Flow profiles in rectangular channels with gradually varying width

Summary We examine here the steady flow with constant discharge in an open channel of uniform bottom slope and with rectangular cross-section of uniformly increasing or decreasing width.The discussion of the dynamic differential equation of the gradually varied flow, using the theory of the singular point, permits us to describe the characteristics of the various flow profiles and also of the corresponding energy lines.This work was presented at the IX Convegno di Idraulica e Costruzioni Idrauliche, Trieste, May 1965.     

关于明渠的一个思考

在《水力学》教学过程中,穿插一些与授课内容有关的人、事、物的高雅而不庸俗、深刻但不高深的题外话,旨在提高学生们学习的兴趣,实践表明,这种做法能收到良好的教学效果.     

On the flow field generated by a gradually varying flow through an orifice

The motion of a vortex ring generated by gradually varied flows through a thin-edged orifice has been investigated experimentally using particle image velocimetry. This flow reproduces the primary characteristics of many biological flows, such as cardiac flows through valves or jellyfish and squid propulsion. Even though vortex ring formation has been extensively studied, there is still interest in gradually varying inflows, i.e. the ones that are mostly found in previous conditions. The main purpose of this paper is to extend the time scaling already proposed in the literature to the entire cycle of vortex ring formation, pinch-off and free motion. To this end, eight inflow time laws have been tested, with different acceleration and deceleration phases. They have been selected in relation to practical applications by their resemblance to the main characteristics of cardiovascular and pulsed locomotion flows. Analysis of measured velocity and vorticity fields suggested a general criterion to establish the instant of vortex pinch-off directly from the imposed velocity program. This allows the proper scaling of the entire time evolution of the vortex ring for all tested inflows. Since it is quite easy to identify this instant experimentally, these results give a simple, practical rule for the computation of scales in vortex ring formation and development in the case of gradual inflows. The “slug model” has been used to test the proposed scaling and to obtain predictions for the vortex position, circulation and vorticity which are in agreement with experimental data.     

On the stability of non-isothermal flow in channels

Summary We study here stability of non-isothermal flow between two closely spaced, heat conducting, infinite parallel flat plates of lengthl and distanceh apart. Fluid enters uniformly alongx = 0 at temperatureT ₁ >T _w the plate temperature. The flow non-uniformity is assumed to occur due to coupling between the energy equation, which describes the heat transfer mechanism between fluid and channel walls, and the flow equation which includes the temperature dependence of viscosity.The model for the flow assumes that similarity profiles exist for velocity and temperature in the flow direction. The stability of the unidirectional flow by a linearized first order perturbation analysis of the proposed model is examined.Notation b rheological parameter of the fluid defined by eq. [4] - B dimensionless viscosity-temperature parameter defined by eq. [11] - C rheological parameter defined by eq. [4] - h distance between the two parallel plates, ft. - H a thermal transfer coefficient (l/h) - l length of the plates, ft. - p pressure - P inlet pressure - G _z Graetz number defined by eq. [11] - t time, h - T mean temperature as defined by eq. [2] - T ₁ inlet temperature - u velocity vector withu _x ,u _y ,u _z as component velocities - v mean velocity vector as defined by eq. [1] - V mean steady state axial velocity - x, y, z Cartesian coordinate system - w refers to wall condition - thermal diffusivity, ft²/h - A effective thermal diffusivity tensor - dimensionlessx coordinate - wave number iny direction - dimensionless wave number iny direction - µ ₀ viscosity of fluid - density of fluid - dimensionless velocity inx direction - growth rate of disturbances - dimensionless growth rate - proportionality constant for heat generation in eq. [5] With 4 figures     

The solitary waves in a gradually varying channel of arbitrary cross-section

In this paper, the solitary waves in an arbitrary cross-section channel which gradually changes in the streamwise have been studied. TheKdV equation with slowly varying coefficients is derived. Thus, we produced the first term of its asymptotic solution, travel speed of soiltary waves and the relation between the amplitude of wave and the geometric size of channel. The results have been applied to the cases of triangular and rectangular channels. For the channel with varying depths and breadths they are fairly consistent with those of Johnson, Shuto and Mile.     

A variational problem in the hydrodynamics of open channels

A solution is found to the problem of the control of the unsteady motion of water in an open channel bounded by a pumping station. The law of variation of the discharge of the pumping station which ensures the least cost of pumping water from the channel is determined. The solution is found by the method of undetermined Lagrangian multipliers. Examples of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 159–163, July–August, 1980.     

Heat conduction problem for tightly compressed adjacent layers of plane plates with gradually varying heat conduction coefficients

In dealing with the heat conduction problem of the tightly compressed adjacent plane plates with different constant thermal conduction coefficients, we conventionally solve it layer by layer requiring to fulfill two requirements of equal temperature and equal heat flux at every two directly contacting intermediate surfaces.In this paper, we consider only such a case that the adjacent plane plates possess a gradually varying thermal conduction coefficients in order to offer a chance to solve such problem once for all instead of pursuiting the old tedious and time-consuming procedure, i.e., by solving the problem layer by layer.     

