Unsteady Pipe Flow Research Articles

Discrepancies between Theory and Experiment for a Laboratory Surge Tank Rig

This paper deals with the discrepancies between experimental results and the simple theory usually used for an undergraduate laboratory surge tank rig. It has been found that most are due to the ‘annular effect’ which occurs in unsteady laminar pipe flow. The results will also be of interest when interpreting physical model tests of full size surge tank systems.

Unsteady roof gutter flow: Development and application of simulation

The flow in a roof gutter during and following a storm is unsteady free surface flow, however design guides concentrate on the determination of safe' flow capacities based on steady state data. The development of partially filled unsteady pipeflow simulations aimed at building drainage design also make possible simulations capable of defining the roof gutter flow response to both time and location dependent lateral inflows by means of a method of characteristics solution of the governing St Venant equations. This paper details the developed simulation and compares its predictions to both historic data and to recently gathered storm data. It concludes that such simulations are advantageous and address a number of important parameters omitted from current design guides, including the need to improve on gutter roughness simulation by means of the Colebrook-White frictional representation and to determine gutter response times to variations in rainfall intensity.

Approximate computational model for sound generation due to unsteady flows in pipes

A model for the generation of sound due to unsteady vortical flow in pipes of varying cross-sectional area is described. The model is based upon Howe’s acoustic analogy and an approximate model for the vorticity field. This formulation allows the sound generation to be specified in terms of properties of the unsteady behavior of the vortical flow and the potential flow solution for the pipe that would exist in the absence of vortical inhomogeneities. The approximate model for vorticity formation and evolution is presented, and the potential flow solution is obtained from the axial duct shape. Computed estimates of the time evolution of both a jet passing through a pipe constriction and the sound this flow produces are presented and compared to experimental data. [Research supported by NSF/ARPA IRI-9314946 and ARPA DAST 63-93-C-0064.]

Test cases for unsteady one-dimensional flow with variable cross-section

The unsteady, inviscid flow generated by the passing of a shock through a nozzle can be computed exactly in the limiting case of a nozzle of zero length connecting two infinitely long pipes. Depending on the values of two parameters (strength of shock and cross-sectional ratio) this problem displays eight different types of smooth flow, and four further types are generated by separated flow through a sudden enlargement. It is shown a) how to compute these variants and b) how to use them for the testing of numerical algorithms for unsteady pipe flow with variable cross-section.

Approximate model for sound generation due to unsteady flows in pipes

An approximate model for the generation of sound due to unsteady vortical flow in pipes of varying cross-sectional area is described. This model, which is being used in speech synthesis research, is based upon Howe’s acoustic analogy. This formulation allows the sound generation to be specified in terms of properties of the unsteady behavior of a vortical flow and the potential flow solution for the pipe that would exist in the absence of vortical inhomogeneities. An approximate model for vorticity formation and evolution is presented, and the potential flow solution is obtained from the axial duct shape. Computed estimates of the time evolution of the sound generated by a confined jet passing through a pipe constriction are presented. These results are compared to experimental data. [Research supported by NSF/ARPA IRI-9314946 and ARPA DAST 63-93-C-0064.]

A perturbation solution to the transient pipe flow problem

The analysis of hydraulic transients in pipes is traditionally carried out numerically. An approximate analytical solution of the nonlinear problem of unsteady flow in pipes is presented herein. The continuity and momentum equations are combined to give a nonlinear second-order hyperbolic partial differential equation whose approximate solution is derived using a perturbation technique called the delta expansion. It is shown that the zeroth-order perturbation solution gives ample accuracy for practical application. Useful series expansions for small values of the waterhammer parameter k are also obtained which compare extremely well with the results of the method of characteristics especially at small times. The explicit and simple form of the solution is suitable for further mathematical analysis and manipulation such as determining the maximal values and their time of occurrence. They are especially useful in determining the minimum time of closure possible while keeping the transient pressure in the pi...

Unsteady axial viscoelastic pipe flows

The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: 1. (a) when the pressure gradient varies exponentially with time; 2. (b) when the pressure gradient is pulsating; 3. (c) a starting flow under a constant pressure gradient. In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e ± α 2 τ . The behaviour of the flow field depends to a large extent on β where β 2 = α 2(1 ± Hα 2) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small | βη|, η being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large | βη| the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance. The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos nτ. Again the flow depends to a large extent on a parameter β ( β 2 = i n − n 2 H). For small values of | βη| the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large | βη| the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period. The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.

