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Sears–Haack Body
The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a given body length and given volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl–Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947. The theory indicates that the wave drag scales as the square of the second derivative of the area distribution, D_\text \sim S''(x)2 (see full expression below), so for low wave drag it is necessary that S(x) be smooth. Thus, the Sears–Haack body is pointed at each end and grows smoothly to a maximum and then decreases smoothly toward the second point. Useful formulas The cross-sectional area of a Sears–Haack body is : S(x) = \frac x(1-x) = \pi R_\text^2 x(1-x), its volume is : V = \frac R_\text^2 L, its radius is : r(x) = R_\text x(1-x), the derivative (sl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Drag
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a ''traveling wave''; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a '' standing wave''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. Waves are often described by a ''wave equation'' (standing wave field of two opposite waves) or a one-way wave equation for single wave propagation in a defined direction. Two types of waves are most commonly studied in classical physics. In a ''mechanical wave'', stress and strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (strain) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prandtl–Glauert Transformation
The Prandtl–Glauert transformation is a mathematical technique which allows solving certain compressible flow problems by incompressible-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases. Mathematical formulation Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation: :\phi_ + \phi_ + \phi_ = M_\infty^2 \phi_ \quad \mbox together with the small-disturbance flow-tangency boundary condition. :V_\infty n_x + \phi_y n_y + \phi_z n_z = 0 \quad \mbox M_\infty is the freestream Mach number, and n_x, n_y, n_z are the surface-normal vector components. The unknown variable is the perturbation potential \phi(x,y,z), and the total velocity is given by its gradient plus the freestream velocity V_\infty which is assumed here to be along x. :\vec = \nabla \phi + V_\infty \hat = (V_\infty + \phi_x) \hat + \phi_y \hat + \phi_z \hat The above formulation is valid only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Haack
Wolfgang Siegfried Haack (24 April 1902 – 28 November 1994) was a German mathematician and aerodynamicist. He in 1941 and William Sears in 1947 independently discovered the Sears–Haack body. Life Wolfgang Haack studied mechanical engineering at the Leibniz University Hannover and mathematics in Jena. He earned his doctorate in 1926 at the Friedrich Schiller University in Jena. After a short study and research period in Hamburg and a job as an assistant at the Technical University of Stuttgart he habilitated in 1929 at the TH Danzig (now Gdańsk). In 1935 he moved to the TH Berlin and in 1937, he followed the call to the TH Karlsruhe. During the Second World War he worked on projectile design. Although the TH Berlin did invite him to work there in 1944, Wolfgang Haack was unable to take up the post because of the war. In 1949 he became the successor to Georg Hamel as Professor of Mathematics and Mechanics at the TU Berlin Department of Mathematics and Mechanics. O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William R
William is a male given name of Germanic origin.Hanks, Hardcastle and Hodges, ''Oxford Dictionary of First Names'', Oxford University Press, 2nd edition, , p. 276. It became very popular in the English language after the Norman conquest of England in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will, Wills, Willy, Willie, Bill, and Billy. A common Irish form is Liam. Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie or the play ''Douglas''). Female forms are Willa, Willemina, Wilma and Wilhelmina. Etymology William is related to the given name ''Wilhelm'' (cf. Proto-Germanic ᚹᛁᛚᛃᚨᚺᛖᛚᛗᚨᛉ, ''*Wiljahelmaz'' > German ''Wilhelm'' and Old Norse ᚢᛁᛚᛋᛅᚼᛅᛚᛘᛅᛋ, ''Vilhjálmr''). By regular sound changes, the native, inherited English form of the name sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slender-body Theory
In fluid dynamics and electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body. Principal applications are to Stokes flow — at very low Reynolds numbers — and in electrostatics. Theory for Stokes flow Consider slender body of length \ell and typical diameter 2a with \ell \gg a, surrounded by fluid of viscosity \mu whose motion is governed by the Stokes flow, Stokes equations. Note that the Stokes' paradox implies that the limit of infinite aspect ratio \ell/a \rightarrow \infty is singular, as no Stokes flow can exist around an infinite cylinder. Slender-body theory allows us to derive an approximate relationship between the velocity of the body at each point along its length and the force per unit length experienced by the body at that point. Let the axis of the body be described by \boldsymbol(s,t), where s is an ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drag Coefficient
