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.     

充分发展槽道湍流的小波数值分析

数值计算了高斯子波变换Navier Stokes(N-S)方程后得到的积分方程.在利用高斯子波得到的以弯曲度为基本量的无穷域中N-S方程的基础上,得到了有界区域内的以弯曲度为基本量的N-S方程.将此N-S方程看作一个特殊的扩散方程,将压力项与对流项看作是源项,得到一个积分方程.利用特征线法对该方程求解,得到通解.并将所得结果运用于对称槽道湍流和非对称槽道湍流的研究中.将计算与实验所得的平均量与实验结果进行了对比.     

湍流边界层中重粒子弥散的随机模型

在重粒子轨道模型的基础上,引入了Saffman力,并考虑了粒子-固壁碰撞和粒子-粒子碰撞的影响,建立了重粒子运动方程,耦合湍流脉动的随机方程,发展了重粒子弥散的随机模型,并在湍流边界层中考察该模型.将数值计算结果与实验结果进行比较,同时考察了Saffman力和粒子碰撞对计算结果的影响.     

An Improved Method to Determine Particle Dispersion Width for Efficient Modeling of Turbulent Two‐Phase Flows

An improved approach is presented for the hybrid Eulerian‐Lagrangian modeling of turbulent two‐phase flows. The hybrid model consists of a nonlinear k–ε model for the fluid flow and an efficient Lagrangian trajectory model for the particulate flow. The improved approach avoids an empirical correlation required to determine the dispersion width for the existing Stochastic‐Probabilistic Efficiency Enhanced Dispersion (SPEED) model. The improved SPEED model is validated using experimental data for a poly‐dispersed water spray interacting with a turbulent annular air jet behind a bluff‐body. Numerical results for the number‐mean and Sauter‐mean droplet diameters, as well as mean and fluctuating droplet velocities are compared with the experimental data and with the predictions of other dispersion models. It is demonstrated that higher computational efficiency and smoother profiles of Sauter‐mean diameter can be obtained with the improved stochastic‐probabilistic model than with the eddy‐interaction model.     

Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method

The energy gradient method has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this method, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transverse gradient of the total mechanical energy which amplifies the disturbance and the energy loss from viscous friction which damps the disturbance, for given imposed disturbance. For a given flow geometry and fluid properties, when the maximum of the function $K$ (a function standing for the ratio of the gradient of total mechanical energy in the transverse direction to the rate of energy loss due to viscous friction in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy gradient analysis, the equation for calculating the energy gradient function $K$ for plane Couette flow is derived. The result indicates that $K$ reaches the maximum at the moving walls. Thus, the fluid layer near the moving wall is the most dangerous position to generate initial oscillation at sufficient high $\operatorname{Re}$ for given same level of normalized perturbation in the domain. The critical value of $K$ at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow when two walls move in opposite directions (anti-symmetry). This value is about the same as that for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded that the critical value of $K$ at turbulent transition is about 370-389 for wall-bounded parallel shear flows which include both pressure (symmetrical case) and shear driven flows (anti-symmetrical case).     

考虑双向耦合效应的格子Boltzmann气固两相湍流模型

在流体粒子概率密度函数输运方程中考虑颗粒对流体的反作用力,发展了考虑双向耦合效应的LB气固两相流模型,引入Smagorinsky亚格子模型模拟高雷诺数气相流场.对经典后台阶气固两相流动进行模拟,气相和颗粒相速度分布与实验结果进行比较,发现考虑双向耦合效应的LB气固两相流模型结果明显优于单向耦合结果.进一步研究不同惯性颗粒在流场中的弥散特性,小颗粒(St~O(0.1))对流体的跟随性较好,在流场中分布较为均匀;而St~O(1)的颗粒难被流场涡卷吸进入涡内,呈现倾向性弥散现象;大颗粒(St~O(10))由于自身惯性进入流场涡,在流场中分布较为均匀.     

三种湍流模式数值模拟直角弯管内三维分离流动的比较

采用有限体积法数值离散雷诺时均Navier Stokes方程,模拟了方形截面90°大曲率弯曲管道内的三维湍流流动.用3种湍流模式(标准k-ε湍流模式、RNG k-ε湍流模式、Realizable k-ε湍流模式)求解该问题.给出了数学模型和计算结果,并与实验数据进行了比较.结果表明,采用RNG k-ε湍流模式并结合两层壁面模型的处理,能有效准确地求解强曲率影响的管道内及近壁区湍流的流动.     

