Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.     

Finite element study of free surface turbulent flow in rotating channels

Under certain conditions of liquid flow through rotating channels, the Coriolis force can induce a free surface to be formed. This problem is of practical importance in a Coriolis wear tester, which is used for determining the sliding wear coefficient of wear materials in slurry handling equipment. A deforming Galerkin finite element method is presented for predicting two‐dimensional turbulent free surface mean flow in rotating channels. Reynolds‐averaged Navier–Stokes (RANS) equations are cast into weak(algebraic) form using primitive variables (velocity and pressure). Eddy viscosity is determined via a mixing length model. Velocity is interpolated biquadratically, while pressure is interpolated bilinearly. The kinematic condition is used to form the Galerkin residual for the free surface. The free surface is represented by Hermite polynomials of zeroeth order for continuity of position and slope. Combined Newton's iteration is used to simultaneously solve for the free surface and the field variables. Results of velocity and pressure fields, as well as the free surface are shown to converge with mesh‐size refinement. There is excellent respect for mass conservation. Results are presented for various values of Rossby number (Ro) and height‐based Reynolds number (Re_H). Parameter continuation in Ro and Re_H space is used to compute solutions at higher values of flow rate and angular velocity. Copyright © 2002 John Wiley & Sons, Ltd.     

NUMERICAL SIMULATION OF COMPLICATED PIPE FLOW WITH k-εMODEL

In this paper, turbulence in a complicated pipe is simulated by using the k-ε model. The ladder-like mesh approximation is used to solve the problem of complicated boundary with the result of numerical simulation favorable. Two computational examples are given to validate the strong adaptability and stability of k-ε model.     

Laser-Doppler measurements of laminar and turbulent flow in a pipe bend

Laser-Doppler measurements are reported for laminar and turbulent flow through a 90° bend of circular cross-section with mean radius of curvature equal to 2.8 times the diameter. The measurements were made in cross-stream planes 0.58 diameters upstream of the bend inlet plane, in 30, 60 and 75° planes in the bend and in planes one and six diameters downstream of the exit plane. Three sets of data were obtained: for laminar flow at Reynolds numbers of 500 and 1093 and for turbulent flow at the maximum obtainable Reynolds number of 43 000. The results show the development of strong pressure-driven secondary flows in the form of a pair of counter-rotating vortices in the streamwise direction. The strength and character of the secondary flows were found to depend on the thickness and nature of the inlet boundary layers, inlet conditions which could not be varied independently of Reynolds number. The quantitative anemometer measurements are supported by flow visualization studies. Refractive index matching at the fluid-wall interface was not used; the measurements consist, therefore, of streamwise components of mean and fluctuating velocities only, supplemented by wall pressure measurements for the turbulent flow. The displacement of the laser measurement volume due to refraction is allowed for in simple geometrical calculations. The results are intenden for use as benchmark data for calibrating flow calculation methods.     

Symbolic computation of flow in a rotating pipe

A perturbation solution of the fully developed flow through a pipe of circular cross-section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ωa²/ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non-linear equations for the axial velocity ω and the streamfunction ? of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ?_n, n ? 1, of the perturbation expansion ? = ∑ Rⁿ?_n in terms of ω_n-1, the (n-1)-th order of the expansion ω = ∑ Rⁿω_n, and of the lower orders ?₁,…,?_{n ? 1}. Then it permits the computation of ω_n from ω₀,…,ω_{n ? 1} and ?₁,…,?;_n. The computation starts from the Hagen–Poiseuille flow ω₀, i.e. the perturbation is around this flow. The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co-ordinates: in the complex co-ordinates chosen the two-dimensional harmonic (Laplace, Δ) and biharmonic (Δ²) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ω_n, ?_n and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co-ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ? 6 of the perturbation expansion.     

