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.     

Direct numerical simulation of a fully developed compressible wall turbulence over a wavy wall

Compressible turbulent channel flow over a wavy surface is investigated by direct numerical simulations using high-resolution finite difference schemes. The Reynolds number considered in the present paper is 3380 based on the bulk velocity, the channel half-width and the kinetic viscosity at the wall. Four test cases are simulated and analysed at Ma_m = 0.33, 0.8, 1.2, 1.5 based on the bulk velocity and the speed of sound at the wall. We mainly focus on the curvature and the Mach number effects on the compressible turbulent flows. Numerical results show that although the wavy wall has effects on the mean and fluctuation quantities, log law still exists in the distribution of the wave-averaged streamwise velocity if the roughness effects are taken into consideration in the scaling of it. Near-wall streaks are broken by the wavy surface and near-wall quasi-streamwise vortices mostly begin at the upslope of the wave and pass over the crest of it. The wavy wall makes the turbulence more active and the flow easier to be blended. From the viewpoint of turbulent kinetic budgets, curvature effects strengthen both the diffusion terms and the dissipation terms. At the same time, they change the properties of the compressibility-related terms and promote more inner energy transferring into turbulent kinetic energy. As the Mach number increases, the reattachment of the mean flow is delayed, which indicates the mean separation bubble becomes larger. Concerning the near-wall coherent structures, the vortices are more sparsely distributed with the increasing of the Mach number. For the supersonic cases, shock waves appear. Though they have little effects on the mean turbulent quantities, they change the structures of the flow fields and induce local separations at the upper wall of the channel.     

纤维悬浮湍流的方程和解及其在管流中的应用

推导了纤维悬浮流的平均运动方程和纤维平均取向角的概率密度函数方程，提出了计算纤维平均取向角分布和悬浮流平均量与脉动量关联函数的逐级迭代方法。将推导的方程和提出的方法用于纤维悬浮管流场的计算并进行了相应的实验，计算和实验结果吻合很好。研究结果表明，在相同的压力降下，与没有纤维的牛顿流相比，纤维悬浮流具有较大的流量，因而纤维在流场中起着减阻的作用，减阻的量随着纤维浓度的增加而增大。在相同情况下，纤维悬浮流的相对湍流强度和雷诺应力比牛顿流小，这说明纤维能够抑制湍流，抑制的量与纤维的质量浓度成正比。     

Orientation distribution of fibres in a channel flow of fibre suspension

The orientation and concentration distributions of fibres in laminar and turbulent channel flows are investigated numerically. The obtained results are in good agreement with the experimental data. In the laminar flow regime, more fibres orient to the flow direction as the Reynolds number increases. The shear rate of fluid around a fibre plays an important role in determining the orientation distribution of fibres, while the fibre density and the fibre aspect-ratio have marginal influence on the orientation distribution. In the turbulent regime, the orientation distribution of fibres becomes more homogeneous with the increase of Reynolds number, and the concentration profile is flatter than that in the laminar regime. The fluctuating intensity of fibre velocity in the downstream direction is larger than that in the lateral directions.     

Effects of spanwise system rotation on turbulent stripes in a plane Poiseuille flow

In the transitional channel flow, the large-scale intermittent structure of localised turbulence, which is called the turbulent stripe pattern, can be found in the form of stripe arrangement. The structure of the turbulent stripe pattern is an oblique laminar–turbulent banded pattern and is inclined with respect to the streamwise direction. We performed direct numerical simulation at a transitional Reynolds number and very low-rotation numbers, and focused on the turbulent stripe pattern in the plane Poiseuille flow subjected to spanwise system rotation. We captured the turbulent stripe pattern in a rotating channel flow and found the augmentation and diminution of the turbulent stripe pattern were affected by the spanwise rotation. The contents of the discussion are the spatial size of the turbulent stripe pattern on the basis of the instantaneous flow fields, the energy spectra, and various statistics relating to the spanwise velocity component that characterise the turbulent stripe pattern. The turbulent stripe pattern was found to contain kinetic energy that was larger in very weakly rotating flows than in the static system. It was also found that the magnitude of the spanwise secondary flow increases, while the quasi-laminar region is wider at a very lowrotation number.     

