Pr05Cf k 02 Pra44c


For the reference cf value required in each correlation, one could use either experimental data (since it is available in this study) or one of the empirical correlations. Since the St correlations are intended for use as a predictive tool, it was deemed most useful to base St on a cf correlation rather than on data. For this reason, the Schlichting cf correlation was employed since it had the best match to the experimental data in the ks > k/2 regime of

Figure 7 (average 3% difference with data). Kays and Crawford suggest C = 1 in their correlation based on a fit to data obtained with closely-packed spheres. Since this value gave extremely large St predictions, the constant C was arbitrarily dropped to a value of 0.35 to better match the present "real" roughness models (both cases are shown on the plot). The Kays and Crawford correlation predictions thus adjusted are not a significant improvement over the Dipprey and Sabersky results which are obtained without any special tailoring. Both predict values of St which are on average 8-11% higher than the data in the k+ > 70 regime. Again, in the region ks < k/2, the ks-based correlations are clearly inappropriate when using Eqn. 7 to determine ks for these "real" roughness models. The more reasonable results using the minimum threshold of ks > k/2 in this region are indicated in Figure 8 as they were in Figure 7.

The fact that using the ks(Ls) correlation appears to be inadequate to span the entire range of real roughness in this study led to a closer examination of this formulation for ks. In order to assess the appropriateness of the Sigal-Danberg correlation, ks was adjusted to the value required to exactly match the Schlichting cf correlation with the experimental data for each of the six panels. This ksadj (non-dimensionalized by k) is plotted vs. Ls in Figure 9 (data for both Reynolds numbers). Also shown are the log fit (Eqn. 7) and the Shlichting [30] and Hosni et al. [18] simulated roughness data compiled in Bogard et al. [20]. The data in the figure show that in the fully rough regime, the real roughness models follow the Eqn. 7 fit to the simulated roughness data. Whereas, for the models with Ls > 100, there is a large discrepancy between "real" roughness and simulated roughness. An alternate log fit to the "real" roughness data is also indicated on the figure:

This substantial difference between real and simulated roughness ks estimates led to the consideration of alternate geometrical dependencies for ks that might unify the entire data set. As such, each of the statistical parameters in Table 1 was correlated with the ksadj values for the six panels. Table 2 contains the correlation coefficient of a least squares fit to the resultant data. Different fitting functions were attempted in each case (linear, logarithmic, and polynomial) and the most successful attempt is included in the table. The results clearly show that Rz (»k), Ra, 1c, Sk, and K do not correlate well with the ksadj parameter. This may explain why Eqn. 7 (which estimates ks as a function of k and Ls) has difficulty for some of the roughness samples. When Ls alone is correlated directly against ksadj, the result is quite good (see Table 2). Surprisingly, a rms and S^/S outperform Ls as parameters of choice for this data set. Of these three, a rms is the easiest to obtain for real turbine roughness, requiring only a handful of 2D traces. While determining S^/S and L s require full 3D surface maps. Figure 10 contains the ksadj vs. arms data and the associated fit:

This parameter was first reported by Acharya et al. [2] and has received only limited attention in the literature. The reader is cautioned to be careful in the application of Eqn. 11 inasmuch as the data set is extremely limited (only six data points thus far).

Also, the surface angles were determined from data acquired using a contact stylus, sampled every 5-40mm. Both of these factors will influence the resultant value of arms . Finally, one can readily construct surfaces with widely disparate roughness heights that could register identical a rms on a single 2D trace (e.g. right-angle cones of 1mm height separated by 2mm vs. right angle cones of 10mm height separated by 20mm). Clearly this cannot mean that the two would have the same ks if both were in a 10mm thick boundary layer. So, this correlation must be considered applicable only in the range of 0.5 < Rz/ 9 < 3 used in this study. It would also be limited to roughness that is locally fairly uniform (such as the panels studies) as opposed to widely-spaced, isolated roughness elements which would register widely varying a rms levels from trace to trace. Despite these limitations, it is encouraging to note that the correlation has the correct physical behavior as arms approaches zero (ks ® 0). Until further validation can be obtained over a wider array of surfaces, it is only presented as a candidate for calculating ks for real roughness.

