Efficacy of single-component MTV to measure turbulent wall-flow velocity derivative profiles at high resolution

Abstract

Physical interpretations and especially analytical considerations benefit from the ability to accurately estimate derivatives of experimentally measured statistical profiles. Toward this aim, experiments were conducted to investigate the efficacy of single-component molecular tagging velocimetry (1c-MTV) to measure mean velocity profiles that can be differentiated multiple times. Critical effects here pertain to finite measurement uncertainty in the presence of high spatial resolution. Measurements acquired in fully developed turbulent channel flow over a friction Reynolds number range from 390 to 1800 are used to investigate these issues. Each measured profile contains about 880 equally spaced data points that span from near the edge of the viscous sublayer to the channel centreline. As a result of the high spatial resolution, even very small levels of uncertainty in the data adversely affect the capacity to produce smooth velocity derivative profiles. It is demonstrated that the present 1c-MTV measurements can be differentiated twice, with the resulting profile remaining smooth and accurate. The experimental mean velocity profiles and their wall-normal derivatives up to second order are shown to convincingly agree with existing DNS data, including the apparent variations with Reynolds number.

Appendix

1.1 Required resolution and displacement accuracy

Overall, the accuracy of the profiles of the terms in the mean momentum equation (or any other mean equation that has derivatives) depends upon the flow situation. In the present study, the mean equation for the canonical flow is examined. In a flow that is perturbed from the canonical state, the nature of the perturbations will generate features in the instantaneous dynamics that impart their signature on the mean profile. Reasonably then, one needs the capacity to resolve these features. Thus, as a general operational rule, measurements must capture derivatives over only a few viscous lengths. This is why we sought to generate profiles containing over 850 data points each, as this will provide the desired resolution, essentially independent of perturbation, over the given Reynolds number range of the facility.

More generally, to quantify the spatial resolution required to obtain accurate derivatives, one can differentiate the data using various step sizes ($\Delta y^+$). For $\delta ^+=955$, the step size varied from 1 to 4 viscous units in the wall-normal direction and at $\delta ^+=1800$, $\Delta y^+$ was varied between 2 and 8. The data shown in Fig. 12 for both $\delta ^+=955,\,1800$ indicate that if $\Delta y^+>2$, the accuracy of the derivatives diminishes in the near-wall region. However, for $y^+ > 100$, the deviation is not pronounced and increasing the step size can smooth the data. In the outer layer, the increase in step size is acting as a filter. As $y^+ \rightarrow \delta ^+$, the gradient approaches zero; therefore, any error in the velocity introduces noise that propagates when determining the Reynolds stress and subsequent gradients.

Not surprisingly, the requisite spatial resolution is a function of wall-normal position. This is made apparent by the resolutions employed in DNS. At $y^+=10,\, \delta ^+=934$, the spacing is about 1 viscous unit, but increases to $\Delta y^+=3.3$ at $y^+=100$. At $\delta ^+=2004$, the spacing is 1.5 viscous units at $y^+=10$ and 4 at $y^+=100$. The experimental data are at these resolutions in the near-wall region (1 and 2 viscous units for $\delta ^+=955$ and 1800) and are constant across the channel half-height. From Fig. 12, when $\Delta y^+$ exceeds that of the DNS, the first derivative loses accuracy. Therefore, the spatial resolution significantly impacts the first derivative, especially for $y^+<100$. Based upon the experimental results and the wall-normal spacing of the DNS, the minimum required spatial resolution at a given $y^+$ can be estimated from the DNS data. The minimum spatial resolution required for the mean momentum balance is based upon the first derivative since the Reynolds stress is obtained from the integrated momentum equation. The second derivative essential follows the first derivative in terms of spatial resolution requirements, as shown in Fig. 13.

To set a baseline on the required velocity accuracy, the $U^+$ profile from the $\delta ^+=934$ DNS was differentiated using a forward finite-difference scheme with the result being compared to the published $\mathrm {d}U^+/\mathrm {d}y^+$ derivative presented in the DNS database. The resulting profiles are presented in Fig. 14. It is clear that any noise in the velocity data results in a noisy derivative, since it is assumed that the spatial resolution of the DNS simulation is sufficient. The standard deviation about the published derivative and the forward-difference calculated at $\delta ^+=934$ is 0.014. To determine the level of accuracy required for the experimental mean velocity data, white noise is added at various signal-to noise ratios. The data presented in Fig. 14 are for signal-to-noise ratios (SNR) from 60 to 40. The SNR for the original profiles is approximately 90. The error increases with decreasing SNR, as shown in Fig. 14, and for the aforementioned SNRs, the standard deviations are 0.021, 0.065, and 0.205, respectively.

Based upon the data presented in Fig. 14, it is not possible to provide a maximum uncertainty in the mean velocity that allows for accurate determination of derivatives. Smoothing derivative schemes can, however, be used to mitigate the effect of noisy data, e.g., noise reducing Richard extrapolations schemes. The effect of noise in the displacement profile increases when calculating the second derivative. Overall, we summarize that accurate velocity data are required across the entire half-channel, but spatial resolution can be relaxed for $y^+ \gtrsim 100$.

Fig. 12

Mean velocity gradient profiles as a function of $\Delta y^+$ for $\delta ^+=955, 1800$. Note that the profiles are shifted vertically by 0.6 between each $\delta ^+$. Solid lines give the experimental profiles and dashed lines give the DNS profiles

Fig. 13

Mean viscous force profiles as a function of $\Delta y^+$ for $\delta ^+=955,1800$. Note that the profiles are shifted vertically by 0.06 between each $\delta ^+$. Solid lines give the experimental profiles and dashed lines give the DNS profiles

Fig. 14

Mean velocity gradient profiles as a function of SNR for $\delta ^+=955$. Note that the profiles are shifted vertically by 0.5 for each SNR. Solid lines give the experimental profiles and dashed/dotted lines give the DNS profiles at $\delta ^+=934$