The Reynolds experiment

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Bernadette Olivia Lynch
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1 Chapter 13 The Reynolds experiment 13.1 Laminar and turbulent flows Let us consider a horizontal pipe of circular section of infinite extension subject to a constant pressure gradient (see section [10.4]). The viscous theory implies a parabolic profile for the velocity field (Fig. 8.3). However, it can be easily observed that such a flow exists only if the pressure gradient does not exceed a certain value. A further increase of the pressure gradient leads to a flow which is no longer laminar, but which shows irregular fluctuations superimposed to an average behavior different from the previous parabolic profile. These fluctuations include different length scales, ranging from the order of the radius of the pipe to a scale so small that the energy dissipation due to viscosity becomes the dominant process. As the pressure gradient further increases, not only the average motion, but the magnitude of the fluctuations increases as well. The flow is said to be turbulent, to distinguish it from the previous regular, laminar flow. The volume rate is still a monotone function of the pressure gradient as in the laminar case, but this function is no longer linear. At first glance nothing happens to justify such a sudden transition between two regimes of motion that are so different. In a turbulent flow, even if the pressure gradient is kept constant in time, the velocity field is not. In general, we 95

2 96 Franco Mattioli (University of Bologna) expect that the properties of the cause, such as the stationarity of the pressure force, should reflect in a similar property of the effect, i.e., the stationarity of the motion. But this is not the case of turbulence, whose behavior seems in sharp contrast with the ordinary laws of physics. Not only, but the velocity fluctuations do not seem governed by any intelligible law. They develop on various space scales and incessantly vary in time, so that they appear as an incomprehensible and unpredictable motion, which is practically impossible to describe in all its details. We refer to the strong irregularity of the velocity fluctuations in a turbulent motion by saying that the motion is disorganized, disordered, erratic, incoherent, random or chaotic. As a consequence, it is not possible to generate in laboratory two flows that are perfectly equal in all the details. We say that a turbulent motion in not physically reproducible. But mean quantities, such as the average velocity, do exist, and are easily reproducible. For this reason we can see the motion as the sum of erratic fluctuations plus an average motion.... u.... Fig. 13.1: In a pipe of circular section the velocity profile is parabolic if the flow is laminar. When the flow is turbulent, the profile of the mean velocity is much more flattened. It follows that the two profiles cross near the walls, even if this detail cannot be detected in the figure. The profile of the mean velocity, however, is flatter than the parabolic behavior of the laminar solution of the Navier Stokes equations (Fig. 13.1). The zero velocity at the solid boundaries is reached in a very thin layer. Due to the presence of small scale fluctuations the energy dissipation in very strong especially near the boundaries, so that the same pressure gradient generates a smaller flow rate.

3 Elements of Fluid Dynamics ( 97 The transition to turbulence for sufficiently high velocities is not a peculiarity of the circular Poiseuille flow, but a general property of many flows. In particular, the shear flows, that is, the flows characterized by a strong transversal variation of the velocity, are affected by this phenomenon. Many of the considerations we will draw for the flow in a circular pipe can be extended with minor changes to these flows as well. For example, the plane Poiseuille flow also becomes turbulent with the increase of the pressure gradient, and the plane Couette flow becomes turbulent as the velocity of the upper plate exceeds a certain value The Reynolds experiment The reason for which the transition to turbulence was observed, and then systematically studied for the first time in a pipe, lies in the simplicity of the experimental apparatus necessary to observe the phenomenon. The Reynolds experiment (Fig. 13.2), after the name of the author that firstly faced this problem in a systematic way, is sufficient to derive the principal features of the phenomenon. In this experiment a horizontal pipe is immersed in a tank filled with water (Fig. 13.2). Just outside the tank the pipe is bended downward in order to provide a sufficient velocity to the flow for the various experiments. For this reason the tank is placed on an elevated platform, since the water is discharged at the level of the floor. At the bottom, a valve connected to the free atmosphere controls the water flow in order to generate the wanted pressure gradients along the horizontal section of the pipe. The valve is regulated by a long lever operated by a technician from the elevated platform. Pipes of different sections are used to see the dependence of the phenomenon on the radius of the pipe and the viscosity of the water is modified by changing its temperature. Accurate readings of the water level permit to measure the flow rate in the pipe. The intake of the pipe is fitted with a trumpet mouthpiece, in order to avoid the formation of vortices along its edges. The regimes of the flow are made visible by introducing a colored tracer, able to provide an image of the velocity field (Fig. 13.3). When the flow is laminar the tracer appears as a straight colored line (Fig a). As soon as the regime becomes turbulent the tracer spreads over the whole cross-section of the pipe (Fig b), so that the fluid changes its color everywhere. For low values of the mean velocity in the pipe the flow is laminar everywhere. Once the mean velocity exceeds a certain critical value, we see the formation of a turbulent motion involving the whole section of the pipe, starting suddenly at a certain distance from the intake. As the pressure gradient increases, the

4 98 Franco Mattioli (University of Bologna) Fig. 13.2: The apparatus of the Reynolds experiment. transition to turbulence occurs at a smaller and smaller distance from the intake, but without never reaching it. Near the intake of the pipe the flow is always laminar. Indeed the transition from laminar to turbulent flow is rather complicated. In fact, just above the critical mean velocity regions of almost laminar flow alternate with regions of turbulent flow. In a point fixed in space the flow is alternatively almost laminar (that is, small perturbations are added to a laminar motion) and fully turbulent. This phenomenon is called intermittency. As the patches of turbulent motion move with the flow, they increase their length, merge and finally extend to the whole domain occupied by the fluid. By still further increasing the velocity of the flow, the transition to complete turbulence occurs in less space, until the transition from laminar to turbulent flow occurs in a single step.

