Skin Friction Correlation in Open Channel Boundary Layers

Abstract

Contributed by the Fluids Engineering Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Fluids Engineering Division February 17, 2000; revised manuscript received May 10, 2001. Associate Editor: J. Katz. In near-wall turbulence research, accurate determination of the wall shear stress, τw, (or skin friction coefficient Cf) is of critical importance due to its practical relevance and also because it determines the friction velocity uτ used by most boundary layer scaling laws. A number of techniques are available to determine the wall shear stress in turbulent boundary layers. If sufficient data are obtained in the linear viscous sublayer y+⩽4, the wall shear stress can be determined from the velocity gradient at the wall. However, it is often difficult to obtain adequate data within the linear sublayer in most experiments, especially if the Reynolds number is high and instruments such as pitot-tube and cross-wire probes are used. Alternatively, a reliable estimate of the wall shear stress can be obtained by fitting polynomial to the near-wall data up to y+⩽15 (George and Castillo 1, Tachie et al. 2). In a high Reynolds number flow, where a well-defined log-region exists, the Clauser plot technique is frequently used to determine the wall shear stress. For a turbulent boundary layer, leading edge geometry as well as freestream turbulence intensity can significantly modify the skin friction characteristics. For such conditions, a formulation that does not implicitly fix the strength of the wake but rather allows its value to be optimized while ensuring a reliable estimate of uτ was proposed by Finley et al. 3 and subsequently used by Granville 4 and Krogstad et al. 5. At sufficiently high Reynolds numbers, the wall shear stress can also be estimated from the peak value of the Reynolds shear stress profile near the wall. Finally, on smooth surfaces wall mounted probes can be used to measure τw directly. Using these methods, a number of correlations (e.g., Schultz-Grunow 6) have been developed to allow the prediction of skin friction for practical purposes. Most of the existing correlations consider moderate to high Reynolds numbers in canonical turbulent boundary layers, whereas the focus of the present work is low to moderately high Reynolds numbers in channel flows. More recently, Osaka et al. 7 reported extensive measurements in a smooth wall turbulent boundary layer for a Reynolds number range of 800⩽Reθ⩽6300, where Reθ is the Reynolds number based on boundary layer momentum thickness, θ. Direct wall shear measurements were made and the skin friction correlation derived from their measurements appears to be the most reliable in the literature for the range of Reθ they considered. Another widely referenced study of low Reynolds number boundary layer flows is that of Purtell et al. 8 who used the velocity gradient at the wall and momentum balance to infer the wall shear stress. In the present study, we are specifically concerned with open channel flows, which show some significant similarities with canonical boundary layers studied in mechanical engineering applications (Tachie et al. 9). In the hydraulic engineering community, it would appear that on occasion rather crude methods have been advocated for the prediction of skin friction. For example, Moody chart using the hydraulic diameter has been recommended for the prediction of the skin friction (ASCE Task Force 10). While this may be adequate for preliminary calculations, more reliable correlations are required. Recently, Schultz and Swain 11 reported a correlation for Cf on a smooth plate in a water tunnel at moderately high Reynolds numbers. Their correlation was typically 6-9 percent higher than that given by Coles 12, and did not consider low Reθ data. The present study proposes a new skin friction correlation for a smooth wall boundary layer in channel flows at low to moderately high Reynolds number. The experimental data used to develop the correlation were obtained from experiments conducted in different facilities under different conditions, and cover a relatively wide range of Reynolds number 100<Reθ<20,000.The data used to develop the present correlation were obtained from the experiments summarized in Table 1. The studies of Tachie et al. 2913, Balachandar and Ramachandran 14, Balachandar and Tachie 15 and Tachie and Balachandar 16 were conducted in the same test facility. Details of the open channel flume and the LDA system are provided in Balachandar and Ramachandran 14 and are not repeated here. These experiments also used the same laser Doppler anemometer, except for Tachie et al. 2 who used an additional beam expansion device to achieve a higher spatial resolution. The freestream velocity Ue in these experiments was in the range 0.01<Uem/s<0.62 while the background turbulence levels varied from 2.0 to 4.0 percent. The turbulence levels are typical of water channel experiments but are significantly higher than typical turbulence intensities reported in wind tunnel studies. The experiment of Schultz and Swain 11 was conducted using a two-component laser Doppler anemometer in a water tunnel at Ue=1.2 to 4.0 m/s. Their turbulence intensity ranged from 2.5 to 3.0 percent. In the first five studies summarized in Table 1, the wall shear stress was obtained from the velocity gradient at the wall y+⩽6 and the maximum uncertainty in Cf was 6 percent. In Tachie et al. 2, the high resolution LDA system together with a slight tilt of the probe towards the wall allowed measurements to be obtained down to y+=1 in some of the tests. Both velocity gradient at the wall y+⩽4 and a fifth-order polynomial fit to the near-wall data y+⩽15 were used to determine the wall shear stress and the uncertainty in Cf was less than 3 percent. Schultz and Swain 11 considered three different methods, namely, the velocity gradient at