Reynolds stresses and eddy viscosities

Laboratory experiments show that the transition from laminar to turbulent motions in an incompressible fluid depends on the Reynolds number

v which is a measure of the relative magnitude of the inertial to viscous forces occurring in the flow (see Eq. [2.7]). Here U is the characteristic velocity of the flow, L is a characteristic length for the problem on hand, and v = x/p is the coefficient of kinematic viscosity. Turbulent flows always occur when the nondimensional number Re exceeds some critical value Rec (say). This critical number is not a universal constant but takes different values for each type of flow. (A laminar flow in a pipe normally becomes turbulent when Re > Rec ^ 2,200.) This explains why the majority of fluid motions in systems with large dimensions and low viscosity are turbulent.

Damping due to molecular viscosity is very small and its effects on the large-scale motions encountered in geophysics and astrophysics is utterly negligible. However, for the very reason that one can make direct measurements in the Earth's atmosphere and in the oceans, it has long been recognized that these systems contain a wide spectrum of eddylike motions that coexist with the largest-scale motions. (As we shall see in Section 3.6, similar small-scale motions exist in stellar interiors, but their existence can be inferred by reasoning only.) Since there is as yet no practical and accurate theory that describes all scales of motion, from the largest to the smallest scales, it is convenient to restrict consideration to the large-scale motions only. Because Eq. (2.7) contains the nonlinear terms v ■ grad v, this isolation can never in fact be complete, with motions on one spatial scale necessarily interacting with motions on other spatial scales. These interactions are often modeled by the inclusion of a large anisotropic eddy viscosity in the momentum equation, of much larger magnitude than the molecular viscosity; the functional form of this frictional force is analogous to that of Eq. (2.9). Unfortunately, because turbulence is not a feature of fluids but of fluid flows, the momentum exchange by eddylike motions only superficially resembles molecular exchange of momentum. Yet, although the empirical concept of eddy viscosity cannot be derived rigorously from first principles alone, it has proven to be both useful and effective in many dynamical problems that demand some frictional forces to be present.

At any given point and time, the physical variables of a system may be expressed in terms of mean values (denoted by overbars) and fluctuating values (denoted by primes). For such a decomposition to make sense, a suitable averaging period has to be found so that the mean values are substantially independent of this averaging period. Here we shall assume that it is possible. Hence, we let v = v + V, (2.52)

and we write similar expressions for the other physical variables. By definition, the components of the mean velocity are given by so that

Note that we have also

which vanishes only in the case of an incompressible fluid. Equation (2.54) ensures that, on the average, there is no transfer of mass due to turbulence and that Eq. (2.6) remains valid for the mean flow. It follows at once that

p Dt

Combining next Eqs. (2.6) and (2.7), we can recast the momentum equation in the form

T- (Pvr) + -7— (Pvr vk) = Pgr - + T— > (2.57)

d t d xk d Xj d xk where the viscous stress tensor is defined in Eq. (2.8). If we suppose the body force to be unaffected by turbulence, the average of Eq. (2.57) is d _ d___ d p 9_

d t d Xk d Xr d Xk since the operations of averaging and differentiation commute. The tensor t is the average of the tensor t . The new tensor a has the components

These six quantities define the Reynolds stresses. Equation (2.58) is identical to Eq. (2.7) with all quantities replaced by their mean values, except for the additional Reynolds stresses. This symmetric tensor represents the flux of momentum due to the eddylike motions. The term div a in Eq. (2.58) thus exchanges momentum between these small-scale motions and the mean flow, even though the three components pv'k of the mean momentum of the turbulent velocity fluctuations are zero. Whenever eddylike motions prevail, the average viscous stresses T are usually negligible compared to the Reynolds stresses a.

The central problem in this representation of small-scale motions lies in the fact that Eq. (2.58) introduces six unknown quantities, namely, the six components of the tensor a. The simplest approach is to draw an analogy with molecular viscosity. Following Boussinesq, we shall assume that the turbulent motions act on the large-scale flow in a manner that mimics the microscopic transfer of momentum between the constitutive particles, when a macroscopic velocity gradient prevails. In order to apply this method to geophysical problems, we shall make use of Cartesian coordinates. The relevant equations for a rotating star will be discussed further in Section 3.6.

In the Earth's atmosphere and in the oceans, the horizontal dimensions of the large-scale motions are much greater than the vertical ones. This anisotropy of the large-scale flows strongly suggests that the turbulent transport of momentum in these two directions cannot be expected to be the same. If the axes are chosen so that the x3 axis is in the vertical direction, a particularly simple expression for the Reynolds stresses is

O11 = 2Ah —— , CT22 = 2Ah —— , O33 = 2Ay -—, (2.60)

3v1 dv2

9vi dv3

9V2 dv3

where A H and Ay are the horizontal and vertical coefficients of eddy viscosity. Neglecting molecular viscosity and omitting the overbars, one can thus rewrite Eq. (2.58) in the form

Dt p p where F is the turbulent viscous force per unit volume, which is the vectorial divergence of the tensor a. Neglecting compressibility effects, one obtains

where we have assumed that AH and Ay are constant quantities. The preferred vertical direction is thus properly taken into account. (Compare with Eqs. [2.7] and [2.9].)

Because the eddy viscosities cannot be calculated from first principles alone, crude measurements of their values in the Earth's atmosphere and in the oceans have been made. Typical atmospheric values of Ky (= Ay/p) lie in the range 104-106 cm2 s-1, whereas one has v * 10-1 cm2 s-1 for air. It follows that

— * 105-107, (2.66) v in the atmosphere (Houghton 1986). For the oceans, estimates of KV range from 1 cm2 s-1 to 102 cm2 s-1. This implies that

v in the oceans, since one has v = 10-2 cm2 s-1 for water. The smaller values go with smaller-scale motions, and conversely (Apel 1987). It is also worth noting that in the Earth's lower atmosphere one has AH/AV < 102, whereas this ratio may be as large as 105 in the surface layer of the ocean where large-scale currents are observed.

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