Romuald Szymkiewicz

Alternative Convolution Approach to Friction in Unsteady Pipe Flow

In the paper some aspects of the unsteady friction in pipe flow expressed by the convolution are analyzed. This additional term introduced into the motion equation involves the accelerations of fluid occurring in the past and a weighting function. The essence of such approach is to assume the appropriate form of weighting function. However, until now no fully reliable formula for this function has been found. To avoid some inconveniences typical for the commonly used weighting functions, an alternative form of the convolution is proposed. Instead of a weighting function an impulse response function in a general form is introduced. This function, defined in the real time domain, having clear physical interpretation and some useful properties is not related to the usually assumed viscosity distribution over the pipe’s cross section. The proposed approach involves two parameters. The convergence of the impulse response function, characterized by the flow memory, is determined by a parameter which can be related to the pressure wave frequency. The second parameter determines the magnitude of the unsteady friction force. The proposed alternative convolution approach was tested basing on the laboratory measurements for a water hammer event initiated by turbulent flow in pipes made of steel. Although the alternative convolution approach causes a very good damping of the pressure wave amplitude, it appears to be unable to ensure appropriate smoothing of the pressure heads. This is because it acts in the dynamic equation as a source/sink term. To ensure the required smoothing of the pressure wave the diffusive term was included into the dynamic equation.

Wave Damping and Smoothing in the Unsteady Pipe Flow

AbstractA modification of the governing equations for unsteady pipe flow is proposed based on the results of many experiments carried out for steel and HDPE pipes of various lengths. Strong damping and smoothing of the pressure wave observed in the experiments suggest that these effects are caused not only by the liquid viscosity, represented in the governing equations by an algebraic term, but also by some other processes that are not described by these equations. The phenomena observed in physical experiments, such as time-varying pressure wave celerity and smoothing of the wave front, cannot be reproduced by the standard unsteady pipe flow model even if various modifications of the formula for shear stress available in the literature are applied. For this reason, an attempt to take into consideration additional dissipative processes was undertaken. On the basis of an approximate model of the elastic behavior of the liquid and wall pipe material, an approach accounting for the variable pressure wave cel...

Numerical Solution of Ordinary Differential Equations

The initial value problem and the boundary value problem for the ordinary differential equations are discussed in this chapter. Derivation of simple numerical methods as well as a general approach for approximating formulas is described. The basic numerical methods applicable for open channel flow (i.e. for non-uniform grid) are developed. The problem of accuracy and stability including A-stability is discussed. Time integration of the systems of ordinary differential equations arising while solving the partial differential equations using the finite element method is described. A couple examples illustrating application of the presented methods for solving some typical problems of open channel hydraulics are included. At the end of chapter the shooting method and the difference method for solving the boundary value problem are presented.

Reservoir Influence on Pressure Wave Propagation in Steel Pipes

AbstractThe relation between the length of steel pipe and the pressure wave period in the system, reservoir-pipe-valve, is discussed. Experiments carried out for the short steel pipes showed that in such a system, the pressure wave celerity estimated using the measured wave cycles systematically increases with the increase of the pipe length. The observed differences can exceed even 90  m/s. As the pressure wave traveling in the pipe with a speed dependent on the pipe and fluid properties is reflected from the reservoir, one can expect that this process is not as ideal as it is assumed, but it needs some time to happen. This causes a wave cycle to increase so that an apparent wave celerity occurs. For determination of the real value of the wave celerity, a modified laboratory installation was designed. This facility is constituted by a pipe closed at its both ends and divided into two parts by the third valve installed midlength. Water in both parts of the pipe is subjected to different hydrostatic pressu...

One-Dimensional Modeling of Flows in Open Channels

In this chapter, modeling of the unsteady open channel flow using one-dimensional approach is considered. As this question belongs to the well-known and standard problems of open channel hydraulic engineering, comprehensively presented and described in many books and publications, our attention is focused on some selected aspects only. As far as the numerical solution of the governing equations is considered, one can find out that essentially there are no problems with provided accuracy. Usually, the implementation of the Saint Venant equations (i.e. the full dynamic wave model) for any case study is successful as long as the basic assumptions introduced during their derivation are fulfilled. Otherwise, some computational difficulties can occur. For this reason we would like to draw the readers’ attention only to such situations when special computational tricks or simplification of the governing equations should be applied.

Methods for Solving Algebraic Equations and Their Systems

This chapter presents some basic numerical techniques to solve nonlinear algebraic equations and systems of linear and nonlinear equations. For non-linear algebraic equations the bisection, false position, Newton, fixed point iteration and some hybrid methods are described. Application of these methods is shown for typical open channel problems, like computation of the normal depth, critical depths or the depth of water over sharp-crested weir. Next the standard methods of solution of the system of linear equations are presented. The last section of the chapter is devoted to the solution of systems of non-linear equations, including the Newton and Picard iterative methods.

Numerical Solution of the Advection Equation

This chapter presents a number of schemes for solution of 1D advection equation, which are based on the finite difference method, the finite element method and the method of characteristics. The roots of numerical errors in the form of numerical diffusion and dispersion generated in the solution of hyperbolic equations are discussed. The method of analysis and graphical presentation of the numerical properties of applied schemes is described. The chapter ends with basic information on the accuracy analysis using the modified equation approach.

Partial Differential Equations of Hyperbolic and Parabolic Type

This chapter is devoted to the partial differential equations applicable in open channel hydraulics, which can be of hyperbolic or parabolic type. The role of characteristics for hyperbolic equations is underlined. The conditions of well posed solution problem for both types of equations are presented. The finite difference and the finite element methods are introduced. The chapter ends with basic information on the convergence, consistency and stability.

Steady Gradually Varied Flow in Open Channels

This chapter begins with derivation of the governing equations. Instead of the ordinary differential equation with regard to depth, commonly used for prismatic channels, the ordinary differential energy equation is proposed. It is showed that the standard step method used for computation of the flow profiles in natural channel is in fact the differential energy equation integrated numerically with the implicit trapezoidal rule. Consequently this approach is applicable for solving the steady varied flow in both prismatic and non-prismatic channels. Analysis of the non-linear equations obtained while solving the energy equation showed that it can have one, two or even three roots. Appropriate choice of the root allows us to obtain all types of the flow profiles occurring in open channels. The same approach based on the energy equation is developed for a single channel as well as for a channel network of both branched and looped types. Several examples illustrate the possibilities of the method.

Numerical Integration of the System of Saint Venant Equations

This chapter begins with brief review of the numerical methods applicable for the Saint Venant equations. Detailed description of the finite difference Preissmann scheme and of the modified finite element method used for a channel with fixed bed is provided. For both methods stability analysis using the Neumann approach and accuracy analysis using the modified equations approach is carried out. The problem of the boundary condition required at the downstream end is discussed. Some practical aspects of solution the unsteady flow equations are underlined. Application of the Saint Venant equations for particular cases as flow in channel with moveable bed and propagation of the steep waves are presented as well. Numerous examples of solution show the flexibility and large area of application of the Saint Venant equations.