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.     

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.     

Observation of vortex packets in direct numerical simulation of fully turbulent channel flow

A number of experimental studies have inferred the existence of packets of inclined, hairpinlike vortices in wall turbulence on the basis of observations made in two-dimensional x−y planes using visualization and particle image velocimetry (PIV). However, there are very few observations of hairpins in existing three-dimensional studies made using direct numerical simulation (DNS), and no such study claims to have revealed packets. We demonstrate, for the first time, the existence of hairpin vortex packets in DNS of turbulent flow. The vortex packet structure found in the present study at low Reynolds number,Re _t=300, is consistent with and substantiates the observations and the results from twodimensional PIV measurements at higher Reynolds numbers in channel, pipe and boundary layer flows. Thus, the evidence supports the view that vortex packets are a universal feature of wall turbulence, independent of effects due to boundary layer trips or critical conditions in the aforementioned numerical studies. Visualization of the DNS velocity field and vortices also shows the close association of hairpin packets with long low-momentum streaks and the regions of high Reynolds shear stress.     

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.     

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.     

Direct numerical simulation of the flow around a wall-mounted square cylinder under various inflow conditions

The flow around a wall-mounted square cylinder of side d is investigated by means of direct numerical simulation (DNS). The effect of inflow conditions is assessed by considering two different cases with matching momentum-thickness Reynolds numbers Re_θ ? 1000 at the obstacle: the first case is a fullyturbulent zero pressure gradient boundary layer, and the second one is a laminar boundary layer with prescribed Blasius inflow profile further upstream. An auxiliary simulation carried out with the pseudo-spectral Fourier–Chebyshev code SIMSON is used to obtain the turbulent time-dependent inflow conditions which are then fed into the main simulation where the actual flow around the cylinder is computed. This main simulation is performed, for both laminar and turbulent-inflows, with the spectral-element method code Nek5000. In both cases the wake is completely turbulent, and we find the same Strouhal number St ? 0.1, although the two wakes exhibit structural differences for x > 3d downstream of the cylinder. Transition to turbulence is observed in the laminar-inflow case, induced by the recirculation bubble produced upstream of the obstacle, and in the turbulent-inflow simulation the streamwise fluctuations modulate the horseshoe vortex. The wake obtained in our laminar-inflow case is in closer agreement with reference particle image velocimetry measurements of the same geometry, revealing that the experimental boundary layer was not fully turbulent in that dataset, and highlighting the usefulness of DNS to assess the quality of experimental inflow conditions.     

Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.     

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.     

Kolmogorov scaling of turbulent flow in the vicinity of the wall

How to scale even the simplest of turbulent flows continues to be a cause for considerable controversy. In the present research, a data base compiling results from channel flow direct numerical simulations and turbulent boundary layer experiments is employed to investigate the properties of shear and normal Reynolds stresses very close to the wall. Two types of scaling based on Kolmogorov length and velocity scales are analyzed. It is shown that it is highly likely that large length scales of the order of the channel half-width or the boundary layer thickness play an important role even in the innermost regions of wall-bounded turbulent flows, which hints at the persistence of Reynolds number effects in even high Reynolds number flows.     

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.     

Investigating asymptotic suction boundary layers using a one-dimensional stochastic turbulence model

The turbulent asymptotic suction boundary layer is studied using a one-dimensional turbulence (ODT) model. ODT is a fully resolved, unsteady stochastic simulation technique. While flow properties reside on a one-dimensional domain, turbulent advection is represented using mapping events whose occurrences are governed by a random process. Due to its reduced spatial dimensionality, ODT achieves major cost reductions compared to three-dimensional (3D) simulations. A comparison to recent direct numerical simulation (DNS) data at moderate Reynolds number (Re = u_∞ / v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively) suggests that the ODT model is capable of reproducing several velocity statistics, i.e. mean velocity and turbulent kinetic energy budgets, while peak turbulent stresses are under-estimated by ODT. Variation of the Reynolds number in the range Re ∈ [333,400,500,1000] shows that ODT can reproduce various trends observed as a result of increased suction in turbulent asymptotic suction boundary layers, i.e. the reduction of Reynolds stresses and enhanced skin friction. While up to Re = 500 our results can be directly compared to recent LES data, the simulation at Re = 1000 is currently not feasible through full 3D simulations, hence ODT may assist the design of future DNS or LES simulations at larger Reynolds numbers.     

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.     

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.     

