Role of data uncertainties in identifying the logarithmic region of turbulent boundary layers

Abstract

Composite expansions based on the log-law and the power-law were used to generate synthetic velocity profiles of zero pressure gradient turbulent boundary layers (TBLs) in the range of Reynolds number $800 \le Re_{\theta } \le 860{,}000,$ based on displacement thickness and freestream velocity. Several artificial errors were added to the velocity profiles to simulate typical measurement uncertainties. The effects of the simulated errors were studied by extracting log-law and power-law parameters from all these pseudo-experimental profiles. Various techniques were used to establish a measure of the deviations in the overlap region. When parameters extracted for the log-law and the power-law are associated with similar levels of deviations with respect to their expected values, we consider that the profile leads to ambiguous conclusions. This ambiguity was observed up to $Re_{\theta }=16{,}000$ for a 4 % dispersion in the velocity measurements, up to $Re_{\theta }=8.6 \times 10^{5}$ for a 400 $\upmu$m uncertainty in probe position (in air flow at atmospheric pressure), and up to $Re_{\theta }=32{,}000$ for 3 % uncertainty in the determination of $u_{\tau }.$ In addition, a new method for the determination of the log-law limits is proposed. The results clearly serve as a further note for caution when identifying either a log or a power-law in TBLs. Together with a number of available studies in the literature, the present results can be seen as a additional reconfirmation of the log-law.

Appendix

The analytical forms of the composite expansions used in the present study are included in this section. The velocity profiles based on a logarithmic description of the overlap region were generated from the composite profile of Chauhan et al. (2009):

$$\begin{aligned} U_{\mathrm{composite}}^{+}=U_{\mathrm{inner}}^{+} +\frac{\exp \left[ - \ln ^{2} \left( y^{+}/30 \right) \right] }{2.85} + \frac{2 \varPi }{\kappa } {\mathcal W} (\eta ). \end{aligned}$$

(9)

The inner profile $U_{\mathrm{inner}}^{+}$ was developed by Musker (1979) and is given by the following:

$$\begin{aligned} U_{\mathrm{inner}}^{+}&= \frac{1}{\kappa } \ln \left( \frac{y^{+}-a}{-a} \right) + \frac{R^{2}}{a(4 \alpha -a)} \nonumber \\&\times \Biggl \{ (4 \alpha +a) \ln \left( -\frac{a}{R} \frac{\sqrt{(y^{+}-\alpha )^{2}+\beta ^{2}}}{y^{+}-a} \right) \nonumber \\&+\frac{\alpha }{\beta }(4 \alpha +5a) \left[ \arctan \left( \frac{y^{+}-\alpha }{\beta } \right) + \arctan \left( \frac{\alpha }{\beta } \right) \right] \Biggl \}, \end{aligned}$$

(10)

where $\alpha =(-1/\kappa -a)/2,$ $\beta =\sqrt{-2 a \alpha - \alpha ^{2}}$ and $R=\sqrt{\alpha ^{2}+\beta ^{2}}.$ The parameters $\kappa =0.384$ and $a=-10.3061$ give $B=4.17.$ The exponential term in Eq. (9) was added to account for an overshoot observed in the experimental data around $y^{+} \simeq 50,$ as shown by Monkewitz et al. (2007).

Following Coles (1956), a wake function ${\mathcal W}(\eta )$ was introduced in Eq. (9) to represent the outer region, with $\varPi$ being a parameter associated with the wake strength. Experimental measurements in the NDF wind tunnel at IIT (Nagib et al. 2004a) were used by Chauhan et al. (2007) to develop and exponential wake function ${\mathcal W}_{\mathrm{exp}}$ of the form:

$$\begin{aligned} {\mathcal W}_{\mathrm{exp}}(\eta )&= \frac{1-\exp \left[ - (5 a_{2}+6 a_{3} + 7 a_{4}) \eta ^{4}/4 + a_{2} \eta ^{5} + a_{3} \eta ^{6} + a_{4} \eta ^{7} \right] }{1-\exp \left[ - (a_{2}+2 a_{3} + 3 a_{4})/4 \right] } \nonumber \\&\times \left[ 1- \frac{1}{2 \varPi } \ln (\eta ) \right] , \end{aligned}$$

(11)

where $a_{2}=132.841,$ $a_{3}=-166.2041$ and $a_{3}=71.9114.$ The wake parameter $\varPi$ is calculated following the method described by Chauhan et al. (2009).

Velocity profiles based on the GC power-law description of the overlap region were generated from the outer profile proposed by George and Castillo (2006), and developed by George (1997, 2007):

$$\begin{aligned} U_{\mathrm{outer}}^{+}&= (0.99-C_{o}) \overline{y} \sin \left( \frac{\pi \overline{y}}{2} \right) \left[ \frac{C_{i}}{C_{o}} (\delta _{99}^{+})^{\gamma } \right] \\&\quad+ C_{i} (y^{+}+a^{+})^{\gamma }, \end{aligned}$$

(12)

where $a^{+}=-16$ and $\overline{y} \equiv y/\delta _{99}.$ This equation, valid in the interval $30<y^{+}<\delta _{99}^{+},$ was enough for the comparison purposes of the present study. The parameters $C_{i},$ $C_{o}$ and $\gamma$ are $Re$-dependent, and they are expressed in terms of the asymptotic values $C_{i \infty }=56.7,$ $C_{o \infty }=0.897$ and $\gamma _{\infty }=0.0362$ through the following empirical relations:

$$\begin{aligned} \gamma&= \gamma _{\infty } + \frac{\alpha A}{\left[ \ln \left( \delta _{99}^{+} \right) \right] ^{1+\alpha }},\end{aligned}$$

(13)

$$\begin{aligned} C_{o}&= C_{o \infty } \left[ 1+0.283 \exp \left( -0.00598 \delta _{99}^{+} \right) \right] ,\end{aligned}$$

(14)

$$\begin{aligned} \frac{C_{o}}{C_{i}}&= \frac{C_{o \infty }}{C_{i \infty }} \exp \left\{ \frac{(1+\alpha )A}{\left[ \ln (\delta _{99}^{+}) \right] ^{\alpha }} \right\} , \end{aligned}$$

(15)

where $\gamma \rightarrow \gamma _{\infty },$ $C_{i} \rightarrow C_{i \infty }$ and $C_{o} \rightarrow C_{o \infty }$ as $Re \rightarrow \infty$. The constants $A$ and $\alpha$ take the values 2.9 and 0.46, respectively.