Simulation of flow and pressure variations in fluid pipelines using finite difference and finite element models can give unrealistic results, corresponding to errors in natural frequencies. A novel finite element model of hydraulic pipelines has been developed, using an interlacing grid system. The grid spacing is non-uniform and is optimized, using a genetic algorithm, to make some or all of the undamped natural frequencies of the model as close as possible to exact theoretical ones for a uniform pipe with the extreme boundary conditions of either constant pressure or no flow. Inaccuracies in the highest natural frequencies may be acceptable because of the effect of frequency-dependent friction and limited system frequency response. The optimized model gives accurate results in time domain simulation, and it allows variable properties and a variable integration step to be readily accommodated.

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Hydraulic pipeline dynamics play an important role in many real systems, for example in Diesel fuel injection, anti-skid braking, under-sea oil production and hydraulic control in general. When investigating the behaviour of such systems by simulation it is necessary to have suitable numerical models for the line. An excessively complicated model wastes computing time, but an inadequate one fails to predict the behaviour satisfactorily. A first step in deciding upon a suitable model is to look at the relationship between the frequency content of transient operations in the system and the natural frequencies of the lines. Rapid transients can be initiated by such things as valve operation, actuators reaching the end of their travel and cavitation and air release. A second important criterion is the amount of damping in the lines caused by fluid friction. This may be steady or frequency dependent friction in laminar or turbulent flow. On the one hand, increasing the complexity of the friction model employed makes the computations more involved. On the other hand, friction may limit the amount of higher frequency motion and consequently make the computations simpler. This paper examines the requirements for line models under different circumstances, and how these requirements can be met. Discussions are illustrated with results on practical systems. NOMENCLATURE Fluid Power. Edited by T. Maeda. (c) 1993 E & FN Spon. ISBN 0 419 19100 3.

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SUMMARYNumerical solutions for two-dimensional or axisymmetric viscous fluid flow problems are usually based on the stream function/vorticity formulation. Frequently, however, the pressure distribution is of prime interest. Difficulties have been reported in the literature with the use of obvious pressure marching schemes. Consequently, several investigators have preferred to use an iterative method which involves solving a Poisson equation with Neumann boundary conditions. In this paper, the fundamental cause of failure of the marching schemes is investigated. The authors introduce the concept of compatible pressure and vorticity schemes and show that lack of compatibility has been the principal reason for the poor results obtained using marching schemes. Compatible pressure marching methods are developed and shown to give good results. Comparisons are made between the Poisson equation method and the compatible marching method. To make the comparisons meaningful, special test cases with analytical solutions have been used.