Transient growth instability cancelation by a plasma actuator array

Abstract

This study investigates an actuation scheme that can be integrated as part of a feedback control system in the laboratory for the purpose of negating the transient growth instability in a Blasius boundary layer and delaying transition. The actuators investigated here consist of a spanwise array of symmetric plasma actuators, which are capable of generating spanwise-periodic counter-rotating vortices. Three different actuator geometries are investigated, resulting in 45, 67 and 70% reduction of the total disturbance energy produced inside the boundary layer by an array of roughness elements. It is demonstrated that the control effectiveness of the actuators can be significantly improved by optimizing the geometry of the array.

Appendix: A uncertainty analysis

The analysis followed to estimate the uncertainty on the measured values of $\overline{\phi_u}$ is summarized here. Since most of the energy is contained in the first mode i = 1 (i.e. for β _f ), (1) suggests that the relative uncertainty on ϕ _u (β _f ) can be approximated by

$$ \frac{\delta\left\langle u^2\right\rangle}{\left\langle u^2\right\rangle} = \frac{\delta \phi_u(\beta_f)}{\phi_u(\beta_f)}, $$

(3)

where δα is the uncertainty on parameter α. Given that u = U − 〈U〉, it can be shown that

$$ \left\langle u^2\right\rangle = \left\langle U^2\right\rangle - \left\langle U\right\rangle^2. $$

(4)

From here, the uncertainty due to bias and precision errors need to be treated separately. Considering the bias errors first, it is clear that δ〈U ²〉 and δ〈U 〉² are correlated such that

$$ \left[\delta\left\langle u^2\right\rangle\right]_b = \left[\delta\left\langle U^2\right\rangle\right]_b - \left[\delta\left\langle U\right\rangle^2\right]_b, $$

(5)

where the subscript b denotes an uncertainty associated with bias error sources. If the relative uncertainty on each U measurements remains constant, which is typically the case in experiments, it can be shown that

$$ \left[\frac{\delta\left\langle U^2\right\rangle}{\left\langle U^2\right\rangle}\right]_b =\left[\frac{\delta\left\langle U\right\rangle^2}{\left\langle U\right\rangle^2}\right]_b = 2 \left[\frac{\delta U}{U}\right]_b. $$

(6)

Therefore,

$$ \left[\delta\left\langle u^2\right\rangle\right]_b = 2\left\langle U^2\right\rangle \left[\frac{\delta U}{U}\right]_b - 2 \left\langle U\right\rangle^2\left[\frac{\delta U}{U}\right]_b, $$

(7)

but by invoking (4), this equation simplifies to

$$ \left[\frac{\delta\left\langle u^2\right\rangle}{\left\langle u^2\right\rangle}\right]_b = 2 \left[\frac{\delta U}{U}\right]_b. $$

(8)

Considering now the precision errors, the uncertainty on δ〈u ² 〉 is conservatively given by

$$ \left[\delta\left\langle u^2\right\rangle\right]^2_p = \left[\delta\left\langle U^2\right\rangle\right]_p^2 + \left[\delta\left\langle U\right\rangle^2\right]_p^2, $$

(9)

where the subscript p denotes an uncertainty associated with a source of precision error. The surrogate of (5) for precision errors is

$$ \left[\frac{\delta\left\langle U^2\right\rangle}{\left\langle U^2\right\rangle}\right]_p =\left[\frac{\delta\left\langle U\right\rangle^2}{\left\langle U\right\rangle^2}\right]_p = \frac{2}{\sqrt{N_z}} \left[\frac{\delta U}{U}\right]_p, $$

(10)

where N _z is the number of independent samples relevant to the spanwise average. Introducing (4) and (10) into (9), the following can be obtained

$$ \left[\frac{\delta\left\langle u^2\right\rangle}{\left\langle u^2\right\rangle}\right]_p = \frac{2}{\sqrt{N_z}}\left(1 + 2 \frac{\left\langle U \right\rangle^2}{\left\langle u^2\right\rangle} + 2\frac{\left\langle U\right\rangle^4}{\left\langle u^2\right\rangle^2}\right)^{1/2} \left[\frac{\delta U}{U}\right]_p. $$

