Efficient Treatment of the Vardy–Brown Unsteady Shear in Pipe Transients

An accurate, simple, and efficient approximation to the Vardy–Brown unsteady friction equation is derived and shown to be easily implemented within a one-dimensional characteristics solution for unsteady pipe flow. For comparison, the exact Vardy–Brown unsteady friction equation is used to model shear stresses in transient turbulent pipe flows and the resulting waterhammer equations are solved by the method of characteristics. The approximate Vardy–Brown model is more computationally efficient (i.e., requires one-sixth the execution time and much less memory storage) than the exact Vardy–Brown model. Both models are compared with measured data from different research groups and with numerical data produced by a two-dimensional turbulence waterhammer model. The results show that the exact Vardy–Brown model and the approximate Vardy–Brown model are in good agreement with both laboratory and numerical experiments over a wide range of Reynolds number and wave frequencies. The proposed approximate model only requires the storage of flow variables from a single time step while the exact Vardy–Brown model requires the storage of flow variables at all previous time steps and the two-dimensional model requires the storage of flow variables at all radial nodes.