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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 Reynolds-averaged Navier–Stokes equations ： ウィキペディア英語版 Reynolds-averaged Navier–Stokes equations The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged〔 The true time average ($\backslash bar$) of a variable ($x$) is defined by : $\backslash bar\; =\; \backslash lim\_\backslash frac\backslash int\_^\; x\backslash ,\; dt.$ For this to be a well-defined term, the limit ($\backslash bar$) must be independent of the initial condition at $t\_0$. In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations. However, assuming the limit exists (which it does for any bounded system, which fluid velocities certainly are), there exists some $T$ such that integration from $t\_0$ to $T$ is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on $T$.〕 equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds.〔 Reynolds, Osborne, 1895: "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion." ''Philosophical Transactions of the Royal Society of London. A'', v. 186, pp. 123-164. Available online from (JSTOR ).〕 The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations. For a stationary, incompressible Newtonian fluid, these equations can be written in Einstein notation as: : $\backslash rho\backslash bar\_j\; \backslash frac$ = \rho \bar_i + \frac \left(- \bar\delta_ + \mu \left( \frac + \frac \right) - \rho \overline \right ). The left hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress $\backslash left(\; -\; \backslash rho\; \backslash overline\; \backslash right)$ owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models. The time-average operator $\backslash overline$ is a Reynolds operator. == Derivation of RANS equations == The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity $u$) into the mean (time-averaged) component ($\backslash overline$) and the fluctuating component ($u^$). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity being equal to zero $(\backslash bar\; =\; 0)$. Thus, :$u(\backslash boldsymbol,t)\; =\; \backslash bar(\backslash boldsymbol)\; +\; u^\backslash prime(\backslash boldsymbol,t)\; \backslash ,$, where $\backslash boldsymbol\; =\; (x,y,z)$ is the position vector. Some authors prefer using $U$ instead of $\backslash bar$ for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term $u^\backslash prime$ is represented instead by $u$. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation $u,\; \backslash bar,\; \backslash mbox\; u^$ will be used to represent the instantaneous, mean, and fluctuating terms, respectively. The properties of Reynolds operators are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid): : $\backslash frac\; =\; 0$ : $\backslash frac\; +\; u\_j\; \backslash frac$ = f_i - \frac \frac + \nu \frac where $f\_i$ is a vector representing external forces. Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, 〔 Splitting each instantaneous quantity into its averaged and fluctuating components yields, : $\backslash frac\; =\; 0$ : $\backslash frac$ + \left( \bar + u_j^\prime\right) \frac = \left( \bar + f_i^\prime\right) - \frac \frac + \nu \frac. Time-averaging these equations yields, : $\backslash overline\}\; =\; 0$ : $\backslash overline\}$ + \overline + u_i^\prime\right)}} = \overline - \frac \overline} + \nu \overline}. Note that the nonlinear terms (like $\backslash overline$) can be simplified to, $\backslash overline$ = \overline + u_i^\prime \right) } = \overline + \baru_i^\prime + u_i^\prime\bar + u_i^\prime u_i^\prime} = \bar\bar + \overline 〕 to yield: : $\backslash frac\; =\; 0$ : $\backslash frac$ + \bar\frac + \overline} = \bar - \frac\frac + \nu \frac. The momentum equation can also be written as,〔 This follows from the mass conservation equation which gives, : $\backslash frac\; =\; \backslash frac\; +\; \backslash frac\; =\; 0$ 〕 : $\backslash frac$ + \bar\frac = \bar - \frac\frac + \nu \frac - \frac. On further manipulations this yields, : $\backslash rho\; \backslash frac$ + \rho \bar \frac = \rho \bar + \frac \left(- \bar\delta_ + 2\mu \bar \right ) where, $$ \bar\left( \frac + \frac \right) is the mean rate of strain tensor. Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving: : $\backslash rho\; \backslash bar\backslash frac$ = \rho \bar + \frac \left(- \bar\delta_ + 2\mu \bar \right ). 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「Reynolds-averaged Navier–Stokes equations」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.