Quantifying the dynamics of flow within a permeable bed using time-resolved endoscopic particle imaging velocimetry (EPIV)

Abstract

This paper presents results of an experimental study investigating the mean and temporal evolution of flow within the pore space of a packed bed overlain by a free-surface flow. Data were collected by an endoscopic PIV (EPIV) technique. EPIV allows the instantaneous velocity field within the pore space to be quantified at a high spatio-temporal resolution, thus permitting investigation of the structure of turbulent subsurface flow produced by a high Reynolds number freestream flow (Re _s in the range 9.8 × 10³–9.7 × 10⁴). Evolution of coherent flow structures within the pore space is shown to be driven by jet flow, with the interaction of this jet with the pore flow generating distinct coherent flow structures. The effects of freestream water depth, Reynolds and Froude numbers are investigated.

Appendix 1: Uncertainty analysis

The uncertainty ($\sigma_{\text{U}}$) of each EPIV instantaneous velocity measurement (valid for both U _x and U _y ) can be evaluated as the combination of the systematic errors of each submeasurement. The uncertainty ($\sigma_{{\overline{\text{U}} }}$) of the time-averaged velocity $\overline{{U}}$ can be estimated as the statistical uncertainty of the time history sample.

1.1 Systematic error

The flow velocity at position x for time t can be defined as:

$$ u = \alpha_{\text{m}} \left( {\frac{\Updelta S}{\Updelta t}} \right) + \delta u $$

(2)

where $\Updelta S = \left| {X_{t + 1} - X_{t} } \right|$ represents the particle displacement measured on the image and δu expresses the uncertainty of the velocity introduced by specific experimental assumptions (i.e. negligible particle lag and 3D effects). The total standard uncertainty of the velocity ($\sigma_{\text{U}}$) can be evaluated by combining all the individual uncertainties of the submeasurements as:

$$ \sigma_{\text{U}} = \sqrt {\left( {\frac{\partial u}{{\partial \alpha_{\text{m}} }}} \right)^{2} \cdot \sigma_{{\alpha_{\text{m}} }}^{2} + \left( {\frac{\partial u}{\partial \Updelta s}} \right)^{2} \cdot \sigma_{\Updelta s}^{2} + \left( {\frac{\partial u}{\partial \Updelta t}} \right)^{2} \cdot \sigma_{\Updelta t}^{2} + \left( {\frac{\partial u}{\partial \delta u}} \right)^{2} \cdot \sigma_{\delta u}^{2} } $$

(3)

where $\sigma_{{g_{i} }}$ represents the standard uncertainty of the physical quantities g _i , and s $\left( {\frac{\partial u}{{\partial g_{i} }}} \right)^{2}$ represents the sensitivity factors. The factors considered in the uncertainty analysis are: (1) the magnification factor (α _m), which is affected by the dot spacing on the calibration target, measurement of the dot spacing, calibration of the target angle and the gap between the measurement plane and target; (2) the displacement (Δs), which can be affected by possible fluctuations in the laser sheet, view angle, sub-pixel accuracy and mismatching errors; (3) the time interval (Δt) that may be affected by synchronisation errors and (4) the velocity lag (δu) that is influenced by the characteristics of the tracer particles and flow.

On the basis of the analysis described above, the uncertainty $\sigma_{\text{U}}$, which for this application was non-uniform over the field of view due to the different levels of optical distortion, was estimated to be in the range of 1–3%.

1.2 Uncertainty of the mean values

This analysis is based upon the assumption that no systematic error is present in the velocity data and the representativeness of the mean value can be evaluated statistically. If the sources of error are assumed to be independent, each value of a sample U _{x(i, j)} (or U _{y(i, j)}) can be considered as a random variable, R. Under these assumptions, the uncertainty $\sigma_{{\overline{\text{U}} }}$, which measures the statistical stability of the estimated mean values, can be estimated by computing the standard deviation, S, of the velocity magnitude (derived through computing the components by Eqs. 2 and 3) and introducing a statistical analysis approach where $\sigma_{{\overline{\text{U}} }}$ is proportional to the ratio $S/\sqrt n$. Using the probability function concept and introducing a confidence interval of 95%, the mean flow estimated with a sample of 500 realizations allows an estimate of both the vertical and horizontal components of the velocity field with an uncertainty no larger than 2.5% of the sample mean.