Alternative Convolution Approach to Friction in Unsteady Pipe Flow

Abstract

In this 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.