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag. The drag coefficient of a lifting airfoil or hydrofoil also includes the effects of lift-induced drag. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag. Definition The drag coefficient c_\mathrm d is defined as c_\mathrm d = \dfrac where: * F_\mathrm d is the drag force, which is by definition the force component in the direction of the flow velo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Thomas Jones (engineer)
Robert T. Jones, (May 28, 1910 – August 11, 1999), was an aerodynamicist and aeronautical engineer for NACA and later NASA. He was known at NASA as "one of the premier aeronautical engineers of the twentieth century". Designer One of Jones' first jobs was with the Nicholas-Beazley Airplane Company. Jones developed the Pobjoy Special air racer prior to the company shutting down in the depression. Research Jones was a researcher at NACA's Langley Research Center in Hampton, Virginia. As a self-trained aerodynamicist and mathematician, he had built up a national, if not international, reputation through his perceptive and original work at Langley. For this work he was given the IAS Sylvanus Albert Reed Award in 1946. Jones spent much of his time at Langley working in the Stability Research Division which pioneered many concepts that were incorporated into U.S. aircraft. In January 1945, Jones developed a theory of the delta wing based on thin-airfoil theory. Others at Lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mach Cone
In fluid dynamics, a Mach wave is a pressure wave traveling with the speed of sound caused by a slight change of pressure added to a compressible flow. These weak waves can combine in supersonic flow to become a shock wave if sufficient Mach waves are present at any location. Such a shock wave is called a Mach stem or Mach front. Thus, it is possible to have shockless compression or expansion in a supersonic flow by having the production of Mach waves sufficiently spaced (''cf.'' isentropic compression in supersonic flows). A Mach wave is the weak limit of an oblique shock wave where time averages of flow quantities don't change; (a normal shock is the other limit). If the size of the object moving at the speed of sound is near 0, then this domain of influence of the wave is called a Mach cone. Mach angle A Mach wave propagates across the flow at the Mach angle ''μ'', which is the angle formed between the Mach wave wavefront and a vector that points opposite to the vector of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area Rule
The Whitcomb area rule, named after NACA engineer Richard Whitcomb and also called the transonic area rule, is a design procedure used to reduce an aircraft's drag at transonic speeds which occur between about Mach 0.75 and 1.2. For supersonic speeds a different procedure called the supersonic area rule, developed by NACA aerodynamicist Robert Jones, is used. Transonic is one of the most important speed ranges for commercial and military fixed-wing aircraft today, with transonic acceleration an important performance requirement for combat aircraft and which is improved by reductions in transonic drag. Description At high-subsonic flight speeds, the local speed of the airflow can reach the speed of sound where the flow accelerates around the aircraft body and wings. The speed at which this development occurs varies from aircraft to aircraft and is known as the critical Mach number. The resulting shock waves formed at these zones of sonic flow cause a sudden increase in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area Rule
The Whitcomb area rule, named after NACA engineer Richard Whitcomb and also called the transonic area rule, is a design procedure used to reduce an aircraft's drag at transonic speeds which occur between about Mach 0.75 and 1.2. For supersonic speeds a different procedure called the supersonic area rule, developed by NACA aerodynamicist Robert Jones, is used. Transonic is one of the most important speed ranges for commercial and military fixed-wing aircraft today, with transonic acceleration an important performance requirement for combat aircraft and which is improved by reductions in transonic drag. Description At high-subsonic flight speeds, the local speed of the airflow can reach the speed of sound where the flow accelerates around the aircraft body and wings. The speed at which this development occurs varies from aircraft to aircraft and is known as the critical Mach number. The resulting shock waves formed at these zones of sonic flow cause a sudden increase in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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