| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 |
|
|---|---|
The contraction or "nozzle" accelerates the flow from the settling chamber into the test section, further reducing percentage variations in velocity. The old-style contraction shape with a small radius of curvature at the wide end and a large radius at the narrow end to provide a gentle entry to the test section is not the optimum. There is a danger of boundary-layer separation at the wide end, or perturbation of the flow through the last screen. Good practice is to make the ratio of the radius of curvature to the flow width about the same at each end.However, too large a radius of curvature at the upstream end leads to slow acceleration and therefore increased rate of growth of boundary-layer thickness, so the boundary layer - if laminar as it should be in a small tunnel - may suffer from Taylor-Goertler "centrifugal'' instability when the radius of curvature decreases. Bell and Mehta discuss boundary-layer predictions for contractions in "Boundary-Layer Predictions for Small Low-Speed Contractions" AIAA J., Vol. 27, p. 372, (1989). T. Morel gives rules for design of contractions with polynomial curves for wall shapes (J. Fluids Engg Vol. 97, p. 225, 1975 for axisymmetric contractions and Vol. 99, p. 371, 1977 for two-dimensional shapes). Brassard in an unpublished project report "Transformation of a Polynomial for a Contraction Wall Profile" describes a generalization of the fifth-order polynomial proposed by Bell and Mehta, to extend the range of shapes and provide different radii of curvature at the two ends.
The contraction area ratio should be "as large as possible", to reduce the total-pressure loss through the screens. In medium-size tunnels, the area ratio is limited by the desirability of easy access to the test section while the operator is standing on the floor of the laboratory, implying that the test section floor should be no more than 4-5 ft. above the floor of the room. The ceiling height imposes another limit. There is no overwhelming reason why a contraction should be symmetrical top and bottom - modern potential-flow codes can easily calculate the pressure distribution on a chosen asymmetric shape - but the only practical examples are "one-sided" contractions looking like half a conventional shape. An area ratio of 9 is acceptable for a typical tunnel with three screens each with a pressure-drop coefficient of 1.2: the contribution to the power factor (inversely proportional to the square of the area ratio)is then 3x1.2/81 = 0.044. The area ratio of 31 used in the RAE 4 ft x 3 ft tunnel (Fig. 4) is unnecessarily large, and led to flow separation in the region of the rapid change of wall angle between the wide-angle diffuser and the contraction.
Effect of a Contraction on Velocity Variations
The contraction is the last component before the test section, and should further reduce the variations of velocity components in time and space that are created in the return circuit or the laboratory and then attenuated by the honeycomb and screens. The effectiveness of a contraction of area ratio N in reducing variations of mean axial velocity over the cross-section can be seen by applying Bernoulli's equation to an incompressible flow with a small region of increased speed somewhere in the cross section. The total pressure of the main stream is pi + (1/2) |
The effect of a contraction on unsteady velocity variations and turbulence is more complicated: the reduction of x-component (axial) fluctuations is greater than that of transverse fluctuations. A simple analysis due to Prandtl predicts that the ratio of root-mean-square axial velocity fluctuation to mean velocity will be reduced by a factor 1/N2, as for mean-velocity variations, while the ratio of lateral rms fluctuations to mean velocity is reduced only by a factor of Note that the measured value of free-stream "turbulence" may depend strongly on the low-frequency limit of the measuring device, because the power spectral density (contribution to the mean-square fluctuation per unit frequency band) of the unsteadiness usually rises rapidly as frequency decreases. An arbitrary but simple rule is to regard fluctuations with a wavelength (flow speed divided by frequency) of more than twice the test-section length as "unsteadiness" rather than turbulence: transverse-component disturbances with such long wavelengths would be forced to zero by the tunnel walls. For a typical tunnel with a 10 ft. long test section, running at 100 ft/sec, this rule implies that the low-frequency limit should be set at 10 Hz if only "real turbulence" is to be recorded. Low-frequency unsteadiness, confined to the longitudinal component of velocity, does not usually have a significant effect on the flow, except for separated flows over stalled airfoils or bluff (non-streamlined) bodies. However it is a nuisance in measuring mean values because longer averaging times are needed. Also, unsteadiness is usually the result of flow separation or incipient separation in the tunnel circuit, or unsteadiness in the entry flow of an open-circuit tunnel, and is therefore often accompanied by high levels of real turbulence, so it should be regarded as a warning signal.
A good wind tunnel will have an axial-component rms fluctuation in the test section of less than 0.1 percent of the mean velocity (this figure is sometimes quoted as the maximum for serious studies of laminar-turbulent transition). Lateral and vertical rms intensities in such a tunnel are likely to be about 0.3 percent
 |
U2, where pi is the initial static pressure, and the total pressure in the high-velocity region, which is assumed to have the same initial static pressure, is pi + (1/2) 
U)2. At the exit, the main stream velocity is approximately NU (we neglect the extra mass flow in the high-speed region) and the static pressure is therefore pi - (1/2)
N: that is, the lateral fluctuations (in m/s, say) increase through the contraction, because of the stretching and spin-up of elementary longitudinal vortex lines. Batchelor, The Theory of Homogeneous Turbulence, Cambridge 1953, gives a more refined analysis, but Prandtl's results are good enough for tunnel design. The implication is that tunnel free-stream turbulence is far from isotropic. The axial-component fluctuation is easiest to measure, e.g. with a hot-wire anemometer, and is the "free-stream turbulence" value usually quoted. However it is smaller than the others, even if it does contain a contribution from low-frequency unsteadiness of the tunnel flow as well as true turbulence.