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Kwasu Function: A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation
(American Institute of Aeronautics and Astronautics, 2018-01-07) Taofiq O. Amoloye
An attempt is made to re ne the classical potential theory of the flow over a circular cylinder by introducing a viscous sink-source-vortex sheet on the surface of the cylinder. These singularities introduced into the flow are modeled as concentric at every location. The vortices are modeled as variations of Lamb-Oseen, Batchelor and Burgers vortices and analytic expressions for their strengths and those of the sinks/sources are obtained from the classical theory. These are employed to obtain a viscous potential function named the Kwasu function which provides a closed form analytic solution to the complete three dimensional unsteady compressible Navier-Stokes equation. Preliminary results of the work show that the theory presented captures important features of a bluff body flow includin flow separation, wake formation, vortex shedding as well as compressibilty effects. The condition at a viscous wall is shown to be transient from slip towards a complete no-slip for a steady freestream flow. It is the hope that the present theory will shed more light on the important phenomenon of turbulence in planned future work in which quantitative analysis of the theory will be carried out.
Analysis of an Unsteady Incompressible Cross􀂧ow on a Stationary Circular Cylinder at Reynolds number 3,900 Using Re􀂦ned Potential Flow Theory
(Research Square, 2022-06-03) Taofiq O. Amoloye
The motion of a fluid around a circular cylinder presents interesting phenomena including flow separation, wake and turbulence. The physics of these are enshrined in the continuity equation and the Navier-Stokes Equations (NSE). Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, computationally, or theoretically. Theoretical studies of a cylinder flow using classical potential flow theory (CPT) have some gaps when compared to experiments. Attempting to bridge these gaps, this article introduces refined potential flow theory (RPT) and employs it on a stationary circular cylinder incompressible crossflow at Reynolds number 3, 900. It leverages experimental observations, physical deductions and some agreements between CPT and experiments in the theoretical development. This results in the incompressible Eulerian Kwasu function which is a quasi-irrotational stream function that satisfies the governing equations and boundary conditions. It captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and static unsteadiness. The Lagrangian form of the function is exploited for the flow pathlines that are used to incorporate dynamic unsteadiness. A gravity analogy is used to predict the separation, transition, and reattachment points. The analogy introduces the perifocal frame of fluid motion. The forces are obtained in this frame with a change of variable. The drag prediction is within the error bound of measured data. The RPT pressure distribution, separation point and Strouhal number are also within acceptable ranges. Energy spectra analyses of the wake velocity display Kolmogorov’s Five-Thirds law of homogeneous isotropic turbulence.
A Refined Potential Theory for the Incompressible Unsteady Subcritical-Reynolds number Flows on Canonical Bluff Bodies
(2020-11-16) Taofiq O Amoloye
The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. In classical potential theory, the steady inviscid incompressible flow over a body can be obtained by the superposition of elementary flows with known analytical solutions. Analytical solutions can offer huge advantages over numerical and experimental solutions in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, the classical potential theory falls short of reconciling the actions of viscosity in an experimentally observed flow with the theoretical analysis of such a flow. As such, it is unable to resolve the boundary layer and predict the especially important flow separation phenomenon that results in the pressure drag experienced by a body in the flow. This has relegated potential theory to idealized flows of little practical importance. Therefore, an attempt is made in this thesis to refine the classical potential theory of the flow over a circular cylinder to bridge the gap between the theory and experimentally observed flows. This is to enhance the ability to predict and/or control the flows' aerodynamic quantities and the evolution of the wake for design purposes. The refinement is achieved by introducing a viscous sink-source-vortex sheet on the surface of the cylinder to model the boundary layer. These vortices, sources and sinks introduced at the cylinder surface are modeled as concentric at every location. The vortices are modeled as Burgers' vortices, and analytic expressions for their strengths and those of the sinks/sources are obtained from the classical theory. These are employed to obtain a viscous and time-dependent stream function that captures critical qualitative features of the flow including flow separation, reattachment, wake formation, and vortex shedding. After that, a viscous potential function, the Kwasu function, with which the pressure field is obtained from the Navier-Stokes equation, is derived from the stream function. It is obtained