Acceleration-dependent unsteady friction revisited

In the vast majority of unsteady flows in pipes and ducts, unsteady friction plays only a minor role. One exception is when the rate of steepening or flattening of steep fronted waves is important - for instance when high frequency waves are used in leak detection or when wavefronts might steepen into shocks in compressible flows. Another is when damping of successive wave reflections is important. It would be highly desirable to have a reliable method of modelling unsteady friction that (i) is sufficiently accurate in the rare cases when unsteady friction matters, (ii) does not have significant time penalties for software run times and (iii) is reasonably simple to code. Weighting function methods of modelling unsteady friction nearly achieve the first of these objectives for high frequency waves, but do not achieve the other two and they do not even achieve the first objective for sustained periods. In comparison, so-called instantaneous-acceleration methods do achieve the second and third objectives and, with suitable tuning, they can also achieve the first objective except at high frequencies. Thus the two methods are, to a degree, complementary. The paper explores features that the two methods have in common and it shows that both can expressed in two very different forms, one of which is mathematically rigorous and one that is not, but that is nevertheless informative. The primary focus of the paper is on instantaneous-acceleration methods and the various deductions include some surprises (for the author, anyway). Two conclusions are: (1) The most common implementations of instantaneous-acceleration models of unsteady friction do not depend on instantaneous acceleration; (2) A simple generalisation of the popular MIAB method has potential for improving the generality of the method without significantly complicating its use.