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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dynamic similarity (Reynolds and Womersley numbers) ： ウィキペディア英語版 Dynamic similarity (Reynolds and Womersley numbers) In fluid mechanics, dynamic similarity refers to the phenomenon that when there are two geometrically similar vessels (same shape, different sizes) with the same boundary conditions (ex. No-slip, center-line velocity) and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.〔Jones, Robert T. "Blood Flow," ''Annual Review of Fluid Mechanics'', 1(1969)223:244.〕 ==Derivation== Reynolds number and Womersley number are the only two physical parameters necessary to solve an incompressible fluid flow problem. Reynolds Number is given by: $N\_R\; =\; \backslash tfrac\backslash ,\backslash !$ The terms of the equation itself represent the following: :$N\_R\; =\; \}\backslash ,\backslash !$. When Reynolds Number is large, it shows that the flow is dominated by convective inertial effects; When Reynolds Number is small, it shows that the flow is dominated by shear effects. Womersley Number is given by: :$N\_W\; =\; L\backslash sqrt\}=\backslash sqrt\backslash ,\backslash !$, which is simply the square-root of Stokes Number, the terms of the equation itself represents the following: :$N\_W\; =\; \}\backslash ,\backslash !$. When Womersley Number is large(around 10 or greater), it shows that the flow is dominated by oscillatory inertial forces and that the velocity profile is flat. When the Womersley parameter is low, viscous forces tend to dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with the driving pressure gradient.〔Ku, David N. "Blood Flow in Arteries," ''Annual Review of Fluid Mechanics'', 1(1969)223:44.〕 Starting with Navier–Stokes equation for Cartesian flow: : $\backslash left\; (\backslash fracu\}t\}+u\backslash fracu\}x\}+v\backslash fracu\}y\}+w\backslash fracu\}z\}\backslash right\; )=\; g-\backslash fracP\}x\}+\backslash left(\; \backslash fracu\}x^2\}+\backslash fracv\}y^2\}+\backslash fracw\}z^2\}\backslash right)\backslash ,\backslash !$. The terms of the equation itself represent the following: : $\backslash text=\backslash text\backslash ,\backslash !$ 〔Fung, Yuan-cheng. "Biomechanics: Circulation," ''Dynamic Similarity'', "New York: Springer", 2(2008)130:134.〕 Ignoring gravitational forces and dividing the equation by density ($\backslash rho$) yields: : $\backslash left\; (\backslash fracu\}t\}+u\backslash fracu\}x\}+v\backslash fracu\}y\}+w\backslash fracu\}z\}\backslash right\; )=\; -\backslash frac\backslash fracP\}x\}+\backslash left(\; \backslash fracu\}x^2\}+\backslash fracv\}y^2\}+\backslash fracw\}z^2\}\backslash right)\backslash ,\backslash !$, where $\backslash nu=/$ is the kinematic viscosity. Since both Reynolds and Womersley numbers are dimensionless, Navier-Stokes must be represented as a dimensionless expression as well. Choosing $V$, $\backslash omega$, and $L$ as a characteristic velocity, frequency, and length respectively yields dimensionless variables: Dimensionless Length Term (same for y' and z'):$x\text{'}\; =\; \backslash ,\backslash !$, Dimensionless Velocity Term (same for v' and w'): $u\text{'}\; =\; \backslash ,\backslash !$, Dimensionless Pressure Term: $P\text{'}\; =\; \}\backslash ,\backslash !$, Dimensionless Time Term: $t\text{'}\; =\; t\backslash ,\backslash !$. Dividing the Navier-Stokes equation by $\backslash tfrac\backslash ,\backslash !$ (Convective Inertial Force term) gives: : $\backslash frac^2\}\; \backslash left\; (\backslash fracu\text{'}\}t\text{'}\}\backslash right)+\backslash left\; (u\text{'}\backslash fracu\text{'}\}x\text{'}\}+v\text{'}\backslash fracu\text{'}\}y\text{'}\}+w\text{'}\backslash fracu\text{'}\}z\text{'}\}\backslash right\; )=\; -\backslash fracP\text{'}\}x\text{'}\}+\backslash frac\backslash left(\; \backslash fracu\text{'}\}x\text{'}^2\}+\backslash fracv\text{'}\}y\text{'}^2\}+\backslash fracw\text{'}\}z\text{'}^2\}\backslash right)\backslash ,\backslash !$, With the addition of the dimensionless continuity equation (seen below) in any incompressible fluid flow problem the Reynolds and Womersley Numbers are the only two physical parameters that are in the two equations :$\backslash fracu\text{'}\}x\text{'}\}+\backslash fracv\text{'}\}y\text{'}\}+\backslash fracw\text{'}\}z\text{'}\}=0\; \backslash ,\backslash !$, 〔van de Vosse, Frans M. "Pulse Wave Propagation in the Arterial Tree.," ''Annual Review of Fluid Mechanics'', 43(2011)467:499.〕 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「Dynamic similarity (Reynolds and Womersley numbers)」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.