24
2. Introduction to Numerical Methods
the car model and blow air a t it - but the floor has t o move at the air speed,
which is difficult to do. It is not difficult to do in a numerical simulation.
Other types of boundary conditions are easily prescribed in computations;
for example, temperature or opaqueness of the fluid pose no problem. If we
solve the unsteady three-dimensional Navier-Stokes equations accurately (as
in direct simulation of turbulence), we obtain a complete data set from which
any quantity of physical significance can be derived.
This sounds t o good to be true. Indeed, these advantages of CFD are
conditional on being able to solve the Navier-Stokes equations accurately,
which is extremely difficult for most flows of engineering interest. We shall
see in Chap. 9 why obtaining accurate numerical solutions of the NavierStokes equations for high Reynolds number flows is so difficult.
If we are unable t o obtain accurate solutions for all flows, we have to determine what we can produce and learn to analyze and judge the results. First
of all, we have to bear in mind that numerical results are always approximate.
There are reasons for differences between computed results and 'reality' i.e.
errors arise from each part of the process used to produce numerical solutions:
The differential equations may contain approximations or idealizations, as
discussed in Sect. 1.7;
Approximations are made in the discretization process;
In solving the discretized equations, iterative methods are used. Unless
they are run for a very long time, the exact solution of the discretized
equations is not produced.
When the governing equations are known accurately (e.g. the NavierStokes equations for incompressible Newtonian fluids), solutions of any desired accuracy can be achieved in principle. However, for many phenomena
(e.g. turbulence, combustion, and multiphase flow) the exact equations are
either not available or numerical solution is not feasible. This makes introduction of models a necessity. Even if we solve the equations exactly, the
solution would not be a correct representation of reality. In order to validate the models, we have to rely on experimental data. Even when the exact
treatment is possible, models are often needed to reduce the cost.
Discretization errors can be reduced by using more accurate interpolation
or approximations or by applying the approximations to smaller regions but
this usually increases the time and cost of obtaining the solution. Compromise
is usually needed. We shall present some schemes in detail but shall also point
out ways of creating more accurate approximations.
Compromises are also needed in solving the discretized equations. Direct
solvers, which obtain accurate solutions, are seldom used, because they are
too costly. Iterative methods are more common but the errors due t o stopping
the iteration process too soon need to be taken into account.
Errors and their estimation will be emphasized throughout this book. We
shall present error estimates for many examples; the need to analyze and
estimate numerical errors can not be overemphasized.
2. Introduction to Numerical Methods
the car model and blow air a t it - but the floor has t o move at the air speed,
which is difficult to do. It is not difficult to do in a numerical simulation.
Other types of boundary conditions are easily prescribed in computations;
for example, temperature or opaqueness of the fluid pose no problem. If we
solve the unsteady three-dimensional Navier-Stokes equations accurately (as
in direct simulation of turbulence), we obtain a complete data set from which
any quantity of physical significance can be derived.
This sounds t o good to be true. Indeed, these advantages of CFD are
conditional on being able to solve the Navier-Stokes equations accurately,
which is extremely difficult for most flows of engineering interest. We shall
see in Chap. 9 why obtaining accurate numerical solutions of the NavierStokes equations for high Reynolds number flows is so difficult.
If we are unable t o obtain accurate solutions for all flows, we have to determine what we can produce and learn to analyze and judge the results. First
of all, we have to bear in mind that numerical results are always approximate.
There are reasons for differences between computed results and 'reality' i.e.
errors arise from each part of the process used to produce numerical solutions:
The differential equations may contain approximations or idealizations, as
discussed in Sect. 1.7;
Approximations are made in the discretization process;
In solving the discretized equations, iterative methods are used. Unless
they are run for a very long time, the exact solution of the discretized
equations is not produced.
When the governing equations are known accurately (e.g. the NavierStokes equations for incompressible Newtonian fluids), solutions of any desired accuracy can be achieved in principle. However, for many phenomena
(e.g. turbulence, combustion, and multiphase flow) the exact equations are
either not available or numerical solution is not feasible. This makes introduction of models a necessity. Even if we solve the equations exactly, the
solution would not be a correct representation of reality. In order to validate the models, we have to rely on experimental data. Even when the exact
treatment is possible, models are often needed to reduce the cost.
Discretization errors can be reduced by using more accurate interpolation
or approximations or by applying the approximations to smaller regions but
this usually increases the time and cost of obtaining the solution. Compromise
is usually needed. We shall present some schemes in detail but shall also point
out ways of creating more accurate approximations.
Compromises are also needed in solving the discretized equations. Direct
solvers, which obtain accurate solutions, are seldom used, because they are
too costly. Iterative methods are more common but the errors due t o stopping
the iteration process too soon need to be taken into account.
Errors and their estimation will be emphasized throughout this book. We
shall present error estimates for many examples; the need to analyze and
estimate numerical errors can not be overemphasized.