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Diffstat (limited to 'source/know/concept/prandtl-equations')
| -rw-r--r-- | source/know/concept/prandtl-equations/index.md | 58 |
1 files changed, 28 insertions, 30 deletions
diff --git a/source/know/concept/prandtl-equations/index.md b/source/know/concept/prandtl-equations/index.md index d82657c..8f5c4d0 100644 --- a/source/know/concept/prandtl-equations/index.md +++ b/source/know/concept/prandtl-equations/index.md @@ -11,13 +11,12 @@ layout: "concept" 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. +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/). +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. @@ -29,9 +28,9 @@ 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 +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 $$\delta$$, @@ -39,8 +38,7 @@ 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 which the fluid is not stationary, but still viscosity-dominated. In 2D, the steady Navier-Stokes equations are as follows, where the flow $$\va{v} = (v_x, v_y)$$: @@ -66,13 +64,13 @@ Let $$\tilde{x}$$ and $$\tilde{y}$$ be dimenionless variables of order $$1$$: $$\begin{aligned} x = L \tilde{x} - \qquad \quad + \qquad \qquad y = \delta \tilde{x} - \qquad \quad + \qquad \qquad \pdv{}{x} = \frac{1}{L} \pdv{}{\tilde{x}} - \qquad \quad + \qquad \qquad \pdv{}{y} = \frac{1}{\delta} \pdv{}{\tilde{y}} \end{aligned}$$ @@ -85,10 +83,10 @@ by [Bernoulli's theorem](/know/concept/bernoullis-theorem/): $$\begin{aligned} v_x = U \tilde{v}_x - \qquad \quad + \qquad \qquad v_y = \frac{U \delta}{L} \tilde{v}_y - \qquad \quad + \qquad \qquad p = \rho U^2 \tilde{p} \end{aligned}$$ @@ -124,15 +122,15 @@ 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: +Some dimensional analysis yields the following estimate: $$\begin{aligned} \delta \approx \sqrt{\frac{\nu x}{U}} - \sim \sqrt{\frac{\nu L}{U}} + \approx \sqrt{\frac{\nu L}{U}} \end{aligned}$$ -We thus insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us: +So we insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us: $$\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} @@ -163,15 +161,14 @@ so we can drop many terms, leaving us with these redimensionalized equations: $$\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdvn{2}{v_x}{y} - \qquad \quad + \qquad \qquad \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$$, +the pressure is the same at the surface as in the ideal main flow $$y > L$$. +In the latter regime, we can use Bernoulli's theorem to rewrite $$p$$, using the *Bernoulli head* $$H$$ and the mainstream velocity $$U(x)$$: $$\begin{aligned} @@ -185,18 +182,19 @@ we arrive at the Prandtl equations: $$\begin{aligned} \boxed{ - v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} - = U \dv{U}{x} + \nu \pdvn{2}{v_x}{y} - \qquad \quad - \pdv{v_x}{x} + \pdv{v_y}{y} - = 0 + \begin{aligned} + v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} + &= U \dv{U}{x} + \nu \pdvn{2}{v_x}{y} + \\ + \pdv{v_x}{x} + \pdv{v_y}{y} + &= 0 + \end{aligned} } \end{aligned}$$ 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. +where the surface in question is a semi-infinite plane. |