by defining the viscous stream function on a principal axis of the flow about which the vorticity vector is identically zero. Strategies have also been developed to account for the finite extent of the cylinder and dynamic unsteadiness of the flow, and to predict the points of separation/reattachment/transition and the boundary layer thickness. Additionally, the strategies are used to obtain forces and apply the solution to arbitrary geometries focusing on spheres and spheroids. These strategies include the gravity analogy that considers a fluid element-cylinder scenario to be like a two-body problem in orbital mechanics. This analogy introduces the perifocal frame of fluid motion and exploits it to resolve the d'Alembert's Paradox. The perifocal frame is also used to predict flow separation/reattachment/transition and explain the observation of sign changes in the shear stress distribution at the rear of a circular cylinder in a crossflow. The refined potential theory is verified against experimental and numerical data on the cylinder in an incompressible crossflow at freestream Re∞=3,900. Its drag prediction is within the error bound of measured data and tHRLES (transitional Hybrid Reynolds-averaged Navier-Stokes Large Eddy Simulation) prediction. The predictions of the pressure distribution, separation point and Strouhal number are also within acceptable ranges. Its prediction of the force coefficients over the range 25≤Re∞<300,000 is validated against experimental and theoretical data on the cylinder in crossflow. There is a good agreement in the magnitude and trend for Re∞>100. For Re∞<100, there is a disparity in magnitude that is unsafe for design purposes. Similarly, it under-predicts the coefficient of drag in some of the explored axial flow configurations. However, at Re∞=96,000 and an aspect ratio of 2, the RPT drag prediction falls within 1.2% of validated computational result. The energy spectra of the wake velocity display the Kolmogorov's Five-Thirds law of homogeneous isotropic turbulence. This verifies and validates the unsteadiness in refined potential theory as turbulent in nature. The drag coefficient of a sphere for 25≤Re∞< 300,000 is explored to demonstrate the application of refined potential theory. Additionally, the flow over a sphere at Re∞=100,000 is explored in detail. A generally good agreement is observed in the prediction of the experimental trend for Re∞≥2,000. The transitional incompressible flows over a 6:1 prolate spheroid at an angle of attack β=45° for Re∞=3,000$ and Re∞=4,000 are also explored. The present theoretical pressure distribution has a close agreement with the DNS (direct numerical simulation) result in the starboard rear of the spheroid. However, the magnitude of the predicted force coefficients are generally less than five times the corresponding DNS results. The asymmetry of the DNS pressure distribution in the meridian plane is not captured. Therefore, further analyses of the spheroid flow including the separation locations are recommended for further studies. It is concluded that the refined potential theory can be used to resolve, explore and/or control the aerodynamic quantities of the flows around canonical bluff bodies as well as the evolution of their wakes.
A Refined Potential Theory for the Incompressible Un- steady Subcritical-Reynolds number Flows on Canonical Bluff Bodies
(Georgia Institute of Technology, 2020-11-19) Amoloye, Taofiq Omoniyi
The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. In classical potential theory, the steady inviscid incompressible flow over a body can be obtained by the superposition of elementary flows with known analytical solutions. Analytical solutions can offer huge advantages over numerical and experimental solutions in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, the classical potential theory falls short of reconciling the actions of viscosity in an experimentally observed flow with the theoretical analysis of such a flow. As such, it is unable to resolve the boundary layer and predict the especially important flow separation phenomenon that results in the pressure drag experienced by a body in the flow. This has relegated potential theory to idealized flows of little practical importance. Therefore, an attempt is made in this thesis to refine the classical potential theory of the flow over a circular cylinder to bridge the gap between the theory and experimentally observed flows. This is to enhance the ability to predict and/or control the flows' aerodynamic quantities and the evolution of the wake for design purposes. The refinement is achieved by introducing a viscous sink-source-vortex sheet on the surface of the cylinder to model the boundary layer. These vortices, sources and sinks introduced at the cylinder surface are modeled as concentric at every location. The vortices are modeled as Burgers' vortices, and analytic expressions for their strengths and those of the sinks/sources are obtained from the classical theory. These are employed to obtain a viscous and time-dependent stream function that captures critical qualitative features of the flow including flow separation, reattachment, wake formation, and vortex shedding. After that, a viscous potential function, the Kwasu function, with which the pressure field is obtained from the Navier-Stokes