11. Efficiency and Accuracy Improvement
The best measure of the efficiency of a solution method is the computational
effort required to achieve the desired accuracy. There are several methods
for improving the efficiency and accuracy of CFD methods; we shall present
three that are general enough to be applied to any of the solution schemes
described in previous chapters.
11.1 Error Analysis and Estimation
The various types of errors which are unavoidable in the numerical solution
of fluid flow problems have been briefly discussed in Sect. 2.5.7. Here we give
a more detailed discussion of the various types of error and discuss how these
can be estimated and eliminated. Issues of code and model validation will
also be addressed.
11.1.1 Description of Errors
Modelling Errors. Fluid flow and related processes are usually described
by integral or partial differential equations that represent basic conservation
laws. The equations may be considered a mathematical model of the problem. Although the Navier-Stokes equations can be considered exact, solving
them is impossible for most flows of engineering interest. Turbulence places
huge demands on computer resources if it is to be simulated directly; other
phenomena like combustion, multi-phase flow, chemical processes etc. are
difficult to describe exactly and inevitably require the introduction of modeling approximations. Newton's and Fourier's laws are themselves only models,
although they are solidly based on experimental observations for many fluids.
Even when the underlying mathematical model is nearly exact, some properties of the fluid may not be exactly known. All fluid properties depend
strongly on temperature, species concentration and, possibly, pressure; this
dependence is often ignored, introducing additional modeling errors (e.g. the
use of the Boussinesq approximation for natural convection, the neglect of
compressibility effects in low Mach-number flows, etc.).
The equations require initial and boundary conditions. These are often
difficult to specify exactly. In other cases, one is forced to approximate them
The best measure of the efficiency of a solution method is the computational
effort required to achieve the desired accuracy. There are several methods
for improving the efficiency and accuracy of CFD methods; we shall present
three that are general enough to be applied to any of the solution schemes
described in previous chapters.
11.1 Error Analysis and Estimation
The various types of errors which are unavoidable in the numerical solution
of fluid flow problems have been briefly discussed in Sect. 2.5.7. Here we give
a more detailed discussion of the various types of error and discuss how these
can be estimated and eliminated. Issues of code and model validation will
also be addressed.
11.1.1 Description of Errors
Modelling Errors. Fluid flow and related processes are usually described
by integral or partial differential equations that represent basic conservation
laws. The equations may be considered a mathematical model of the problem. Although the Navier-Stokes equations can be considered exact, solving
them is impossible for most flows of engineering interest. Turbulence places
huge demands on computer resources if it is to be simulated directly; other
phenomena like combustion, multi-phase flow, chemical processes etc. are
difficult to describe exactly and inevitably require the introduction of modeling approximations. Newton's and Fourier's laws are themselves only models,
although they are solidly based on experimental observations for many fluids.
Even when the underlying mathematical model is nearly exact, some properties of the fluid may not be exactly known. All fluid properties depend
strongly on temperature, species concentration and, possibly, pressure; this
dependence is often ignored, introducing additional modeling errors (e.g. the
use of the Boussinesq approximation for natural convection, the neglect of
compressibility effects in low Mach-number flows, etc.).
The equations require initial and boundary conditions. These are often
difficult to specify exactly. In other cases, one is forced to approximate them