Biomedical Engineering Reference
In-Depth Information
system will remain constant. The above equation is the continuity equation for
2-D flow.
X-Momentum
(8)
Y-Momentum
(9)
where,
ρ
is density and g is acceleration due to gravity. In the preceding
equations,
ρ
is assumed to be constant. Equation (8) and (9) are called the
equations of momentum in x and y directions, respectively. They are the
mathematical representation of the law of conservation of momentum that is
also called Newton's second law. This law states that the rate of change in the
system's linear momentum is balanced by the resultant of the forces acting on
it. Consequently a mass of fluid originally at rest is set in motion (accelerates)
only when there is a net resultant force applied to it. The mass, then, will move
in the direction of the applied resultant force.
Residuals
Ideally, the above equations on being solved should result in a null value.
It is difficult to find an exact solution to the above equations. However, an
approximate solution is found by the CFD solver and the closer it is to zero,
the more reliable is the flow characteristic predicted by the CFD model. The
closeness to the expected null value is determined by observing the residual
plots generated by the CFD software. Residuals are the difference between the
expected value for a given set of flow parameters and the predicted value that
is generated by the solver (Figure 28).
The Navier-Stokes' equations are a set of partial differential equations.
The solution to partial differentiation equations is obtained numerically at
discrete points. These discrete points are obtained by subdividing the flow
domain into small computational cells. The resulting collection of these cells is
called a computational mesh. Distinct points on these cells represent the
computational points at which the solution is obtained. The size and
distribution of the cells determines also the accuracy of the solution.
