the mean friction velocity and channel height. To investigate the grid dependency of the LES results caused by the SGS model, three grid resolutions were tested under the same definition of a roughness shape by using the immersed boundary method. The results obtained were compared with direct numerical simulation data with and without the wall roughness and those without the extra anisotropic term. The primary focus was on how the present anisotropic SGS model with coarser grid resolutions can properly provide the effects of roughness on the mean velocity and turbulent stresses, leading to a considerable reduction of the computational cost of LES.     

Predictions of canonical wall-bounded turbulent flows via a modified k–ω equation

A major challenge in computation of engineering flows is to derive and improve turbulence models built on turbulence physics. Here, we present a physics-based modified k–ω equation for canonical wall-bounded turbulent flows (boundary layer, channel and pipe), predicting both mean velocity profile (MVP) and streamwise mean kinetic energy profile (SMKP) with high accuracy over a wide range of Reynolds number (Re). The result builds on a multi-layer quantification of wall flows, which allows a significant modification of the k–ω equation. Three innovations are introduced: first, an adjustment of the Karman constant to 0.45 is set for the overlap region with a logarithmic MVP; second, a wake parameter models the turbulent transport near the centreline; third, an anomalous dissipation factor represents the effect of a meso-layer in the overlap region. Then, a highly accurate (above 99%) prediction of MVPs is obtained in Princeton pipes, improving the original model prediction by up to 10%. Moreover, the entire SMKP, including the newly observed outer peak, is predicted. With a slight change of the wake parameter, the model also yields accurate predictions for channels and boundary layers.     

A Web services accessible database of turbulent channel flow and its use for testing a new integral wall model for LES

The output from a direct numerical simulation (DNS) of turbulent channel flow at Re_τ ≈ 1000 is used to construct a publicly and Web services accessible, spatio-temporal database for this flow. The simulated channel has a size of 8πh × 2h × 3πh, where h is the channel half-height. Data are stored at 2048 × 512 × 1536 spatial grid points for a total of 4000 time samples every 5 time steps of the DNS. These cover an entire channel flow-through time, i.e. the time it takes to traverse the entire channel length 8πh at the mean velocity of the bulk flow. Users can access the database through an interface that is based on the Web services model and perform numerical experiments on the slightly over 100 terabytes (TB) DNS data on their remote platforms, such as laptops or local desktops. Additional technical details about the pressure calculation, database interpolation, and differentiation tools are provided in several appendices. As a sample application of the channel flow database, we use it to conduct an a-priori test of a recently introduced integral wall model for large eddy simulation of wall-bounded turbulent flow. The results are compared with those of the equilibrium wall model, showing the strengths of the integral wall model as compared to the equilibrium model.     

Anisotropic Reynolds stress tensor representation in shear flows using DNS and experimental data

Tensorial decompositions and projections are used to study the performance of algebraic non-linear models and predict the anisotropy of the Reynolds stresses. Direct numerical simulation (DNS) data for plane channel flows at friction Reynolds number (Re_τ = 180, 395, 590, 1000), and for the boundary layer using both DNS (Re_τ = 359, 830, 1271) and experimental data (Re_τ = 2680, 3891, 4941, 7164) are used to build and evaluate the models. These data are projected into tensorial basis formed from the symmetric part of mean velocity gradient and non-persistence-of-straining tensor. Six models are proposed and their performances are investigated. The scalar coefficients for these six different levels of approximations of the Reynolds stress tensor are derived, and made dimensionless using the classical turbulent scales, the kinetic turbulent energy (κ) and its dissipation rate (ε). The dimensionless coefficients are then coupled with classical wall functions. One model is selected by comparing the predicted Reynolds stress components with experimental and DNS data, presenting a good prediction for the shear and normal Reynolds stresses.     

A deconvolution enhancement of the Navier–Stokes-αβ model

A deconvolution enhancement of the Navier–Stokes-αβ model for turbulent flow is introduced. The energy and energy-dissipation rate for the enhanced model are derived. It is also shown that the consistency error, relative to the Navier–Stokes equations, and the microscale of the enhanced model are less than those of the Navier–Stokes-αβ model. The proposed model is used to simulate the Taylor–Green vortex problem and results show a qualitatively improved representation of the mean-square vorticity when compared to the Navier–Stokes-αβ model. Numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three-dimensional turbulent flows with Re = 200 are used to explore the utility of the model. A benchmark problem of a two-dimensional channel flow over a step for Re = 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions.