Unsteady Pipe Flow Research Articles

The stability of developing pipe flow at high Reynolds number and the existence of nonlinear neutral centre modes

AbstractThe high-Reynolds-number stability of unsteady pipe flow to axisymmetric disturbances is studied using asymptotic analysis. It is shown that as the disturbance amplitude is increased, nonlinear effects first become significant within the critical layer, which moves away from the pipe wall as a result. It is found that the flow stabilizes once the basic profile has become sufficiently fully developed. By tracing the nonlinear neutral curve back to earlier times, it is found that in addition to the wall mode, which arises from a classical upper branch linear stability analysis, there also exists a nonlinear neutral centre mode, governed primarily by inviscid dynamics. The centre mode problem is solved numerically and the results show the existence of a concentrated region of vorticity centred on or close to the pipe axis and propagating downstream at almost the maximum fluid velocity. The connection between this structure and the puffs and slugs of vorticity observed in experiments is discussed.

Numerical modelling of unsteady turbulent flow in smooth-walled pipes

An efficient numerical model for turbulent friction has been developed for smooth-walled pipe flow. The aim was to develop a new approach to the numerical modelling, eliminating some important approximations and sources of error, such that the method can be applied reliably under a wide range of conditions. A simple two-region model of effective viscosity is used. For short timescales, the turbulence level and effective viscosity distribution are ‘frozen’ in time. The velocity profile is determined numerically for a range of frequencies and viscosity distributions, and this is used to determine the frequency-dependent friction. This is then approximated using simple weighing functions. This turbulence model can be implemented readily in several types of numerical model for pipe flow, including simple lumped parameter models, finite difference/finite element methods, and the method of characteristics.

The linear stability of oscillatory Poiseuille flow in channels and pipes

The linear stability of confined, periodic, parallel fluid flows is examined. The flow fields considered consist of a steady pressure gradient-driven velocity field combined with a purely oscillatory component generated by either an oscillatory pressure gradient or by harmonically oscillating bounding surfaces. Plane channel and circular pipe geometries are studied and all possible combinations of the steady and oscillatory flow components investigated. Neutral stability curves and critical conditions for instability are computed for a selection of steady–unsteady velocity ratios, channel half-widths and pipe radii. The results obtained confirm previous investigations into the effects of small amounts of periodic modulation on the linear stability of the underlying steady flow, but provide much more comprehensive information on the linear stability regions of unsteady parallel flows in channels and pipes.

Unsteady Elbow Pipe Flow to Develop a Flow-Induced Vibration Evaluation Methodology for Japan Sodium-Cooled Fast Reactor

Unsteady Elbow Pipe Flow to Develop a Flow-Induced Vibration Evaluation Methodology for Japan Sodium-Cooled Fast Reactor

A computational study of viscous dissipation and entropy generation in unsteady pipe flow

Numerical simulations of transient entropy generation in a reservoir-pipe-valve system are presented. The flow transient is initiated through sudden closure of the downstream valve. An unsteady two-dimensional water hammer model is adopted. Time integration is performed using the classical fourth-order Runge–Kutta method while the spatial terms are discretized using central difference expressions. Entropy generation is shown to depend on a non-dimensional parameter representing the ratio of the viscous diffusion time scale to the pipe period. For small values of the non-dimensional parameter, entropy generation is rapidly attenuated from its steady-state value to zero while for large values, entropy generation persists for a much longer time. Moreover, for large values of the non-dimensional parameter, excessive entropy generation rates are realized during the transient which are several orders of magnitude higher than the steady-state rate. Such a behavior is attributed to elevated transient shear stress values in the near wall region which result in excessive viscous dissipation and hence higher entropy generation rates. Finally, it is shown that during the transient, the location of maximum entropy generation is no longer restricted to the pipe wall.

Flow Rate Measurement in Unsteady Pipe Flow through a Plate Orifice

Flow Rate Measurement in Unsteady Pipe Flow through a Plate Orifice

Evaluation of Unsteady Wall Shear Stress by Zielke’s Method

The method used in the classical paper by Zielke to estimate the unsteady component of shear stress in unsteady pipe flows is revisited. It is found that the method is undesirably sensitive to the size of the integration time step. The sensitivity is shown to be caused dominantly by the first term in the integration when inadequate allowance is made for the infinite value of the weighting function. A simple method of avoiding the error without requiring the use of small grid sizes is presented.