the wall y+⩽7, Reynolds stress method and Bradshaw’s method, to estimate the wall shear stress. The values of Cf determined using the velocity gradient at the wall and Bradshaw’s method are used in the present work. For these methods, the uncertainty in Cf was 7 percent or less. Figure 1 shows a plot of the skin friction coefficient Cf with Reynolds number Reθ for the studies summarized in Table 1. The Reynolds number varied from low (150) to moderately high (15,000) values, and the range is two orders of magnitude. The skin friction correlation developed in the present work is as follows: (1)Cf=4.13×10−2−2.68×10−2log Reθ+6.528×10−3log Reθ2−5.54×10−4log Reθ3Equation (1) and 7 percent error bands at some selected values of Reθ are shown in Fig. 1. The choice of 7 percent reflects the maximum uncertainty in Cf for the data sets used in developing the correlation. An assessment of goodness-of-fit using a chi-square distribution at 95 percent confidence level indicated that Eq. (1) correlates the experimental data well. The skin friction data of Purtell et al. 8 obtained in a canonical turbulent boundary layer at 465⩽Reθ⩽5,100, as well as the correlation developed by Osaka et al. 7 are also shown. In the Reynolds number range for which the data of Purtell et al. 8 and the correlation of Osaka et al. 7 overlap, there is a reasonable agreement between the experimental data and the correlation. Figure 1 also shows the data of Tachie et al. 17 obtained using pitot-tube in a wind tunnel for Ue=17 m/s to 20 m/s. The turbulence intensity in the wind tunnel was 0.6 percent, which is lower than the values reported in the water channel experiments. However, the wake parameter was found to be Π≈0.1 and this is comparable to the values obtained in open channel flow experiments, e.g., Π=0.08∼0.1 in Tachie et al. 9. The low value of Π in the wind tunnel experiment was attributed to leading edge effects. For this set of data, the defect profile was correlated using the formulation proposed by Finley et al. 3. The following observations can be made from Fig. 1. The skin friction obtained in the open channel flow is comparable to the wind tunnel data reported by Purtell et al. 8 for Reθ<1000. On the other hand, the Cf values for the wind tunnel data of Purtell et al. 8 and the correlation developed by Osaka et al. 7 are lower than the open channel data and the water channel data of Schultz and Swain 11 at higher Reθ. The older but widely-used correlation of Coles 12 is also shown in the figure. As noted in the figure, Coles’ 12 correlation describes the present data for Reθ<2000, but would under predict the open channel data for Reθ>2000. Furthermore, the difference between the water channel and wind tunnel data increases as Reθ increases. For example, compared to the Cf values predicted from the correlation of Osaka et al. 7, prediction from the present correlation is 10 and 15 percent higher at Reθ=1000 and 6000, respectively. The similarities and differences noted above can be explained as follows. The turbulence level in the wind tunnel experiments of Purtell et al. 8 and Osaka et al. 7 is an order of magnitude lower than the water channel data. One effect of high freestream turbulence intensity on the mean flow is to reduce the wake parameter (Π) which in turn increases Cf (e.g., Hoffman and Mohammadi 18, White 19). For the experiments conducted in open channel, the values of Π were less than 0.15. Measurements, as well as results obtained from direct numerical simulation for canonical turbulent boundary layers, showed that the values of Π are low at low Reynolds number but Π increases as the Reynolds number increases. Typical asymptotic values for Π in wind tunnel experiments varied from 0.55 to 0.62. Since values of the outer wake parameter at low Reθ in wind tunnel do not differ much from those obtained in the open channel experiments the good agreement between the data reported by Purtell et al. 8 and the open channel data at Reθ<1000 is to be expected even though the turbulence levels are distinctly different. On the other hand, the lower Cf values for the wind tunnel data of Purtell et al. 8 and the correlation developed by Osaka et al. 7 in comparison to the open channel data and the water channel data of Schultz and Swain 11 can be attributed to the much stronger wake components (Π) for the wind tunnel experiments. It is interesting to observe that although the data reported by Tachie et al. 17 were obtained in a wind tunnel, they are adequately described by the present correlation but not by the wind tunnel correlation of Osaka et al. 7. While the turbulence intensity for this experiment (i.e., Tachie et al. 17) is similar to those reported by Purtell et al. 8, the strength of the outer wake parameter was 0.1. This is significantly lower than values obtained in wind tunnel experiments at similar Reθ but comparable to open channel experiments as noted previously. This observation provides further evidence that the skin friction characteristics are strongly affected by the strength of the wake. Therefore, techniques such as that proposed by Finley et al. 3 that implicitly account for the role of the outer flow showed be used to determine the wall shear stress. A new skin friction correlation for open channel boundary layers was developed using skin friction data obtained from a variety of experiments, mostly in low Reynolds number open channel flows. The range of Reθ varied from 150 to 15,000, which covers most of the Reθ experiments available in the literature. The present correlation describes the existing data to within ±7 percent. It is observed that skin friction correlations developed for canonical turbulent boundary layers, e.g., Osaka et al. 7, only slightly under-predict the skin friction coefficient in open channel boundary layers at low Reynolds numbers, i.e., Reθ<2,000. The level of under-prediction increases at higher Reynolds numbers.