旋转湍流中雷诺应力模型压力应变关联项的修正研究

利用张量的不变量理论,推导得出传统雷诺应力模型中压力应变关联项模型应用于旋转湍流模拟中的一些基本问题,即在纯旋转条件下,传统模型所描述的初始各向异性的湍流中雷诺应力张量演化规律是一个无衰减振荡过程,而快速畸变理论推导结果显示,其演化应是一个阻尼振荡衰减的过程。以衰减雷诺应力为目的,构造出包含旋转率张量高阶量的关联项。然后,结合变形率张量的高阶项,将修正模型扩展至椭圆形流线类型流动。最后,将修正模型应用于轴向旋转圆管内湍流流场的模拟,并将结果与实测结果进行了对比。     

Computational intellectual analytical theory of computational analytical approach to rotating flow of non-Newtonian fluid

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Three-dimensional channel flow of second grade fluid in rotating frame

An analysis is performed for the hydromagnetic second grade fluid flow between two horizontal plates in a rotating system in the presence of a magnetic field.The lower sheet is considered to be a stret...     

Effect of the pattern of initial perturbations on the steady pipe flow regime

The influence of the inlet flow formation mode on the steady flow regime in a circular pipe has been investigated experimentally. For a given inlet flow formation mode the Reynolds number Re* at which the transition from laminar to turbulent steady flow occurred was determined. With decrease in the Reynolds number the difference between the resistance coefficients for laminar and turbulent flows decreases. At a Reynolds number approximately equal to 1000 the resistance coefficients calculated from the Hagen-Poiseuille formula for laminar steady flow and from the Prandtl formula for turbulent steady flow are equal. Therefore, we may assume that at Re > 1000 steady pipe flow can only be laminar and in this case it is meaningless to speak of a transition from one steady pipe flow regime to the other. The previously published results [1–9] show that the Reynolds number at which laminar goes over into turbulent steady flow decreases with increase in the intensity of the inlet pulsations. However, at the highest inlet pulsation intensities realized experimentally, turbulent flow was observed only at Reynolds numbers higher than a certain value, which in different experiments varied over the range 1900–2320 [10]. In spite of this scatter, it has been assumed that in the experiments a so-called lower critical Reynolds number was determined, such that at higher Reynolds numbers turbulent flow can be observed and at lower Reynolds numbers for any inlet perturbations only steady laminar flow can be realized. In contrast to the lower critical Reynolds number, the Re* values obtained in the present study, were determined for given (not arbitrary) inlet flow formation modes. In this study, it is experimentally shown that the Re* values depend not only on the pipe inlet pulsation intensity but also on the pulsation flow pattern. This result suggests that in the previous experiments the Re* values were determined and that their scatter is related with the different pulsation flow patterns at the pipe inlet. The experimental data so far obtained are insufficient either to determine the lower critical Reynolds number or even to assert that this number exists for a pipe at all.     

Hydromagnetic rotating flow of third grade fluid

This work investigates the flow of a third grade fluid in a rotating frame of reference. The fluid is incompressible and magnetohydrodynamic （MHD）. The flow is bounded between two porous plates, the lower of which is shrinking linearly. Mathematical modelling of the considered flow leads to a nonlinear problem. The solution of this nonlinear problem is computed by the homotopy analysis method （HAM）. Graphs are presented to demonstrate the effect of several emerging parameters, which clearly describe the flow characteristics.     

An investigation of possible mechanisms of heterogeneous drag reduction in pipe and channel flows

