Author(s): Alastair Barnett

Linked Author(s):

Keywords: Scalar 3D; Energy Modelling; Hydraulic jump; Shock Losses; Resistance Losses

Abstract: Flood flows arrive through channels in response to inflow increases or surge waves. Natural channels are irregular, so must be defined in a model by 3D description of the channel bed. Such 3D models may be physical (as in a scale model in a laboratory), or computational, but in both cases water boundaries at the open ends of a channel reach and any free surfaces complete the definition of a control element used as the conceptual basis of the model. Within each element, a balance equation will express a conservation law if inflow – outflow = change in storage. Well-known conservation laws balance energy, momentum and mass in this way. Energy and mass are scalar, producing only one 3D balance equation each, while momentum is a vector quantity, producing three 3D balance equations, one for each of three vector components. In Newtonian mechanics, mass balances are always accurate, but both energy and momentum balances are inaccurate, requiring correction by the introduction of friction. Mechanical energy balances also need to allow for conversions to and from non-mechanical energy. For example, electrical energy may be converted from mechanical energy through a dynamo, then returned to mechanical energy through an electric pump. After such allowances, the residual corrections required to make up the balance are called “energy losses” in hydraulics. These losses divide into two categories: resistance losses and shock losses. Resistance energy losses have a direct analogy with bed and wall shear in momentum balances, as both are considered to arise from the action of the channel perimeter on the passing flow. The transfer of shear stress from a fixed perimeter to turbulent flow is the subject of continuing research. Semi-empirical rules such as the Manning formula have been developed with reasonable success, unless an attempt is made to subdivide the control element laterally or vertically by conceptual surfaces to create 2D or 3D sub-elements. Then modelling shear stress faces the additional challenge of arbitrarily divided turbulent flows. Forces normal to the channel bed and walls do no work, and so make no contribution to the energy balance. However, in an irregular channel, normal force components may have a significant effect on the streamwise momentum balance. Proper accounting for the details of these becomes more difficult as the channel topography becomes more complex. Therefore, energy (scalar) modelling has a strong validity advantage over momentum (vector) modelling, except where shock losses occur. This will not happen where resistance losses dominate shock losses, so a criterion can be set up for choosing between vector and scalar modelling based on the ratio of resistance losses to shock losses. If this validity ratio Nv is greater than unity, scalar modelling is more accurate.

DOI: https://doi.org/10.3850/IAHR-39WC252171192022936

Year: 2022