円筒絞りの非定常キャビテーション特性

As regards fluid power systems, cavitation under unsteady flow conditions plays an important role regarding their dynamic characteristics. To clarify the unsteady cavitation mechanisms, as the first step, cavitation of long orifices in unsteady flows was studied. Sinusoidal and stepwise flows were obtained from a variable displacement piston pump. Long orifices made of steel were 2.0mm and 2.5mm in diameter, and the length/diameter ratios were 5.0 and 4.0 respectively. The fluid tested was ISO VG 32.Gavitation was detected mainly in the pressure ripples obtained by the pressure sensor positioned on the outlet side of the orifice. Also, a strobo-light and TV camera was used to observe the cavitation clouds.The test results, related to inception and desinence of cavitation choking, were mainly shown as the relationship between cavitation number and the amplitude and frequency (0.0-1.0Hz) in case of the sinusoidal flow, and the step height of the stepwise flow. In conclusion, it could be shown that the cavitation in an unsteady flow occurs much easier than that in a steady flow and discontinuous cavitation choking with stepwise flow input could be explained by the numerical results for the unsteady pipe flows.

An implicit method for water flow modelling in channels and pipes

An implicit time integration method for the simulation of steady and unsteady flow in pipes and channels is presented. It is based on the theory of Total Variation Diminishing (TVD) methods. A conservative linearization leads to a block tridiagonal system of equations which can be cheaply solved by means of a non-iterative matrix decomposition method. It keeps the advantages of classical implicit schemes, and properly deals with all kinds of flow. It is specially suited for steady flows including phenomena such as hydraulic jumps and also gives satisfactory results for time dependent problems. Several test cases are shown to illustrate the performance of this implicit technique in single channels and networks.

A direct theory of viscous fluid flow in pipes I. Basic general developments

This paper is concerned w ith the construction by a direct approach of a fairly general nonlinear theory of viscous fluid flow in both straight and curved pipes. First, a new procedure is used to establish a 1-1 correspondence between the lagrangian and eulerian form ulations of the integral balance laws of a general thermom echanical theory ofdirected(orCosserat) curve in the presence of a number of directors. (Such correspondence between lagrangian and eulerian formulations was previously possible in the special case of two directors w ith the resulting theory appropriate, for example, only for free surface jets.) The utility of the basic theory, which is valid for both compressible and incompressible linear viscous fluids, is then demonstrated with reference to the unsteady flow of an incompressible viscous fluid in straight pipes of varying elliptical cross section. A general solution is obtained for unsteady flow in straight pipes of elliptical cross section, and is applied to the cases in which a swirling motion is superposed on a uniform axial flow and a flow which is symmetric about the m ajor and minor axes of the elliptical cross section.

Transformation calculus of the variables in an unsteady pipe flow.

If the flow is laminar and axially symmetrical, transfer functions which relate variables in unsteady pipe flow can be obtained in the form of functions which contain the Bessel function. The author has presented the approximate functions of the transfer functions in another report. The present report explains the transformation calculus which is the technique for the estimation of one variable from another variable in unsteady pipe flow. By applying the approximate transfer function, the transformation calculus becomes quite efficient. The technique of transformation calculus explained in this paper is expected to be applied to the indirect measurement of the variables in unsteady pipe flow.

Approximate transfer function relating to variables in an unsteady pipe flow.

For fluid control and measurement in an unsteady pipe flow, it is useful to estimate one variable precisely from others which can be obtained briefly. This is done using the transformation calculus. In the case of laminar and axially symmetrical flow, transfer functions which relate pressure gradient, mean flow velocity and local velocity at a specified point on the same cross section of pipe contain the Bessel function. In this paper, good approximate functions of the transfer functions are obtained in the simple form of linear combination of the proportional element, the first order lag element and the element based on the second order lag element. by the use of approximate transfer functions, an efficient transformation calculus of the variables in unsteady pipe flow can be realized.

管内非定常流れの解析と計測に関する調査研究分科会報告

管内非定常流れの解析と計測に関する調査研究分科会報告

Applicability of the Colebrook-White Formula to Represent Frictional Losses in Partially Filled Unsteady Pipeflow.

