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Coastal Engineering: Theory and Practice
Navier and George Gabriel Stokes). These describe the motion of fluid
particles.
The forces constitute of body forces, acting on the volumétrie mass of
the element and surface forces acting on the boundary of the fluid element,
due to pressure distribution and shear stress distribution on the surface.
For incompressible Newtonian fluids under laminar flow, the momentum
équations are written as,
P —u + u • Vu
dt
= —\7p + /uV2u + pg
(10.3)
The dérivation of analytic solution for Eqs. (10.2) and (10.3) is feasible
only for simple domain geometries and for the linear force components. In
most of the real field conditions, a direct analytic solution is not possible
due to complex geometries and flow conditions. Hence, approximate solutions through numerical discretization is always sought to analyse for flow
characteristics.
10.4 Discretization
Numerical approximation provide solutions only at discrète points in the
domain rather than continuous variation of the variable in the domain as
in the case of analytical solutions. These discrète points are named as grid
points or nodes. The grids may be structured or unstructured; structured
grid refers to that grid which follows certain geometrical regularity, whereas
the unstructured grid is the one in which the grid points are placed randomly. Discretization methods can be broadly classified as Explicit and
Implicit methods. Briefly, an explicit method obtains the successive values
at say, tn+1 parametrically in terms of given or previously computed quantifies at time tn, the advantage of it being simplified solution algorithms,
as it involves direct computation without solving any System of équations.
An implicit method is the one which calculâtes the values at tn+i by using
unknown values at tn+i and has to be solved simultaneously at ail the
grid points as a System of algebraic équations. The latter usually involves
inversion of a matrix équation either directly or iteratively at each time
step.
The discretization is an art bounded by the consistency and convergence requirements. Further, under the control of conservation laws, the
stability of the prescribed numerical scheme dépends on the discretization.
A discretization scheme is consistent if the différence between partial differential équations (PDEs) and its finite incrément minimizes as the grid
Coastal Engineering: Theory and Practice
Navier and George Gabriel Stokes). These describe the motion of fluid
particles.
The forces constitute of body forces, acting on the volumétrie mass of
the element and surface forces acting on the boundary of the fluid element,
due to pressure distribution and shear stress distribution on the surface.
For incompressible Newtonian fluids under laminar flow, the momentum
équations are written as,
P —u + u • Vu
dt
= —\7p + /uV2u + pg
(10.3)
The dérivation of analytic solution for Eqs. (10.2) and (10.3) is feasible
only for simple domain geometries and for the linear force components. In
most of the real field conditions, a direct analytic solution is not possible
due to complex geometries and flow conditions. Hence, approximate solutions through numerical discretization is always sought to analyse for flow
characteristics.
10.4 Discretization
Numerical approximation provide solutions only at discrète points in the
domain rather than continuous variation of the variable in the domain as
in the case of analytical solutions. These discrète points are named as grid
points or nodes. The grids may be structured or unstructured; structured
grid refers to that grid which follows certain geometrical regularity, whereas
the unstructured grid is the one in which the grid points are placed randomly. Discretization methods can be broadly classified as Explicit and
Implicit methods. Briefly, an explicit method obtains the successive values
at say, tn+1 parametrically in terms of given or previously computed quantifies at time tn, the advantage of it being simplified solution algorithms,
as it involves direct computation without solving any System of équations.
An implicit method is the one which calculâtes the values at tn+i by using
unknown values at tn+i and has to be solved simultaneously at ail the
grid points as a System of algebraic équations. The latter usually involves
inversion of a matrix équation either directly or iteratively at each time
step.
The discretization is an art bounded by the consistency and convergence requirements. Further, under the control of conservation laws, the
stability of the prescribed numerical scheme dépends on the discretization.
A discretization scheme is consistent if the différence between partial differential équations (PDEs) and its finite incrément minimizes as the grid