Many geological flows involve turbulence, wherein the velocity field involves complex, fluctuating motions superimposed on a mean motion. Flows in natural river channels are virtually always turbulent. Magma flow in dikes and sills, and lava flows, can be turbulent. Atmospheric flows involving eolian transport are turbulent. The complex, convective overturning of fluid in a magma chamber or geyser is a form of turbulence. Thus, a description of the basic qualities of these complex flows is essential for understanding many geological flow phenomena. Turbulent flows generally are associated with large Reynolds numbers. Recall from Chapter 5 that the Reynolds number Re is a measure of the ratio of inertial to viscous forces acting on a fluid element, . . . Re = ρUL/μ . . . . . . (14.1) . . . where the characteristic velocity U and length L are defined in terms of the particular flow system. Thus, turbulence is typically associated, for given fluid density ρ and viscosity μ, with high-speed flows (although we must be careful in applying this generality to thermally driven convective motions; see Chapter 16). A simple, visual illustration of this occurs when smoke rises from a cigar within otherwise calm, surrounding air. The smoke acts as a flow tracer. Smoke molecules at the cigar tip start from rest, since they are initially attached to the cigar. Upward fluid motion, as traced by the smoke, initially is of low speed, and viscous forces have a relatively important influence on its behavior. The flow is laminar; smoke streaklines are smooth and locally parallel. But as the flow accelerates upward, it typically reaches a point where viscous forces are no longer sufficient to damp out destabilizing effects of growing inertial forces, and the flow becomes turbulent, manifest as whirling, swirling fluid motions (see Tolkien [1937]). Throughout this chapter we will consider only incompressible Newtonian fluids. Unfortunately, the complexity of turbulent fluid motions precludes directly using the Navier–Stokes equations to describe them. Instead, we will adopt a procedure whereby the Navier–Stokes equations are recast in terms of temporally averaged or spatially averaged values of velocity and pressure, and fluctuations about these averages.
Finite-scale equations for compressible fluid flow