Influence of time-dependent viscosity on wall shear stresses in unsteady pipe flows

The behaviour of wall shear stresses in unsteady flows in pipes is investigated for flows in which the effective kinematic viscosity varies in time. Such variations can arise through the effects of temperature, pressure, shear or chemical change. The theoretical approach is founded on an analytical method of solution using finite Hankel transforms. It is well suited to studying trends arising in flows with viscosity that is uniform spatially, but varies in time. The method is applied to a range of flows driven by pulsed pressure gradients. The overall wall shear stress is found to follow trends similar to those followed by the mean velocity, but the unsteady component of the wall shear stress is much more strongly influenced by the acceleration. The kinematic behaviour tends to be influenced more strongly by the instantaneous viscosity (or, strictly, by its local temporal average) than by the local rate of change of viscosity.

Nonconservative Formulation of Unsteady Pipe Flow Model

A new approach to numerical modeling of water hammer is proposed. An unsteady pipe flow model incorporating Brunone’s unsteady friction model is used, but in contrast to the standard treatment of the unsteady friction term as a source term, the writers propose a nonconservative formulation of source term. Second-order flux limited and high order weighted essentially nonoscillating numerical schemes were applied to the proposed formulation, and results are in better agreement with measurements when compared with results obtained with standard form.

Unsteady turbulent pipe flow due to water hammer using k–θ turbulence model

The behaviour of unsteady turbulent pipe flow resulting from water hammer is herein numerically studied. An accurate k–θ turbulence model for two-dimensional boundary layers under adverse and favorable pressure gradients (Wilcox 1994) was applied. The results of the numerical method are in good agreement with test data obtained from both steady and unsteady turbulent pipe flows. Also, the behaviour and the structure of turbulence in unsteady turbulent pipe flows caused by water hammer were investigated. A non-dimensional parameter P as the time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed introduced by Ghidaoui et al. (2002) was used. The results show that for P ≈ 1, the turbulence structure is significantly changed in the first cycle of water hammer, which is important for water quality modeling, whereas these variations are small for P ≫ 1.

Unsteady turbulent pipe flow due to water hammer using k-ω turbulence model

Unsteady turbulent pipe flow due to water hammer using k-ω turbulence model

Unsteady flows in pipes with finite curvature

Motivated by the study of blood flow in a curved artery, we consider fluid flow through a curved pipe of uniform curvature, δ, driven by a prescribed oscillatory axial pressure gradient. The curved pipe has finite (as opposed to asymptotically small) curvature, and we determine the effects of both the centrifugal and Coriolis forces on the flow. In addition to δ, the flow is parameterized by the Dean number, D, the Womersley number, α, and a secondary streaming Reynolds number, Rs. Asymptotic solutions are developed for the case when δ≪1, α≪1 and the magnitude of the axial pressure gradient is small, using regular perturbation techniques. For intermediate values of the governing parameters, a pseudospectral code is used to obtain numerical solutions. For flows driven by a sinusoidal pressure gradient (D=0), we identify three distinct classes of stable solutions: 2π-periodic symmetric, 2π-periodic asymmetric, and asymmetric solutions that are either quasi-periodic, or periodic with period 2πk for k∈ $\mathbb{N}$. The transition between solutions is dependent on the value of δ; thus pipes with finite curvature may exhibit qualitatively different transitions between the solution classes as the governing parameters are varied from those of curved pipes with asymptotically small curvature. When α≫1, matched asymptotic expansions are used to simplify the system, and the resulting equations are solved analytically for Rs≪1, δ≪1 and numerically for larger parameter values. We then determine the effect of a non-zero steady component of the pressure gradient (D≠0), and show that, for certain parameter values, when D is above a critical value the periodic asymmetric solutions regain spatial symmetry. Finally, we show that the effects of finite curvature can lead to substantial quantitative differences in the wall shear stress distribution and discuss briefly the physiological implications of the results for blood flow in arteries.

Temperature variations in the simulation of high-pressure injection-system transient flows under cavitation

Temperature variations and their effects on the simulation of unsteady pipe flows, in the presence of pressure-wave induced cavitation, were investigated with reference to high-pressure fuel injection systems. The thermal effects due to the compressibility of the liquid and to the thermodynamic process in the cavitating flow mixture were analyzed. To that end, the energy conservation equation was applied, in addition to the mass-continuity and momentum-balance equations, along with the constitutive state equation of the fluid. In particular, for the liquid, the physical properties (i.e., bulk modulus of elasticity, density, isothermal speed of sound, thermal expansivity, kinematic viscosity, specific heat at constant pressure) were implemented as functions of pressure and temperature in a closed analytical form matching carefully determined experimental data. Consistent with virtually negligible combined effects of heat transfer and viscous power losses involved in the flow process, the equation of energy was reduced to a state relation among the fluid thermodynamic properties, leading to a barotropic flow model. A comparison between isentropic and isothermal evolutions in the pure liquid regions was carried out for evaluating the influence of the temperature variation simulation on the macroscopic results given by local pressure time-histories. Besides, for cavitation analysis, different thermodynamic transformations of the vapor–liquid mixture were considered and compared.A recently developed conservative numerical model of general application, based on a barotropic flow model, was applied and further assessed through the comparison of prediction and measurement results on injection-system performance.A conventional pump-line-nozzle system was considered for this purpose, being relevant to model evaluation for its pressure-wave dynamics and also because it was subject to severely cavitating flow conditions at part loads. Predicted time-histories of injector-needle lift and pressure at two pipe locations were compared to experimental results. This substantiated the validity and robustness of the conservative model taking temperature variation effects into account, in the simulation of high-pressure injection-system transient flows with great degree of accuracy, even in the presence of cavitation induced discontinuities. The thermal effects due to the temperature variations in the liquid fuel and in the cavitating mixture were analyzed and discussed.