The use of Manning's n as a friction factor is shown to be unsuitable in the case of small bore (less than about one meter diameter) partially filled pipeflow, particularly for relatively smooth materials such as glass and cast-iron. The Colebrook-White equation with the roughness coefficient k is presented in a form suitable for inclusion in a computer program to solve the partially filled unsteady pipeflow equations by means of the method of characteristics. Results are presented which show that the Colebrook-White equation provides substantially improved predictions of the wave velocity along the pipe. It provides slightly improved predictions for the maximum depth of flow along the pipe.

COMPUTATION OF UNSTEADY PIPE FLOW WITH RESPECT TO VISCO-ELASTIC MATERIAL PROPERTIES

Many of the materials used in pipe manufacturing, particularly plastics, prove to be visco-elastic. As the propagation velocity of pressure waves primarily depends on the fluid bulk modulus and the elasticity of the pipe wall the above material property is of great influence in unsteady pipe flow. Furthermore the damping of the waves is caused by frictional losses in the flow as well as by the non-elasticity of the pipe material. The applicability of the method of characteristics - although being a powerful tool for the analysis of a wide range of problems - is limited in the sense that purely elastic strain behaviour has to be assumed. In order to take into account the special properties of visco-elastic pipes two methods for the computation of various types of unsteady flow were developed. By means of the impedance method it is possible to calculate a stable periodic pipe flow and, in particular, the resonance behaviour of conduits. The impulse-responsemethod based on the theory of linear systems is use...

Structure of the transient wall-friction law in one-dimensional models of laminar pipe flows

The problem of describing an unsteady cylindrical pipe flow with one-dimensional equations is investigated, and an exact method for obtaining a closure relationship is proposed for the transient shear stress in a laminar flow submitted to an arbitrary transient pressure gradient. Extensive comparisons are given for a step or a harmonic pressure gradient between the approximate solution derived from this method, some results of the literature and exact solutions of the Navier–Stokes equations.

On elastic effects in unsteady pipe flows

Theoretical consideration is given to two unsteady pipe flows. In the first, a combined steady and oscillatory shear flow is generated by a pulsatile pressure gradient in a stationary pipe. In the second, the pressure gradient is constant, but the pipe wall executes axial vibrations.

A finite difference scheme for unsteady pipe-flows

A description of a new method finite differences for unsteady pipe flows is presented. Boundary conditions are fixed by making small approximations without much loss in accuracy leading to a simplified computer programs for calculations concerning even complicated flows. The method of integration used is the Runge-Kutta fourth or third order numerical scheme.

Turbulent mass transfer to a wall in unsteady pipe flows.

The wall mass transfer in unsteady turbulent pipe flows was studied experimentally. Dynamic responses of mass-transfer coefficient to changes in the wall shear stress were measured by the electrochemical method. The results show that there exists an element of dead time and that the frequency response is universally correlated on the Bode diagram by a dimensionless angular frequency with Schmidt number as a parameter. For an analytical prediction, it is useful to assume pseudo-steady eddy diffusivity which includes an empirical dead time. A detailed investigation of the dead time was also made through studies on temporal characteristics of turbulence in both steady and transient flows. As a result, the occurrence of dead time is found to be due to the temporal coherence of the structure of turbulence.

Unsteady Effects in Flow Rate Measurement at the Entrance of a Pipe

Unsteady flow in pipes and nozzles occur frequently in engineering applications and they pose special problems of measurement and calibration. When the Reynolds number is high the entrance region of a pipe (following a smooth contraction) is characterized by a thin boundary layer and the unsteady effects are then bound up in the unsteady behavior of the boundary layer. Woblesse and Farrell [1]2 have recently considered unsteady effects in laminar pipe entrance flows that start from rest by an integral method. Periodic disturbances also arise which require a different treatment. The primary interest of the present work is for thin entrance boundary layers subject to peridodic disturbances. In either case the ratio of the average velocity to the velocity in the potential core is V[sub]avg/V[sub]core = 1- 2[delta]*/R [equation 1] where [delta]* is the usual displacement thickness and R is the pipe radius. In steady flow this ratio is just the discharge coefficient, c[sub]d. In unsteady flow it is very desirable to know how this ratio changes with time because many of the presently available experimental methods enable one to measure V[sub]core but not V[sub]avg readily. In this brief note we will estimate the unsteady effects of a periodic, fluctuating main flow on the displacement thickness of a laminar, flat plate boundary layer. It is assumed that the boundary layer is sufficiently thin compared to the radius of a pipe so that the pressure gradient caused by this effect in a pipe can be neglected; the results should then be directly applicable to equation (I).