Model Experiments in the Case of Fully Developed Turbulent Flow

P> Pi

FIGURE 12.27 Relative change in the volume expansion coefficient for air and water.

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.

Figure 12.28 shows as an example how the dimensionless maximum velocity in a room wrm/w0 is constant at different supply velocities, U(), for Reynolds num­bers larger than 45 000 in the case of isothermal flow. This value is called the crit­ical Reynolds number, ReLJ or the threshold Reynolds number for the maximum velocity in the room. A model experiment in the room (Fig. 12.28) with the pur­pose of determining the maximum velocity in the occupied zone can therefore be carried out at any Reynolds number larger than Rec if the full-scale flow has a Reynolds number larger than Rec. It is also obvious that the Reynolds number must have the same value in the full scale and in the model for Reynolds numbers smaller than Rec. It is not given that all quantities in the flow are Reynolds num­ber-independent, although the dimensionless maximum velocity in the occupied zone has reached that level, but increasing velocity will of course make all quanti­ties Reynolds number-independent at a given velocity level; see, e. g., ref. 9.

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.10 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.

0.03 — 0.02 If0 0.0 X

20 000 40 000 60 000 80 000

FIGURE

Re

12.28 Normalized velocity versus Reynolds number in a ventilated room.4

It is necessary to study the Reynolds number independence in the full scale and in the model as part of a complete model experiment. The threshold or the critical Reynolds number, Rec, 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 Rec if the Reynolds number in the full scale is larger than Rec for the problem considered.

Fully developed nonisothermal flow may also be similar at different Rey­nolds numbers, Prandt! 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.11 Sixteen experiments carried out in the rooms 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 T — 4WH/{2W + 2H) And U = q0/WH, where W Is The width of the room and H is the height of the room.

The neglect of a low turbulence effect and a laminar flow is not justified in regions close to solid surfaces where the turbulent velocity fluctuations

U/u0 uIuq

FIGURE 12.29 Velocity distribution and turbulence intensity in the occupied zone of a room at two different Reynolds numbers. H is the height of the room.

7600

4.7S

3130 13200j 2410 36900! 1180 152000^ 589 242000) 459

0.53

76900] 57 150000! 40205000! 34.3

Hu.

105

MO’ Ar

FIGURE 12.30 Model experiments with nonisothermal flow in three different models with ceiling heating.

Must tend to zero. In experiments where the correct level of heat transfer and mass transfer through the boundary layer is important, it is necessary to take all dimensionless numbers involved into full consideration. It is possible to draw a parallel to the situation in computational fluid dynamics where a K-e Turbulence model describes the main flow clear of surfaces as a fully developed turbulent flow used together with wall functions, which take the low turbu­lent effect and laminar flow close to surfaces into consideration.

Model experiments where free convection is the important part of the flow are expressed by the Grashof number instead of the Archimedes number, as in Eq. (12.61). The general conditions for scale-model experiments are the use of identical Grashof number, Gr, Prandtl number, Pr, and Schmidt number, Sc, in the governing equations for the room and in the model.

A practical approach is to simulate cold or hot surfaces with replacement jets which match the airflow in the model to the flow in the full-size room. This method is described by Nevrala and Probert12.

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