5 Elements of Fluid Dynamics ( 99 Fig. 13.3: In (a) the flow is laminar. In (b) the flow is turbulent. In (c) the transition from laminar to turbulent motion is seen in detail, with the formation of curls in the tracer before the mixing with the other fluid becomes complete. By illuminating the flow by a spark, we see that the tracer just before the region where turbulence starts shows swirls formed by elongated filaments that become finer and finer, until they mix completely with the surrounding fluid (Fig c). This suggests the presence of continuously changing eddies in the flow. For this reason, a property of a turbulent flow is often labeled by the adjective eddy. Indeed, the transition to turbulence depends in a crucial way on the structure of the adopted experimental apparatus. If the flow is perturbed, the transition to turbulence occurs at lower velocities. Perturbations can be generated in the flow outside the pipe, along the edges of its intake or inside the pipe itself. Two kinds of experiments are possible. On one side, one can try to eliminate these perturbations in order to keep the flow laminar as much as possible. This is the case of (Fig. 13.2) where a trumpet mouthpiece is adopted. On the other side, we might want to explore the limits under which the flow remains laminar even after strong perturbations have been introduced in it.

6 100 Franco Mattioli (University of Bologna) In the former case the critical mean velocity is much less defined than in the latter. More exactly, in the former case the transition to turbulence can be obtained for mean velocities up to 50 times greater by an improved equipment able to prevent the onset of perturbations. In the latter case the transition to turbulence occurs for mean velocities that vary only of the order of 10% even for very different perturbations. Let us consider in more detail the physical meaning of a perturbation of the basic state. When we perform an experiment there are always small perturbations that interfere with it: the vibrations due to traffic, to an engine present in the room, to the steps of people walking in the room. Furthermore, the apparatus is not made of perfectly smooth surfaces, and the junctions between its different parts can present small invisible cracks. All these imperfections can generate perturbations, making a given experiment not exactly reproducible Stable and unstable flows Let us search for some general (and generic) explanation of what occurs in a turbulent motion. The sensitivity of the transition to turbulence to the disturbances of the flow strongly suggests that the turbulence itself is determined by the growth of initially small disturbances. We have already pointed out in the derivation of the Euler equations that the advective terms of the total derivative are responsible for great difficulties in the solution of a problem involving the momentum equation. Such difficulties are in part of a mathematical kind, but in part are related to the problem we will now describe, but only from a qualitative point of view. Let us suppose that we have mathematically solved a given problem of fluid dynamics governed by a set of equations, some of which are nonlinear. Let us call this solution, for the time being, basic solution. Let us now construct a new possible solution, adding an infinitesimal perturbation to this basic solution in such a way as to satisfy the boundary conditions of the problem. If we introduce this new solution in the equations of motion, what we obtain is a set of equations for the perturbation. There are three possibilities. The perturbation can decrease, maintain the same amplitude or increase in time. In the first case, the basic solution is said to be stable, in the second one, neutral and in the third one, unstable. When the perturbations decay in time, so that the flow is stable, the basic solution is also a physical solution that can be easily observed. But if the perturbations grow over time so that the flow is unstable, the basic flow is only a

7 Elements of Fluid Dynamics ( 101 mathematical solution, which does not even correspond to the average flow. The perturbations, as small as they are at the initial time, grow until they reach a large amplitude, carrying along with them a significant fraction of the energy of the flow. On the other hand, a turbulent flow as a whole is very stable. A disturbance introduced in the flow is rapidly damped. In general, to change the structure of a turbulent flow by an external forcing is much more exacting than for a laminar flow The various kinds of instability An instability mechanism does not necessarily lead to turbulence. In certain phenomena the instability of the basic flow gives rise to perturbations which grow with regular shapes. Sometimes the resulting motion can show a very complicated structure, without being turbulent. Only in a limited number of circumstances, as in the Reynolds experiment, the result of the instability is a turbulent motion. Unfortunately, only in a limited number of cases it has been possible to demonstrate mathematically the stability or instability of a flow. Sometimes, a flow can be stable for perturbations of small (i.e., infinitesimal) amplitude, but unstable for perturbations of larger (i.e., finite) amplitude. Studies of this kind present many subtle mathematical difficulties. Nevertheless, at present, this interpretation of the physical mechanism giving rise to turbulence is not brought into question. The Poiseuille flow in a pipe is shown to be stable to infinitesimal perturbations. But in a real experiment we have only pipes of finite length, and the stationarity of the flow is reached only after a transient. The problem to discover the mechanism that gives rise to unstable perturbations is still open. From a mathematical point of view the responsibility for the instability of a flow lies in the presence of nonlinear terms in the equations of motion. In their absence, all the flows would be stable because of the principle of superposition of the effects (see section [E.2]). A small change in the initial state can only lead to strongly correlated changes in the solution at a later time. Therefore, the nonlinearity is responsible not only for the strong technical difficulties in the solution of many problems of fluid dynamics, but also for the various mechanisms of instability. The presence of instabilities shows that a solution of the Navier Stokes equations might or might not correspond to a real solution present in nature. An

8 102 Franco Mattioli (University of Bologna) unstable motion, in spite of being an exact solution of the equations of motion, will never be observed. At first glance, more mathematical solutions of the Navier Stokes equations are possible for a given macroscopic problem. Indeed, the Navier Stokes equations do have a unique solution. But this solution depends on uncontrollable and insignificant details, such as the exact behavior of the surface of a rough wall or the exact evolution of the flow in all the previous instants of time. The choice of the actual motion depends on the particular structure of such subtle details. In any case the validity of the Navier Stokes equations is never put in discussion.

Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 17 Laminar and Turbulent flows

Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 17 Laminar and Turbulent flows Welcome back to the video course on fluid mechanics. In