Schiller (1932), suggested that a roughness element affects transition when the Reynolds number, based on the height of roughness element and the velocity in the boundary layer at the height of the top of the roughness element, exceeds a certain critical value. Experimental work (see for example, Gregory & Walker 1950; Dryden 1953; Klebanoff, Schubauer & Tidstrom 1955 and Tani 1961) showed that a marked difference existed between the effects of two-dimensional and three-dimensional roughness elements. Measurements on the effects of two-dimensional and three- dimensional elements on boundary layer transition were carried out in a wind tunnel by Klebanoff, Schubauer & Tidstrom (1955) who observed the relationship between Ret and k/5i (where Ret is the Reynolds number at transition based on distance from the leading edge, k is the roughness height and 5i is the boundary layer displacement thickness). The two dimensional roughness elements (cylindrical rods) and the three dimensional roughness elements (spheres ranging in size from 0.838mm to 4.763mm) were cemented to the flat plate and the point of transition was observed using a hot-wire anemometer. They observed that three-dimensional roughness elements are not as effective as two dimensional disturbances in inducing transition for a given k/§i but their behaviour is of a much more critical nature. This is seen in figure 6.1 from the very narrow change in k/ôi covered by a given size, indicating a rapid change in transition Reynolds number (Ret) for a small change in wind speed. Transition caused by a two- dimensional roughness element will move towards the disturbance, gradually, with increasing velocity. In contrast, a three-dimensional roughness element is more straightforward in producing three-dimensional disturbances, causing transition to move very quickly towards the roughness location when a critical velocity has been
o X <u a: 2.8 2 .4 — -IN C H AT 2 FEET INCH AT I FOOT - ^ - I N C H AT 2 FEET 2.0 - L - i N C H AT 2 FEET 4 - -INCH AT I FOOT SPAONG EFFECT . 2 DIAMETERS V 8 DIAMETERS 0.8 2-DIMENSIONAL ROUGHNESS--- (CYLINDRICAL RODS) 0 .4 0.0 0.0 0 .4 0.8 1.2 1.6 2.0 k/8,
Figure 6.1: Graph showing the relationship between two and three dimensional roughness elements and the transition Reynolds number (Ret). Reproduced from Klebanoff, Schubauer & Tidstrom (1955).
reached.
Klebanoff et al. (1955) observed that a three-dimensional isolated roughness element may produce immediate transition at the roughness. This is indicated by a wedge shaped region of turbulent flow originating at the disturbance and extending downstream. When the free stream velocity or the roughness size is reduced, however, turbulence begins some distance downstream of the disturbance indicating that the initial perturbation generated needs to grow in size and magnitude as it is convected downstream prior to developing into a turbulent spot.
The critical ffee-stream velocity or roughness height that just causes turbulence to originate from the fixed disturbance is governed by the roughness Reynolds number, the existence of which was first proposed by Schiller (1932). This critical Reynolds number is based on the roughness height, k, and the velocity in the laminar boundary layer at the height at the top of the roughness element, Uk. Numerous investigations on roughness Reynolds number for three-dimensional elements have been carried out, and parametric data exists for a variety of geometrical shapes, such as spheres, cones, cylinders and hemispheres (see for example Klebanoff et al, 1955; Smith & Clutter, 1958; Acarlar & Smith, 1987; Klebanoff et a l, 1992). It has also been found that the shape of the roughness has a major effect on the value of the critical-roughness parameters, although insufficient data has as yet been produced to draw general conclusions regarding the detailed influence of shape.
The present investigation observes the transition process triggered by a 'spherical cap'
roughness element. This roughness type is comparable to that investigated by Acarlar & Smith (1987a) and Klebanoff et al. (1992), which used hemispherical elements to induce transition. The use of spherical caps, however, permit the variation of height, k,
with diameter, d, and hence in the present tests the same experiment can be carried out for roughness elements which possess different k/d values (where k is the roughness height and d is the roughness diameter; k/d for a hemisphere is fixed at 0.5).
