fluid_mechanics

8.3.4 Turbulence Modeling
8.4 Dimensional Analysis of Pipe Flow 409
Although it is not yet possible to theoretically predict the random, irregular details of turbulent
flows, it would be useful to be able to predict the time-averaged flow fields 1pressure, velocity, etc.2
directly from the basic governing equations. To this end one can time average the governing Navier–
Stokes equations 1Eqs. 6.31 and 6.1272 to obtain equations for the average velocity and pressure.
However, because the Navier–Stokes equations are nonlinear, the resulting time-averaged differential
equations contain not only the desired average pressure and velocity as variables, but also
averages of products of the fluctuations—terms of the type that one tried to eliminate by averaging
the equations! For example, the Reynolds stress ru¿v¿ 1see Eq. 8.262 occurs in the timeaveraged
momentum equation.
Thus, it is not possible to merely average the basic differential equations and obtain governing
equations involving only the desired averaged quantities. This is the reason for the variety of
ad hoc assumptions that have been proposed to provide “closure” to the equations governing the
average flow. That is, the set of governing equations must be a complete or closed set of equations—the
same number of equation as unknowns.
Various attempts have been made to solve this closure problem 1Refs. 1, 322. Such schemes
involving the introduction of an eddy viscosity or the mixing length 1as introduced in Section
8.3.22 are termed algebraic or zero-equation models. Other methods, which are beyond the scope
of this book, include the one-equation model and the two-equation model. These turbulence
models are based on the equation for the turbulence kinetic energy and require significant computer
usage.
Turbulence modeling is an important and extremely difficult topic. Although considerable
progress has been made, much remains to be done in this area.
Chaos theory may
eventually provide a
deeper understanding
of turbulence.
8.3.5 Chaos and Turbulence
Chaos theory is a relatively new branch of mathematical physics that may provide insight into the complex
nature of turbulence. This method combines mathematics and numerical 1computer2 techniques
to provide a new way to analyze certain problems. Chaos theory, which is quite complex and is currently
under development, involves the behavior of nonlinear dynamical systems and their response to
initial and boundary conditions. The flow of a viscous fluid, which is governed by the nonlinear Navier–
Stokes equations 1Eq. 6.1272, may be such a system.
To solve the Navier–Stokes equations for the velocity and pressure fields in a viscous flow, one
must specify the particular flow geometry being considered 1the boundary conditions2 and the condition
of the flow at some particular time 1the initial conditions2. If, as some researchers predict, the
Navier –Stokes equations allow chaotic behavior, then the state of the flow at times after the initial
time may be very, very sensitive to the initial conditions. A slight variation to the initial flow conditions
may cause the flow at later times to be quite different than it would have been with the original,
only slightly different initial conditions. When carried to the extreme, the flow may be “chaotic,” “random,”
or perhaps 1in current terminology2, “turbulent.”
The occurrence of such behavior would depend on the value of the Reynolds number. For
example, it may be found that for sufficiently small Reynolds numbers the flow is not chaotic 1i.e.,
it is laminar2, while for large Reynolds numbers it is chaotic with turbulent characteristics.
Thus, with the advancement of chaos theory it may be found that the numerous ad hoc turbulence
ideas mentioned in previous sections 1i.e., eddy viscosity, mixing length, law of the wall,
etc.2 may not be needed. It may be that chaos theory can provide the turbulence properties and
structure directly from the governing equations. As of now we must wait until this exciting topic
is developed further. The interested reader is encouraged to consult Ref. 4 for a general introduction
to chaos or Ref. 33 for additional material.
8.4 Dimensional Analysis of Pipe Flow
As noted previously, turbulent flow can be a very complex, difficult topic—one that as yet has
defied a rigorous theoretical treatment. Thus, most turbulent pipe flow analyses are based on
experimental data and semi-empirical formulas. These data are expressed conveniently in dimensionless
form.