Effect of the driving algorithm on the turbulence generated by a random jet array

Abstract

Different driving algorithms for a large random jet array (RJA) were tested and their performance characterized by comparing the statistics of the turbulence generated downstream of the RJA. Of particular interest was the spatial configuration of the jets operating at any given instant (an aspect that has not been documented in previous RJAs studies), as well as the statistics of their respective on/off times. All algorithms generated flows with nonzero skewnesses of the velocity fluctuation normal to the plane of the RJA (identified as an inherent limitation of the system resulting from the unidirectional forcing imposed from only one side of the RJA), and slightly super-Gaussian kurtoses of the velocity fluctuations in all directions. It was observed that algorithms imposing spatial configurations generated the most isotropic flows; however, they suffered from high mean flows and low turbulent kinetic energies. The algorithm identified as RANDOM (also referred to as the "sunbathing algorithm") generated the flow that, on an overall basis, most closely approximated zero-mean-flow homogeneous isotropic turbulence, with variations in horizontal and vertical homogeneities of RMS velocities of no more than ±6 %, deviations from isotropy (w _RMS/u _RMS) in the range of 0.62–0.77, and mean flows on the order of 7 % of the RMS velocities (determined by averaging their absolute values over the three velocity components and three downstream distances). A relatively high turbulent Reynolds number (Re _T = u _T ℓ/ν = 2360, where ℓ is the integral length scale of the flow and u _T is a characteristic RMS velocity) was achieved using the RANDOM algorithm and the integral length scale (ℓ = 11.5 cm) is the largest reported to date. The quality of the turbulence in our large facility demonstrates the ability of RJAs to be scaled-up and to be the laboratory system most capable of generating the largest quasi-homogeneous isotropic turbulent regions with zero mean flow.

Appendix: Sources of error and uncertainty analysis

The purpose of this section is to describe the potential sources of error and quantify their effect on the results. It is important to note that the total uncertainty arises from the (1) ADV uncertainty, and (2) propagation of the ADV uncertainties due to the corrections performed. The analysis that follows considers each potential source of error in the velocity measurements conducted by acoustic Doppler velocimetry. The uncertainty analysis models of Voulgaris and Trowbridge (1998) and Taylor (1997) are used to estimate the uncertainty. The sources of error are then combined to calculate the uncertainty in each component of the velocity. Finally, the total relative uncertainty resulting from the propagation of the ADV uncertainty is calculated.

1.1 Acoustic Doppler velocimetry uncertainties

Voulgaris and Trowbridge (1998) identified three sources of error for the total velocity along each receiver beam (σ _t): (1) sampling error (σ _m), caused by the inability of the system to resolve the phase shift of the return pulse, (2) Doppler noise (σ _D), due to random scatter motions within the sample volume, and (3) error resulting from the mean velocity shear within the sampling volume (σ _u ). σ _u becomes important in the presence of sharp velocity gradients (e.g., in boundary or mixing layers). However, in homogeneous flows like those produced by our algorithms, the mean velocity gradients are negligible far enough from the RJA. Thus, the mean velocity shear error was neglected in our calculations. In the following sections, the sampling and Doppler errors are calculated individually. The calculations are made for the RANDOM algorithm at the three downstream positions investigated (similar results are obtained for the other algorithms, but not presented herein).

1.1.1 Sampling error

Sampling error results from the inaccuracy of the A/D converter (in the ADV system) in resolving the changes in phase of the return pulse and the noise induced by the electronics. This error is independent of the flow and depends on the velocity range employed when operating the ADV. During our experiments, the ADV’s ±0.3 m/s velocity range was used for the velocity measurements at x/M = 5.5 and 6.7, while the ±0.1 m/s velocity range was used for the measurements at x/M = 9.3. The sampling error can be calculated as (Voulgaris and Trowbridge 1998):

$$\sigma_{\text{m}}^{2} = \frac{{c^{2} }}{4}\frac{1}{{f^{2} }}\frac{1}{{4\pi^{2} }}K^{2} \sigma_{\text{s}}^{2} \frac{1}{\tau }\frac{1}{{\left( {T - t_{0} } \right)}} ,$$

where c is the speed of sound in water (1481 m/s at 20 °C), ƒ is the operating frequency of the ADV (10 MHz), K is an empirical constant (1.4, Zedel et al. 1996), σ ²_s is the system’s uncertainty to resolve the phase (1.08 and 0.63 for the ±0.3 and ±0.1 m/s velocity range, respectively), τ is the time between transmissions (4.35 and 5.55 ms for the ±0.3 and ±0.1 m/s velocity range, respectively), T is the inverse of the sampling frequency (0.04 s at a sampling frequency of 25 Hz), and t ₀ is the time required by the system to carry out the necessary conversions (2 ms). The calculated sampling error is shown in Table 5.

