A boundary layer method for obtaining an approximate equation for the unsteady velocity-time history of flow startup in a pipe

An incompressible fluid volume is initially at rest in a pipe. A valve is suddenly opened and the fluid accelerates from rest to the steady state velocity. The equation describing the accelerating motion is a first order nonlinear equation containing friction, pressure drop and kinetic energy terms. The solution to this equation yields a relationship between average velocity and time. The traditional solution method involves assuming friction factor is a constant, and equal to the steady state value. The equation can be solved numerically using a difference scheme, which does not involve assuming a constant friction factor. In this study, a method of generating a velocity-time equation in polynomial form is developed to obtain an alternative solution. The method is used to solve two specific examples and the results obtained are compared with the numerical solution method. The comparison yields a small error, and so the polynomial approximation method can be used as an alternative technique to describe the motion of an accelerating fluid volume