Overview of the present work

The effects of grooves on flow responses in laminar channel flows have been analyzed and are presented in this dissertation. Grooves with an arbitrary shape and an arbitrary orientation with respect to the flow direction have been considered. The groove geometry has been modelled using spectral techniques and therefore the analysis has been limited to the shapes that can be expressed by Fourier expansions. Discretization has been performed using Fourier expansions in the periodic directions and Chebyshev expansions

in direction across the channel. The difficulty associated with the enforcement of the boundary conditions on irregular surfaces has been dealt with by implementing either the immersed boundary conditions (IBC) concepts or the domain transformation (DT) method. The former method relies on employment of a fixed computational domain extending in the direction across the channel far enough that it completely encloses the grooved channel. The boundary conditions form internal constraints that provide closing conditions for the field equations. In the latter method the physical irregular domain is analytically mapped onto a regular computational domain which enables classical enforcement of the boundary conditions. Various tests have been conducted to show the performance of these algorithms and to prove that they provide spectral accuracy.

Analysis of drag generation in conduits with transverse corrugated walls has been carried out analytically using long wavelength approximation. Three mechanisms for generation of additional pressure losses have been identified, i.e. the additional shear drag due an increase of the wetted area and the re-arrangement of the shear stress distribution, the pressure form drag associated with the mean pressure gradient, and the pressure interaction drag associated with the phase difference between the surface geometry and the periodic part of the pressure field.

Detailed analyses of the effects of small-amplitude grooves on pressure losses have been performed for pressure-driven flows. It has been shown that losses can be expressed as a superposition of two parts, one associated with change in the mean positions of the walls and one induced by flow modulations associated with the geometry of the grooves. While the former effect can be determined analytically, the latter effect has to be determined numerically. Reduced-order geometry model generated by projection of the wall shape onto a Fourier space has been used to capture the modulation effects. The results demonstrate a strong dependence of the pressure losses on the groove orientation. Comprehensive examinations of the extreme cases, i.e. transverse and longitudinal grooves, have been carried out. The effects of each of contributing factors on drag formation have been studied. Drag-reducing laminar grooves have been identified in the case of long wavelength longitudinal grooves. For sufficiently short wavelength grooves, it has been shown that the wall shear can be eliminated from the majority of the wetted

surface area regardless of the groove orientation, thus exhibiting the potential for the creation of drag reducing surfaces. Such surfaces can become practicable if a method for elimination of the undesired pressure and shear peaks through proper groove shaping can be found.

Optimal shapes of laminar, drag-reducing longitudinal grooves in a pressure driven flow have been determined. It has been demonstrated that the optimal shapes can be characterized using reduced-order geometry models involving just a few Fourier modes. Two classes of grooves have been considered, i.e. equal-depth grooves, which have the same height and depth, and unequal-depth grooves. It has been shown that the optimal grooves in the former cases are characterized by a certain universal trapezoid. There exists an optimum depth in the latter case and this depth, combined with the corresponding groove shape, defines the optimal geometry; this shape is well- approximated by a Gaussian function. The maximum possible drag reduction has been determined for the optimal shapes. The analysis has been extended to kinematically- driven flows. It has been shown that in this case the longitudinal grooves always increase flow resistance regardless of their shape.

Effects of longitudinal grooves on the flow resistance in a channel flow driven by a combination of the pressure gradient and the movement by one of the walls have been studied. Three distinct zones leading to an increased flow rate have been identified depending on the pressure gradient and the groove wavenumber. Two of these zones correspond to grooves with long wavelengths and one to grooves with short wavelengths. Optimization has been used to determine shapes that provide the largest increase of the flow rate. It has been shown that no optimal shape exists in the latter case if the groove amplitude is less than a certain well defined limit as the shortest admissible wavelength dominates the system performance. There exists the most effective wavenumber for the taller grooves but the optimal shapes could not be determined due to numerical limitations. Conclusions regarding the optimal shapes for long wavelength grooves are similar to those of pressure-driven flows discussed above. Two distinct zones emerged when the reduction of the force acting on the upper wall was used as the performance criterion. The best performance for both of these cases was associated to the short

wavelength grooves and the system response was qualitatively similar to that found in the case of the flow rate increase.