翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Prandtl-Meyer expansion fan ： ウィキペディア英語版 Prandtl–Meyer expansion fan A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, is a centred expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. In case of a smooth, circular corner, these waves can be extended backwards to meet at a point. Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave because this would violate the second law of thermodynamics.〔 Impossibility of expanding a flow through a single "shock" wave: Consider the scenario shown in the adjacent figure. As a supersonic flow turns, the normal component of the velocity increases ($w\_2\; >\; w\_1$), while the tangential component remains constant ($v\_2\; =\; v\_1$). The corresponding change is the entropy ($\backslash Delta\; s\; =\; s\_2-s\_1$) can be expressed as follows, :$\backslash begin\; \backslash frac$ & = ln \bigg(\bigg( \frac \bigg)^ \bigg( \frac \bigg)^ \bigg ) \\ & \approx \frac \bigg( \frac \bigg)^3 \\ & \approx \frac \bigg(\frac \bigg( 1-\frac\bigg) \bigg )^3 \end where, $R$ is the universal gas constant, $\backslash gamma$ is the ratio of specific heat capacities, $\backslash rho$ is the static density, $p$ is the static pressure, $s$ is the entropy, and $w$ is the component of flow velocity normal to the "shock". The suffix "1" and "2" refer to the initial and final conditions respectively. Since $w\_2\; >\; w\_1$, this would mean that . Since this is not possible it means that it is impossible to turn a flow through a single shock wave. The argument may be further extended to show that such an expansion process can occur only if we consider a turn through infinite number of expansion waves in the limit $\backslash Delta\; s\; \backslash rightarrow\; 0$. Accordingly an expansion process is an isentropic process. 〕 Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. total pressure and total temperature) remain constant across the fan. == Flow properties == The expansion fan consists of infinite number of expansion waves or Mach lines.〔 Mach lines (cone) and Mach angle: Mach lines are a concept usually encountered in 2-D supersonic flows (i.e. $M\; \backslash ge\; 1$). They are a pair of bounding lines which separate the region of disturbed flow from the undisturbed part of the flow. These lines occur in pair and are oriented at an angle :$\backslash textstyle\; \backslash mu\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\; \backslash right)\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\; \backslash right)$ with respect to the direction of motion (also known as the Mach angle). In case of 3-D flow field these lines form a surface known as Mach cone, with Mach angle as the half angle of the cone. To understand the concept better, consider the case sketched in the figure. We know that when an object moves in a flow, it causes pressure disturbances (which travel at the speed of sound, also known as Mach waves). The figure shows an object moving from point A to B along the line AB at supersonic speeds ($u\; >\; c$). By the time the object reaches point B, the pressure disturbances from point A have travelled a distance c·t and are now at circumference of the circle (with centre at point A). There are infinite such circles with their centre on the line AB, each representing the location of the disturbances due to the motion of the object. The lines propagating outwards from point B and tangent to all these circles are known as Mach lines. ''Note:'' These concepts have a physical meaning only for supersonic flows ($u\; \backslash ge\; c$). In case of subsonic flows the disturbances will travel faster than the source and the argument of the $\backslash arcsin(\; )$ function will be greater than one. 〕 The first Mach line is at an angle $\backslash mu\_1\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\; \backslash right)$ with respect to the flow direction, and the last Mach line is at an angle $\backslash mu\_2\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\; \backslash right)$ with respect to final flow direction. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic.〔 This simplifies the calculations of the flow properties significantly. Since the flow is isentropic, the stagnation properties like stagnation pressure ($p\_0$), stagnation temperature ($T\_0$) and stagnation density ($\backslash rho\_0$) remain constant. The final static properties are a function of the final flow Mach number ($M\_2$) and can be related to the initial flow conditions as follows, :$\backslash begin$ \frac &= & \bigg( \fracM_1^2}M_2^2} \bigg) \\ \frac &= & \bigg( \fracM_1^2}M_2^2} \bigg)^ \\ \frac &= &\bigg( \fracM_1^2}M_2^2} \bigg)^. \end The Mach number after the turn ($M\_2$) is related to the initial Mach number ($M\_1$) and the turn angle ($\backslash theta$) by, :$\backslash theta\; =\; \backslash nu(M\_2)\; -\; \backslash nu(M\_1)\; \backslash ,$ where, $\backslash nu(M)\; \backslash ,$ is the Prandtl–Meyer function. This function determines the angle through which a sonic flow (M = 1) must turn to reach a particular Mach number (M). Mathematically, : $\backslash begin\; \backslash nu(M)$ & = \int \fracM^2}\frac \\ & = \sqrt} \cdot \arctan \sqrt (M^2 -1)} - \arctan \sqrt. \\ \end By convention, $\backslash nu(1)\; =\; 0.\; \backslash ,$ Thus, given the initial Mach number ($M\_1$), one can calculate $\backslash nu(M\_1)\; \backslash ,$ and using the turn angle find $\backslash nu(M\_2)\; \backslash ,$. From the value of $\backslash nu(M\_2)\; \backslash ,$ one can obtain the final Mach number ($M\_2$) and the other flow properties. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Prandtl–Meyer expansion fan」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.