壁面在展向作周期运动的槽道湍流的大涡模拟

分别采用3种亚格子模式:传统的Smagorinsky模式、动力Smagorinsky模式和Cui(2004)基于Kolmogorov方程所提出的新模式,对壁面在展向作周期运动的槽道湍流进行了大涡模拟,以考察这3种模式对平均运动为三维、非定常的湍流流动的模拟能力.通过对湍流基本统计量的分析,发现动力模式和新模式都可以较好地预测这种三维非定常的湍流流动;对相位平均的湍流统计量,动力模式的结果略优于新模式;传统的Smagorinsky模式对这种流动的预测结果是最差的.     

各向异性颗粒PDF输运方程及其颗粒湍流扩散

根据Minier的模型,获得颗粒所见流体脉动速度的朗之万模型,并推导颗粒所见流体湍流脉动速度自关联函数,该函数具有各向异性的特征,进一步由流体脉动速度朗之万模型和颗粒运动方程获得相应的颗粒PDF输运方程.为描述在网格生成湍流中的颗粒扩散运动,将颗粒PDF输运方程作相应的简化,获得一个具有颗粒湍流各向异性的解析解,将方程的解与Wellsand Stock的颗粒湍流扩散实验结果进行比较,获得了很好的结果.揭示了具有各向异性特征的颗粒PDF输运方程的优越性.     

甲醇喷雾湍流反应流的JPDF数值模拟

喷雾湍流燃烧过程中,液滴、湍流和化学反应之间强烈耦合,物理化学机理非常复杂。本文将速度-标量-频率联合概率密度函数JPDF输运方程方法应用于两相喷雾湍流反应流问题,利用火焰面模型解耦流动和化学反应动力学的耦合关系,建立起相应的数值计算模型。采用Monte-Carlo数值计算方法,针对澳大利亚悉尼大学Masri等人以甲醇为燃料所进行的湍流喷雾燃烧值班火焰这一试验进行了数值模拟,通过与Fluent下的计算结果及试验结果的对比分析,验证了本文所建模型的准确性。     

Turbulent gas-dispersed flow in a pipe with sudden expansion: numerical simulation

A mathematical model was developed to simulate two-phase gas-dispersed flow moving through a pipe with axisymmetric sudden expansion. In the model, the two-fluid Euler approach was used. The model is based on solving Reynolds-averaged Navier — Stokes equations for a two-phase stream. In calculating the fluctuating characteristics of the dispersed phase, equations borrowed from the models by Simonin (1991), Zaichik et al. (1994), and Derevich (2002) were used. Results of a comparative analysis with previously reported experimental and numerical data on two-phase flows with separation past sudden expansion in a plane channel and in a pipe are given. This work was supported by the President of the Russian Federation through the Foundation for Young Candidates of Sciences under Grant MK-186.2007.8 and by the Russian Foundation for Basic Research (Grants Nos. 05-08-33586 and 06-08-00967).     

管流中注入粒子的非定常波系的数值研究

使用二维有限体积方法,对在管流中固定位置处注入静止固体粒子的可压缩含灰气体流动进行数值模拟,讨论流场和粒子在过程中的耦合,研究质量增加和热量变化所产生的非定常波系,分析物理参数在过程中的变化.结果表明,在添质和加热过程中,流场会产生不同类型和不同强度的非定常波,在分析其物理规律的同时,讨论添质和加热相互作用导致的波系间转换,最后求解流场中各区域的热力学参数,得到不同的流场速度和粒子温度情况下各非定常波波强的相图,定量解释改变参数引起的非定常波变化规律.     

高超声速可压缩流中粗糙壁热流研究

用计算流体力学（CFD）数值模拟和理论方法对高超声速可压缩湍流中粗糙壁面热增量进行研究. 着重考虑粗糙单元密度和粗糙单元形状对粗糙表面热流的影响. 结果表明： ①粗糙单元密度变化时, CFD数值方法计算所得的粗糙单元等效热流随着粗糙单元密度的降低而增加, 理论方法预测结果的规律随方法不同而不同;②在相同粗糙单元密度和高度时,如果粗糙单元形状改变,CFD计算结果也随之发生改变. 理论预测方法所得结果不发生变化.     

