The discrete-element method for predicting skin friction for turbulent flow over rough surfaces considers the drag on the surface to be the sum of the skin friction on the flat part of the surface and the drag on the individual roughness elements that protrude into the boundary layer. The discrete-element method considers heat transfer from a rough surface to be the sum of convection through the fluid on the flat part of the surface and the convection from each of the roughness elements. The discrete-element method has been widely used and validated for roughness composed of sparse, ordered, and deterministic elements. Modifications made to the discrete-element roughness method to extend the validation to real surface roughness are detailed. These modifications include accounting for the deviation of the roughness element cross sections from circular configurations, determining the location of the computational "surface" that differs from the physical surface, and accounting for temperature changes along the height of the roughness elements. Two randomly-rough surfaces found on high-hour gas-turbine blades were characterized using a Taylor-Hobson Form Talysurf Series 2 profilometer. A method for using the three-dimensional profilometer output to determine the geometry input required in the discrete-element method for randomly-rough surfaces is presented. Two randomly-rough surfaces, two elliptical-analog surfaces, and two cone surfaces were generated for wind-tunnel testing using a three-dimensional printer. The analog surfaces were created by replacing each random roughness element from the original randomly-rough surface with an elliptical roughness element with the equivalent planorm area and eccentricity. The cone surfaces were generated by placing conical roughness elements on a flat plate to create surfaces with equivalent values of centerline-averaged height or root-mean-square (RMS) height as the randomly-rough surfaces. The results of the wind tunnel skin friction coefficient and Stanton number measurements and the discrete-element method predictions for each of the six surfaces are presented and discussed. For the randomly-rough surfaces studied, the discrete-element method predictions are within 7% of the experimentally measured skin friction coefficients. The discrete-element predictions are within 16% of the experimentally measured Stanton numbers for the randomly-rough surfaces.

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We study the spontaneous edge current of chiral superconductors with high chirality both in the absence and presence of Meissner screening. We compute the edge current from a self-consistent solution to a set of coupled equations: quasiclassical Eilenberger equation, superconducting gap equation, and Maxwell equation. We find that the spatial dependent chiral edge current is largely suppressed and has more nodes for higher chirality pairings. In the absence of Meissner screening, the integrated current at T=0 is zero for all higher chirality pairings; while it is substantial for chiral p-wave. This conclusion is consistent with previous studies. In contrast, at finite T, the integrated current is non-zero even for higher chiral pairings. It turns out that the spatial varying order parameter is crucial to understand this finite T behavior of the edge current. When Meissner screening is included, the magnitude of the edge currents is reduced for all chiral pairings; however, the reduction is much weaker in higher chirality cases. We conclude that the Meissner effect is not that important for higher chiral pairings. We also consider the effect of the rough surface on the edge current. The edge current of even chiral pairings is inverted by the strong surface roughness; however, that of the odd chiral pairings is not. The sub-dominant order parameters, induced by the surface, are the key to understanding this current inversion. / Thesis / Master of Science (MSc)

The presence of roughness on a surface subject to high Reynolds number flows promotes the formation of a turbulent boundary layer and the generation of a fluctuating pressure field imposed on the surface. While numerous studies have investigated the wall pressure fluctuations over zero-pressure gradient smooth walls, few studies have examined the effects of surface roughness on the wall pressure field. Additionally, due to the difficulties in obtaining high Reynolds number flows over fully rough surfaces in laboratory settings, an even fewer number of studies have investigated this phenomenon under flow conditions predicted to be fully free of transitional effects that would ensure similarity laws could be observed. This study presents the efforts to scale and describe the wall pressure spectrum of a rough wall, high Reynolds number turbulent boundary layer free of transitional effects. Measurements were taken in the Virginia Tech Stability Wind Tunnel for both smooth and rough walls. A deterministic roughness fetch composed of 3-mm hemispheres arranged in a 16.5-mm square array was used for the rough surface. Smooth and rough wall flows were examined achieving Reynolds numbers up to Re_θ = 68700 and Re_θ = 80200 respectively, with the rough wall flows reaching roughness based Reynolds numbers up to k_g⁺ = 507 with a simultaneous blockage ratio of δ/k_g = 76. A new roughness based inner variable scaling is proposed that provides a much more complete collapse of the rough wall pressure spectra than previous scales had provided over a large range of Reynolds numbers and roughness configurations. This scaling implies the presence of two separate time scales associated with the near wall turbulence structure generation. A clearly defined overlap region was observed for the rough wall surface pressure spectra displaying a frequency dependence of Ï ^-1.33, believed to be a function of the surface roughness configuration and its associated transport of turbulent energy. The rough wall pressure spectra were shown to decay more rapidly, but based on the same function as what defined the smooth wall decay. / Master of Science

