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A backwater method for trans-critical flows
Published in R. A. Falconer, K. Shiono, R. G. S. Matthew, Hydraulic and Environmental Modelling: Estuarine and River Waters, 2019
P. G. Samuels, K. S. Chawdhary
The computation of the steady flow profile for water level and Froude number have been carried out for some test cases examined by Priestley (1990). The geometry represents a rectangular channel 10 km long with width varying smoothly from 10m at either end to 5m in the centre, see Fig 4. The bed slope is uniform except for the central 1 km where the bed gradient is doubled, see Fig 5, this forces transitions between sub- and super-critical flow for some bed gradients. In each test the value of Manning’s n was 0.03, the discharge was 20 m3/s and a step size of 25 m was used. Figs 6, 7 & 8 show the water depth and Froude number for the bed slopes of magnitude 0.002, 0.01 and 0.02 respectively. The results show no oscillation and capture the transitions in depth cleanly. In all cases the calculations proceeded from the downstream end of the channel and passed through the transitions (Fig 7) without any special action to locate these points. The results are nearly indistinguishable from the corresponding plots given by Priestley (1990). Priestley studied the conservative form of the unsteady flow equations using the Riemann invariants of this hyperbolic set of PDEs and produced steady state solutions by applying steady boundary conditions.
Saturation of generalized partially hyperbolic attractors
Published in Dynamical Systems, 2019
A hyperbolic set is a compact invariant set over which the tangent bundle splits into two invariant sub-bundles, one is contracting and the other one is expanding. The Lebesgue measure (volume) of hyperbolic sets is an interesting subject considered in many articles. The scenario begun by the seminal works of Bowen in 1970s. Bowen proved in [5] that a hyperbolic invariant set of positive volume of a -diffeomorphism does contain some stable and unstable manifolds. On the other hand, he showed in [4] the existence of a -diffeomorphism admitting a totally disconnected hyperbolic set of positive volume. The issue of the volume and the interior of a hyperbolic set followed by many authors. For instance, it is shown in [1] that a transitive hyperbolic set which attracts a set with positive volume necessarily attracts a neighbourhood of itself. It is also proved in [1] that there are no proper transitive hyperbolic sets with positive volume for diffeomorphisms whose differentiability is higher than one. In the context of volume preserving diffeomorphisms, it is proved in [6] that a volume preserving diffeomorphism with a hyperbolic set of positive volume should be an Anosov diffeomorphism (see also [11]). The main point in this context is the saturation of an invariant set of positive volume with stable or unstable leaves. In light of the rich consequences on the volume of a hyperbolic set, a lot of studies has been undertaken in a more general landscape which are non-uniform hyperbolicity, partial hyperbolicity and maps with some discontinuities. For instance, Alves and Pinheiro proved in [1] the non-existence of horseshoe-like partially hyperbolic sets with positive volume for -diffeomorphims. Extending their result, Zhang proved in [12] that the hyperbolic lamination over the limit sets of s-density points of a partially hyperbolic set with positive volume of a -diffeomorphism should be contained in the partially hyperbolic set. The aim of this article is to study the volume of invariant sets in the context of non-uniform and partially hyperbolic dynamics with some discontinuities. First, we prove that a generalized hyperbolic attractor of a -diffeomorphism with positive volume should contain the stable and unstable laminations. Then inspired by the Bowen's example of a fat horseshoe, we build a -flow whose time one map admits a horseshoe-like generalized partially hyperbolic attractor with positive volume. Our construction is based on a modification of a classical Lorenz flow using the Bowen map. We begin by recalling some essential notions.