266
9. Turbulent Flows
is useful when chemical mixing or heat transfer are needed; both of these
may be increased by orders of magnitude by turbulence. On the other hand,
increased mixing of momentum results in increased frictional forces, thus
increasing the power required to pump a fluid or to propel a vehicle; again,
an increase by an order of magnitude is not unusual. Engineers need to be
able to understand and predict these effects in order to achieve good designs.
In some cases, it is possible to control the turbulence, at least in part.
In the past, the primary approach to studying turbulent flows was experimental. Overall parameters such as the time-averaged drag or heat transfer
are relatively easy to measure but as the sophistication of engineering devices
increases, the levels of detail and accuracy required also increase, as does cost
and the expense and difficulty of making measurements. To optimize a design,
it is usually necessary to understand the source of the undesired effects; this
requires detailed measurements that are costly and time-consuming. Some
types of measurements, for example, the fluctuating pressure within a flow,
are almost impossible to make at the present time. Others cannot be made
with the required precision. As a result, numerical methods have an important role to play.
Before proceeding to the discussion of numerical methods for these flows,
it is useful t o introduce a classification scheme for the approaches t o predicting turbulent flows. According to Bardina et al. (1980) there are six categories, most of which can be divided in sub-categories.
The first involves the use of correlations such as ones that give the friction
factor as a function of the Reynolds number or the Nusselt number of
heat transfer as a function of the Reynolds and Prandtl numbers. This
method, which is usually taught in introductory courses, is very useful but
is limited to simple types of flows, ones that can be characterized by just
a few parameters. As its use does not require the use of a computer, we
shall say no more about it here.
0 The second uses integral equations which can be derived from the equations
of motion by integrating over one or more coordinates. Usually this reduces
the problem t o one or more ordinary differential equations which are easily
solved. The methods applied to these equations are those for ordinary
differential equations which are discussed in Chap. 6.
0 The third is based on equations obtained by averaging the equations of
motion over time (if the flow is statistically steady), over a coordinate in
which the mean flow does not vary, or over an ensemble of realizations
(an imagined set of flows in which all controllable factors are kept fixed).
This approach is called one-point closure and leads t o a set of partial differential equations called the Reynolds-averaged Navier-Stokes (or RANS)
equations. As we shall see later, these equations do not form a closed set so
this method requires the introduction of approximations (turbulence models). Some of the turbulence models in common use today and a discussion
9. Turbulent Flows
is useful when chemical mixing or heat transfer are needed; both of these
may be increased by orders of magnitude by turbulence. On the other hand,
increased mixing of momentum results in increased frictional forces, thus
increasing the power required to pump a fluid or to propel a vehicle; again,
an increase by an order of magnitude is not unusual. Engineers need to be
able to understand and predict these effects in order to achieve good designs.
In some cases, it is possible to control the turbulence, at least in part.
In the past, the primary approach to studying turbulent flows was experimental. Overall parameters such as the time-averaged drag or heat transfer
are relatively easy to measure but as the sophistication of engineering devices
increases, the levels of detail and accuracy required also increase, as does cost
and the expense and difficulty of making measurements. To optimize a design,
it is usually necessary to understand the source of the undesired effects; this
requires detailed measurements that are costly and time-consuming. Some
types of measurements, for example, the fluctuating pressure within a flow,
are almost impossible to make at the present time. Others cannot be made
with the required precision. As a result, numerical methods have an important role to play.
Before proceeding to the discussion of numerical methods for these flows,
it is useful t o introduce a classification scheme for the approaches t o predicting turbulent flows. According to Bardina et al. (1980) there are six categories, most of which can be divided in sub-categories.
The first involves the use of correlations such as ones that give the friction
factor as a function of the Reynolds number or the Nusselt number of
heat transfer as a function of the Reynolds and Prandtl numbers. This
method, which is usually taught in introductory courses, is very useful but
is limited to simple types of flows, ones that can be characterized by just
a few parameters. As its use does not require the use of a computer, we
shall say no more about it here.
0 The second uses integral equations which can be derived from the equations
of motion by integrating over one or more coordinates. Usually this reduces
the problem t o one or more ordinary differential equations which are easily
solved. The methods applied to these equations are those for ordinary
differential equations which are discussed in Chap. 6.
0 The third is based on equations obtained by averaging the equations of
motion over time (if the flow is statistically steady), over a coordinate in
which the mean flow does not vary, or over an ensemble of realizations
(an imagined set of flows in which all controllable factors are kept fixed).
This approach is called one-point closure and leads t o a set of partial differential equations called the Reynolds-averaged Navier-Stokes (or RANS)
equations. As we shall see later, these equations do not form a closed set so
this method requires the introduction of approximations (turbulence models). Some of the turbulence models in common use today and a discussion