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.     

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.     

横向粗糙元壁面槽道湍流的大涡模拟研究

采用大涡模拟和浸没边界法相结合对不同高度和不同间距横向粗糙元壁面槽道湍流进行了模拟,得到了光滑壁面和粗糙壁面湍流的流向平均速度分布,雷诺剪切应力,脉动速度均方根和近壁区拟序结构。结果发现横向粗糙元降低了流向平均速度,增大了流动阻力,粗糙壁面湍流的雷诺剪切应力大于光滑壁面。粗糙元降低了流向脉动速度,增强了展向和法向脉动速度。粗糙元高度越高,对湍流流动影响越大,而粗糙元间距对湍流统计特性的影响不大。粗糙壁面仍然存在着和光滑壁面类似的条带结构。     

Measurements of Reynolds stress and turbulence in the boundary plasma of the HT-7 tokamak

Measurements of electric field fluctuations, Reynolds stress and poloidal flow have been performed in the boundary region of the HT-7 tokamak using a Langmuir probe array.Sheared radial electric field and poloidal flow have been found in the vicinity of the limiter and the turbulence has been clearly modified in this region. Furthermore,the electrostatic Reynolds stress component shows a radial gradient close to the velocity shear layer location.All results here indicate that the radial gradient of Reynolds stress may play an important role in the driving of poloidal flows in the plasma boundary region.     

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.     

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.     

A mixed-time-scale low-Reynolds-number one-equation turbulence model

In this paper, a new low-Reynolds-number (LRN) one-equation turbulence model for eddy viscosity is proposed. A mixed time scale, representing a combination of three time scales: two time scales made of strain-rate parameter S and vorticity parameter Ω and the turbulent time scale k/?, is introduced into this model. The proposed model is derived from an LRN k?? two-equation model where the mixed time scale has been proved to be very effective for predicting local flows over complex terrains. In the transport equation of the model, the mixed time scale is included in the production and the dissipation terms. The new model is evaluated in channel flows at various Reynolds numbers, boundary layer flows with or without pressure gradient and backward-facing step flows with different expansion ratios and Reynolds numbers. Then the grid convergence of the model is investigated. Finally, the model performance for different values of the weighting constant C_s in the mixed time scale is assessed. The results show that the proposed model reproduces the correct wall-limiting behaviour of turbulent quantities and performs well in the near-wall region of turbulent flows. The model could be expected to be adopted in hybrid Reynolds averaged Navier–Stokes/large eddy simulation methodology for complex wall-bounded flows at high Reynolds numbers.     

Velocity,temperature, and Reynolds-stress scaling in the wall region of turbulent boundary layer on a permeable surface

A finite total number of flow parameters in the wall region of a turbulent boundary layer points to universal behavior of turbulent shear stress as a function of mean-velocity gradient and turbulent heat flux as a function of both mean-velocity and mean-temperature gradients. Combined with dimensional arguments, this fact is used to reduce the momentum and heat equations to first-order ordinary differential equations for temperature and velocity profiles amenable to general analysis. Scaling laws for velocity and temperature in boundary layer flows with transpiration are obtained as generalizations of well-known logarithmic laws. Scaling relations are also established for shear stress and rms transverse velocity fluctuation. The proposed method has substantial advantages as compared to the classical approach (which does not rely on fluid-dynamics equations [1–3]). It can be applied to establish scaling laws for a broader class of near-wall turbulence problems without invoking closure hypotheses.     

Experimental measurement of wall shear stress in strongly disrupted flows

Mean and fluctuating wall shear stress is measured in strongly disrupted cases generated by various low-porosity wall-mounted single- and multi-scale fences. These grids generate a highly turbulent wake which interacts with the wall-bounded flow modifying the wall shear stress properties. Measurement methods are validated first against a naturally growing zero pressure gradient turbulent boundary layer showing accuracies of 1% and 4% for extrapolation and direct measurement of the mean shear stress respectively. Uncertainty associated with the root mean square level of the fluctuations is better than 2% making it possible to measure small variations originating from the different fences. Additionally, probability density functions and spectra are also measured providing further insight into the flow physics. Measurement of shear stress in the disrupted cases (grid+TBL) suggest that the flow characteristics and turbulence mechanisms remain unaltered far from the grid even in the most disrupted cases. However, a different root mean square level of the fluctuations is found for different grids. Study of the probability density functions seem to imply that there are different degrees of interaction between the inner and outer regions of the flow.     

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.     

Analytical model of the time developing turbulent boundary layer

An analytical model for the time-developing turbulent boundary layer (TD TBL) over a flat plate is presented. The model provides explicit formulae for the temporal behavior of the wall-shear stress and both the temporal and spatial distributions of the mean streamwise velocity, the turbulence kinetic energy and Reynolds shear stress. The resulting profiles are in good agreement with the DNS results of spatially-developing turbulent boundary layers at momentum thickness Reynolds numbers equal to 1430 and 2900 [5–7]. Our analytical model is, to the best of our knowledge, the first of its kind for TD TBL. The text was submitted by the authors in English.     

Homoclinic tangle on the edge of shear turbulence

Experiments and simulations lend mounting evidence for the edge state hypothesis on subcritical transition to turbulence, which asserts that simple states of fluid motion mediate between laminar and turbulent shear flow as their stable manifolds separate the two in state space. In this Letter we describe flows homoclinic to a time-periodic edge state that display the essential properties of turbulent bursting. During a burst, vortical structures and the associated energy dissipation are highly localized near the wall, in contrast with the familiar regeneration cycle.     

