Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.The coupled motion—between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics—is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian particle path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani et al. (J Fluid Struct 59:432–460, 2015) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.This work is supported by the EPSRC under Grant number EP/K008188/1

Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.A class of augmented approximate Riemann solvers due to George (2008) is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx-efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL). A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge-Kutta integrator for the vessel motion.The research reported in this paper is supported by the EPSRC under Grant number EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]. The authors are grateful to both referees for their valuable comments

Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordA numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible 1D shallow-water equations in a moving vessel with a rigid-lid with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale et al. (2002). The method splits the jump in the fluxes and source terms including the pressure gradient at the rigid-lid into waves propagating away from each grid cell interface. For the influx-efflux boundary conditions the time dependent source terms are handled via a fractional step approach. In the linear case the numerical solutions are validated by comparison with the exact analytical solutions. Numerical solutions presented for the nonlinear case include shallow-water sloshing waves due to prescribed surge motion of the vessel.The research reported in this paper is supported by the Engineering and Physical Sciences Research Council Grant EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]

Breakdown of the shallow water equations due to growth of the horizontal vorticity

In an oceanographic setting, the shallow water equations are an asymptotic approximation to the full Euler equations, in the limit ϵ =h0/L→ 0, withh0being the vertical length scale andLa horizontal length scale associated with the fluid layer. However, in arriving at the shallow water equations an additional key step in the derivation is the condition that at some reference time (e.g.t= 0) the thin-layer horizontal vorticity field is identically zero, which corresponds to the horizontal fluid velocity field being independent of the vertical coordinate,z, att= 0. With this condition in place, the ‘thin-layer equations’ reduce exactly to the shallow water equations. In this paper, we show that this exact condition may be unstable: small, even infinitesimal, perturbations of the thin-layer horizontal vorticity field can grow without bound. When the thin-layer horizontal vorticity grows to be of order 1, the shallow water equations are no longer asymptotically valid as a model for shallow water hydrodynamics, and the ‘thin-layer equations’ must be adopted in their place.</jats:p

Therapeutic aims of drugs offering only progression-free survival are misunderstood by patients, and oncologists may be overly optimistic about likely benefits

PURPOSE: The use of novel and often expensive drugs offering limited survival benefit in advanced disease is controversial. Treatment recommendations are influenced by patient characteristics and trial data showing overall response rates (ORR), progression-free survival (PFS) and overall survival (OS). PFS is frequently the primary outcome in licencing studies. PATIENTS AND METHODS: As part of a longitudinal study Assessing the 'VALue' to patients of PROgression Free Survival (AVALPROFS), oncologists completed checklists at baseline following consultations with patients. Questions probed perceived clinical benefits of the drugs to populations in general. Patients completed study-specific interview schedules at baseline, 6 weeks into treatment, and at withdrawal due to toxicity or progression. Patients also completed tumour- and treatment-specific quality of life questionnaires monthly for their time in the study. Only baseline results are reported here. RESULTS: Thirty-two UK oncologists discussed management options with 90 patients with heterogeneous advanced cancers. Oncologists' estimates of medical benefit in general from treatment varied between 10 and 80 %. They expected 46/90 (51 %) of their patients to derive some clinical benefit from the prescribed treatment but were either unsure or expected none for 44/90 (49 %). Predictions of life expectancy were variable but 62 % (56/90) of patients were expected to survive longer with treatment. A majority of patients 51/90 (57 %) had 'no idea' or were 'unclear' what PFS meant and 45/90 (50 %) thought extension of life was the primary therapeutic aim of treatment. CONCLUSION: Discussions between doctors and patients with metastatic disease about future management plans and likely therapeutic gains are challenging. Factors influencing decisions about putative benefits of novel drugs are often applied inconsistently can be overly optimistic and may even contradict published data