Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Aerodynamics (15)
- Supersonic aerodynamics (13)
- Fluid dynamics (12)
- Turbulent boundary layer (10)
- Turbulence (9)
-
- Boundary layer (8)
- Particle image velocimetry (8)
- Wind tunnels (7)
- Aerofoils (6)
- Computational fluid dynamics (6)
- Drag (Aerodynamics) (6)
- Aeroelasticity (5)
- Cascades (Fluid dynamics) (5)
- Computational Fluid Dynamics (5)
- Flow visualization (5)
- Laminar flow (5)
- Aerodynamic load (4)
- Heat--Transmission (4)
- Hypersonic aerodynamics (4)
- Unsteady flow (Fluid dynamics) (4)
- Wind tunnels--Flow visualization (4)
- Axial flow compressors--Blades (3)
- Base flow (Aerodynamics) (3)
- Compressibility (3)
- Drone aircraft (3)
- Fluid dynamic measurements (3)
- Flutter (Aerodynamics) (3)
- Hypersonic (3)
- Micro air vehicles (3)
- Reynolds number (3)
- Publication Year
- Publication Type
Articles 121 - 150 of 171
Full-Text Articles in Engineering
Numerical And Experimental Investigation Of The Flowfield Near A Wrap-Around Fin, Carl P. Tilmann
Numerical And Experimental Investigation Of The Flowfield Near A Wrap-Around Fin, Carl P. Tilmann
Theses and Dissertations
A wall-mounted semi-cylindrical model fitted with a single wrap-around fin (WAF) has been investigated both numerically and experimentally, with the objective of characterizing the mean and turbulent flowfield in the vicinity of the fin. Numerical and experimental results are used to determine the nature of the flowfield and quantify the effects of fin curvature on the character of the flow near WAFs. This research has been motivated by the need to identify possible sources of a rolling moment reversal observed at high speeds in sub-scale flight tests. Detailed mean flow and turbulence measurements were obtained in the AFIT Mach 3 …
Experimental Investigation Of Compressible Boundary Layers Under The Influence Of Pressure Gradients, Raymond C. Wier
Experimental Investigation Of Compressible Boundary Layers Under The Influence Of Pressure Gradients, Raymond C. Wier
Theses and Dissertations
This study examined the effect of mild pressure gradients on the mean and turbulent flow of high-speed boundary layers. Three Mach numbers (1.7, 3.0 and 5.0) were investigated. Three pressure gradients were examined; a zero pressure gradient (ZPG), a favorable pressure gradient (FPG), and a combined pressure gradient (CPG). The CPG consisted of an adverse pressure gradient followed by a favorable pressure gradient. Conventional pressure probes, hot-wire and particle image velocimetry (PIV) were used to examine the flow. Measurement included mean velocity, velocity turbulence intensity, mass flux turbulence intensity and energy spectra. Instantaneous (10 nsec) Mie scattering flow visualizations were …
Experimental Investigation Of Transverse Supersonic Gaseous Injection Enhancement Into Supersonic Flow, Mark P. Wilson
Experimental Investigation Of Transverse Supersonic Gaseous Injection Enhancement Into Supersonic Flow, Mark P. Wilson
Theses and Dissertations
In pursuit of more efficient and effective fuel-air mixing for a SCRAMJET combustor, this study was conducted to investigate relative near field enhancements of penetration and mixing of a discrete low-angled (25°) injected air jet into a supersonic (M=2.9) cross flow. The enhancements were achieved by injecting the transverse air jet parallel to the compression face of eight different ramp geometries. The jet-ramp interactions created collinear shock structures, baroclinic torque vorticity enhancement, ramp spillage enhanced vorticity, magnus effect penetration enhancement, and increased total pressure loss. Shadowgraph photography was used to identify the shock structures and interactions in the flow field. …
Direct Measurement Of Skin Friction In High Temperature And Impulsively Started Supersonic Flowfields, David M. Hazelton
Direct Measurement Of Skin Friction In High Temperature And Impulsively Started Supersonic Flowfields, David M. Hazelton
Theses and Dissertations
This study further developed and improved a miniature plastic cantilever floating element gauge for direct measurement of skin friction in shock tunnel tests. Gauge durability and usability were improved, and potential sources of error were quantified. A high frequency 30 kHz gauge (compared to the baseline 10 kHz gauge) was constructed by using an I-beam cantilever rather than a round beam. A directional gauge was developed and demonstrated by measuring the flowfield about a fin in the AFIT shock tunnel. The 10 kHz and 30 kHz gauges were used in two scramjet test sequences conducted at Mach 14 enthalpy (stagnation …
Nonlinear Analysis Of Airfoil Flutter At Transonic Speeds, Scott A. Morton
Nonlinear Analysis Of Airfoil Flutter At Transonic Speeds, Scott A. Morton
Theses and Dissertations
Hopf-bifurcation analysis is used to determine flutter boundaries of a pitch and plunge airfoil (PAPA) at transonic Mach number conditions. The PAPA model is a coupling of the Euler equations and a two-degree-of-freedom structural model composed of linear and torsional springs. The Euler equations are discretized using an upwind total variation diminishing scheme (TVD) of Harten and Yee. Equilibrium solutions of the PAPA model are computed using Newton's method and dynamic solutions are explicitly integrated in time with first-order accuracy. The Hopf-bifurcation point, which models the flutter condition, is computed directly by solving an extended system of equilibrium equations following …
Compressible Turbulence Measurements In A Supersonic Boundary Layer With Impinging Shock Wave Interaction, Michael J. Meyer
Compressible Turbulence Measurements In A Supersonic Boundary Layer With Impinging Shock Wave Interaction, Michael J. Meyer
Theses and Dissertations
No abstract provided.
An Investigation Into The Effects Of Lateral Aerodynamic Asymmetries, Lateral Weight Asymmetries, And Differential Stabilator Bias On The F-15 Directional Flight Characteristics At High Angles Of Attack, David R. Evans
Theses and Dissertations
The F-15 is a stable aircraft throughout most of its flight envelope. However, it still exhibits an uncommanded yawing and rolling tendency at high angles-of-attack. Identified influencing factors of this uncommanded motion are lateral aerodynamic asymmetries, lateral weight asymmetries, and differential stabilator bias. Previous research into the effects of these influencing factors has been qualitative. This thesis quantifies the effects and then identifies a symmetric F-15 configuration. The quantifying metric presented is net yaw acceleration. This thesis used both computer simulation and experimental flight test to quantify these effects. A discussion of each influencing factors effects on the F-15B high …
Riblet Effects On Gortler Vortex Development Over A Concave Surface, James A. Rothenflue
Riblet Effects On Gortler Vortex Development Over A Concave Surface, James A. Rothenflue
Theses and Dissertations
Riblet effects on Goertler vortex development in laminar and transitional flows were measured using three-dimensional laser-Doppler anemometry. Three freestream velocities were tested such that riblet spacing corresponded to Goertler vortex (GV) wavelengths which presumably would be either strongly amplified, weakly amplified, or attenuated by centrifugal forces on a curved surface. Experiments revealed the development of paired, counterrotating vortices within the riblet valleys over both flat and concave surfaces. For riblet spacings equivalent to weakly or non-amplified GV wavelengths, GV development on a curved surface was above the riblets and was delayed by the addition of riblets. The GV wavelength was …
Mach 2.9 Investigation Into The Flow Structure In The Vicinity Of A Wrap-Around Fin, Richard E. Huffman Jr.
Mach 2.9 Investigation Into The Flow Structure In The Vicinity Of A Wrap-Around Fin, Richard E. Huffman Jr.
