Application of the discrete-element to fast-transient three-dimensional incompressible flow with free-surface conditions in regions of arbitrary geometry

A fast-transient, three-dimensional, discrete-element computational model is presented for simulating free-surface flow conditions of water (constant density fluid) in regions with geometrically complex shoreline boundaries.

The geometrical discretization technique of the general three-dimensional discrete-element method (DEM) considers the interior elements as rectangular cells and the boundary elements as truncated rectangular cells which are defined by the impermeable shoreline boundaries of the flow region. Therefore, the rates of mass and momentum transport along any orthogonal axis of a variable-size grid in the Cartesian coordinate system is always across an enclosure surface area that is normal to the axis. Hence, the computational efficiency of the discrete-element method (DEM) is equivalent to the finite-difference method (FDM) which is significantly superior to the finite-element method (FEM); and it's spatial resolution accuracy is equivalent to the finite-element method (FEM), since the actual geometrical characteristics of the boundary elements are completely retained in the formulation.

The development of the discrete-element computational system utilizes directly the integral forms of the mass conservation and momentum principles, without resorting to the continuum partial differential equations that are used in the development of the conventional numerical solution methods.

The discrete-element formulation of the momentum principle considers two half-elements, within each element, for the calculation of one component of velocity in the horizontal (x,y) plane. This new formulation requires the computation of the velocity components at approximately twice as many solution locations in comparison to the solution locations required for the computation of pressure. The new formulation exhibits excellent numerical stability characteristics comparable to the conventional staggered mesh systems. Furthermore, it enables the geometrical discretization of the flow region according to a reasonably course spatial resolution grid system for pressure (water-surface elevation) but with twice as high spatial resolution for computationally more critical velocity conditions.

The computational algorithm of the transient, three-dimensional discrete-element method uses an explicit timewise-integration technique, which does not require the implicit solution of any elliptic system of equations. The numerical integration method is based on an explicit, composite time-spliting algorithm which requires only the satifaction of CFL (Courant, Friedrichs and Lewy) stability criterion for free-surface flow conditions.

The general development of the three-dimensional discrete-element method clearly identifies the individual terms associated with the convection-defect transport and with eddy-turbulent transport which collectively constitute the general nonconvective momentum transport. The general discrete-element computational system is independent of the particular phenomenological nonconvective transport model for the Reynolds stress components which is generally required for the closure of the conventional mathematical system. The present formulation of the nonconvective momentum transport (Reynolds stress) considers a local eddy-turbulence model based on element Reynolds number and a Blasius power-law form for the evaluation of the momentum eddy diffusivity tensor components on the enclosure surfaces of the elements and half-elements. The particular formulation enables the establishment of a pseudo unified-eddy-turbulence model applicable to different turbulence scales, including geophysical-meso-scale and laboratory-model-scale flow conditions in different regions.

The results of application of the associated computer code HYDROL of the fast-transient, three-dimensional, discrete-element hydrodynamic model to a test case for the simulation of the flow conditions in a symmetric laboratory-model-scale flow region are presented to illustrate the accuracy of the computational algorithm. The computer simulation results indicate that the required symmetry and asymmetry conditions associated with five different test cases were all satisfied in the application of the model. In all test cases the numerical solutions also exhibit all the expected characteristics based on considerations of the physical conditions. The results of application of the associated computer code HYDROL to the simulation of the flow conditions in the actual laboratory-model-scale flow region of the Browns Ferry physical model are also presented. The computer simulation results for the Browns Ferry physical model also illustrate the special capabilities of the fast-transient, three-dimensional, discrete-element model in predicting the occurrence of oscillatory surface-gravity-wave type flow conditions in flow regions with closed enclosure boundaries.