When concentrated polymer solutions are injected into the core-region of a turbulent pipe or channel flow, the injected polymer solution forms a thread which preserves its identity far beyond the injection point. The resulting drag reduction is called heterogeneous drag reduction.This study presents experimental results on the mechanism of this type of drag reduction. The experiments were carried out to find out whether this drag reduction is caused by small amounts of polymer removed from the thread and dissolved in the near-wall region of the flow or by an interaction of the polymer thread with the turbulence. The friction behavior of this type of drag reduction was measured for different concentrations in pipes of different cross-sections, but of identical hydraulic diameter. The parameters of the injection, i.e. injector geometry as well as the ratio of the injection to the bulk velocity, were varied. In one set of experiments the polymer thread was sucked out through an orifice and the friction behavior in the pipe was determined downstream of the orifice. In another experiment, near-wall fluid was led into a bypass in order to measure its drag reducing properties. Furthermore, the influence of a water injection into the near-wall region on the drag reduction was studied.The results provide a strong evidence that heterogeneous drag reduction is in part caused by small amount of dissolved polymer in the near-wall region as well as by an interaction of the polymer thread with the turbulence.Nomenclature a channel height - b channel width - c _p concentration of the injected polymer solution - c _R effective polymer concentration averaged over the cross-section - d pipe or hydraulic diameter - d _i injector diameter - DR drag reduction - f friction factor - l downstream distance from injector - L length of a pipe segment - P polymer type - p differential pressure - Re Reynolds number - U bulk velocity - u ^* ratio of injection to bulk velocity - y ⁺ dimensionless wall distance - v kinematic viscosity - density of the fluid - _w wall shear stress     

液固两相槽道流中湍流调制的数值研究

结合雷诺应力模型 (Reynolds Stress Model, RSM) 和混合模型 (Mixture Model) 对槽道湍流向下流动中的颗粒调制湍流问题进行了研究．该模型考虑了颗粒流的动能理论和颗粒对湍流的反馈作用．着重分析了颗粒对湍流的调制作用,以及颗粒参数变量（如颗粒密度和质量载荷）对湍流调制的影响．结果表明：(1)在颗粒抑制湍流的范围内,当颗粒密度小于载流体密度时,湍流强度的改变量与颗粒密度成反比;当颗粒密度大于载流体密度时,湍流强度的改变量与颗粒密度成正比;(2)在一定范围内,颗粒抑制湍流的能力随颗粒质量载荷增加而变强．     

An analytical and experimental investigation of flow characteristics generated by rotating porous disk

This paper is concerned with investigations of the gas flow around and in the cell-porous rotating disk. A simple 1D model of gas flow is presented. Usually the cell-porous materials are applied in heat exchangers and stationary filters. On the other hand, peculiarities of the flow revealed under theoretical analysis and some experimental observations demonstrate that rotating porous disks may also be effectively exploited in the shear-force machines for the gas transport purposes.     

Non-Equilibrium Mixing of Turbulence Scales Using a Continuous Hybrid RANS/LES Approach: Application to the Shearless Mixing Layer

A new subgrid-scale (SGS) model based on partially integrated transport method (PITM) is applied to the case of a turbulent spectral non-equilibrium flow created by the mixing of two turbulence fields of differing scales: the shearless mixing layer. The method can be viewed as a continuous hybrid RANS/LES approach. In this model the SGS length scale is no longer given by the size of the discretization step, but is dynamically estimated using an additional transport equation for the dissipation rate. The results are compared to those corresponding to the classical model of Smagorinsky and to the experimental data of Veeravalli and Warhaft. A method for creating an anisotropic analytical pseudo-random field for inflow conditions is also proposed. This approach based on subgrid-scale transport modelling combined with anisotropic inlet conditions gives better results for the prediction of the shearless mixing layer.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.     

Motion equations of the dynamic theory of consolidation from the point of view of the theory of transient pipe flows

An elastic fluid-saturated porous medium is modeled as a bundle of parallel cylindrical tubes aligned in a direction parallel to the fluid movement. The pore space is filled with viscous compressible liquid. A cell model and the theory of transient pipe flow are used to derive one-dimensional governing equations in such media. All macroscopic constants in these equations are defined by the individual material constants of the fluid and solid. The interaction force includes an additional term not found in Biot's theory.     

An analytical solution and analysis of characters for viscoelastic fluid flow in annular pipe

I.Introducti0nLuiCiqunandHuangJunqiI'I(l989),ZhuWeihuiandLuiCiquri1'l(l992)sequentiallystudiedtheaxialflowofsecondorderandMaxwellfluidsandanalyzedtheflowcharactersofthesefluids.Thispaperwillstudyunsteadyrotat0ryflowofsecondorderandMaxwellfluidsinannularpi…