Numerical studies on non-linear free surface flow using generalized Schwarz-Christoffel transformation

Chuang, J. M.

doi:10.1002/(sici)1097-0363(20000415)32:7<745::aid-fld981>3.0.co;2-6

Cited by 7 publications

(4 citation statements)

References 19 publications

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“…Among the bottom topographies commonly considered is the half-cylinder (Forbes & Schwartz 1982;Forbes 1988;Zhang & Zhu 1996), the triangular obstacle (Dias & Vanden-Broeck 1989;Chuang 2000;Binder & Vanden-Broeck 2007) and the step (King & Bloor 1987;Binder et al 2006). Typical of these problems is the way in which their asymptotic behaviour depends on the criticality of the flow; a notable example (which falls outside the scope of the present paper) is the case of an hydraulic fall (Dias & Vanden-Broeck 1989;Forbes 1988).…”

Section: Introductionmentioning

confidence: 98%

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Sheared free-surface flow over three-dimensional obstructions of finite amplitude

Akselsen

Ellingsen

2019

J. Fluid Mech.

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When shallow water flows over uneven bathymetry, the water surface is modulated. This type of problem has been revisited numerous times since it was first studied by Lord Kelvin in 1886. Our study analytically examines currents whose unperturbed velocity profile U (z) follows a power-law z q , flowing over a three-dimensional uneven bed. This particular form of U , which can model a miscellany of realistic flows, allows explicit analytical solutions. Arbitrary bed shapes can readily be imposed via Fourier's theorem provided their steepness is moderate.Three-dimensional vorticity-bathymetry interaction effects are evident when the flow makes an oblique angle with, a sinusoidally corrugated bed. Streamlines are found to twist and the fluid particle drift is redirected away from the direction of the unperturbed current.Furthermore, a perturbation technique is developed which satisfies the bottom boundary condition to arbitrary order also for large-amplitude obstructions which penetrate well into the current profile. This introduces higher-order harmonics of the bathymetry amplitude. States of resonance for first and higher order harmonics are readily calculated. Although the method is theoretically restricted to bathymetries of moderate inclination, a wide variety of steeper obstructions are satisfactorily represented by the method, even provoking occurrences of recirculation.All expressions are analytically explicit and sequential fast Fourier transformations ensure quick and easy computation for arbitrary three-dimensional bathymetries. A method for separating near and far fields ensures computational convergence under the appropriate radiation condition. † Email address for correspondence: andreas.h.akselsen@ntnu.no arXiv:1909.05678v1 [physics.flu-dyn]

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Sheared free-surface flow over three-dimensional obstructions of finite amplitude

Akselsen

Ellingsen

2019

J. Fluid Mech.

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“…The two-dimensional version of this problem has an extensive history. Typically, complex variable methods are used to reformulate the problem in terms of a integral equation, which can be solved numerically using collocation (Binder et al 2013(Binder et al , 2006Chuang 2000;Dias & Vanden-Broeck 2002;Forbes & Schwartz 1982;Hocking et al 2013;King & Bloor 1990;Pethiyagoda et al 2018b;Zhang & Zhu 1996b). With these schemes, the effects of bottom topography and nonlinearity on the downstream waves can be explored in some detail.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional free-surface flow over arbitrary bottom topography

Pethiyagoda

Moroney

et al. 2018

J. Fluid Mech.

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We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton Krylov (JFNK) method is utilised. Using a blockbanded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes. †

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“…En esta situación, H. Schwarz y E. Chistoffel demostraron, independientemente (Schwarz conocía el resultado en 1864 y lo publicó en 1866, Christoffel publicó su resultado en 1867), que la aplicación abierta de Riemann que transforma el disco unitario D en G, llamada transformación de Schwarz-Christoffel, viene dada por z = g(w) = a + c Son muchas las aplicaciones de la transformación de Schwarz-Christoffel, en primer lugar, mencionamos que, con un argumento de continuidad,ésta fórmula permite dar una prueba alternativa del Teorema de la Transformación de Riemann (ver [11]) para regiones simplemente conexas G cuya frontera sea una curva de Jordan continua a trozos. En elámbito de la física, esta transformación ha resultado una herramienta poderosa para analizar fenómenos relacionados con el magnetismo (ver [10]), y con el flujo de un fluido a través de un tubo (ver [4,5]). En matemáticas, la transformación de Schwarz-Christoffel se puede usar para hallar soluciones a la ecuación de Laplace con ciertas condiciones de fronteras (ver [8]) y más recientemente como herramienta en la probabilidad aplicada (ver [14]).…”

Section: Introductionunclassified

Determinacion de valores de los parametros en la transformacion de Schwarz-Christoffel

Fernández¹,

Rivas²

2008

B Soc Paran Mat

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Numerical studies on non-linear free surface flow using generalized Schwarz-Christoffel transformation

Cited by 7 publications

References 19 publications

Sheared free-surface flow over three-dimensional obstructions of finite amplitude

Sheared free-surface flow over three-dimensional obstructions of finite amplitude

Three-dimensional free-surface flow over arbitrary bottom topography

Determinacion de valores de los parametros en la transformacion de Schwarz-Christoffel

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