Micro-machined super-miniature hot-film multi-array probe for field experiments with sub-Kolmogorov resolution

This is a report on the first successful experiments with a unique micro-hot-film multi-array probe. The main aim of this part of the project was to design, fabricate, implement and test in laboratory conditions a micro-hot-film multi-array probe for measurements in turbulent flows in field experiments with sub-Kolmogorov resolution. The key feature is that the probe is enabling to access, along with all three components of turbulent velocity fluctuations, also all nine components of the spatial velocity gradients tensor, including the option of obtaining the stream-wise velocity derivatives without employing the Taylor hypothesis, as well as velocity temporal derivatives. Hence the multi-array probe (typically five arrays) is built with each array consisting of four hot-film sensors, similarly to that of larger multi-array probe, based on arrays with four hot wires used in the atmospheric surface layer at Taylor micro-scale Reynolds number up to Re_λ ≈ 10⁴. This part of work relates to Reynolds numbers Re_λ < 500 and employs several laboratory flows including flow in a channel of rectangular cross section, in its bulk and in the boundary layers in the proximity of its smooth and rough walls, and in the confined jet of circular cross section.     

Characteristic boundary conditions for direct simulations of turbulent counterflow flames

Improved Navier–Stokes characteristic boundary conditions (NSCBC) are formulated for the direct numerical simulations (DNS) of laminar and turbulent counterflow flame configurations with a compressible flow formulation. The new boundary scheme properly accounts for multi-dimensional flow effects and provides nonreflecting inflow and outflow conditions that maintain the mean imposed velocity and pressure, while substantially eliminating spurious acoustic wave reflections. Applications to various counterflow configurations demonstrate that the proposed boundary conditions yield accurate and robust solutions over a wide range of flow and scalar variables, allowing high fidelity in detailed numerical studies of turbulent counterflow flames.     

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.     

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.     

Simulations of turbulent asymptotic suction boundary layers

A series of large-eddy simulations of a turbulent asymptotic suction boundary layer (TASBL) was performed in a periodic domain, on which uniform suction was applied over a flat plate. Three Reynolds numbers (defined as ratio of free-stream and suction velocity) of Re = 333, 400 and 500 and a variety of domain sizes were considered in temporal simulations in order to investigate the turbulence statistics, the importance of the computational domain size, the arising flow structures as well as temporal development length required to achieve the asymptotic state. The effect of these two important parameters was assessed in terms of their influence on integral quantities, mean velocity, Reynolds stresses, higher order statistics, amplitude modulation and spectral maps. While the near-wall region up to the buffer region appears to scale irrespective of Re and domain size, the parameters of the logarithmic law (i.e. von Kármán and additive coefficient) decrease with increasing Re, while the wake strength decreases with increasing spanwise domain size and vanishes entirely once the spanwise domain size exceeds approximately two boundary-layer thicknesses irrespective of Re. The wake strength also reduces with increasing simulation time. The asymptotic state of the TASBL is characterised by surprisingly large friction Reynolds numbers and inherits features of wall turbulence at numerically high Re. Compared to a turbulent boundary layer (TBL) or a channel flow without suction, the components of the Reynolds-stress tensor are overall reduced, but exhibit a logarithmic increase with decreasing suction rates, i.e. increasing Re. At the same time, the anisotropy is increased compared to canonical wall-bounded flows without suction. The reduced amplitudes in turbulence quantities are discussed in light of the amplitude modulation due to the weakened larger outer structures. The inner peak in the spectral maps is shifted to higher wavelength and the strength of the outer peak is much less than for TBLs. An additional spatial simulation was performed, in order to relate the simulation results to wind tunnel experiments, which – in accordance with the results from the temporal simulation – indicate that a truly TASBL is practically impossible to realise in a wind tunnel. Our unique data set agrees qualitatively with existing literature results for both numerical and experimental studies, and at the same time sheds light on the fact why the asymptotic state could not be established in a wind tunnel experiment, viz. because experimental studies resemble our simulation results from too small simulation boxes or insufficient development times.     

Direct numerical simulations of second-order Stokes wave driven smooth-walled oscillatory channel: investigation of net current formation

In wall-bounded time-periodic flows, nonlinearity, associated with higher harmonic term(s) in velocity and/or acceleration outside the boundary layer, can significantly change the wall turbulence compared with that in the linear Stokes Boundary Layer. A significant feature of a nonlinear wall-bounded turbulent time-periodic flow is the formation of a net current which has not yet been mechanistically explained. This study investigates the effects of asymmetric velocity outside the boundary layer on wall turbulence and net current formation through Direct Numerical Simulations of a smooth-walled planar channel driven by the Second-order Stokes Wave. Simulation results suggest that net current characteristics depend on whether developed turbulence is present. When turbulence is developed, asymmetric viscous length scale is found to be the primary reason of the net current whereby a vertical offset between negative and positive Reynolds shear stress profiles, associated with forward and reverse flows, respectively, is created in a cycle. After averaging over a cycle, residual Reynolds shear stress, which drives the net current, is observed to be within the offset layer.