Thus far in this course, the focus has been on laminar or streamline flow, where fluids move in "layers". This type of flow is dominant in pressure driven flows only at low flow rates. At higher flow rates, the streamlines are disrupted by eddies moving in all directions. This is turbulent flow. Most industrial flows are turbulent , but analytical treatment is complex and difficult. As a result, empirical methods (such as friction factors) are commonly used.
The goal of this section is to describe turbulence and some of the methods and techniques used to try and understand turbulent flows.
Turbulent eddies form from contact of the fluid with a solid boundary or from two fluid layers moving at different speeds. As these eddies grow, the laminar flow becomes unstable and velocities and pressures in the flowing fluid no longer have constant or smoothly varying values.
In pipes, relatively large rotational eddies form in regions of high shear near the pipe wall. These degenerate into smaller eddies as energy is dissipated by action of viscosity. The presence of eddies means that the local velocity is not the same as the bulk velocity and that there are components of velocity in all directions.
Fluids with low viscosity and high density tend to support turbulence.
Turbulence makes convective transport dominant. Need Figure -- similar to Text 6.6
The transition between laminar and turbulent flow is fuzzy. The behavior depends on entrance conditions and distance from the inlet. Often it is useful to speak of a transition region for flows that are neither laminar nor fully turbulent.
Osborne Reynolds did flow experiments and discovered that the transition between laminar flow and turbulent flow depended on diameter, velocity, density, and viscosity and that these could be arranged in a dimensionless group:
Laminar flows have Reynolds numbers below 2100. Fully turbulent flows will have Reynolds numbers greater than 10000 (although values between 4000 and 10000 are often effectively turbulent). The magnitude of the Reynolds number is independent of the system of units.
Consider a single particle of fluid moving in a turbulent stream. The particle will have its own instantaneous velocity with components in all directions, often moving differently from the bulk flow of the stream. Turbulent flow is an example of a chaotic, nonlinear function.
Need figure like Text 6.5
In turbulent flow, all three velocity components will look similar, unlike in laminar flow when only one component is nonzero. Detailed treatment of turbulence requires a statistical approach. The frequency distribution of a velocity component is Gaussian.
In turbulent flow, the flow and fluid variables -- especially velocity -- vary with time and position. In order to eliminate the time dependence, a time averaged velocity is generally used:
The time averaged velocity can be averaged again (over the area) to obtain the bulk velocity used previously.
Need figure showing bulk vs. instantaneous
The time averaged velocity does a good job of describing bulk flow, but doesn't precisely account for instantaneous behavior. This is done by expressing the instantaneous quantities as the sum of the average value and the instantaneous deviation from the average.
This velocity can be used in any of the equations we have been discussing, and allow us to express all properties in terms of the time-averaged values and the deviation from that value. Note that, by definition, the time average of the deviation term is zero (the perturbations cancel out over time). For this model, laminar flow corresponds to zero perturbation.
The intensity (or level) of the turbulence is calculated as the "root mean square" of the instantaneous velocity divided by the time averaged velocity, or
Typical values for the intensity are between 0.01 and 0.1. Boundary layer transition and separation, heat and mass tranfer rates all depend on the intensity of the turbulence. To conduct a model test, the Reynolds number and the intensity of the turbulence must be reproduced.
Random molecular motion transfers momentum, heat, and mass between layers of fluid, so it seems reasonable that the larger scale motion of turbulent eddies should result in additional transfer. Additionally, The velocity components in turbulent flow are always correlated, so any change in one component will effect the other two.
The equations that have been used apply to all flow regimes, as long as the appropriate velocity is used. Consider, for instance the general property balance equation.
The shear stress in turbulent flow can be found by substituting the perturbation form of the velocity into the equation of motion and time averaging the result. The total shear stress for turbulent flow then becomes
This term is one element of the Reynolds stress tensor given by
Analytical prediction of Reynold's stress terms is almost impossible, so experimental measurements and correlations based on fitted data are the normal approach to turbulent flow.
References:
R.M. Price
Original: 6/1/99
Revised: 10/1/99; 10/15/99
Copyright 1999 by R.M. Price -- All Rights Reserved