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diff --git a/content/know/concept/prandtl-equations/index.pdc b/content/know/concept/prandtl-equations/index.pdc new file mode 100644 index 0000000..00f7773 --- /dev/null +++ b/content/know/concept/prandtl-equations/index.pdc @@ -0,0 +1,211 @@ +--- +title: "Prandtl equations" +firstLetter: "P" +publishDate: 2021-05-29 +categories: +- Physics +- Fluid mechanics +- Fluid dynamics + +date: 2021-05-10T18:41:20+02:00 +draft: false +markup: pandoc +--- + +# Prandtl equations + +In fluid dynamics, the **Prandtl equations** or **boundary layer equations** +describe the movement of a [viscous](/know/concept/viscosity/) fluid +with a large [Reynolds number](/know/concept/reynolds-number/) $\mathrm{Re} \gg 1$ +close to a solid surface. + +Fluids with a large Reynolds number +are often approximated as having zero viscosity, +since the simpler [Euler equations](/know/concept/euler-equations) +can then be used instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/). + +However, in reality, a viscous fluid obeys the *no-slip* boundary condition: +at every solid surface the local velocity must be zero. +This implies the existence of a **boundary layer**: +a thin layer of fluid "stuck" to solid objects in the flow, +where viscosity plays an important role. +This is in contrast to the ideal flow far away from the surface. + +We consider a simple theoretical case in 2D: +a large flat surface located at $y = 0$ for all $x \in \mathbb{R}$, +with a fluid *trying* to flow parallel to it at $U$. +The 2D treatment can be justified by assuming that everything is constant in the $z$-direction. +We will not solve this case, +but instead derive general equations +to describe the flow close to a flat surface. + +At the wall, there is a very thin boundary layer of thickness $\delta$, +where the fluid is assumed to be completely stationary $\va{v} = 0$. +We are mainly interested in the region $\delta < y \ll L$, +where $L$ is the distance at which the fluid becomes practically ideal. +This the so-called **slip-flow** region, +in which the fluid is not stationary, +but still viscosity-dominated. + +In 2D, the steady Navier-Stokes equations are as follows, +where the flow $\va{v} = (v_x, v_y)$: + +$$\begin{aligned} + v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} + &= - \frac{1}{\rho} \pdv{p}{x} + \nu \Big( \pdv[2]{v_x}{x} + \pdv[2]{v_x}{y} \Big) + \\ + v_x \pdv{v_y}{x} + v_y \pdv{v_y}{y} + &= - \frac{1}{\rho} \pdv{p}{y} + \nu \Big( \pdv[2]{v_y}{x} + \pdv[2]{v_y}{y} \Big) + \\ + \pdv{v_x}{x} + \pdv{v_y}{y} + &= 0 +\end{aligned}$$ + +The latter represents the fluid's incompressibility. +We non-dimensionalize these equations, +and assume that changes along the $y$-axis +happen on a short scale (say, $\delta$), +and along the $x$-axis on a longer scale (say, $L$). +Let $\tilde{x}$ and $\tilde{y}$ be dimenionless variables of order $1$: + +$$\begin{aligned} + x + = L \tilde{x} + \qquad \quad + y + = \delta \tilde{x} + \qquad \quad + \pdv{x} + = \frac{1}{L} \pdv{\tilde{x}} + \qquad \quad + \pdv{y} + = \frac{1}{\delta} \pdv{\tilde{y}} +\end{aligned}$$ + +Furthermore, we choose velocity scales +to be consistent with the incompressibility condition, +and a pressure scale inspired +by [Bernoulli's theorem](/know/concept/bernoullis-theorem/): + +$$\begin{aligned} + v_x + = U \tilde{v}_x + \qquad \quad + v_y + = \frac{U \delta}{L} \tilde{v}_y + \qquad \quad + p + = \rho U^2 \tilde{p} +\end{aligned}$$ + +We insert these scalings into the Navier-Stokes equations, yielding: + +$$\begin{aligned} + \frac{U^2}{L} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \frac{U^2}{L} \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} + &= - \frac{U^2}{L} \pdv{\tilde{p}}{\tilde{x}} + + \nu \Big( \frac{U}{L^2} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{U}{\delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) + \\ + \frac{U^2 \delta}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{U^2 \delta}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} + &= - \frac{U^2}{\delta} \pdv{\tilde{p}}{\tilde{y}} + + \nu \Big( \frac{U \delta}{L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{U}{L \delta} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) +\end{aligned}$$ + +For future convenience, +we multiply the former equation by $L / U^2$, and the latter by $\delta / U^2$: + +$$\begin{aligned} + \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{x}} + + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{L}{U \delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) + \\ + \frac{\delta^2}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\delta^2}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{y}} + + \nu \Big( \frac{\delta^2}{U L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) +\end{aligned}$$ + +We would like to estimate $\delta$. +Intuitively, we expect that higher viscosities $\nu$ give thicker layers, +and that faster velocities $U$ give thinner layers. +Furthermore, we expect *downstream thickening*: +with distance $x$, viscous stresses slow down the slip-flow, +leading to a gradual increase of $\delta(x)$. +Some dimensional analysis thus yields the following estimate: + +$$\begin{aligned} + \delta + \approx \sqrt{\frac{\nu x}{U}} + \sim \sqrt{\frac{\nu L}{U}} +\end{aligned}$$ + +We thus insert $\delta = \sqrt{\nu L / U}$ into the Navier-Stokes equations, giving us: + +$$\begin{aligned} + \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{x}} + + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{1}{\nu} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) + \\ + \frac{\nu}{U L} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\nu}{U L} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{y}} + + \nu \Big( \frac{\nu}{U^2 L^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) +\end{aligned}$$ + +Here, we recognize the definition of the Reynolds number $\mathrm{Re} = U L / \nu$: + +$$\begin{aligned} + \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{x}} + + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \pdv[2]{\tilde{v}_x}{\tilde{y}} + \\ + \frac{1}{\mathrm{Re}} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} + &= - \pdv{\tilde{p}}{\tilde{y}} + + \frac{1}{\mathrm{Re}^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_y}{\tilde{y}} +\end{aligned}$$ + +Recall that we are only considering large Reynolds numbers $\mathrm{Re} \gg 1$, +in which case $\mathrm{Re}^{-1} \ll 1$, +so we can drop many terms, leaving us with these redimensionalized equations: + +$$\begin{aligned} + v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} + = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdv[2]{v_x}{y} + \qquad \quad + \pdv{p}{y} + = 0 +\end{aligned}$$ + +The second one tells us that for a given $x$-value, +the pressure is the same at the surface +as in the main flow $y > L$, where the fluid is ideal. +In the latter regime, we apply Bernoulli's theorem to rewrite $p$, +using the *Bernoulli head* $H$ and the mainstream velocity $U(x)$: + +$$\begin{aligned} + p + = \rho H - \frac{1}{2} \rho U^2 + = p_0 - \frac{1}{2} \rho U^2 +\end{aligned}$$ + +Inserting this into the reduced Navier-Stokes equations, +we arrive at the Prandtl equations: + +$$\begin{aligned} + \boxed{ + v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} + = U \dv{U}{x} + \nu \pdv[2]{v_x}{y} + \qquad \quad + \pdv{v_x}{x} + \pdv{v_y}{y} + = 0 + } +\end{aligned}$$ + +A notable application of these equations is +the [Blasius boundary layer](/know/concept/blasius-boundary-layer/), +where the surface in question +is a semi-infinite plane. + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. |