Chapter 8 - Equations of Motion With Viscosity

Chapter 8 Contents

8.3 Calculation of Reynolds Stress

The Reynolds stresses such as < u' w' >/z are called virtual stresses (cf. Goldstein, 1965: 69 & 81) because we assume that they play the same role as the viscous terms in the equation of motion. To proceed further, we need values or functional form for the Reynolds stress. Several approaches are used.

By Analogy with Molecular Viscosity
Let's return to the simple example shown in Figure 8.1, which shows a boundary layer above a flat plate in the x, y plane. Prandtl, in a revolutionary paper published in 1904, stated that turbulent effects are only important in a very thin layer close to the surface, the boundary layer. Prandtl's invention of the boundary layer allows us to describe very accurately turbulent flow of wind above the sea surface, or flow at the bottom boundary layer of the ocean. See the box below.

To calculate flow in a boundary leyer, we assume that flow is constant in the x, y direction, that the statistical properties of the flow vary only in the z direction, and that the mean flow is steady. Therefore /t = /x = /y = 0, and (8.9) can be written:

(8.11)

We now assume, in analogy with (8.2)

(8.12)

where Az is an eddy viscosity which replaces the molecular viscosity in (8.2). With this assumption,

(8.13)

where we have assumed that Az is either constant or that it varies more slowly in the z direction than U/z. Later, we will assume that Az z.

The x and y momentum equations for a homogeneous, steady-state, turbulent boundary layer above or below a horizontal surface are:

(8.14a)
(8.14b)

where f = 2ω sin φ is the Coriolis parameter, and we have dropped the molecular viscosity term because it is much smaller than the turbulent eddy viscosity. Note, (8.14b) follows from a similar derivation from the y-component of the momentum equation. We will need (8.14) when we describe flow near the surface.

The assumption that an eddy viscosity Az can be used to relate the Reynolds stress to the mean flow works well for describing the flow near a horizontal surface where U is a function of distance z from the surface, and W, the mean velocity perpendicular to the surface is zero (See the box Turbulent Boundary Layer Over a Flat Plate). This is the approach first described in 1925 by Prandtl, who introduced the concept of a boundary layer, and by others. Please notice that a value for Az cannot be obtained from theory. Instead, it must be calculated from data collected in wind tunnels or measured in the surface boundary layer at sea. See Hinze (1975, 5-2 and 7-5) and Goldstein (1965: 80) for more on the theory of turbulence flow near a flat plate.

The Turbulent Boundary Layer Over a Flat Plate

The theory for the mean velocity distribution in a turbulent boundary layer over a flat plate was worked out independently by G.I.Taylor (1886-1975), L. Prandtl (1875-1953), and T. von Karman (1818-1963) from 1915 to 1935. Their empirical theory, sometimes called the mixing-length theory predicts well the mean velocity profile close to the boundary. Of interest to us, it predicts the mean flow of air above the sea. Here's a simplified version of the theory applied to a smooth surface.

We begin by assuming that the mean flow in the boundary layer is steady and that it varies only in the z direction. Within a few millimeters of the boundary, friction is important and (8.2) has the solution

(8.15)

and the mean velocity varies linearly with distance above the boundary. Usually (8.15) is written in dimensionless form:

(8.16)

where u*2 Tx /ρ is the friction velocity.

Further from the boundary, the flow is turbulent, and molecular friction is not important. In this regime, we can use (8.12), and

(8.17)

Prandtl and Taylor assumed that large eddies are more effective in mixing momentum than small eddies, and therefore Az ought to vary with distance from the wall. Karman assumed that it had the particular functional form Az = kzu*, where k is a dimensionless constant. With this assumption, the equation for the mean velocity profile becomes

(8.18)

Because U is a function only of z, we can write dU = u*/(kz) dz, which has the solution

(8.19)

where z0 is distance from the boundary at which velocity goes to zero.

For airflow over the sea, k = 0.4 and z0 is given by Charnock's (1955) relation z0 = 0.0156 u*2/g. The mean velocity in the atmospheric boundary layer just above the sea surface described in §4.3 fits well the logarithmic profile of (8.19), as does the mean velocity in the upper few meters of the sea just below the sea surface. Furthermore, using (4.1) in the definition of the friction velocity, then using (8.19) gives Charnock's form of the drag coefficient as a function of wind speed in Figure 4.6

Prandtl's theory based on assumption (8.12) works well only where friction is much larger than the Coriolis force. This is true for air flow within tens of meters of the sea surface and for water flow within a few meters of the surface. The application of the technique to other flows in the ocean is less clear. For example, the flow in the mixed layer at depths below about ten meters is less well described by the classical turbulent theory. Tennekes and Lumley (1970: 57) write:

Mixing-length and eddy viscosity models should be used only to generate analytical expressions for the Reynolds stress and mean-velocity profile if those are desired for curve fitting purposes in turbulent flows characterized by a single length scale and a single velocity scale. The use of mixing-length theory in turbulent flows whose scaling laws are not known beforehand should be avoided.

Problems with the eddy-viscosity approach:

1. Except in boundary layers a few meters thick, geophysical flows may be influenced by several characteristic scales. For example, in the atmospheric boundary layer above the sea, at least three scales may be important:

i) the height above the sea z,
ii) the Monin-Obukhov scale L discussed in 4.3, and
iii) the typical velocity U divided by the Coriolis parameter U / f.

2. The velocities u', w' are a property of the fluid, while Az is a property of the flow;

3. Eddy viscosity terms are not symmetric:

< u' v' > = < v' u' >; but

From a Statistical Theory of Turbulence
The Reynolds stresses can be calculated from various theories which relate < u 'u' > to higher order correlations of the form < u' u' u' >. The problem then becomes: How to calculate the higher order terms? This is the closure problem in turbulence. There is no general solution, but the approach leads to useful understanding of some forms of turbulence such as isotropic turbulence downstream of a grid in a wind tunnel (Batchelor 1967). Isotropic turbulence is turbulence with statistical properties that are independent of direction.

The approach can be modified somewhat for flow in the ocean. In the idealized case of high Reynolds flow, we can calculate the statistical properties of a flow in thermodynamic equilibrium. Because the actual flow in the ocean is far from equilibrium, we assume it will evolve towards equilibrium. Holloway (1986) provides a good review of this approach, showing how it can be used to derive the influence of turbulence on mixing and heat transports. One interesting result of the work is that zonal mixing ought to be larger than meridional mixing.

Summary
The turbulent eddy viscosities Ax, Ay, and Az cannot be calculated accurately for most oceanic flows.

  1. They can be estimated from measurements of turbulent flows. Measurements in the ocean, however, are difficult; and measurements in the lab, although accurate, cannot reach Reynolds numbers of 1011 typical of the ocean.
  2. The statistical theory of turbulence gives useful insight into the role of turbulence in the ocean, and this is an area of active research.

Some Values for Viscosity

νwater = 10-6m2/s
νtar at 15°C = 106m2/s
νglacier ice = 1010m2/s
Ay = 104m2/s

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