« PreviousContinue »
Consider a fluid that has been set into smooth and uniform motion in a circular trough (Fig. 3). It has zero total (vector) momentum: as much is moving east as is moving west at every instant. Tangential shearing drag on the walls slows the motion so that mean-flow kinetic energy is lost. Where does the energy go-to turbulence or to heat? The competition is fierce, and heat always wins in the end, but fluids yield themselves to the inevitable only grudgingly. If at all possible, they transform at least part of their kinetic energy to turbulence as an intermediate step along the way (Fig. 4). Let's replace this animistic description with physics. The conversion of meanflow kinetic energy directly to heat is limited by the viscosity of the fluid and by the steepness of the mean-flow velocity gradients. For example, consider fluid flow between two plates moving in parallel but opposite directions (Fig. 5). Although a variety of flow-velocity profiles could have been depicted, the one shown has the smallest fluid kinetic energy of any flow profile with that same momentum between the moving plates. This profile is thus the flow distribution to which all others inevitably tend.
Suppose we now examine a flow profile at the opposite extreme-one in which the gradient at the midpoint between the plates is very sharp (Fig. 6(a)). Both this distribution and the stable one in Fig. 5 have the same total fluid momentum (namely zero); however, in the distribution in Fig. 6(a), every fluid element has the same speed (uo), whereas in the stable distribution, most elements are moving slower than uo. Thus the fluid in Fig. 6(a) possesses an excess of kinetic energy compared to the fluid with the stable profile and will lose part of this energy as it transforms toward the stable configuration. Will turbulence be an intermediate state in this evolution? To answer this question we must dig deeply into the competitive processes of dissipation and instability.
Consider first the dissipation of mean-flow kinetic energy into heat. Let H be heat energy per unit volume and du/dy describe some measure of the mean-flow velocity gradient in a fluid with molecular viscosity μm. Then the rate at which heat is generated is given by
To estimate the rate at which turbulence energy is generated, we return to the flow described in Fig. 6(a), which is susceptible to a destabilizing process called the KelvinHelmholtz instability. The presence of such an instability is easily demonstrated for an incompressible fluid if we arbitrarily assume that the slip interface between the upper and lower halves of the flow profile is distorted by a sinusoidal wave of wavelength A and amplitude A (Fig. 6(b)). Because the fluid is incompressible, wherever the flow area is constricted the fluid has to move faster than average, and wherever the flow area is expanded the fluid has to move more slowly (Fig. 6(c)). What is the associated behavior of the pressure? Each cycle in the perturbation is like a Venturi nozzle, for which Bernoulli's law says the pressure is less in the constricted region where fluid speed is higher and is greater in the expanded region where fluid speed is lower. Thus, there is a pressure difference across the perturbed slip plane, acting in exactly the right direction to enhance the perturbation amplitude.
The growth in amplitude is dA/dt, so the kinetic energy per unit volume involved in this turbulent-like motion is
K = (2340) 2 PADA
dA ΡΑ dt
With w = (2πuo/X), a solution of Eq. 2 is A = Age, and
where p is the density of the fluid. Differentiating Eq. 3 and substituting Eq. 2, we see that turbulence energy K, in turn, grows as
The essence of these results is that dK /dt increases with time (w is positive), whereas, because viscosity smears out the sharp velocity transition, thus decreasing du/dy, dH/dt decreases with time (Eq. 1). Whenever the amplitude (scale) of the disturbance is large enough, turbulence creation will dominate.
The important dimensionless quantities involved in the competition between turbulence energy creation and heat dissipation can be illustrated by taking the ratio of the growth rates of turbulence energy and heat energy at t = 0. Using Eqs. 1 and 6 and setting du/dy ≈ 2uo/A at t = 0 we find
As long as dH/dt dominates, the mean-flow kinetic energy dissipates to heat, and the intermediate turbulent stage is bypassed; we say that the mean flow is stable. If dK/dt dominates, then the mechanism driving the instability draws the excess kinetic energy into turbulence. We can thus formulate a stability criterion, based on the Reynolds number, in which molecular viscosity plays a central role. For large viscosity, dH/dt is able to exactly balance the loss rate for mean-flow kinetic energy. Decreasing the viscosity eventually drops dH/dt below the mean-flow loss rate, and the flow becomes unstable.
As a corollary, note that conservation of total energy raises an interesting question about mean-flow dynamics. What mechanism accounts for destruction of mean-flow kinetic energy at exactly the required rate to ensure conservation? The answer is viscosity-molecular viscosity and turbulence viscosity.
For the case of only direct viscous dissipation to heat, viscous drag between the opposing currents causes each to slow down, and the corresponding loss rate for kinetic energy exactly accounts for the dissipative heating. For the case of transfer of mean-flow kinetic energy to turbulence, a directly analogous process occurs in which turbulence viscosity produces drag. More precisely, the presence of turbulence induces a fluid shear stress, the Reynolds stress, that is independent of the molecular viscosity of the fluid. Expression of the components of the Reynolds stress tensor in terms of readily measured flow quantities (such as pressure and mean-flow velocity) lies at the heart of our theoretical work and is discussed in detail in the next section.
