Reynolds experiment

Figure 3.2. Break-up of laminar thread in Reynolds experiment...

Now return to a view of the nature of flow in the boundary layer. It has been called laminar, and so it is for values of the Reynolds number below a critical value. But for years, beginning about the time of Osborne Reynolds experiments and revelations in the field of fluid flow, it has been known that the laminar property disappears, and the flow suddenly becomes turbulent, when the critical VUv is reached. Usually flow starts over a surface as laminar but after passing over a suitable length the boundary layer becomes turbulent, with a thin laminar sublayer thought to exist because of damping of normal turbulent components at the surface. See Fig. 6. 

Figure 2.5-1. Reynolds experiment for different types of flow (a) laminar flow, (b)...

Figure 2-7. Reynolds experiment—transition and turbulent flow.

We have considered a one-dimensional flow case for a Newtonian fluid (Newton s Law of Viscosity) as well as a phenomenological consideration of fluid dynamics (the Reynolds experiment, the Reynolds number, velocity profiles). Now, let us direct our attention to the concepts of the multidimensional cases. 

In turbulent flow the fluid has a chaotic pattern as shown by the Reynolds experiment. Because of the nature of turbulent flow, the velocity will actually fluctuate (see Figure 3-1). It is possible to time-smooth these fluctuations such that... 

The conclusions that can be drawn from the Reynolds experiment are ... 

The phenomenon of thermal transpiration was discovered by Osborne Reynolds [82], who gave a clear and detailed description of his experiments, together with a theoretical analysis, in a long memoir read before the Royal Society in February of 1879. He experimented with porous plates of stucco, ceramic and meerschaum and, in the absence of pressure gradients, found that gas passes through the plates from the colder to the hotter side. His experimental findings were summarized in the following "laws" of thermal transpiration. 

Calculations of relative partition coefficients have been reported using the free energy perturbation method with the molecular dynamics and Monte Carlo simulation methods. For example, Essex, Reynolds and Richards calculated the difference in partition coefficients of methanol and ethanol partitioned between water and carbon tetrachloride with molecular dynamics sampling [Essex et al. 1989]. The results agreed remarkably well with experiment... 

Suppose that an experiment were set up to determine the values of drag for various combinations of O, p, and ]1. If each variable is to be tested at ten values, then it would require lO" = 10, 000 tests for all combinations of these values. On the other hand, as a result of dimensional analysis the drag can be calculated by means of the drag coefficient, which, being a function of the Reynolds number Ke, can be uniquely determined by the values of Ke. Thus, for data of equal accuracy, it now requires only 10 tests at ten different values of Ke instead of 10,000, a remarkable saving in experiments. 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

Equations (6-236) to (6-239) are based on experiments on cube-oc tahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity f ranges from 0.906 to 0.670, respectively. See also Chft, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with y as a parameter may be found in Brown, et al. (Unit Operations, Whey, New York, 1950) and in Govier and Aziz. 

Coefficient A and exponent a must be evaluated experimentally. Experiments have shown that A and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a... 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

It is necessary to study the Reynolds number independence in the hill scale and in the model as part of a complete model experiment. The threshold or the critical Reynolds number, Re, for the problem considered should be found by the experiments, and measurements should be made at either the same Reynolds number as in the full scale or at a Reynolds number equal to or larger than Re if the Reynolds number in the full scale is larger than Re. for the problem considered. 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

The model experiments were carried out in a model on the scale tjf I to iO. Experience with measurements on flow from wall-mounted dif(users for displacement ventilation indicates that it is possible to ignore the level of the Reynolds number at the given dimensions, which will enable reasonable temperature differences in the model experiments. 

A reduced scale of the model requires an increased velocity level in the experiments to obtain the correct Reynolds number if Re < Re for the prob lem considered, but the experiment can be carried out at any velocity if Re > RCj.. The influence of the turbulence level is shown in Fig. 12.40. A velocity u is measured at a location in front of the opening and divided by the exhaust flow rate in order to obtain a normalized velocity. The figure show s that the normalized velocity is constant for Reynolds numbers larger than 10 000, which means that the flow around the measuring point has a fully developed turbulent structure at that velocity level. The flow may be described as a potential flow with a normalized velocity independent of the exhaust flow rate at large distances from the exhaust opening— and far away from surfaces. 

The assumption of a self-similar flow (Reynolds number-independent flow) simplifies full-scale experiments and is also a useful tool in the formulation of simple measuring procedures. This section will show two examples of self-similar flow where the Archimedes number is the only important parameter. 

Figure 12.41 shows the results of three experiments with a similar Archimedes number and different Reynolds numbers. The figure shows vertical temperature profiles in a room ventilated by displacement ventilation. The dimensionless profiles are similar within the flow rates shown in the figure, although the profile may involve areas with a low turbulence level in the middle of the room. A test of this type could indicate that further experiments can be performed independently of the Reynolds numbers. 

Experience gives values of f for various geometries, but St is found to be a weak function of Reynolds number, so in practice there is relatively little variation in cooling efficiency (0.6 < tjc[Pg.72]

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