Theses and Dissertations
A ceiling-mounted semi-cylindrical model containing a single wrap-around fin (WAF) was tested in the AFIT Mach 2.9 test facility. Flow visualization using oil- low streaklines, schlieren images and shadowgraph photography revealed a A-shock at the fin-body juncture and the development of an asymmetric bow-shock about the fin. Quantitative measurements were taken with a 100 cone-static pressure probe, a Pitot pressure probe and two cross-wire hot-film probes (u-v and u-w components, respectively). Measurements were made at cutting-planes from the inlet of the test section to aft of the model, with emphasis placed in the vicinity of the WAF. Results include cutting-plane …
Investigation Of Aerodynamic Alterations For Improving The Kc-135 Boom Performance During Aerial Refueling, Debra A. Nawrocki
Investigation Of Aerodynamic Alterations For Improving The Kc-135 Boom Performance During Aerial Refueling, Debra A. Nawrocki
Theses and Dissertations
This study investigated the eligibility of the KC-135 air refueling boom for improved capabilities in the areas of control and performance. By using a thick airfoil cross-section for the boom tube, rather than the current circular cross-section, the ability to increase the lift characteristics was verified Prior compiled test data was used for comparison against analytical computer solutions. The possibility and effects of control frequency oscillations were also examined due to the unstable nature of the flow at test conditions. Additionally, the effect of other cross-section shapes, such as the blunt fairing, on the size of the flyable envelope for …
Influence Of A Moving Endwall On The Tip Clearance Vortex In An Axial Compressor Cascade, Lawrence J. Peter
Influence Of A Moving Endwall On The Tip Clearance Vortex In An Axial Compressor Cascade, Lawrence J. Peter
Theses and Dissertations
This experiment involved the design, construction, validation and testing of a new facility for the investigation of vortices generated by compressor rotor blade tip clearance with a moving endwall. A five-tube pressure rake placed downstream of the trailing edge of a cascade of blades measured the pressure field for flow coefficients ranging from 20 to 1.66 and tip clearances of 0.33, 1.0, 1.7 and 2.4 percent chord. Contour plots of mass averaged pressure loss coefficient appear to show the no-flow tip vortex becomes entrained and diffused by the moving wall boundary layer. The high loss region near the moving wall …
Experimental Investigation Of Non-Adiabatic Supersonic Slot Injection Into A Supersonic Cross-Flow, Kelly C. Tucker
Experimental Investigation Of Non-Adiabatic Supersonic Slot Injection Into A Supersonic Cross-Flow, Kelly C. Tucker
Theses and Dissertations
Turbulence and mean flow measurements were acquired downstream of a normal, supersonic (M=1.6), two-dimensional, slot injection into Mach 3.0 flow (Re/m = 15x106). Two non-adiabatic cases Ti/T∞ = 1.11 and Ti/T∞ = 0.93) were studied. Multiple overheat anemometry and conventional mean flow analysis, as well as shadowgraph and schlieren photography, were utilized to study the turbulent mixing layer. The mean flow data included the Pitot, cone-static, and Mach number profiles. The measured turbulence data included shear stresses, mass flux, total temperature turbulence intensities, and turbulent heat fluxes. The temperature and density variation …
Compressible Turbulence Measurement In The Mixing Layer Of An Adiabatic Normal Slot Injection Into Supersonic Flow, Christopher D. Whitcomb
Compressible Turbulence Measurement In The Mixing Layer Of An Adiabatic Normal Slot Injection Into Supersonic Flow, Christopher D. Whitcomb
Theses and Dissertations
In this study mean flow and compressible turbulence measurements were taken at a station x = 72W downstream of the injection, where W is the injector throat width, of an adiabatic 2-D Mach 1.6 normal slot injection into a Mach 2.9 flow. Data were collected using a conventional Pitot probe, a cone-static probe, and multiple overheat cross-wire anemometry. In addition, schlieren and shadowgraph flow visualization was used to investigate the flow structure at both the injection point and at the downstream data collection point. From these measurements, mass flux component turbulence intensities of 8% to 10% were seen. The total …
An Experimental Investigation Of A Sting-Mounted Finite Circulation Control Wing, Lorenzo C. Bradley Iii
An Experimental Investigation Of A Sting-Mounted Finite Circulation Control Wing, Lorenzo C. Bradley Iii
Theses and Dissertations
