La Conspiración

The presented friction factor diagrams and correlations are all based on the flow being steady-state. But how can we calculate the friction when the flow varies over time? As shown by Zielke (1968), Trikha (1975), Ham (1982), and Bratland (1986), there are several methods available for calculating transient friction very accurately for laminar flow, and doing so must now be considered relatively trivial.

Transient turbulent friction, a much more frequently encountered phenomenon, is generally not as well understood, but it is clear that the transients can make the friction several orders of magnitude higher than the steady-state friction. Brekke (1984) has developed a frequency-domain model for improved stability analysis of hydro power plants, and it has a built-in transient friction estimator which seems to work well for pulsations superimposed on a relatively high average flow. Somewhat more general models have been presented by Zarzycki (2000), Vardy et al. (1994) and several others, and their models represent clear improvements compared to using steady-state friction factors directly. But the subject is complicated, and no one has succeeded in developing a general, well documented and practical theory.

Instead of going into details about the available models, let us try to give some rules for when we can get away with using steady-state friction rather than going to greater sophistication.

Figure 2.14.1. Straight pipe with fast-closing valve at the end.

To illustrate which mechanisms affect the friction factor, consider first a pipe with laminar flow. By using Newton‟s law, equation 2.2.1, it is relatively easy to show that steady-state laminar velocity profiles are described by:

( )

. / (2.14.1)

Suppose a valve at the end of the pipe is suddenly closed. One may at first expect this to stop all the fluid immediately after the closure so all particles near the valve come to rest. For that to happen, however, particles near the center of the pipe, which has a higher initial velocity than those closer to the wall, would have to be stopped simultaneously. Instead, the instantaneous pressure step resulting from the closure leads to the same velocity step for all fluid particles in each cross section, and continuity means the average velocity becomes zero. Those particles close to the center continue at a reduced speed towards the closed valve, while those close to the wall are forced backwards away from it. Therefore, closure does not mean the fluid just upstream from the valve comes to rest immediately.

This is illustrated in figure 2.14.2. Curve 1 is the velocity profile before closure and curve 2 the profile just after. Curve 2 is very similar to curve 1, but it has been moved to the left so that the flow is zero. This leads to a very sharp velocity gradient near the wall, and the instantaneous friction can become very high just after closure. Gradually, friction removes more and more of the kinetic energy, and after awhile, the velocity

v

profile looks like curve 3, before the fluid eventually comes to rest. The curves are actual simulations of how flow develops, not mere illustrations. The results have been shown to agree with measurements carried out by Holboe & Rouleau (1967), (Bratland 1986).

What is of interest here, though, is observing that velocity gradients near the wall can become very steep after fast changes in the average velocity. In this particular example, the average velocity obviously becomes zero after closure, but the friction does not. The Darcy-Weisbach friction factor f would have to be infinite to describe the non-zero friction! It is also possible for the fluid closest to the wall to temporarily go in the opposite direction of the average velocity, leading to a negative f.

Figure 2.14.2. Velocity profiles near the valve at different times after closure, laminar flow.

In turbulent flow, both the steady-state and the transient velocity profiles appear slightly different than these laminar profiles, but the main mechanisms remain the same: Transients deform the velocity profile, and that makes f differ from what steady-state correlations predict. The frequency-dependent friction, as it is sometimes called, tends on average to be much higher than steady-state friction.

If Re is constant over a relatively long period of time, the friction factor approaches the steady-state value. Accurate criteria for how long it takes to reach steady-state do not exist. But one may compare with how a velocity profile develops at a pipe‟s inlet, where it is generally accepted that the profile is fully developed around 30 diameters into the pipe. It is therefore reasonable to assume steady-state friction once the average velocity

(2.14.2)

where is the time it takes for from the velocity, , becomes constant until the friction has become steady-state, too, meaning the normal Darcy-Weisbach friction calculations have become valid.

In very long pipelines, it takes hours and even days to change the velocity significantly, and under such conditions, transient friction can safely be neglected. The friction factor may simply be computed from equation 2.9.4 and updated every time the Reynolds number changes. That is good news to the engineer since it means frequency-dependent friction is not something to worry about in most cases. The exceptions are systems where fast changes in Reynolds number occur, and where those changes can have significant effect on the problems of interest. There are at least two important situations when transient friction cannot be neglected:

1. In some cases we are interested in studying hydraulic noise attenuation, for instance downstream of pumps. The noise corresponds to velocity transients and may be in the range of hundreds or thousands of Hz. Under such conditions,

the frequency-dependent part of the friction may be orders of magnitude higher than the steady-state part, and the noise ripples are damped much faster than the steady-state friction would suggest. Accurate prediction of the damping is obviously not possible by using steady-state friction. In the case of laminar flow, Ham‟s frequency-domain model may offer the simplest and most effective approach (Ham, 1982), but as mentioned above, time-domain models are also available.

2. In many hydropower plants, regulators maintain constant turbine speed during varying load conditions by controlling the water flow through each turbine. If, for instance, a consumer switches off his air conditioner, the regulator responds to the reduced load by reducing the nozzle or guide vane opening. The water in the penstock may have high enough inertia for the retardation to lead to a significant pressure increase at first, and that in turn gives higher rather than lower power output. Stabilizing such regulators is therefore difficult, and large, expensive surge tanks are often needed to ensure proper operation. The frequency-dependent

In long pipelines with slow pressure and mass flow variations the friction can be calculated as if it were

steady-state.

friction works to improve stability, and accurate prediction and utilization of it enables cheaper designs.

Much of the theoretical work done on transient turbulent friction has focused on instantaneous valve closure and how the reflected pressure surges gradually dissipate after the first, initial surge. Such cases represent a well-defined reference, but paradoxically, they are of little interest in most engineering situations. The maximum pressure occurring in first surge is not much affected by the friction‟s frequency dependence, and the later reflections are typically smaller than the first one, making the exact damping of those of little concern. When the average flow is non-zero, which is what we have to deal with when studying noise propagation or highly dynamic regulators, the turbulent eddies responsible for most of the friction are constantly re-energized, and it is not obvious that the newer, relatively fragile models give adequate results under such conditions.