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<p> An implicit finite difference technique has been developed for the solution of the steady two dimensional boundary layer equations. </p> <p> The numerical method is free of stability limitations and similarity assumptions. Use has been made of Wu-type starting profiles which enable one to start the calculation from the leading edge. </p> <p> Attractive features of the technique are its simplicity, flexibility and applicability to a wide range of boundary layer problems. In addition, results obtained from several case studies indicate that the numerical procedure is both accurate and fast. </p> / Thesis / Master of Engineering (MEngr)

Hydrodynamic slip is known to suppress vorticity production at the solid-fluid boundary in bluff body flows. This suppression combined with the enhanced vorticity convection results in a substantial reduction in the unsteady vortex shedding and the hydrodynamic loads experienced by the bluff body. Here, using combined theoretical and computational techniques, we investigate the effect of slip on three-dimensional wake dynamics and convective scalar transport from a circular cylinder placed in the uniform cross-flow of a Newtonian incompressible fluid over Reynolds numbers ranging from 0.1 to 1000. We find the wake patterns to be strongly influenced by the degree of the slip, quantified through the non-dimensional slip length in the Naiver slip model, with the asymptotic slip lengths of zero and infinity characterizing no-slip and no-shear boundaries, respectively. With increasing slip length, the wake three-dimensionality, that is observed in the case of a no-slip surface for Re > 190, is gradually suppressed and eventually eliminated completely. For each Reynolds number, we identify the critical slip length beyond which the three-dimensionality is completely suppressed and the wake becomes two-dimensional, on the basis of the total transverse entropy present in the flow field. Over the Reynolds number range considered in this work, we find the critical slip length to be an increasing function of Reynolds number. For sufficiently large slip lengths, we observe suppression of two-dimensional vortex shedding leading to formation of a steady separated wake. Further increments in slip length lead to reduction in the intensity and size of the recirculating eddy pair eventually resulting in its complete disappearance for a no-shear surface for which the flow remains attached all along the cylinder boundary. Next, we quantify the effect of hydrodynamic slip on convective transport from an isothermal circular cylinder placed in the uniform cross flow of an incompressible fluid at a lower temperature. For low Reynolds and high P´eclet numbers, theoretical analysis based on Oseen and thermal boundary layer equations allows us to obtain explicit relationships for the dependence of transport rate on the prescribed slip length. We observe that the non-dimensional transport coefficients follow a power law scaling with respect to the P´eclet number, with the scaling exponent increasing gradually from the lower asymptotic limit of 1/3 for the no-slip surface to 1/2 for a no-shear boundary. Results from our simulations at finite Reynolds number indicate that the local time-averaged transport rates for a no-shear surface exceed the one for the no-slip surface all along the cylinder except in the neighbourhood of the rear stagnation region, where flow separation and reversal augment the transport rates substantially.

