Representación de las preguntas

The drag reduction results presented in this thesis for both the passivity-based and_H∞loop-shaping controllers are unlikely to resemble what would be observed

if these control methodologies were applied and tested in the real-world. This is due to the following:

• All controllers were designed for and tested on idealised channel flows which are periodic in both the streamwise and spanwise spatial directions. In reality, it is not possible to enforce periodic boundary conditions in a plane channel. To obtain a channel flow which is approximately homogeneous in these directions would require a wide channel and accurate enforcement of fixed mass flow. The channel would need to be wide to ensure that the testing portion of the channel was sufficiently far from the walls such that the effect of the walls was negligible. The strict enforcement of mass flow rate is necessary to ensure a constant bulk velocity.

• DNS testing of the controllers assumed sensors and actuators were infinitesi- mally small, collocated and densely distributed on the walls. In reality, none of these assumptions are feasible. Modern MEMS sensors and actuators have characteristic length scales up to the order of 1mm [49]. Therefore, there will be a finite number of sensors and actuators which can be arranged on the walls of a real-world channel. Finding the optimal number and arrangements of the sensors and actuators would be an additional task.

• Real-world actuators have a finite bandwidth, i.e. have a maximum fre- quency that they can operate at. All controllers tested in this thesis incor- porate low-pass filters into their dynamics which ensures they have a finite

CHAPTER 6. DISCUSSION, CONCLUSIONS AND FUTURE WORK 146 FFT IFFT

..

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..

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Sensor Processor Actuator Controller

Computer Processor

Figure 6.1: Block diagram depicting how the controllers developed in this thesis could be implemented on a real channel flow.

bandwidth. However, they all use relatively small time-constants and there- fore have a large bandwidth. Depending on the actuator used, the controllers may have to use larger time-constants which could affect the performance of the controllers.

• In real-world wind tunnel and water tunnel experiments, there are many exogenous disturbances inputs affecting the flow which were absent in the DNS testing of the controllers. Examples include acoustic noise, changes in ambient temperature and noise from sensors and actuators. The advantage of using feedback control is that it can handle such exogenous uncertainty to a degree. However, only testing would show if the controllers could stabilise the flow in the presence of such uncertainty.

Figure 6.1 shows a block diagram depiction of how the controllers developed in this thesis could be implemented upon a real-world channel flow. Each wall of the channel would be densely populated with sensors and actuators; prior DNS testing could be used to find their optimal arrangement. All sensors would be con- nected to a sensor processor unit which would convert the analogue sensor signals to digital and using interpolation, generate an equispaced grid of measurements for each wall. This information would then be sent to a computer processor which would perform a fast Fourier transform (FFT) on the measurement grids. The measurement Fourier components would then be fed to the respective controllers, the output of which would be low-pass filtered. The computer processor would then inverse fast Fourier transform (IFFT) the low-pass filtered control signals and send these to an actuator controller unit in the form of an actuation grid for each wall. The actuation controller would then use interpolation to decide which

6.1. DISCUSSION 147

Control αmax βmax kmax ∆S∗

Passivity 0 10 10 2π_10Nh∗

s

H∞ loop-shaping ±2 12 ≈ 12 2π₁₂h

∗ Ns

Figure 6.2: Table of minimum dimensional sensor/actuator spacings ∆S∗ based on

channel half-height h∗ _{and sensor/actuator resolution per wavelength N} s.

actuation signals to send to each actuator and then convert the digital control sig- nals to voltages. This process would be repeated for every control loop iteration. The required population density of sensors and actuators is dictated by the small- est scales of the flow which are being controlled. For the passivity-based controllers the smallest controlled spatial scale is (α = 0, β = 10), and for the _H∞ loop-

shaping controllers they are (α =_{±2, β = 12). The dimensionless wavelength, λ,} corresponding to a wavenumber pair (α, β) is found from:

k =pα2_{+ β}2 ₌ 2π

λ . (6.1)

The dimensional wavelength for a channel with half-height h∗ is: λ∗ = h∗λ. If we define Ns as the number of sensors/actuators per dimensionless wavelength, then

the dimensional sensor/actuator spacing is found from: ∆S∗ := 2π

k h∗ Ns

. (6.2)

A table summarising the minimum dimensional sensor/actuator spacings can be found in Figure 6.2. For example, for a channel of total height 1 metre, and with a sensible wavelength resolution of Ns = 10, the minimum sensor/actuator spacing

for the Kα2,β12 H∞ loop-shaping controller would be ∆S∗ = 2.6 cm. This spacing

is sufficiently large enough for MEMS sensors and actuators to be implemented. The required bandwidth of the actuators is determined by the time-constants of the low-pass filters implemented in the controllers. The passivity-based con- trollers use a time-constant of τφ = 0.01 which corresponds to a cut-off frequency

of ωmax = 100 rad/T, and the H∞ loop-shaping controllers use a time-constant

of τφ = 0.1 which corresponds to a cut-off frequency of ωmax = 10 rad/T. The

cut-off frequency can be used as the maximum required bandwidth of the actu- ators. These frequencies are non-dimensionalised by channel half-height h∗ and maximum laminar centreline velocity Ucl∗. Therefore, time is non-dimensionalised

as t = t∗/T∗, where T∗ = h∗/U∗_cl. Using the definition for maximum laminar cen- treline velocity Reynolds number Re, T∗ can also be written as:

T∗ := h

∗2

CHAPTER 6. DISCUSSION, CONCLUSIONS AND FUTURE WORK 148

Control τφ Air/Water ω∗_max(rad/s) F_max∗ (Hz)

Passivity 0.01 Air 3.35(h∗)−2 0.53(h∗)−2

Passivity 0.01 Water 0.22(h∗₎−2 _0.04(h∗₎−2

H∞ loop-shaping 0.1 Air 0.33(h∗)−2 0.05(h∗)−2

H∞ loop-shaping 0.1 Water 0.02(h∗)−2 0.004(h∗)−2

Figure 6.3: Table of dimensional actuator bandwidths for controlling air (ν = 1.5_×10−5₎ and water (ν = 1× 10−6_{) Re = 2230 channel flows.}

Therefore, the dimensional bandwidth of the actuators is found using: ω∗_max = ωmax/T∗. The table in Figure 6.3 shows a table of required dimensional actuator

bandwidths for controlling Re = 2230 air and water channel flows. When con- trolling air channel flow, the actuators need to have higher bandwidths due to the fluid having a higher kinematic viscosity. When controlling a Re = 2230 air channel flow with total channel height 1 metre using passivity-based control, the maximum bandwidth of the actuators would need to be F_max∗ = 2.12 Hz. This is relatively low and it is likely that MEMS actuators could be produced which could operate within this bandwidth.

It is unlikely that either the passivity-based or _H∞ loop-shaping controllers de-

veloped in this thesis could be practically applied to reducing skin-friction drag on real-world air or marine craft. This is due to the high Reynolds numbers that most air and marine vehicles operate in. For both, Reynolds numbers of the order 106 _{and greater are typical. The controllers developed in this thesis were designed}

for and tested on relatively low Reynolds number flows. As Reynolds number in- creases, the required bandwidth of the actuation increases and it is unlikely that actuators with the required bandwidth exist for such high Reynolds numbers.