明渠气二相流的双流体模型

从解决平均二相流基本方程的封闭问题出发,分析了水利工程中气水二相流的特点,介绍了据此提出的可用于水利工程稀疏气泡流计算的双流体模型,并对明渠气水二相流进行了算例验证。     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Hydraulic calculation of steady uniform flows in trapezoidal compound open channels

Hydraulic calculation of steady uniform flows in trapezoidal compound open channels is studied. Based on the force balance of water in each sub-section, the average velocities of the main channel, side slope, and floodplain are derived. The lateral momentum exchanges between the sub-sections are expressed by using the apparent shear stress. To verify the model, seven groups of UK Flood Channel Facility （UK-FCF） measured data with a relative water depth between the floodplain and the main channel varying from 0.057 to 0.4 are used for comparison. The result shows that the calculated velocity is larger than the measured data when the relative water depth is small, while it is less than or close to the measured value in the case of a larger relative water depth. The influence of the apparent shear stress on the calculation of velocity on the floodplain is not obvious, while it is much greater on the main channel. The three-stage model is compared with Liu’s two-stage model, showing that the former can give a better prediction for a three-stage trapezoidal compound channel. Finally, the apparent shear stress is calculated and compared with the measured data. The result shows that the chosen values of the momentum transfer coefficients are appropriate.     

Numerical simulation of turbulent flows in trapezoidal meandering compound open channels

Three‐dimensional (3D) numerical study is presented to investigate the turbulent flow in meandering compound open channels with trapezoidal cross‐sections. The flow simulation is carried out by solving the 3D Reynolds‐averaged continuity and Navier–Stokes equations with Reynolds stress equation model (RSM) for steady‐state flow. Finite volume method (FVM) is applied to numerically solve the governing equations of fluid flow. The velocity magnitude, tangential velocity, transverse velocity and Reynolds stresses are calculated for various flow conditions. Good agreement between the simulated and available laboratory measurements was obtained, indicating that the RSM can accurately predict the complicated flow phenomenon. Comparison of the calculated secondary currents of four cases (one being inbank flow and other three being overbank flow) with different water depths reveals that (i) the inbank flow exhibits different flow behaviors from that of the overbank flow does and (ii) the water depth has significant effects on the magnitude and direction of secondary currents. Copyright © 2010 John Wiley & Sons, Ltd.     

Geometric stability criteria for certain non-linear time varying systems

Explicit geometric criteria for the exponential stability of a non-linear feedback system (consisting of a time-invariant block G in the forward path and a non-linear time varying gain φ.k(t) in the feedback path) are presented when φ(.) belongs to certain classes of non-linear functions. The resulting bound on [$\text{(}\text{dk}\text{dt}\text{)}\text{k}$] is less stringent than those found in the literature.     

Mixed convection heat transfer in open ended channels with protruding heaters

Mixed convection heat transfer has been studied in vertical channels, open at the bottom and top, with protruding discrete heaters installed on one side. The flow is assumed to be steady, laminar and two-dimensional. The Boussinesq approximation is used to account for the density variation. Non-dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the SIMPLER method. Heat transfer through the top and the right are calculated as functions of the Rayleigh number (0≤Ra≤10⁷), the Reynolds number (0≤Re≤200), various aspect ratios (1≤A≤6). The effect of the entrance and exit lengths and that of the position of the electronic components in the channel are also examined. Flow and temperature fields for various cases are produced, and the temperature variations in the electronic components are calculated. Received on 2 March 1998     

Flow characteristics of rectangular open channels with compound vegetation roughness

An idealized parallel flow caused by a lateral bed roughness difference due to the partial vegetation across a channel is investigated. Similar to the flow in a compound channel, there are mixing layers adjacent to the interface between the vegetation and the non-vegetation lanes, and a lateral momentum exchange occurs between the slow-moving water in the former lane and the fast-moving water in the latter lane. Under a uniform flow condition, the three-dimensional (3D) instantaneous velocities of two cases with different discharges and water depths are measured with a 16MHz acoustic Doppler velocimeter (micro ADV). The longitudinal variation of the streamwise velocity and the vertical variation of the Reynolds stress are analyzed. A quadrant analysis is carried out to investigate the outward and inward interaction, ejection, and sweep phenomenon caused by the vegetation variation across the channel. The results show that the flow characteristics in the vegetation lane are similar to those in an open channel fully covered with submerged vegetation, and the flow characteristics in the smooth non-vegetation lane are similar to those in a free open channel. For the cases studied here, the width of the mixing region is about 10% of the channel width, and the mixing region is mainly on the non-vegetation half.     

Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions

The dynamic stability of axially accelerating plates is investigated. Longitudinally varying tensions due to the acceleration and nonhomogeneous boundary conditions are highlighted. A model of the plate combined with viscoelasticity is applied. In the viscoelastic constitutive relationship, the material derivative is used to take the place of the partial time derivative. Analytical and numerical methods are used to investigate summation and principal parametric resonances, respectively. By use of linear models for the transverse behavior in the small displacement regime, the plate is confined by a viscous damping force. The generalized Hamilton principle is used to derive the governing equations, the initial conditions, and the boundary conditions of the coupled planar vibration. The solvability conditions are established by directly using the method of multiple scales. The Routh-Hurwitz criterion is used to obtain the necessary and sufficient condition of the stability. Numerical examples are given to show the effects of related parameters on the stability boundaries. The validity of longitudinally varying tensions and nonhomogeneous boundary conditions is highlighted by comparing the results of the method of multiple scales with those of a differential quadrature scheme.