The foregoing discussion has been focused exclusively on matching cf. Of course, it is of interest to see if the same selection of ksadj (matching the Schlichting cf correlation) also brings the experimental St data in line with the corresponding St correlations. Figure 11 shows the St correlation values of the six panels where both cf and k+ correspond to the ksadj values. Of the three St correlations considered in Figure 8, only Dipprey and Sabersky's is shown in the figure since it showed the most promise in Figure 8 (without tailoring). It is also the oldest and most comprehensive data set of the three. (Both Kays and Crawford and Wassel and Mills based their correlations at least partially on the Dipprey and Sabersky findings.) This correlation still overpredicts five of the six panels by an average of 11%, while the #2 surface is matched to within 1%. This curious result prompted a closer inspection of the model surfaces. Each of the six surfaces is quite unique (as shown by the traces in Figure 1), but surface #2 has an order or regularity that makes it clearly distinct. As mentioned earlier, this intermittently debonded TBC surface has an undulating surface character not unlike that of closely-packed spherical segments. As such, it is most like the simulated roughness surfaces using cones or spheres or the close-packed sandgrain surfaces on which the correlations (like Dipprey-Sabersky) are based. Perhaps these correlations consistently overpredict the "real" roughness St precisely because they were developed with simulated (vs. real) surface roughness. So, even if the ks of the "real" roughness surface is adjusted to match the Schlichting cf correlation value, the Dipprey-Sabersky St correlation based on the same ks will always be high for "real" roughness. Whereas if ks is determined for a simulated rough surface with ordered roughness elements, both the cf and St correlations would be accurate.

To test this hypothesis, a 7th model surface was fabricated consisting of densely packed cones with the following dimensions: height = 2mm, base diameter = 5mm, spacing = 5mm. The Ls and a rms values for this surface are 20.2 and 24.6° respectively, a close match to surface #6 (22.1 and 25.3°). When the ks for this new cone surface is adjusted to match the Schlichting cf correlation, the St correlation of Dipprey-Sabersky matches the data to within 3% (Figure 12). This startling result provides at least a preliminary indication that the hypothesis proposed above is worthy of further investigation. If proven to be more generally true, it would suggest that simulated rough surfaces with ordered roughness elements can be used to model either the heat transfer behavior or the skin friction behavior of "real" turbine roughness, but not both simultaneously. A physical explanation for this may be related to the 3D vortical secondary flow patterns that are generated by regular roughness elements. These secondary flows are known to enhance St with little or no effect on cf [33]. The randomness and wide range of scales present in "real" roughness may serve to break-up these secondary flows and reduce St for the same cf. A possibly related mechanism was reported by Pinson and Wang [34] regarding roughness-induced boundary layer transition. They noted that when large roughness elements were followed by small roughness elements, boundary layer transition was actually suppressed compared to the case of large roughness elements followed by a smooth surface. Their explanation was that the smaller roughness elements break up flow disturbances generated by the larger roughness elements, thus suppressing their amplification and subsequent transition. This same mechanism may partially inhibit the heat transfer augmentation of secondary flows induced by large roughness peaks on real roughness surfaces.

If the first constant in the denominator of the Dipprey-Sabersky St correlation is increased from 5.19 to 6.08, the match with the "real" roughness panels is within ±2% (also shown on Fig. 12). The new correlation would then be,

Of course, this correlation now underpredicts the St for the two ordered surfaces (#2 and #7) by approximately 8%.

Incidentally, the ksadj for surface #7 does fall in line with the ks vs. a rms correlation shown in Figure 10 (Eqn. 11), further suggesting that a rms may be a suitable candidate for unifying simulated and real roughness correlations for ks insofar as cf is concerned. Given the above findings, even if such a unifying correlation were derived and verified, there would still be a discrepancy with regard to St.

Though the above findings are presented for Rex = 900,000 only, similar results were obtained at the lower Reynolds value. Only minor modifications to the ksadj values were required (Figure 9) and the St overprediction for the "real" roughness surfaces was nearly identical. The Rex = 500,000 data also fall along the Eqn. 11 curve fit for ks vs.

Effect of Freestream Turbulence:

Boundary layer velocity profiles taken at midspan near the test section leading edge are shown in Figure 13 for three levels of freestream turbulence: 1%, 5%, and 11%. Also shown in the figure are the turbulence levels in the boundary layer. Note that in the highest turbulence case, the turbulence level in the freestream approaches the level of turbulence associated with the near wall peak. This feature has been determined by other researchers to significantly alter the momentum and energy transport of the turbulent boundary layer.

