Unstable equilibrium : modelling waves and turbulence in water flow

This thesis develops a one-dimensional version of a new data driven model of turbulence that uses the KL expansion to provide a spectral solution of the turbulent flow field based on analysis of Particle Image Velocimetry (PIV) turbulent data. The analysis derives a 2nd order random field over the whole flow domain that gives better turbulence properties in areas of non-uniform flow and where flow separates than the present models that are based on the Navier-Stokes Equations. These latter models need assumptions to decrease the number of calculations to enable them to run on present day computers or super-computers. These assumptions reduce the accuracy of these models. The improved flow field is gained at the expense of the model not being generic. Therefore the new data driven model can only be used for the flow situation of the data as the analysis shows that the kernel of the turbulent flow field of undular hydraulic jump could not be related to the surface waves, a key feature of the jump. The kernel developed has two parts, called the outer and inner parts. A comparison shows that the ratio of outer kernel to inner kernel primarily reflects the ratio of turbulent production to turbulent dissipation. The outer part, with a larger correlation length, reflects the larger structures of the flow that contain most of the turbulent energy production. The inner part reflects the smaller structures that contain most turbulent energy dissipation. The new data driven model can use a kernel with changing variance and/or regression coefficient over the domain, necessitating the use of both numerical and analytical methods. The model allows the use of a two-part regression coefficient kernel, the solution being the addition of the result from each part of the kernel. This research highlighted the need to assess the size of the structures calculated by the models based on the Navier-Stokes equations to validate these models. At present most studies use mean velocities and the turbulent fluctuations to validate a models performance. As the new data driven model gives better turbulence properties, it could be used in complicated flow situations, such as a rock groyne to give better assessment of the forces and pressures in the water flow resulting from turbulence fluctuations for the design of such structures. Further development to make the model usable includes; solving the numerical problem associated with the double kernel, reducing the number of modes required, obtaining a solution for the kernel of two-dimensional and three-dimensional flows, including the change in correlation length with time as presently the model gives instant realisations of the flow field and finally including third and fourth order statistics to improve the data driven model velocity field from having Gaussian distribution properties. As the third and fourth order statistics are Reynolds Number dependent this will enable the model to be applied to PIV data from physical scale models. In summary, this new data driven model is complementary to models based on the Navier-Stokes equations by providing better results in complicated design situations. Further research to develop the new model is viewed as an important step forward in the analysis of river control structures such as rock groynes that are prevalent on New Zealand Rivers protecting large cities