This study investigated the lift, drag and pitching moment performance of a circulation control wing in the AFIT 5-ft wind tunnel. The experimental wing model was a 20 percent thick, 8.5 percent camber, partial elliptical cross-section, single blowing slot, rectangular planform wing. The aspect ratios tested were 3.99, 3.77 and 3.75. The variables in the investigation included the slot blowing rate and model configuration. The model was modified by adding a leading edge nose droop, a trailing edge splitter plate and wing tip fences to improve flow at the leading edge, reduce separation effects, and encourage attached flow on the …
Experimental Investigation Of A Supersonic Turbulent Boundary Layer With Adverse Pressure Gradient, Chad S. Hale
Experimental Investigation Of A Supersonic Turbulent Boundary Layer With Adverse Pressure Gradient, Chad S. Hale
Theses and Dissertations
Laser Doppler Velocimetry (LDV) measurements were made to quantify the effect of adverse pressure gradient on the compressible turbulent flow structure in a Mach 2.9 boundary layer (Re/m= 1.75 x 107). Measurements included profiles of 2-D mean velocities, turbulence intensities, Reynolds shear stresses, intermittency, flatness and skewness. In addition, mean strain rates were also measured. The boundary layer measurements were acquired for both flat plate and compression ramp models. LDV measurements were made at two locations on the compression ramp model at 68 cm and 71 cm downstream of the nozzle throat. At these locations, Β …
Numerical Simulation Of Supersonic Turbulent Boundary Layer Flow Under The Influence Of Mild Pressure Gradients, Eric T. Fick
Numerical Simulation Of Supersonic Turbulent Boundary Layer Flow Under The Influence Of Mild Pressure Gradients, Eric T. Fick
Theses and Dissertations
Mach 2.9 boundary layer flow (Re/m ≈ 1.75x107) under the influence of mild pressure gradients is studied numerically. Baldwin-Lomax and k - ω turbulence models are incorporated into a cell centered finite volume flow solver and the results are compared with hot wire anemometry and Laser Doppler Velocimetry (LDV) measurements obtained for the same geometries in the AFIT Mach 2.9 wind tunnel. Agreement between the present simulations obtained with the k - ω turbulence model and experimental velocity profiles is excellent in all test sections. Nondimensional turbulent shear stress predictions closely match experimental data in the …
Development And Testing Of A New Optimum Design Code For Hypersonic Wind Tunnel Nozzles, Including Boundary Layer, Turbulence, And Real Gas Effects, Ralph C. Tolle
Theses and Dissertations
A robust and efficient optimization code is developed and validated. The code is used to redesign an existing Mach 12 wind tunnel nozzle and utilizes response surface methodology (RSM) techniques. Explicit, globally second-order, flux-difference-splitting algorithms are used to solve the Navier-Stokes (NS) and Parabolized Navier-Stokes (PNS) flow solvers incorporated into the optimizer code. Either the Baldwin-Lomax or the Yang-Shih k-s turbulence model may be employed in the optimization code. First, 2-D/axisymmetric NS and PNS flow solvers are developed/modified and account for perfect gas/nonequilibrium chemically reacting flows. All solvers are validated against Computational Fluid Dynamics (CFD) and experimental data. The optimization …
Compressible Turbulence Measurement In Low-Angle Injection Into A Supersonic Flow, Gregory J. Mccann
Compressible Turbulence Measurement In Low-Angle Injection Into A Supersonic Flow, Gregory J. Mccann
Theses and Dissertations
Mean flow and compressible turbulence measurements have been obtained at two stations downstream of low-angle supersonic injection into a Mach 3.0 flow (Re/m = 15 x 106). Data were collected using conventional Pitot and cone-static probes, single and multiple overheat cross-wire anemometry, and flow visualization techniques (shadowgraphs and schlierens). A direct measure of total Reynolds shear stress was accomplished using a turbulence transformation of the Reynolds-averaged Navier-Stokes equations. Results show compressibility effects, indicated by density fluctuations, to be large relative to the velocity fluctuations and on the same order in all three components. Compressibility appears to account for …
An Investigation Of The Afit 2-Inch Shock Tube As A Flow Source For Supersonic Testing, Kevin M. Vlcek
An Investigation Of The Afit 2-Inch Shock Tube As A Flow Source For Supersonic Testing, Kevin M. Vlcek
Theses and Dissertations
An investigation of the AFIT high pressure shock tube was conducted to determine how closely it followed ideal shock tube theory and to determine the available test times for an attached Mach 3 nozzle. The driver section was five feet (1.52 m) long and the driven section was 25 feet (7.62 m) long. The driver gas used for this study was helium while the driven gas was atmospheric air. The pressure rise measured behind the incident shock wave was, on average, 30% lower than predicted by the ideal shock tube relations. Behind the reflected shock, the pressure rise was 65% …
Flow Visualization Of A Turbulent Shear Flow Using An Optical Wavefront Sensor, Daniel W. Jewell
Flow Visualization Of A Turbulent Shear Flow Using An Optical Wavefront Sensor, Daniel W. Jewell
Theses and Dissertations
The research reported here investigated the use of a shearing interferometer (SI) wavefront sensor to determine the effects of shear-layer turbulence on an optical wavefront. A collimated helium-neon laser beam was propagated through a plane shear-layer produced by mixing helium and nitrogen at different velocities. Since the gases have different indices of refraction, the optical wavefront was distorted by different amounts by each gas. The SI measured the wavefront slope across the sampled area of the wavefront. The shear-layer was viewed from two orthogonal directions. This document contains shadow graphs, interference patterns imaged by each of the SI's six cameras, …
An Experimental Investigation Of The Time-Dependent Separation Of Tangent Bodies In Supersonic Flow, Neal A. Mosbarger
An Experimental Investigation Of The Time-Dependent Separation Of Tangent Bodies In Supersonic Flow, Neal A. Mosbarger
Theses and Dissertations
An experimental, time-dependent separation of tangent bodies was performed in a supersonic wind tunnel (Mach 1.5 and 1.9) to investigate the significance of transient effects and the suitability of using steady-state assumptions to predict a dynamic even. The model configurations consisted of two bodies placed in a near tangent position. A stationary body, plate or ogive, was instrumented to obtain dynamic surface pressures, while a second body, a wedge attached to an air cylinder, was plunged in a constrained motion away from and towards the stationary model. Three-dimensional flow expansion around the edge of the wedge reduced the strength of …
An Experimental Study Of Planar Heterogeneous Supersonic Confined Jets, Frederick J. Tanis Jr.
An Experimental Study Of Planar Heterogeneous Supersonic Confined Jets, Frederick J. Tanis Jr.
Theses and Dissertations
The effects of varying the exit pressure of a supersonic helium jet exhausting coaxially with two parallel supersonic air streams into a constant area duct were investigated. The method used to evaluate the mass entrainment rate was to measure helium molar concentration profiles and mass flux across the duct using a binary gas probe then calculate the mass entrainment into the helium jet. In order to conduct this study a novel binary gas probe was developed which allowed helium concentration and mass flux data to be obtained during continuous traverses across the supersonic flowfield. High exit pressure ratio (EPR) led …
Analysis Of Hypersonic Boundary Layer Second Mode Instability Over A 7° Cone, Richard A. Dennery
Analysis Of Hypersonic Boundary Layer Second Mode Instability Over A 7° Cone, Richard A. Dennery
Theses and Dissertations
This paper presents the results of the analysis of Mach 8.0 flow over a seven degree half-angle cone. The purpose of this analysis was to develop techniques to examine boundary layer transition at hypersonic velocities. The specific objectives were to look for second mode instability waves characteristic of the transition process and to quantify the percentage of turbulent flow. Two sets of data were used in this analysis. The first set of data was taken at several axial positions at a freestream Reynolds number 4.265 million per meter. This data was used to develop the analysis techniques. The second set …
Investigation Of Boundary Layer And Performance Effects Of Transpiration Cooling Through A Porous Plate In A Rocket Nozzle, David N. Keener
Investigation Of Boundary Layer And Performance Effects Of Transpiration Cooling Through A Porous Plate In A Rocket Nozzle, David N. Keener
Theses and Dissertations
This study used a range of low blowing ratios with air through an area of porous material in a Mach 2.0 nozzle to quantity the change in boundary layer thickness as a function of blowing ratio. Also, exit and wall Mach number profiles at each blowing ratio were collected to characterize performance losses as a result of blowing. Performance in terms of specific impulse was also studied. As expected, the boundary layer thickness increased and nozzle performance in terms of exit Mach number decreased with increasing blowing ratio.
Compressible Turbulence Measurements In A Supersonic Flow With Adverse Pressure Gradient, Jon W. Dotter
Compressible Turbulence Measurements In A Supersonic Flow With Adverse Pressure Gradient, Jon W. Dotter
Theses and Dissertations
Mean flow and compressible turbulence measurements have been obtained upstream and within a shock boundary interaction and a compression ramp in Mach 3 flow. Compressible turbulence models have met with little success in the accurate prediction of high-speed flows involving complicated shock boundary interactions and adverse pressure gradients because of a crucial lack of experimental data. Data were collected using conventional Pitot and cone static probes, single overheat cross-wire anemometry, multiple overheat cross-wire anemometry, and flow visualization techniques. Direct measurements of the total Reynolds shear stress were obtained using a turbulence transformation. Results indicate that compressibility effects, as evidenced by …
The Design And Development Of A Low-Speed, Low-Force Wind Tunnel Data Acquisition System With An Application To Bicycle Front Fork Spacing, Alan W. Elledge
The Design And Development Of A Low-Speed, Low-Force Wind Tunnel Data Acquisition System With An Application To Bicycle Front Fork Spacing, Alan W. Elledge
Theses and Dissertations
This thesis encompassed the design, development, and application of a low-speed, low-force data acquisition system for the AFIT 5-ft Wind Tunnel. The new data acquisition system was designed and developed to measure approximately 0.045 N (about 0.01 lbf) at nearly 64.4 km/h (40 mph). Verification of the capability was obtained through testing of a cylindrical model. As part of a technology transition program with Wright Laboratory, this new system was used to investigate bicycle front fork blade spacing in order to determine the optimal spacing for minimum drag. As for the minimum drag of a bicycle with an …
Compressible Turbulence Measurements In A Supersonic Boundary Layer Including Favorable Pressure Gradient Effects, Raymond S. Miller
Compressible Turbulence Measurements In A Supersonic Boundary Layer Including Favorable Pressure Gradient Effects, Raymond S. Miller
Theses and Dissertations
The effect of a favorable pressure gradient on the turbulent flow structure in a Mach 2.9 boundary layer (Re/m approx. 1.5 x 107) is investigated experimentally. Conventional flow and hot film measurements of turbulent fluctuation properties have been made upstream of and along an expansion ramp. Upstream measurements were taken in a zero pressure gradient boundary layer 44 cm from the nozzle throat in a 6.35 cm square test section. Measurements are obtained in the boundary layer, above the expansion ramp, 71.5 cm from the nozzle throat. Mean flow and turbulent flow characteristics are measured in …
A New Drag Measurement System For Wind Tunnel Testing Of The Racing Bicycle And Rider To Determine A Low Drag Configuration, Brian A. Parker
A New Drag Measurement System For Wind Tunnel Testing Of The Racing Bicycle And Rider To Determine A Low Drag Configuration, Brian A. Parker
Theses and Dissertations
This study investigated the application of splitter plate effects to reduce the aerodynamic drag of the racing bicycle and rider system. A sensitive, low-force, beam-type, single-component balance was developed to provide drag measurements accurate to within 0.053 N (0.012 lbf). The performance of the new system was verified by comparing the measured drag on a three-dimensioaal, right-circular cylinder model, 0.127 m (5.0 in) in diameter and 0.610 m (24 in) long, with the results from a commercial balance and a other similar data. The bicycle and rider model consisted of a full-scale mannequin comprising only the hips, legs …
Performance Characterization Of A Highly-Offset Diffuser With And Without Blowing Vortex Generator Jets, Michael B. Senseney
Performance Characterization Of A Highly-Offset Diffuser With And Without Blowing Vortex Generator Jets, Michael B. Senseney
Theses and Dissertations
The effect of blowing vortex generator jets (VGJs) on the performance of a highly-offset (s-duct) diffuser was investigated experimentally. VGJs are pitched, skewed jets which generate streamwise vortices as well as injecting high-momentum fluid into the boundary layer. Diffuser performance with and without VGJs was measured for an inlet Mach number of 0.6 (Re/x = 1.27x107 per cm). Pitot static and hot-film instrumentation was used to measure flow properties at the diffuser inlet and exit planes. Without blowing, the flow on the lower surface of the diffuser was massively separated. Blowing at 0.48% mass flow ratio through three lower-surface …
A Numerical Analysis Of Smoothed Particle Hydrodynamics, David A. Fulk
A Numerical Analysis Of Smoothed Particle Hydrodynamics, David A. Fulk
Theses and Dissertations
This dissertation studies the numerical method of Smoothed Particle Hydrodynamics SPH as a technique for solving systems of conservation equations. The research starts with a detailed consistency analysis of the method. Higher dimensions and non-smooth functions are considered in addition to the smooth one dimensional case. A stability analysis is then performed. Using a linear technique, an instability is found. Solutions are proposed to resolve the instability. Also a total variation stability analysis is performed leading to a monotone form of SPH. The concepts of consistency and stability are then used in a convergence proof. This proof uses lemmas derived …