尺度自适应模拟和大涡模拟的关联性分析

采用理论分析和数值模拟相结合的方法,系统研究了尺度自适应模拟（scale-adaptive simulation,SAS）和大涡模拟（large-eddy simulation,LES）的关联性问题.在理论分析方面,对比分析了系综平均和滤波的定义、Spalart-Allmaras（SA）湍流模型和动态亚格子（subgrid-scale,SGS）模型关于湍流黏性系数的求解方式.理论分析结果表明,系综平均等价于盒式直接滤波,SAS和LES的控制方程在数学形式上具有一致性;SAS存在过多的湍流耗散,主要来自于SA输运方程中的扩散项.在数值模拟方面,选取来流Mach数0.55,Reynolds数2×105的圆柱可压缩绕流为分析算例.计算结果表明,SAS和LES预测的大尺度平均流场信息几乎一致,SAS预测的湍流脉动信息略低于LES.SAS在圆柱近尾迹区的湍流耗散过大,而在稍远的尾迹区几乎能够完全等效于LES.      

Use of Lagrangian statistics for the analysis of the scale separation hypothesis in turbulent channel flow

Turbulence models often involve Reynolds averaging, with a closure providing the Reynolds stress tensor as function of mean velocity gradients, through a turbulence constitutive equation. The main limitation of this linear closure is that it rests on an analogy with kinetic theory. For this analogy to be valid there has to be a scale separation between the mean velocity variations and the turbulent Lagrangian free path whose mean value is the turbulent mixing length. The aim of this work is to better understand this hypothesis from a microscopic point of view. Therefore, fluid elements are tracked in a turbulent channel flow. The flow is resolved by direct numerical simulation (DNS). Statistics on particle trajectories ending on a certain distance y₀ from the wall are computed, leading to estimations of the turbulent mixing length scale and the Knudsen number. Comparing the computed values to the Knudsen number in the case of scale separation, we may know in which region of the flow and to what extent the turbulence constitutive equation is not verified. Finally, a new non-local formulation for predicting the Reynolds stress is 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.     

A fully discrete,kinetic energy consistent finite-volume scheme for compressible flows

A robust, implicit, low-dissipation method suitable for LES/DNS of compressible turbulent flows is discussed. The scheme is designed such that the discrete flux of kinetic energy and its rate of change are consistent with those predicted by the momentum and continuity equations. The resulting spatial fluxes are similar to those derived using the so-called skew-symmetric formulation of the convective terms. Enforcing consistency for the time derivative results in a novel density weighted Crank–Nicolson type scheme. The method is stable without the addition of any explicit dissipation terms at very high Reynolds numbers for flows without shocks. Shock capturing is achieved by switching on a dissipative flux term which tends to zero in smooth regions of the flow. Numerical examples include a one-dimensional shock tube problem, the Taylor–Green problem, simulations of isotropic turbulence, hypersonic flow over a double-cone geometry, and compressible turbulent channel flow.     

Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) [22, 30–33] in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number $k − ε$ model with Launder and Sharma [23] and Abe et al. [1] closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF's. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin's fractional step method [11]. The adaptive upwinding technique, proposed by Lin and Atluri [27], is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura [20] for Reynolds number 4494, based on the channel width, and Grashof number 9.6 × 10⁵. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.     

Extended local balance model of turbulence and couette-taylor flow

An extended local balance model of turbulence, based on a new transport equation for the dissipation rate with a negative diffusion coefficient, is presented. Analytical solutions for the mean velocity and the dissipation rate for the turbulent Couette-Taylor problem are derived. The dependence of torque on the Reynolds number is obtained. These solutions depend only on two constants k=0.4 and C=9.5 of the turbulent boundary layer and, within the limits of a narrow channel, are reduced to the well-known von Karman's solutions for planar Couette flow. Strange attractor behavior in this limit is also observed.     

Effects of variable specific heat on energy transfer in a high-temperature supersonic channel flow

An energy transfer mechanism in high-temperature supersonic turbulent flow for variable specific heat (VSH) condition through turbulent kinetic energy (TKE), mean kinetic energy (MKE), turbulent internal energy (TIE) and mean internal energy (MIE) is proposed. The similarities of energy budgets between VSH and constant specific heat (CSH) conditions are investigated by introducing a vibrational energy excited degree and considering the effects of fluctuating specific heat. Direct numerical simulation (DNS) of temporally evolving high-temperature supersonic turbulent channel flow is conducted at Mach number 3.0 and Reynolds number 4800 combined with a constant dimensional wall temperature 1192.60 K for VSH and CSH conditions to validate the proposed energy transfer mechanism. The differences between the terms in the two kinetic energy budgets for VSH and CSH conditions are small; however, the magnitude of molecular diffusion term for VSH condition is significantly smaller than that for CSH condition. The non-negligible energy transfer is obtained after neglecting several small terms of diffusion, dissipation and compressibility related. The non-negligible energy transfer involving TIE includes three processes, in which energy can be gained from TKE and MIE and lost to MIE. The same non-negligible energy transfer through TKE, MKE and MIE is observed for both the conditions.     