Critical Reynolds number data has been published for a number of disturbance shapes. Some results are shown below in table 6.1. It is difficult, however, to carry out a direct comparison between critical roughness Reynolds numbers, due to differences in roughness shape, experimental conditions and the method of determining the point of transition. It can be noted however that the Reynolds numbers presented in table 6.1 are of the same order of magnitude. This is probably due to the fact that all the roughness shapes tested produce downstream separation and an attached wake.
Authors Type of roughness
Critical Roughness Reynolds No.
Klanfer & Owen (1953) conical ^ 440
Klebanoff et al. (1955) spherical ^ ^ 500-800
Tani et al. (1962) cylindrical 600-1000
Hall (1967) spherical |k 585-655
Klebanoff et al. (1992) h em isp h ericar"(2 Z 3 ï. 325 Table 6.1: Critical roughness Reynolds number data
The fixed 'spherical cap' disturbances were created using two different methods. The first (the simplest, but most time consuming) was to machine the disturbances from UP VC to the correct dimensions. These were then temporarily secured at the correct position. A line of small dye injection holes was drilled into the spherical caps for dye visualization of hairpin vortices and their subsequent breakdown. The dye injection method is similar to the one used by Acarlar & Smith (1987a), in which an extemal reservoir supplies the dye towards the injection holes. A schematic of the setup is shown in figure 6.2. This setup allowed excellent visualization of the hairpin vortices shed from the roughness elements and was therefore used for the majority of the dye visualization tests.
The second method, which involved carefully inflating the flexible perturbation diaphragm to a predetermined height, as described below was generally used for the LDA and hydrogen bubble visualization tests because it was easier to set-up. It was observed early on in these experiments that a small variation in height can cause a large variation on the 'point of transition' position. An accurate method of fixing disturbance heights was required and this is briefly outlined below:
The channel was turned on and allowed to settle. The travelling microscope crosshairs were then set at a predetermined height above the bed using the method described in section 4.2. The perturbation diaphragm was manually inflated, via a flexible tube, until its top surface was slightly higher than the crosshairs and the tube was clamped so that no air was allowed to escape. Whilst looking through the microscope, the clamp was slowly opened, allowing only a little amoimt of air to escape at a time. This slowly
a) Dye Reservoir Sphencal Cap Roughness Water Level 2 ____ Dye Injection Holes Glass Plate Valve for adjusting ^ dye flow Channel Bed b) c)
Figure 6.2: (a) Schematic of the dye visualization setup; (b) Side view image showing "spherical cap' roughness element with injection holes to allow dye visualization of hairpin vortices; (c) Hairpin vortices shed from the roughness element.
reduced the height of the perturbation diaphragm. The valve was clamped again when the perturbation height exactly reached the same level as the microscope crosshairs, leaving a permanently inflated spherical cap.
Prior to these experiments, the flat plate was covered with a white self-adhesive Fablon sheeting. A 20mm grid and the distance from the perturbation were marked on the Fablon sheeting using a fine permanent marker. Dye crystals were placed in the channel upstream of the fixed disturbance. After only a few minutes the dye crystals diluted enough to cover the whole plate and the point of transition was clearly visually indicated by the point where the dye first violently diffused. Its distance from the perturbation was observed using the marked grid. Ten 'point-of-transition' positions were observed for each case and the average position was calculated. These positions varied only slightly, if at all, for each case and the author believes that the averaged results provide an accurate representation of the point of transition.
The fixed spherical-cap disturbance tests aimed first at determining the critical Reynolds numbers for an extensive range of parameters and second, determining the hairpin vortex shedding frequency characteristics and resolving what effect these vortices and their shedding frequency had on the transition process. In Chapters 7 & 8 of this thesis, data obtained from fixed disturbance tests will be compared with results obtained during the transient disturbance experiments.