(11)

The terms in the round brackets represent the effect of the signal to noise ratio. Since 〈U〉²/〈u ²〉 ≫ 1, (11) can be approximated as

$$ \left[\frac{\delta\left\langle u^2\right\rangle}{\left\langle u^2\right\rangle}\right]_p = 2\sqrt{\frac{2}{N_z}} \frac{\left\langle U\right\rangle^2}{\left\langle u^2\right\rangle} \left[\frac{\delta U}{U}\right]_p. $$

(12)

The degradation of the uncertainty on 〈u ²〉, and thus ϕ _u (β _f ), due to the relative magnitude of the velocity disturbance compared to the mean velocity profile (i.e. the signal to noise ratio) is clear from (12). Therefore, the precision uncertainty for a given case depends on the strength of the resulting disturbance.

Given (2), $\overline{\phi_u}$ can be approximated as

$$ \overline{\phi_u}\simeq \frac{1}{5\delta}\sum_i \phi_u^i(\beta_f) \Updelta y_i, $$

(13)

where i denotes a variable measured at the ith location above the wall and $\Updelta y_i = y_i-y_{i-1}.$ Considering precision errors, it is reasonable to assume that the spanwise measurements at each wall-normal locations are independent and that the precision uncertainty on the wall-normal location of these measurements are negligible. Therefore, it follows that

$$ \left[\delta\overline{\phi_u}\right]_p^2 = \frac{1}{(5\delta)^2}\sum_i \left[\delta\phi_u^i(\beta_f)\right]_p^2 \delta y_i^2, $$

(14)

where the subscript p denotes the precision uncertainty. For simplicity, let us assume that $\Updelta y_i =\Updelta y$ for all i. Noting that $5\delta = N_y \Updelta y,$ (14) simplifies to

$$ \left[\delta\overline{\phi_u}\right]^2_p = \frac{\overline{\left[\delta\phi_u^i(\beta_f)\right]_p^2}}{N_y}, $$

(15)

where N _y is the number of spanwise scans for 0 < y ≤ 5δ and the overbar denotes an average over the boundary layer thickness. Combining (3) and (12), and since $\phi_u^i(\beta_f)\simeq \left\langle u^2\right\rangle,$ yields after simplication

$$ \overline{\left[\delta\phi_u^i(\beta_f)\right]_p^2} = \frac{8}{N_z} \overline{\left\langle U\right\rangle^4}\left[\frac{\delta U}{U}\right]_p^2. $$

(16)

Therefore, one can write (15) as

$$ \left[\frac{\delta\overline{\phi_u}}{\overline{\left\langle u^2\right\rangle}}\right]^2_p = \frac{8}{N_yN_z}\frac{\overline{\left\langle U\right\rangle^4}}{\overline{\left\langle u^2\right\rangle}^2}\left[\frac{\delta U}{U}\right]^2_p. $$

(17)

Clearly, the precision uncertainty will diminish as a result of the averaging over the boundary layer thickness due to its inverse dependence on $\sqrt{N_y}.$ In the case of bias errors, the uncertainty on the measurements at each wall-normal locations are not independent. Therefore,

$$ \left[\delta\overline{\phi_u}\right]^2_b =\overline{\left[\delta\phi_u^i(\beta_f)\right]_b^2}, $$

(18)

which, when combined with (8), becomes

$$ \left[\frac{\delta\overline{\phi_u}}{\overline{\left\langle u^2\right\rangle}}\right]_b = 2 \left[\frac{\delta U}{U}\right]_b. $$

(19)

The above equations imply that the uncertainty due to these errors cannot be reduced by increasing the number wall-normal measurements. The overall uncertainty is given by

$$ \left[\delta\overline{\phi_u}\right]^2 = \left[\delta\overline{\phi_u}\right]_b^2 + \left[\delta\overline{\phi_u}\right]_p^2. $$

(20)