In this paper the momentum flux coefficient (ß) is used for the modelling of steady and unsteady head losses in turbulent pipe flows. For this purpose a transport equation is derived, which describes the change of ß in space and time, due to the inertia-driven shift and the shear-driven recovery of velocity profiles. It is demonstrated that ß represents steady friction and it is made plausible thatß represents the turbulence intensity. The key assumption is that the unsteady wall shear stress may be described as a logical extension of the steady wall shear stress and as such is correlated to unsteady head losses. The governing equation for ß is added to the two classical water-hammer equations, where the quasi-steady Darcy-Weisbach friction coefficient is replaced by the dynamic momentum flux coefficient ß (x,t). Thus a convenient friction model is proposed in which steady and unsteady friction are mathematically described in the same manner. Numerical simulations show that the effect of ß as momentum correction factor is negligible, but that its use in an unsteady Darcy-Weisbach type of friction term results in significant damping and rounding of pressure waves.
|Title of host publication
|Proceedings 10th International Conference on Pressure Surges (Edinburgh, UK, May 14-16, 2008)
|Place of Publication
|BHR Group Limited
|Published - 2008