Table 5 Calculation of sampling error (σ ²_m )

1.1.2 Doppler noise

Doppler noise (an intrinsic feature in Doppler acoustic systems) is caused by: (1) the finite residence time of the particles in the sampling volume, (2) turbulence within the sampling volume, and (3) beam divergence. Voulgaris and Trowbridge (1998) presented the following equations for the calculation of the Doppler noise (σ ²_D ):

$$\sigma_{\text{D}}^{2} = \frac{{\pi^{ - 1/2} }}{16}\frac{{c^{2} B_{\text{D}} }}{{f^{2} M_{\text{ADV}} \tau }} ,$$

where M _ADV (=11) is the number of acoustic pulses averaged for the calculation of the radial velocity, and B _D is the total Doppler bandwidth broadening. B _D is the RMS of the three individual contributions of the bandwidth broadening due to the (as mentioned above) finite residence time (B _r), turbulence within the sample volume (B _t), and the beam divergence (B _d):

$$B_{\text{D}}^{2} = B_{\text{r}}^{2} + B_{\text{t}}^{2} + B_{\text{d}}^{2} .$$

B _r, B _t, and B _d can be calculated using the following expressions:

$$B_{\text{r}} = 0.2\frac{{U_{\text{h}} }}{d},$$

where U _h is the mean horizontal speed (i.e., $U_{\text{h}} = \, (\langle U\rangle^{2}\rangle + \langle V\rangle^{2} )^{1/2} )$ and d is the transverse size of the sampling volume.

$$B_{\text{t}} = 2.4\frac{{f\left( {\varepsilon d} \right)^{1/3} }}{c},$$

where ε is the dissipation rate of turbulent kinetic energy (estimated from u ³/ℓ, where u is the characteristic velocity and ℓ is the integral length scale, both calculated in Sect. 5).

$$B_{\text{d}} = 0.84\sin \left( {\varDelta \theta } \right)\frac{{fU_{\text{c}} }}{c} ,$$

where ∆θ is the angle bisector between the transmitter and receiver (15˚ for our system), and U _c is the cross-beam or transverse velocity component (〈V〉). Table 6 summarizes the quantities used to ultimately calculate σ ²_D .

Table 6 Calculation of the Doppler noise (σ ²_D )

1.1.3 Uncertainty for the acoustic Doppler velocimetry measurements

The total uncertainty of the ADV measurements is calculated assuming random independent errors. It can be calculated using the following equation (Taylor 1997):

$${\text{Total}}\,{\text{Uncertainty}} = \sqrt {\left( {{\text{Uncertainty}}_{1} } \right)^{2} + \left( {{\text{Uncertainty}}_{2} } \right)^{2} + \left( {{\text{Uncertainty}}_{3} } \right)^{2} + \cdots }$$

The total velocity uncertainty (σ ²_t-RMS ) for the RMS velocity along each receiver beam of the ADV can then be calculated as the sum of the sampling error (σ ²_m ) and the Doppler noise (σ ²_D ). Recall that the error due to the mean velocity shear is negligible, thus $\sigma_{\text{t - RMS}}^{2} = \sigma_{\text{m}}^{2} + \sigma_{\text{D}}^{2}$, which is tabulated in Table 7.

Table 7 Calculation of the total velocity uncertainty for the RMS along each receiver (σ ²_t-RMS )

We can assume that σ ²_t is the same along each receiver beam if the receiver transducers are identical and ideal. Under this assumption, the uncertainty of the RMS velocity for each velocity component (σ ² _i-RMS ) can be calculated using the ADV’s transformation matrix. Table 8 presents the uncertainties for the three RMS velocity components.

Table 8 Calculation of the total uncertainty for each velocity component (σ ²_i-RMS )

We note the smaller uncertainty for the z-component of velocity (w), as previously discussed, as well as in Khorsandi et al. (2012).

1.2 Propagation of uncertainties

As mentioned in Sect. 3, some corrections were applied to the calculated RMS velocities. The corrections included in some cases the combination of the three calculated RMS velocities resulting in a propagation of the uncertainties. The RMS velocity in the z-direction was not modified and thus the corresponding uncertainty remains the same as calculated in “Uncertainty for the acoustic Doppler velocimetry measurements” in Appendix (i.e., w _RMS = [〈w ²〉 + σ _z-RMS ²]^1/2). The RMS velocity in the y-direction was assumed to be the same as that of the z-direction due to the axisymmetric nature of the flow; then, the uncertainty in v is the same as that of w (z-direction). On the other hand, the corrected RMS velocity in the x-direction was calculated using the three calculated RMS velocity components u _RMS, v _RMS, and w _RMS (each one with a corresponding uncertainty). The expression used in the correction of u _RMS (process described in Sect. 3) is the following:

$$u_{\text{c-RMS}} = \left[ {u_{\text{RMS}}^{2} - \frac{{\left( {a_{11}^{2} + a_{12}^{2} + a_{13}^{2} + a_{14}^{2} } \right)}}{{\left( {a_{21}^{2} + a_{22}^{2} + a_{23}^{2} + a_{24}^{2} } \right)}}\left( {v_{\text{RMS}}^{2} - w_{\text{RMS}}^{2} } \right)} \right]^{1/2} ,$$

where u _c-RMS is the corrected RMS velocity, and a _ij are elements of the transformation matrix. The total relative uncertainties for the RMS velocities were calculated performing a step-by-step analysis of uncertainty propagation (Taylor 1997). The final results are presented in Table 9.

Table 9 Calculation of the total relative uncertainties for the RMS velocity components

The table above shows that the total relative uncertainty for the RMS velocity is less than 5 % in any given direction, being the largest for the u-component of the velocity.