非均衡湍流研究进展

Richardson-Kolmogorov能量级串理论是湍流研究中最重要的基础理论,其中一个推论是能量的传输和耗散应当是均衡的,对应于耗散系数${C_\mathit{\boldsymbol{\epsilon}}}$为常数.然而近些年的实验及数值模拟都发现了不符合Richardson-Kolmogorov能量级串理论的非均衡耗散律,即使在此区域内Reynolds数足够高,能谱满足Kolmogorov的-5/3标度律,${C_\mathit{\boldsymbol{\epsilon}}}$也不为常数,而满足${C_\mathit{\boldsymbol{\epsilon}}}$~Re_Im/Re_Ln,其中m≈1≈n,Re_I为入口Reynolds数,Re_L为以积分尺度为特征长度的当地Reynolds数.近三年来又发现流向速度梯度扭率S_k和Lagrange速度梯度自相关的时间演化Φ'_ijij也可以用来度量非均衡湍流现象,为非均衡湍流的研究开辟了新路.      

Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow

A simple theoretical analysis and direct numerical simulations on 512(3) grid points suggest that the velocity correlation spectrum tensor in the inertial subrange of homogeneous turbulent shear flow at high Reynolds number is given by a simple form that is an anisotropic function of the wave vector. The tensor is determined by the rate of the strain tensor of the mean flow, the rate of energy dissipation per unit mass, the wave vector, and two nondimensional constants.     

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.     

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.     

VELOCITY DISTRIBUTION AND TURBULENCE STRUCTURE OF FULLY DEVELOPED FLOW IN A CURVED TUBE

Turbulent structures in a curved circular tube were measured for three values of the curve radius to the tube radius ratio, R. Distributions of turbulent energy and axial and radial velocities are obtained in a fully developed flow region by a hot wire anemometer. Turbulent energy and Reynolds stress, become maximum in the core region, where turbulence generation becomes maximum. When R is small, turbulent generation does not balance with turbulent energy dissipation, due to a convective effect induced by secondary flow. Experimental results for Reynolds stress and turbulent energy are compared with theoretical results. Due to the secondary flow effect, for small R value flow, the ratio of these two values was not uniform in the core region.     

槽道流POD重构及湍流动能耗散率分析

采用二维粒子图像测速仪（2DPIV）对槽道内涡波流场进行实验研究,用POD技术对2DPIV瞬态速度矢量场进行主导模态重构,得到槽道内的平均流速和湍流动能分布;采用大涡PIV方法对湍流动能耗散率分布进行计算.结果表明：重构流场表征了原始流场的主导结构,剔除了噪声等干扰信息;大涡PIV方法能有效地估算动能耗散率的分布;湍流动能在壁面附近较小,在接近槽道中心区域湍流动能越来越大,呈现出射流的特征;动能耗散率的峰值出现在壁面附近和槽道中心区域,动能耗散率随着远离壁面程度的增加先降低后逐渐增加直至达到峰值.     

Dynamic analysis of bubble-driven liquid flows using time-resolved particle image velocimetry and proper orthogonal decomposition techniques

Abstract  

An experimental study to evaluate dynamic structures of flow motion and turbulence characteristics in bubble-driven water flow in a rectangular tank with a varying flow rate of compressed air is conducted. Liquid flow fields are measured by time-resolved particle image velocimetry (PIV) with fluorescent tracer particles to eliminate diffused reflections, and by an image intensifier to acquire enhanced clean particle images. By proper orthogonal decomposition (POD) analysis, the energy distributions of spatial and temporal modes are acquired. Time-averaged velocity and turbulent kinetic energy distributions are varied with the air flow rates. With increasing Reynolds number, bubble-induced turbulent motion becomes dominant rather than the recirculating flow near the side wall. Detailed spatial structures and the unsteady behavior of dominant dynamic modes associated with turbulent kinetic energy distributions are addressed.     

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.