Scaling and self-similarity in one-dimensional unsteady open channel flow

Ercan A., Kavvas M. L., Haltas I.

HYDROLOGICAL PROCESSES, vol.28, no.5, pp.2721-2737, 2014 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 28 Issue: 5
  • Publication Date: 2014
  • Doi Number: 10.1002/hyp.9822
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2721-2737
  • Keywords: scaling, self-similarity, Saint Venant equations, scale-invariance, similitude, INVARIANCE, SPACE, TIME
  • Middle East Technical University Affiliated: No


The conditions under which the Saint Venant equations system for unsteady open channel flow, as an initial-boundary value problem, becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. One of the advantages of this methodology is that the self-similarity conditions due to the initial and boundary conditions can also be investigated thoroughly in addition to the conditions due to the governing equation. The obtained self-similarity conditions are compared with the scaling relationships that are derived through the Froude similitude. It is shown that the initial-boundary value problem of a one-dimensional unsteady open channel flow process in a prototype domain can be self-similar with that of several different scaled domains. However, the values of all the flow variables (at specified time and space) under different scaled domains can be upscaled to the same values in the prototype domain (at the corresponding time and space), as shown in this study. Distortion in scales of different space dimensions has been implemented extensively in physical hydraulic modelling, mainly because of cost, space and time limitations. Unlike the traditional approach, the distinction is made between the longitudinal-horizontal and transverse-horizontal length scales in this study. The scaled domain obtained by the proposed approach, when scaling ratios of channel width and water depth are equal, is particularly important for the similarity of flow characteristics in a cross-section because the width-to-depth ratio and the inclination angles of the banks are conserved in a cross-section. It is also shown that the scaling ratio of the roughness coefficient under distorted channel conditions depends on that of hydraulic radius and longitudinal length. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal and economical flexibility in setting up physical hydraulic models. Copyright (c) 2013 John Wiley & Sons, Ltd.