Analogous to molecular viscosity, turbulence viscosity depletes mean-flow kinetic energy at precisely the same rate that turbulence energy is growing. A direct consequence is that turbulence contributes to the effective viscosity of the fluid, enhancing the rate of momentum diffusion from one part of the fluid to another as it simultaneously destroys the excess mean-flow kinetic energy. As we shall see, turbulence diffuses anything imbedded in the fluid-momentum, heat, dye, dust particles, dissolved salts.
yo design and test proposed largescale equipment, such as airfoils or entire aircraft, it is often much more practical to experiment with scaled-down versions. If such tests are to be successful, however, dynamic similitude must exist between model and field equipment, which, in turn, implies that geometric, inertial, and kinematic similitude must exist.
can be thought of as a comparative measure of inertial and viscous (diffusive) effects within the flow field. To achieve dynamic similitude in two different laminarflow situations, the Reynolds numbers for both must be identical.
What happens if we increase the flow speed to the point that viscous dissipation can no longer stabilize the flow, and the macroscopic balance between meanflow inertia and viscous effects breaks down? At this point there is a transition from purely laminar flow to turbulence. In similar flows, the transition occurs at a specific Reynolds number characteristic of the flow geometry. For instance, any fluid traveling inside a circular piperegardless of the specific fluid or conduit
being used experiences the onset of turbulence at R = 2000.
At or near this "critical" Reynolds number, inertial contributions to mean-flow momentum that cannot be dissipated by viscous stresses must be absorbed by newly formed turbulent eddies. The presence of turbulence energy is often described in terms of an effective turbulence viscosity, defined as the ratio of the turbulence-shear, or Reynolds, stress to the mean-flow strain rate. With this in mind, an effective turbulence Reynolds number-one that includes molecular viscous effects-is
Note that the molecular kinematic viscosity m is retained in this definition. The choice of molecular viscosity to characterize the dissipative mechanisms responsible for tearing eddies apart is based on the ultimate transformation of turbulence into heat energy. Molecular processes are, in the end, dominant at the smallest scales, and R, is a relative measure of the loss of kinetic energy from an eddy of a given size to heat. For the smallest eddies in a flow system, R, 1; that is, all the energy of the eddy is dissipated into heat.■
What then can we deduce from this example about the features necessary for the creation of turbulence?
• A mean-flow profile richer in kinetic energy than other momentum-conserving states to which it can transform (such as the profile in Fig. 6(a) that can transform to the one in Fig. 5).
• A viscosity low enough that dissipation to heat cannot absorb all the mean-flow energy during the transition to the low-energy profile.
• A driving mechanism for enhancement of the inevitable microscopic perturbations (such as the Kelvin-Helmholtz instability in Fig. 6).
However, the energy of turbulence frequently comes from sources (Fig. 7) other than a velocity profile rich in mean-flow kinetic energy. For example, turbulence can be fed directly from potential energy as when a Rayleigh-Taylor instability develops at the interface between, say, water overlying a less dense layer of oil or cold air overlying warm air. The latter instance, called buoyancy-driven turbulence, produces the dancing air currents that can be seen by looking across the surface of a sunlit roof on a cold day. Similarly, turbulence can be fed by accelerative forces as when a Richtmyer-Meshkov instability develops at the deformable interface between two materials that are perturbed by, say, a passing shock wave or the sudden acceleration of the entire system.
Droplets, particles, or bubbles projected through a liquid or gaseous fluid with some relative velocity likewise can serve as a good source of turbulence energy. The momentum-conserving transition induced by drag tends always to bring such entities and the fluid to the same velocity. Competition for the center-of-mass kinetic energy results in a partition into both heat and turbulence-the winner again depending on the level of viscosity.
Likewise, if a quiescent suspension is subjected to a pressure gradient or shock, a differential acceleration occurs that is in proportion to the difference in densities between the suspended entities and the surrounding medium. Turbulence often gleans a significant share of the resulting interpenetrational energy.
Turbulence Sinks. So far we have been discussing only sources for turbulence and the manner in which the turbulence decays. Here we must return to what constitutes turbulence and, in particular, reaffirm that the existence of turbulence depends on the observer's point of view. Mean flow is that part of the dynamics whose structure is comparable in size to the region being measured; it is capable of being reproducibly duplicated or monitored—at least in some statistical sense. Finer dynamical scales of a capricious nature arising from random initial, boundary, or bulk perturbations constitute the fluid's turbulence. But the mean flow for one observer may simply be the larger scales of a turbulence spectrum for an observer whose field of view encompasses a somewhat larger domain. Thus, the source of turbulence seen by one observer becomes the energy sink for the decay of turbulence at the larger scales of another observer.
This principle and its generalizations have powerful consequences for our mathematical modeling of turbulence dynamics, leading to the concept of a turbulence cascade. In this process turbulence energy is transferred to progressively smaller and smaller fluctuational scales with the source of energy for each scale coming from the mean-flow velocity contortions of the next larger scale (Fig. 8). At each stage, there is competition for the energy, part going into heat and part going into even smaller