Modelling free surface flow with curvilinear streamlines by a non-hydrostatic model

A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

Towards non-intrusive reduced order 3D free surface flow modelling

Nonlinear solutions of free surface flow under a sluice gate are studied. Upstream, the fluid is assumed to be infinitely deep, and the gate is at an angle to the horizontal axis. The flow emerges from the gate and produces a uniform flow far downstream. The problem is solved numerically by a boundary element method derived from an integral equation along the free surface. An analytical function is constructed, related to the upstream flow, so that the integral equation is solvable. As a result, a free surface flow with smooth detachment from the edge of the gate is obtained for relatively large upstream Froude numbers. As the Froude number decreases, a free surface with back flow near the edge of the gate occurs, until at a certain Froude number a stagnation point occurs. References D. D. Frangmeier, T. S. Strelkoff, Solution for gravity flow under a sluice gate, ASCE J. Engng. Mech. Div. , 94 , 1968, 153--176. B. E. Lorock, Gravity-affected flow from planer sluice gate, ASCE J. Engng. Mech. Div. , 96 , 1969, 1211--1226. Y. K. Chung, Solution of flow under sluice gates, ASCE J. Engng. Mech. Div. , 98 , 1972, 121--140. J. Asavanant, J.-M. Vanden-Broeck, Nonlinear free surface flows emerging from vessels and flows under a sluice gate, J. Austral. Math. Soc. , 38 , 1996, 63--86. doi:10.1017/S0334270000000473 J.-M. Vanden-Broeck, Numerical calculations of the free-surface flow under a sluice gate, J. Fluid Mech. , 330 , 1996, 339--347. http://journals.cambridge.org/abstract_S0022112096003849 B. J. Binder, J.-M. Vanden-Broeck, Free surface flows past surfboards and sluice gates, Eur. J. Appl. Maths. , 16 , 2005, 601--619 doi:10.1017/S0956792505006443 B. J. Binder, J.-M. Vanden-Broeck, The effect of disturbances on the flows under a sluice gate and past an inclined plate, J. Fluid Mech. , 576 , 2007, 475--490 doi:10.1017/S0022112007004806 J.-M. Vanden-Broeck, E. O. Tuck, Flow near the interaction of a free surface with a vertical wall, SIAM J. Appl. Math. , 54 , 1994, 1--33. doi:10.1137/S0036139992240838 L. H. Wiryanto, J. Widjaja, B. Supriyanto, Free-surface flow under a sluice gate from deep water, Bul. Malay. Math. Soc. , 34 , 2011, 601--609. http://www.emis.de/journals/BMMSS/vol34_3_17.html L. H. Wiryanto, E. O. Tuck, A back-turning jet formed by a uniform shallow stream hitting a vertical wall, Proc. Int. Conf. on Dif. Eqs. , Kluwer Academic Press, 1997. L. H. Wiryanto, E. O. Tuck, An open-channel flow meeting a barrier and forming one or two jets, J. Austral. Math. Soc. , 41 , 2000, 458--472. doi:10.1017/S0334270000011760 L. H. Wiryanto, Small Froude number solutions of flow caused by a line source, J. Indones. Math. Soc. , 6 , 2000, 1--6. G. C. Hocking, L. K. Forbes, A note on the flow induced by a line sink beneath a free surface, J. Austral. Math. Soc. B , 32 , 1991, 251--260. doi:10.1017/S0956792507006924 G. C. Hocking, L. K. Forbes, Subcritical free-surface flow caused by a line source in a fluid of finite depth, J. Eng. Math. , 26 , 1992, 455--466. doi:10.1007/BF00042763 L. H. Wiryanto, Zero gravity of a jet emerging from a slit, J. Indones. Math. Soc. , 12 , 2006, 89--98. L. M. Milne-Thomson, Theoretical hydrodynamics . Dover Publications, Inc., 1968, 274--281.

This paper deals with a phenomenon often occurring in sewers during storm events - transition from free surface to pressurized flow and vice versa. This transition is also possible in sewers where the discharge is controlled by some control devices, like gates. Experiments were carried out at a test rig consisting of a circular pipe with gates at the upstream and the downstream ends. Because of the relatively steep slope of the pipe and upstream boundary condition (flow below the gate), free surface flow at the upstream end was always supercritical, so that a hydraulic jump was always present during transitions (transcritical flow). Experimental results were used for verification of a numerical model based on a shock capturing method, the McCormack explicit finite difference scheme.