湍流扩散火焰局部熄火和再燃现象的PDF模拟

对一个值班湍流CH4／O2／N2射流扩散火焰(Sandia Flame D)进行了数值模拟研究．所采用的数学物理模型包括双尺度的k—ε湍流模型,标量联合的概率密度函数(PDF)输运方程方法,甲烷氧化的ARM简化化学反应机理(包含16种组分,12步总包反应)和欧几里德最小生成树(EMST)小尺度混合模型．将计算结果和实验数据进行了比较,不仅对于平均量,对于标量的散点分布和条件概率密度分布也是如此．计算结果表明文中采用的模型不仅能够预测宏观的火焰结构,而且预测了湍流燃烧中复杂的局部熄火和再燃过程．     

Investigations on the Effect of Backward-Facing and Forward-Facing Steps on Turbulent Mixed-Convection Flow Over a Flat Plate

The effects of backward-facing and forward-facing steps on a turbulent buoyancy-dominated mixed-convection flow over a flat plate are examined experimentally. Air velocity and temperature distributions and their turbulent fluctuations are measured simultaneously by using a two-component laser-Doppler velocimeter and a cold wire anemometer, respectively. The experiment was carried out for a step (backward-facing/forward-facing) height of 22 mm, a temperature difference, ΔT, of 30°C between the heated walls and the free-stream air (corresponding to a local Grashof number Gr _xi = 6.45 × 10¹⁰), and a free-stream velocity of 0.48 m/s. It was found that the introduction of backward- and forward-facing steps increases the turbulence intensity of the velocity and temperature fluctuations downstream of the step. The present results also reveal that the maximum local Nusselt number occurs in the vicinity of the reattachment zone, and it is approximately twice for the case of the backward-facing step and three times for the case of the forward-facing step than that of the flat plate value at similar flow and thermal conditions.     

Turbulent vortex rings in weakly turbulent and in turbulent ambient flow

Turbulent vortex rings were investigated in weakly turbulent flow and in three different grid generated turbulent flows to clarify the reciprocal action of the vortex ring with defined external turbulence. Assuming self-similarity and turbulent viscosity as proportional to V₀D₀ the equations for the ring diameter D(t) and the velocity of propagation V(t) were derived. The time difference Δt between the virtual origins of 1/V(t) respectively D²(t) led to an invariant term. The equation of momentum is fulfilled. – Position and diameter of the vortex rings were determined till their decay by means of an optical system, which did not disturb the vortex rings. The experimental results in weakly turbulent ambient flow obtained by the author and by others confirm the theory very well. The ambient turbulence was nearly constant in the measuring region; its effect could be described by simply adding its viscosity to the vortex ring’s internal turbulent viscosity. The results could be represented in unified non-dimensional diagrams. Moreover, an explanation was found as to why the mean internal turbulent viscosity is constant.     

Experimental Study of CuO/Water Nanofluid Turbulent Convective Heat Transfer in Square Cross-section Duct

Ducts with a square cross-section are widely used in many industrial applications because of their high compactness, easy forming, and low pressure drop. But the thermal performance of a duct will be reduced when the circular cross-sectional shape is not used. In this study, the convective heat transfer for a CuO/water nanofluid through a square cross-section duct in the turbulent flow regime has been investigated. The Nusselt number of nanofluids for different nanoparticle concentrations, as well as various Peclet numbers, was obtained. The results show considerable enhancement in the heat transfer coefficient and Nusselt number by increasing the nanoparticle concentrations as well as the Peclet number.     

Heat Transfer and Pressure Drop of Nanofluid with Rod-like Particles in Turbulent Flows through a Curved Pipe

Pressure drop, heat transfer, and energy performance of ZnO/water nanofluid with rodlike particles flowing through a curved pipe are studied in the range of Reynolds number 5000 ≤ Re ≤ 30,000, particle volume concentration 0.1% ≤ Φ ≤ 5%, Schmidt number 10⁴ ≤ Sc ≤ 3 × 10⁵, particle aspect ratio 2 ≤ λ ≤ 14, and Dean number 5 × 10³ ≤ De ≤ 1.5 × 10⁴. The momentum and energy equations of nanofluid, together with the equation of particle number density for particles, are solved numerically. Some results are validated by comparing with the experimental results. The effect of Re, Φ, Sc, λ, and De on the friction factor f and Nusselt number Nu is analyzed. The results showed that the values of f are increased with increases in Φ, Sc, and De, and with decreases in Re and λ. The heat transfer performance is enhanced with increases in Re, Φ, λ, and De, and with decreases in Sc. The ratio of energy PEC for nanofluid to base fluid is increased with increases in Re, Φ, λ, and De, and with decreases in Sc. Finally, the formula of ratio of energy PEC for nanofluid to base fluid as a function of Re, Φ, Sc, λ, and De is derived based on the numerical data.     

普通单模光纤色散补偿传输的实验研究

利用自行设计的悬臂梁调节机构,对１２ｃｍ长的均匀光纤光栅进行中心波长和啁啾的同时线性调谐后,将其应用于２０Ｇｂ／ｓ、１００ｋｍ普通单模学纤的色散补偿实验中,取得了良好的补偿效果。