Modelling velocity structures in turbulent floods using proper orthogonal decomposition

Abstract

In New Zealand, there are over 100 communities that are flood prone therefore require good models to predict their vulnerability and also the likely flood depths. Previous work has shown that numerical models under predict water levels in the centre of the flow. One of the causes is the wave action that is prevalent in floodwater flows. Therefore it is important to develop a model of wave action.

The approach taken was to develop a model that included the underlying velocity field associated with the waves. To do this the turbulent structures needed to be understood. This meant that the analysis may also provide data about the interaction of the water flow and ground surface and therefore the flow resistance.

There are various techniques available to model the turbulence structure. Turbulence of incompressible water flow (without sediment) can be modelled completely using the Navier-Stokes equations. However for even a small part of a river channel, the size of the problem is many orders of magnitude larger than even the world’s largest super-computers can handle. This means that other techniques are necessary to describe the turbulence. One of these is Proper Orthogonal Decomposition (POD). POD provides a way of decomposing the velocity vectors int modes of different scales in each direction similar to a Fourier series. Therefore it requires detailed data. Particle Image Velocity (PIV) is an ideal method to obtain such data.

An undular hydraulic jumps was chosen as an ideal structure to model as it is a strong wave in water flow and similar to the rooster tail typical of New Zealand rivers shown in figure 1. In this paper, we explain the application of POD to PIV image data of a slice through an undular hydraulic jump or type of standing wave. The data is two-dimensions, in the direction of flow and the vertical.

This is the first time PIV flow data from an undular hydraulic jump has been analysed using POD. We identify flow structures and develop covariance functions from velocity correlations across this random flow field. We discuss the structures within the context of this data and their significance.

This will lead to the development of a model based on the Navier-Stokes equations using the covariance functions to simulate the turbulent fluctuations. As there are a wide range of structures within the hydraulic jump a wide variety of flow situations can be modelled. Improvements could be made by analysis of further PIV data especially from higher Reynolds Numbers.

The model can therefore be run for the various types of situations in a river or a flood plain to predict wave heights which can be used with the results from a hydrodynamic model such as Hydro2de or Mike21. In addition the flow resistance values for these areas may also be able to be calculated to use in a hydrodynamic model. These results can be put into a database of a local authority where it can be used to give building floor levels in flood plains.