A mathematical model of three-dimensional (3D) free surface flows has been applied to simulate the curved channel flows and mass transport. In the horizontal plane, a channel-fitted curvilinear coordinate system is used, whereas in the vertical plane, the σ-transformation is adopted to track the free surface and variable bed topography. To reduce the numerical diffusion, the second-order upwind scheme of Roe is incorporated to discretize the convection terms. The standard k-ε model has been modified to take account of some anisotropic effects appearing in shallow curved channels, i.e., streamline curvature and the damping effects of free surface and solid walls. The governing equations are solved in a collocated grid system by a fractional three-step implicit algorithm. Two test cases from curved flumes have been studied: (1) A 270° channel bend with a sloped outer bank; and (2) a meandering channel with mass transport. The results are compared with the available data, which shows generally good agreement.

Advances in direct numerical simulation for MHD modeling of free surface flows

Unsteady viscoelastic free surface flows are common in many industrial processes and a variety of numerical techniques have been employed to simulate these flows. The majority of constitutive models employed are based on differential equations to define the extra stress tensor. However, for some complex fluids, for instance, Boger fluids, integral models are more adequate to approximate the nonlinear behaviour of these fluids. This work deals with the numerical solution of the integral constitutive model KBKZ-PSM for two-dimensional unsteady free surface flows. The proposed numerical method is a numerical technique that employs finite differences to simulate moving free surface flows that interact with solid walls. The main features of the method are: numerical solution of the momentum and mass equations by an implicit method; the pressure condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times t is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by the modified Euler method. This new technique is verified by solving shear and uniaxial elongational flows. Moreover, an analytic solution for channel flows is obtained that is used in the verification and convergence analysis of the proposed methodology. For free surface flows, the assessment of convergence lies on the mesh refinement on the simulation of a jet impinging on a flat surface and the extrudade swell problem. Finally, the new method is applied to investigate the jet buckling phenomenon and extrudate swell of KBKZ-PSM fluids.

Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity

In this paper we will present an algorithm to perform free surface flow simulations with the lattice Boltzmann method on adaptive grids. This reduces the required computational time by more than a factor of three for simulations with large volumes of fluid. To achieve this, the simulation of large fluid regions is performed with coarser grid resolutions. We have developed a set of rules to dynamically adapt the coarse regions to the movement of the free surface, while ensuring the consistency of all grids. Furthermore, the free surface treatment is combined with a Smagorinsky turbulence model and a technique for adaptive time steps to ensure stable simulations. The method is validated by comparing the position of the free surface with an uncoarsened simulation. It yields speedup factors of up to 3.85 for a simulation with a resolution of 4803 cells and three coarser grid levels, and thus enables efficient and stable simulations of free surface flows, e.g. for highly detailed physically based animations of fluids.

The problem is a free-surface flow of a fluid, emerging from a semi-infinite container. The fluid is assumed to be inviscid, incompressible and the flow to be two dimensional and irrotational. When surface tension is neglected the free surface leaves the wall of the container tangentially. We show that when surface tension is taken into account, there is, in general, a train of waves on the free surface and a discontinuity in slope where the free surface separates from the wall of the container. These new solutions include, as particular cases, previously obtained solutions for which the free surface is waveless in the far field. Although the calculations are presented for a special flow configuration, the results are general and apply to other potential free surface flows where a free-surface intersects a rigid wall.

Numerical solution of the PTT constitutive equation for unsteady three-dimensional free surface flows

The gravity-driven free surface flow problems for which both the solid and free surface boundaries are highly curved are very difficult to solve. A computational scheme using a variable domain and a fixed domain natural element method (NEM) is developed in the present study for the computation of the free surface profile, velocity and pressure distributions, and the flow rate of a 2D gravity fluid flow through a conduit and under a radial gate. The problem involves two highly curved unknown free surfaces and arbitrary curved-shaped boundaries. These features make the problem more complicated than flow under a sluice gate or over a weir. The fluid is assumed to be inviscid and incompressible and the results obtained are confirmed by conducting a hydraulic model test. The results are in agreement with other flow solutions for free surface profiles and pressure distributions.

The effect of viscosity on free surface flow inside an angularly oscillating rectangular tank

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.

Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation