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Transonic
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=== Mathematical analysis=== [[File:Streamline Patterns for Flow Regimes.png|thumb|Streamlines for three airflow regimes (black lines) around a nondescript blunt body (blue).<ref name=":13" />]] Prior to the advent of powerful computers, even the simplest forms of the [[Compressible flow|compressible flow equations]] were difficult to solve due to their [[Nonlinear system|nonlinearity]].<ref name=":23" /> A common assumption used to circumvent this nonlinearity is that disturbances within the flow are relatively small, which allows mathematicians and engineers to [[Linearization|linearize]] the compressible flow equations into a relatively easily solvable set of [[differential equation]]s for either wholly subsonic or supersonic flows.<ref name=":23" /> This assumption is fundamentally untrue for transonic flows because the disturbance caused by an object is much larger than in subsonic or supersonic flows; a flow speed close to or at Mach 1 does not allow the [[Streamlines, streaklines, and pathlines|streamtubes]] (3D flow paths) to contract enough around the object to minimize the disturbance, and thus the disturbance propagates.<ref name=":13">{{Cite book|last=Ramm|first=Heinrich J.|url=https://www.worldcat.org/oclc/228117297|title=Fluid dynamics for the study of transonic flow|date=1990|publisher=Oxford University Press|isbn=1-60129-748-3|location=New York|pages=|oclc=228117297}}</ref> Aerodynamicists struggled during the earlier studies of transonic flow because the then-current theory implied that these disturbances– and thus drag– approached infinity as local Mach number approached 1, an obviously unrealistic result which could not be remedied using known methods.<ref name=":23" /> One of the first methods used to circumvent the nonlinearity of transonic flow models was the [[hodograph]] transformation.<ref name=":0" /> This concept was originally explored in 1923 by an Italian mathematician named [[Francesco Tricomi]], who used the transformation to simplify the compressible flow equations and prove that they were solvable.<ref name=":0" /> The hodograph transformation itself was also explored by both [[Ludwig Prandtl]] and O.G. Tietjen's textbooks in 1929 and by [[Adolf Busemann]] in 1937, though neither applied this method specifically to transonic flow.<ref name=":0" /> Gottfried Guderley, a German mathematician and engineer at [[Technical University of Braunschweig|Braunschweig]], discovered Tricomi's work in the process of applying the hodograph method to transonic flow near the end of World War II.<ref name=":0" /> He focused on the nonlinear thin-airfoil compressible flow equations, the same as what Tricomi derived, though his goal of using these equations to solve flow over an airfoil presented unique challenges.<ref name=":0" /><ref name=":23" /> Guderley and Hideo Yoshihara, along with some input from Busemann, later used a singular solution of Tricomi's equations to analytically solve the behavior of transonic flow over a [[Supersonic airfoils|double wedge airfoil]], the first to do so with only the assumptions of thin-airfoil theory.<ref name=":0" /><ref name=":23" /> Although successful, Guderley's work was still focused on the theoretical, and only resulted in a single solution for a double wedge airfoil at Mach 1.<ref name=":0" /> [[Walter G. Vincenti|Walter Vincenti]], an American engineer at [[Ames Research Center|Ames Laboratory]], aimed to supplement Guderley's Mach 1 work with numerical solutions that would cover the range of transonic speeds between Mach 1 and wholly supersonic flow.<ref name=":0" /> Vincenti and his assistants drew upon the work of [[Howard Wilson Emmons|Howard Emmons]] as well as Tricomi's original equations to complete a set of four numerical solutions for the drag over a double wedge airfoil in transonic flow above Mach 1.<ref name=":0" /> The gap between subsonic and Mach 1 flow was later covered by both [[Julian Cole]] and [[Leon Trilling]], completing the transonic behavior of the airfoil by the early 1950s.<ref name=":0" />
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