The levels of cf and St augmentation produced on the smooth test plate due to the two levels of freestream turbulence are shown in Figures 14 & 15 and tabulated in Table 3. The table also contains the efficiency factor, h, defined as the ratio of St augmentation to cf augmentation, (St/Slo)/(cf/cfo). The St augmentation range with turbulence is comparable to that generated by the "real" roughness in the previous portion of this report. The cf augmentation, on the other hand, is 50% less than the St augmentation in this case. This is in stark contrast to the augmentation results with roughness where cf/cfo was up to four times St/Sto. The physical reason for this is that increases in cf due to roughness are primarily due to form drag on the individual roughness elements. There is no heat transfer analog to this form drag component of cf augmentation, thus cf/cfo is 2 to 4 times greater than St/St, for rough surfaces.

In the case of freestream turbulence, augmentation occurs due to increased momentum and energy exchange with the freestream. There are no alternate mechanisms for surface drag in this case. The finding that freestream turbulence favors heat transfer augmentation over cf augmentation is consistent with data presented by other researchers. Pedisius et al. [35] reported efficiency factors (h) from 1.10 to 1.15 for grid generated turbulence up to 8% over a smooth plate. Maciejewski and Moffat [36] reported St/St, values up to 1.80 for their free jet turbulence with over 15% Tu. The associated cf/cfo was estimated at 1.10, yielding h @ 164. Blair [37] proposed a correlation to capture this preference for St over cf using a correction to the popular Reynolds analogy (2St/cf @ 1). His correlation (developed for grid turbulence with up to 7% Tu) is:

where the freestream turbulence level (Tu) is in percent. This correlation underestimates the present results by approximately 50%. A more appropriate second constant would be 0.031. Baskaran et al. [38] likewise found an up to 100% increase in this turbulence coefficient in their grid-generated turbulence data.

In addition to the increase in Tu that accompanies the change in turbulence generation mechanism from the square-bar grid to the opposing jets in the current study, the integral lengthscale also more than doubles (from 3.5cm to 8cm). Blair [37] and Simonich and Bradshaw [39] have both studied the role of turbulence scale in heat transfer augmentation. The physical explanation offered is that larger scale turbulence has difficulty influencing the momentum and energy exchange near the wall due to damping at the solid surface. Thus, Simonich and Bradshaw reported reductions in St augmentation as lengthscale increased at constant Tu. Blair tried to capture this effect by plotting ASt/St, and Acf/cfo vs. Tu/(ab) where a and b are defined as:

Leu is the dissipation lengthscale which is typically 50% greater than the integral lengthscale quoted above for this facility. Using these correlations, the results in Table 3 show good agreement (within 8%) with Blair's data for Acf/cfo, but the heat transfer results are again significantly higher than those reported by Blair (50% higher at 5% Tu and 85% higher for 11% Tu). This difference may be attributed to the high level of Tu in this study (Blair reported on grid data up to 8% Tu only) and the mechanism for its generation (jets vs. grids). The results of Maciejewski and Moffat (h @ 1.64) give the indication that the effect of freestream turbulence on heat transfer is not linear with Tu. Different mechanisms may come into play at the higher Tu levels and longer lengthscales associated with non-grid generated turbulence that may explain the sometimes 50-100% differences between various experiments. As such, the smooth plate data with freestream turbulence reported in this study appear to be within the range of results reported in previous studies.

There are a number of different methods to assess the combined effects of roughness and freestream turbulence on cf and St. Of critical importance to the designer is whether results with freestream turbulence alone can be simply added to results with roughness alone to approximate the effect when both are present. If true, this would imply a lack of synergy between the two augmentation mechanisms. This is attractive to the designer because it allows correction factors to be simply superposed without additional parametric testing. One way to determine the degree to which these two mechanisms are synergistic is to compare the augmentation results obtained with both mechanisms present to that achieved by either adding or compounding their individual effects. For example, if the smooth plate cf augmentation due to freestream turbulence was 20% and the cf augmentation of a rough plate (without turbulence) was 30%, the additive prediction method would result in a combined effect of 50%. For the same case, the compound method would predict a combined effect of 56%. These three methods of comparison are outlined algebraically for cf as follows:



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