A new technique is presented to study radio propagation and rough surface scattering problems based on a reformulation of the Magnetic Field Integration Equation (MFIE) called the Single-Scatter Subtraction (S^3) method. This technique amounts to a physical preconditioning by separating the single- and multiple-scatter currents and removing the single-scattering contribution from the integral term that is present in the MFIE. This requires the calculation of a new quantity that is the kernel of the MFIE integral call the kernel integral or Gbar. In this work, 1-dimensional deterministically rough surfaces are simulated by surfaces consisting of single and multiple cosines. In order to truncate the problem domain, a beam illumination is used as the source term and it is shown that this also causes the kernel integral to have a finite support. Using the Single Scatter Subtraction method on these surfaces, closed-form expressions are found for the kernel integral and thus the single-scatter current for a well defined region of validity of surface parameters which may then be efficiently radiated into the far field numerically. Both the closed-form expressions, and the computed radiated fields are studied for their physical significance. This provides a clear physical intuition for the technique as an augmentation to existing ones as a bent-plane approximation as shown analytically and also validated by numeric results. Further analysis resolves a controversy on the nature of Bragg scatter which is found to be a multiple-scatter phenomenon. Error terms present in the kernel integral also raise new questions on the effect of truncation for any MFIE-based solution. Additionally, a dramatic enhancement of backscatter predicted by this new approach versus the Kirchhoff method is observed as the angle of incidence increases due to the error terms. / Doctor of Philosophy / A new technique is presented to study the interaction of electromagnetic waves with rough surfaces. Building on the technique called the Magnetic Field Integral Equation (MFIE) which allows the solution for the electromagnetic fields scattered from the surface by considering only the induced electric and magnetic currents on the surface, the Single-Scatter Substraction (S 3 ) method separates the surface currents into those that interact with the surface only once or single-scatter, and those that interact multiple times called multiple-scatter. Since this is the introduction of this technique, only the former is investigated. In this study, a new quantity which is an integral of one of the components of the standard MFIE is studied and closed-form approximations are presented along with bounds of validity. This provides closed form solutions for the single-scattering currents, from which the radiated fields may be efficiently found numerically. Since they are closed form, the expressions provide insight into the nature of the physical scattering process. Numerical results of these expressions are compared to the standard approximate technique as well as the ”exact” solution found by numerically solving the MFIE. Compared to the standard approximate technique which approximates the surface by a tangent plane at each point on the surface, the single-scatter currents approximate the surface with a bent-plane at each point. This shifts the scattered fields from certain directions to others, and highlights where single- and multiple-scattering have an effect.

The problem of rough surface scattering and propagation over rough terrain in ducting environments has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. In this method, the Helmholtz wave equation is replaced by a PWE under the assumption of predominant forward propagation and scattering. The resulting PWE subjected to the appropriate boundary condition(s) is then solved, given an initial field distribution, using marching techniques such as the split-step Fourier algorithm. As is obvious from the assumption on which it is based, the accuracy of the PWE approximation deteriorates in situations involving appreciable scattering away from the near-forward direction, i.e. when the terrain under consideration is considerably rough. The backscattered field is neglected in all PWE-based models. An alternative and more rigorous method for modeling the problem under consideration is the boundary integral equation (BIE) method, which is formulated in two steps. The first step involves setting up an integral equation (the magnetic field integral equation, MFIE, or the electric field integral equation EFIE) governing currents induced on the rough surface by the incident field and solving for these currents numerically. The resulting currents are then used in the appropriate radiation integrals to calculate the field scattered by the surface everywhere in space. The BIE method accounts for all orders of multiple scattering on the rough surface and predicts the scattered field in all directions in space (including the backscattering direction) in an exact manner. In homogeneous media, the implementation of the BIE approach is straightforward since the kernel (Green's function or its normal derivative) which appears in the integral equation and the radiation integrals is well known. This is not the case, however, in inhomogeneous media (ducting environments) where the Green's function is not readily known. Due to this fact, there has been no attempt, up to our knowledge, at using the BIE (except under the parabolic approximation) to model the problem under consideration prior to the work presented in this thesis. In this thesis, a closed-form approximation of the Green's function for a two-dimensional ducting environment formed by the presence of a linear-square refractivity profile is derived using the asymptotic methods of stationary phase and steepest descents. This Green's function is then modified to more closely model the one associated with a physical ducting medium, in which the refractivity profile decreases up to a certain height, beyond which it becomes constant. This modified Green's function is then used in the BIE approach to study low grazing angle (LGA) propagation over rough surfaces in the aforementioned ducting environment. The numerical method used to solve the MFIE governing the surface currents is MOMI, which is a very robust and efficient method that does not require matrix storage or inversion. The proposed method is meant as a benchmark for people studying forward propagation over rough surfaces using the parabolic wave equation (PWE). Rough surface scattering results obtained via the PWE/split-step approach are compared to those obtained via the BIE/MOMI approach in ducting environments. These comparisons clearly show the shortcomings of the PWE/split-step approach. / Ph. D.