equation, is derived from the stream function. It is obtained by defining the viscous stream function on a principal axis of the flow about which the vorticity vector is identically zero. Strategies have also been developed to account for the finite extent of the cylinder and dynamic unsteadiness of the flow, and to predict the points of separation/reattachment/transition and the boundary layer thickness. Additionally, the strategies are used to obtain forces and apply the solution to arbitrary geometries focusing on spheres and spheroids. These strategies include the gravity analogy that considers a fluid element-cylinder scenario to be like a two-body problem in orbital mechanics. This analogy introduces the perifocal frame of fluid motion and exploits it to resolve the d'Alembert's Paradox. The perifocal frame is also used to predict flow separation/reattachment/transition and explain the observation of sign changes in the shear stress distribution at the rear of a circular cylinder in a crossflow. The refined potential theory is verified against experimental and numerical data on the cylinder in an incompressible crossflow at freestream Re∞=3,900. Its drag prediction is within the error bound of measured data and tHRLES (transitional Hybrid Reynolds-averaged Navier-Stokes Large Eddy Simulation) prediction. The predictions of the pressure distribution, separation point and Strouhal number are also within acceptable ranges. Its prediction of the force coefficients over the range 25≤Re∞<300,000 is validated against experimental and theoretical data on the cylinder in crossflow. There is a good agreement in the magnitude and trend for Re∞>100. For Re∞<100, there is a disparity in magnitude that is unsafe for design purposes. Similarly, it under-predicts the coefficient of drag in some of the explored axial flow configurations. However, at Re∞=96,000 and an aspect ratio of 2, the RPT drag prediction falls within 1.2% of validated computational result. The energy spectra of the wake velocity display the Kolmogorov's Five-Thirds law of homogeneous isotropic turbulence. This verifies and validates the unsteadiness in refined potential theory as turbulent in nature. The drag coefficient of a sphere for 25≤Re∞< 300,000 is explored to demonstrate the application of refined potential theory. Additionally, the flow over a sphere at Re∞=100,000 is explored in detail. A generally good agreement is observed in the prediction of the experimental trend for Re∞≥2,000. The transitional incompressible flows over a 6:1 prolate spheroid at an angle of attack β=45° for Re∞=3,000$ and Re∞=4,000 are also explored. The present theoretical pressure distribution has a close agreement with the DNS (direct numerical simulation) result in the starboard rear of the spheroid. However, the magnitude of the predicted force coefficients are generally less than five times the corresponding DNS results. The asymmetry of the DNS pressure distribution in the meridian plane is not captured. Therefore, further analyses of the spheroid flow including the separation locations are recommended for further studies. It is concluded that the refined potential theory can be used to resolve, explore and/or control the aerodynamic quantities of the flows around canonical bluff bodies as well as the evolution of their wakes.
Wind Engineering: A Review of the Eurocode provisions for the Wind Loading on Low-rise Buildings
(Cranfield University, 2012-09) Amoloye, Taofiq Omoniyi
Building codes such as the Eurocode have usually been used as a cheaper alternative to wind tunnel studies in the consideration of wind loading on a structure. It is often the case that very tall buildings and large structures have enough economic justification for expensive wind tunnel studies in their design stage. Such wind tunnel studies, as per state-of-the-art, feature simultaneous scanning of and acquisition of loading data from hundreds of pressure tappings with subsequent high-speed computer data processing and analysis. This is not the case for low-rise buildings which do not find their way into the wind tunnel except in the case where they are unusual edifices. Low-rise buildings, however, are the most damaged in wind storms. In addition, in the present times, their shapes are increasingly losing touch with the traditional and generic forms dealt with in the Eurocode. Therefore, the question is: How well does the Eurocode, which was put together with information from wind tunnel studies performed in the 50s and 70s using currently outdated data acquisition techniques, deal with present building shapes? The study was based on models of a simple cuboidal building; a quasi-rectangular building with inset faces in its plan; and a building plan featuring a re-entrant corner possessing curved surfaces at the internal and external junctions of its wings. It was concluded from the results of the study that adapting the Eurocode wind loading provisions to irregular building plans characteristic of modern times gives very unsafe solutions. The variations of pressure with wind direction on the internal walls of the wings of and the curved surface at the internal junction of the re-entrant corner were observed to follow coherent wave forms which are mutually similar. These call for further research.

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