Generation of Unsteady Pipe Flow with Arbitrary Waveform by Butterfly Valve

Generation of Unsteady Pipe Flow with Arbitrary Waveform by Butterfly Valve

Approximation of Turbulent Wall Shear Stresses in Highly Transient Pipe Flows

Theoretical predictions of wall shear stresses in unsteady turbulent flows in pipes are developed for all flow conditions from fully smooth to fully rough and for Reynolds numbers from 103 to 108 . A weighting function approach is used, based on a two-region viscosity distribution in the pipe cross section that is consistent with the Colebrook–White expression for steady-state wall friction. The basic model is developed in an analytical form and the resulting weighting function is then approximated as a sum of exponentials using a modified form of an approximation due to Trikha. A straightforward method is presented for the determination of appropriate values of coefficients for any particular Reynolds number and pipe roughness ratio. The end result is a method that can be used relatively easily by analysts seeking to model unsteady flows in pipes and ducts.

On the comparison of the methods used for the solutions of the governing equation for unsteady unidirectional flows of second grade fluids

In order to solve the governing equation for unsteady unidirectional flows of second grade fluids, the use of the Laplace and the Fourier transform methods are discussed. Three characteristic examples which are unsteady flow between two parallel plates, unsteady pipe flow and unsteady flow over a plane wall are considered. It is found that the solution for unsteady flow in bounded regions obtained by the Laplace and the Fourier transform methods are exactly the same as the case of the unsteady flow of a Newtonian fluid. It is shown that the Laplace transform method for small values of time is useful for flows of Newtonian fluids but it is not convenient for flows in unbounded regions of second grade fluids. Furthermore, it is explained that for some unsteady flows of second grade fluids, the solution obtained by using the Laplace transform does not satisfy the initial condition and therefore the Fourier transform method is used.

Numerical aspects of improvement of the unsteady pipe flow equations

AbstractThe paper presents an analysis of some recently proposed improvements of the water hammer equations, which concern the friction term in the momentum equation. A comparison of the experimental data and numerical results shows that the required damping and smoothing of the pressure wave cannot be obtained by modification of the friction factor only.In order to evaluate the significance of the introduced improvements into the momentum equation, the accuracy of the numerical solution has been analysed using the modified equation approach. The analysis shows why the physical dissipation process observed in the water hammer phenomenon cannot be reproduced with the commonly used source term in Darcy–Weisbach form, representing friction force in the momentum equation. Therefore, regardless of the proposed form of the friction factor for unsteady flow, the model of water hammer improved in such a way keeps its hyperbolic character. Consequently, it cannot ensure the expected effects of damping and smoothing of the calculation head oscillations. Copyright © 2007 John Wiley & Sons, Ltd.

Perturbation dynamics in unsteady pipe flows

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

Runge–Kutta time‐stepping schemes with TVD central differencing for the water hammer equations

AbstractIn the present study, Runge–Kutta schemes are used to simulate unsteady flow in elastic pipes due to sudden valve closure. The spatial derivatives are discretized using a central difference scheme. Second‐order dissipative terms are added in regions of high gradients while they are switched off in smooth flow regions using a total variation diminishing (TVD) switch. The method is applied to both one‐ and two‐dimensional water hammer formulations. Both laminar and turbulent flow cases are simulated. Different turbulence models are tested including the Baldwin–Lomax and Cebeci–Smith models. The results of the present method are in good agreement with analytical results and with experimental data available in the literature. The two‐dimensional model is shown to predict more accurately the frictional damping of the pressure transient. Moreover, through order of magnitude and dimensional analysis, a non‐dimensional parameter is identified that controls the damping of pressure transients in elastic pipes. Copyright © 2006 John Wiley & Sons, Ltd.

計測融合シミュレーションを用いた非定常管内流れ場のモニタリング

計測融合シミュレーションを用いた非定常管内流れ場のモニタリング