Strengthened opposition control for skin-friction reduction in wall-bounded turbulent flows

An opposition control scheme with strengthened control input is proposed and tested in turbulent channel flows at friction Reynolds number Re_τ = 180 by direct numerical simulations. When the detection plane is located at less than 20 wall units, the drag reduction rate can be greatly enhanced by increasing the control amplitude parameter. The maximum drag reduction rate achieved in the present study is around 33%, which is much higher than the best value of 25% reported in literature. The strengthened control can be more efficient to attain a given drag reduction rate. Based on the total shear stress at the virtual wall established between the real wall and the detection plane by the control, a new friction velocity is proposed and the corresponding coordinate transform is made. Scaled by the proposed friction velocity, the wall-normal velocity fluctuation and the Reynolds shear stress of the controlled flows are collapsed well with those of the uncontrolled flow in the new coordinate. Based on the similarity, a relation between drag reduction rate and the effectiveness of the virtual wall is deduced, which disclosed that the elevation and residual Reynolds shear stress at the virtual wall are the key parameters to determine the drag reduction rate. The conclusion are also validated at Re_τ = 395 and 590. The decrease of the drag reduction rate with the increase of the Reynolds number is attributed to the enhanced residual Reynolds shear stress at the virtual wall.     

An explicit algebraic model of the planetary boundary layer turbulence: test computation of the neutrally stratified atmospheric boundary layer

An explicit algebraic model of Reynolds stresses and the turbulent heat flux vector for the planetary boundary layer in a neutrally stratified boundary layer of the atmosphere above a homogeneous rough surface is tested. The version of the algebraic model under consideration is constructed on the physical principles of the RANS (Reynolds-averaged Navier?Stokes) approximation for describing stratified turbulence, it employs three forecasting equations, and a correct reproduction of the main characteristics of a neutral atmospheric boundary layer — the components of the mean wind velocity, the wind turn angle, and the turbulent statistics is shown. Test computations show that the proposed model may be used for goal-oriented investigations of the atmospheric boundary layer.     

Surface roughness effects on the turbulence statistics in a low Reynolds number channel flow

A high-resolution particle image velocimetry was used to characterize a low Reynolds number turbulent flow in a channel. Experiments were conducted over a sand grain-coated surface of large relative roughness, and the results were compared with measurements over a smooth surface. The roughness perturbation significantly modified the outer layer. Even though the streamwise Reynolds stress shows less sensitivity in the outer layer to the boundary condition, significant enhancements were observed in the wall-normal Reynolds stress and the Reynolds shear stress. These modifications were considered as footprints of the larger-scale eddies transporting intense wall-normal motions away from the rough wall. A quadrant decomposition shows that strong and more frequent ejections are responsible for the larger values of the mean Reynolds shear stress over the rough wall. The results also indicate that spanwise vortex cores with mean vorticity of the same sign as the mean shear are the dominant smaller-scale vortical structures over the smooth and rough walls. A linear stochastic estimation-based analysis shows that the average larger-scale structure associated with these vortices is a shear layer that strongly connects the outer layer flow to the near-wall flow. A proper orthogonal decomposition of the flow suggests that the large-scale eddy is more energetic for the rough wall, and contributes more significantly to the resolved turbulent kinetic energy and the Reynolds shear stress than the smooth wall.     

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

The localized artificial diffusivity method is investigated in the context of large-eddy simulation of compressible turbulent flows. The performance of different artificial bulk viscosity models are evaluated through detailed results from the evolution of decaying compressible isotropic turbulence with eddy shocklets and supersonic turbulent boundary layer. Effects of subgrid-scale (SGS) models and implicit time-integration scheme/time-step size are also investigated within the framework of the numerical scheme used. The use of a shock sensor along with artificial bulk viscosity significantly improves the scheme for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The proposed combination of Ducros-type sensor with a negative dilatation sensor removes unnecessary bulk viscosity within expansion and weakly compressible turbulence regions without shocks and allows it to localize near the shocks. It also eliminates the need for a wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. For the numerical schemes used, better results are obtained without adding an explicit SGS model than with SGS model at moderate Reynolds number. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity results in additional damping of the resolved turbulence. However, investigations at higher Reynolds numbers suggest the need for an explicit SGS model. The flow statistics obtained using the second-order implicit time-integration scheme with three sub-iterations closely agrees with the explicit scheme if the maximum Courant–Friedrichs–Lewy is kept near unity.     

High-order generalisation of the diagnostic scaling for turbulent boundary layers

The diagnostic scaling concept, introduced for the streamwise turbulence intensity in wall-bounded turbulent flows (Alfredsson, Segalini and Örlü, Phys. Fluids 2011;23:041702), is here extended and generalised not only for the higher even-order central statistical moments, but also for the odd moments and thereby the probability density distribution of the streamwise velocity fluctuations. Turbulent boundary layer data up to a friction Reynolds number of 60,000 are employed and demonstrate the feasibility of the diagnostic scaling for the data throughout the logarithmic and wake regions. A comparison with the generalised logarithmic law for even-order moments by Meneveau and Marusic (J. Fluid Mech. 2013;719:R1) based on the attached-eddy hypothesis, is reported. The diagnostic plot provides an apparent Reynolds-number-independent scaling of the data, and is exploited to reveal the functional dependencies of the constants needed in the attached-eddy-based model. In particular, the invariance of the lowest order diagnostic scaling poses an intriguing incompatibility with the asymptotic constancy of the Townsend–Perry constant.