This thesis was aimed developing a fundamental understanding of the boundary layer response to arbitrary motion. In this context arbitrary motion was defined as the unsteady translation and rotation of an object. Research objectives were developed from the gaps in knowledge as defined during the literature survey. The objectives were divided into three main activities; mathematical formulations for non-inertial bulk flow and boundary layer equations, implementation of said formulations in a numerical solver and simulations for various applications in arbitrary motion. Mathematical formulations were developed for the bulk flow and boundary layer equations in arbitrary motion. It was shown that the conservation of momentum and energy equations remains invariant in the non-inertial forms. The conservations of momentum equation can at most have six fictitious terms for unsteady arbitrary motion. The origin of the terms were found to be from transformation of the material derivative to the non-inertial frame. All fictitious terms were found to be present in the boundary layer equations, none could be eliminated during an order of magnitude analysis. The vector form of the non-inertial equations were implemented in a novel OpenFOAM solver. The non-inertial solver requires prescribed motion input and operate on a stationary mesh. Validation of the solver was done using analytical solutions of a steady, laminar flat plate and rotating disk respectively. Numerical simulation were done for laminar flow on a translating plate, rotating disk and rotating cone in axial flow. A test matrix was executed to investigated various cases of acceleration and deceleration over a range of 70 g to 700 000g. The boundary layer profiles, boundary layer parameters and skin friction coefficients were reported. Three types of boundary layer responses to arbitrary motion were defined. Response Type I is viscous dominant and mimics the steady state velocity profile. In Response Type II certain regions of the boundary layer are dominated by viscosity and others by momentum. Response Type III is dominated by momentum. In acceleration the near-wall velocity gradient increases with increasing acceleration. In deceleration separation occurs at a result of momentum changes in the flow. The mechanism that causes these responses have been identified using the developed boundary layer equations. In acceleration the relative frame fictitious terms become a momentum source which results in an increase in velocity gradient at the wall. In deceleration the relative frame fictitious terms become a momentum sink that induced an adverse pressure gradient and subsequently laminar separation. / Hierdie tesis is gerig op die ontwikkeling van 'n fundamentele begrip aangaande die grenslaag reaksie op arbitrêre beweging. In hierdie konteks word arbitrêre beweging gedefinieer as die ongestadigde translasie en rotasie van 'n voorwerp. Navorsingsdoelwitte is ontwikkel uit die gapings soos omskryf in die literatuuroorsig. Die doelwitte is verdeel in drie hoof aktiwiteite; wiskundige formulerings vir ongestadigde vloei en grenslaag vergelykings, implementering van hierdie formulerings in 'n numeriese kode en simulasies vir verskeie gevalle van arbitrêre beweging. Wiskundige formulerings is ontwikkel vir die vloei en grenslaag vergelykings in arbitrêre beweging. Daar is bewys dat die behoud van massa en energie vergelykings onveranderd in die nie-inertiële vorms bly. Die behoud van momentum vergelyking kan hoogstens ses fiktiewe terme vir ongestadigde, arbitrêre beweging hê. Die oorsprong van die terme is vanuit die transformasie van die ongestadigde en adveksie terme (aan die linker kant van die momentum vergelyking) na die nie-inertiële raam. Alle fiktiewe terme is teenwoordig in die grenslaag vergelykings. Die vektor vorm van die nie-inertiële vergelykings is in 'n nuwe OpenFOAM oplosser geïmplementeer. Die nie-inertiële oplosser vereis voorgeskrewe beweging insette en werk op 'n stilstaande rooster. Die oplosser is getoets teen analitiese oplossings van 'n gestadigde, laminêre plaat plaat en 'n roterende skyf, onderskeidelik. Numeriese simulasies is gedoen vir laminêre vloei op 'n translerende plaat, roterende skyf en roterende konus in aksiale vloei. 'n Toets matriks is gebruik om ondersoek in te stel na gevalle van versnelling en vertraging oor 'n verskeidenheid van 70 g tot 700 000 g. Die grenslaag profiele, grenslaag parameters en oppervlak wrywingskoëffisiënte is aangemeld nie. Drie tipes grenslaag reaksies op arbitrêre beweging is gedefinieer. Reaksie Tipe I is viskeus dominant en boots die bestendige snelheidsprofiel na. In reaksie Tipe II sekere dele van die grenslaag is oorheers deur viskositeit en ander deur momentum. Reaksie Tipe III word in totaliteit oorheers deur momentum. In versnelling die snelheid helling teen die objek neem toe met toenemende versnelling. In vertraging is 'n negatiewe snelheidsprofiel waargeneem as gevolg van momentum veranderinge in die vloei. Die meganisme wat hierdie reaksies veroorsaak is geïdentifiseer deur die grenslaag vergelykings. In versnelling word die fiktiewe terme 'n bron van momentum. Dit lei tot 'n toename in snelheid helling op die objek. In vertraging word die fiktiewe terme 'n momentum gebruiker wat 'n negatiewe drukgradiënt veroorsaak en gevolglik laminêre vloei wegbreking veroorsaak. / Thesis (PhD)--University of Pretoria, 2016. / Mechanical and Aeronautical Engineering / PhD / Unrestricted