Chapter 8  Equations of Motion With Viscosity
8.2 Turbulence If molecular viscosity is important only over distances of a few millimeters, and if it is not important for most oceanic flows, unless of course you are a zooplankton trying to swim in the ocean, how then is the influence of a boundary transferred into the interior of the flow? The answer is: through turbulence. Turbulence arises from the nonlinear terms in the momentum equation (U ∂U/∂x, etc.). The importance of these terms is given by a nondimensional number, the Reynolds Number Re, which is the ratio of the nonlinear terms to the viscous terms:
where, U is a typical velocity of the flow and L is a typical length describing the flow. You are free to pick whatever U, L might be typical of the flow. For example L can be either a typical crossstream distance, or an alongstream distance. Typical values in the open ocean are U = 0.1m/s and L = 1 megameter, so Re = 10^{11}. Because nonlinear terms are important if Re > 101000, they are certainly important in the ocean. The ocean is turbulent. The Reynolds number is named after Osborne Reynolds (18421912) who conducted experiments in the late 19th century to understand turbulence. In one famous experiment (Reynolds 1883), he injected dye into water flowing at various speeds through a tube (Figure 8.2). If the speed was small, the flow was smooth. This is called laminar flow. At higher speeds, the flow became irregular and turbulent. The transition occurred at Re = VD/2000, where V is the average speed in the pipe, and D is the diameter of the pipe.
As Reynolds number increases above some critical value, the flow becomes more and more turbulent. Note that flow pattern is a function of Reynold's number. All flows with the same geometry and the same Reynolds number have the same flow pattern. Thus flow around all circular cylinders, whether 1mm or 1m in diameter, look the same as the flow at the top of figure 8.3 if the Reynolds number is 20. Furthermore, the boundary layer is confined to a very thin layer close to the cylinder, in a layer too thin to show in the figure.
Turbulent Stresses: The Reynolds Stress
To see how these stresses might arise, consider the momentum equation for a flow with mean and a turbulent components of flow:
where the mean value U is calculated from a time or space average:
The nonlinear terms in the momentum equation can be written:
The second equation follows from the first because <U ∂u'/∂x> = 0 and <u' ∂U/∂x> = 0, which follow from the definition of U: <U ∂u'/∂x> = U∂<u'>/∂x = 0. Using (8.7), the continuity equation splits into two equations:
The xcomponent of the momentum equation becomes:
Thus the additional force per unit mass due to the turbulence is:
The terms ρ < u 'u' >, ρ < u' v' >, and ρ < u' w' > transfer eastward momentum (ρ u) in the x, y, and z directions. For example, the termρ < u' w' > gives the downward transport of eastward momentum across a horizontal plane. Because they transfer momentum, and because they were first derived by Osborne Reynolds, they are called Reynolds Stresses.


Department of Oceanography, Texas A&M University Robert H. Stewart, stewart@ocean.tamu.edu All contents copyright © 2005 Robert H. Stewart, All rights reserved Updated on September 26, 2008 