Landau–Squire jet

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File:Landau010.svg

File:Landau100.svg

The problem is described in spherical coordinates $(r,\backslash theta,\backslash phi)$ with velocity components $(u,v,0)$. The flow is axisymmetric, i.e., independent of $\backslash phi$. Then the continuity equation and the incompressible Navier–Stokes equations reduce to

:$$

\begin{align}

& \frac{1}{r^2} \frac{\partial }{\partial r}(r^2u) + \frac{1}{r\sin\theta}\frac{\partial }{\partial \theta}(v\sin\theta) = 0 \\[8pt]

& u\frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} - \frac{v^2}{r}= - \frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left(\nabla^2 u - \frac{2u}{r^2} - \frac{2}{r^2} \frac{\partial v}{\partial \theta} - \frac{2v\cot\theta}{r^2} \right) \\[8pt]

& u\frac{\partial v}{\partial r} + \frac{v}{r}\frac{\partial v}{\partial \theta} + \frac{uv}{r} = -\frac{1}{\rho r} \frac{\partial p}{\partial \theta} + \nu \left(\nabla^2 v + \frac{2}{r^2}\frac{\partial u}{\partial \theta} - \frac{v}{r^2\sin^2\theta}\right)

\end{align}

where

:$\backslash nabla^2\; =\; \backslash frac\{1\}\{r^2\}\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash left(r^2\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}\backslash right)\; +\; \backslash frac\{1\}\{r^2\backslash sin\backslash theta\}\backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash theta\}\backslash left(\backslash sin\backslash theta\backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash theta\}\backslash right).$

A self-similar description is available for the solution in the following form,Sedov, L. I. (1993). Similarity and dimensional methods in mechanics. CRC press.

:$u\; =\; \backslash frac\{\backslash nu\}\{r\backslash sin\backslash theta\}\; f\text{'}(\backslash theta),\backslash quad\; v\; =\; -\backslash frac\{\backslash nu\}\{r\backslash sin\backslash theta\}\; f(\backslash theta).$

Substituting the above self-similar form into the governing equations and using the boundary conditions $u=v=p-p\_\backslash infty=0$ at infinity, one finds the form for pressure as

:$\backslash frac\{p-p\_\backslash infty\}\{\backslash rho\}\; =\; -\backslash frac\{v^2\}\{2\}\; +\; \backslash frac\{\backslash nu\; u\}\{r\}\; +\; \backslash frac\{c\_1\}\{r^2\}$

where $c\_1$ is a constant. Using this pressure, we find again from the momentum equation,

:$-\backslash frac\{u^2\}\{r\}\; +\; \backslash frac\{v\}\{r\}\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash theta\}\; =\; \backslash frac\{\backslash nu\}\{r^2\}\; \backslash left[2u\; +\; \backslash frac\{1\}\{\backslash sin\backslash theta\}\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash theta\}\; \backslash left(\backslash sin\backslash theta\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash theta\}\backslash right)\backslash right]\; +\; \backslash frac\{2c\_1\}\{r^3\}.$

Replacing $\backslash theta$ by $\backslash mu=\backslash cos\backslash theta$ as independent variable, the velocities become

:$u\; =\; -\backslash frac\{\backslash nu\}\{r\}\; f\text{'}(\backslash mu),\backslash quad\; v\; =\; -\backslash frac\{\backslash nu\}\{r\}\; \backslash frac\{f(\backslash mu)\}\{\backslash sqrt\{1-\backslash mu^2\}\}$

(for brevity, the same symbol is used for $f(\backslash theta)$ and $f(\backslash mu)$ even though they are functionally the same, but takes different numerical values) and the equation becomes

:$f\text{'}^2\; +\; ff$ = 2f' + [(1-\mu^2)f]' - 2c_1.

After two integrations, the equation reduces to

:$f^2\; =\; 4\backslash mu\; f\; +\; 2(1-\backslash mu^2)\; f\text{'}\; -\; 2(c\_1\backslash mu^2\; +\; c\_2\; \backslash mu\; +\; c\_3),$

where $c\_2$ and $c\_3$ are constants of integration. The above equation is a Riccati equation. After some calculation, the general solution can be shown to be

:$f\; =\; \backslash alpha(1+\backslash mu)\; +\; \backslash beta(1-\backslash mu)\; +\; \backslash frac\{2(1-\backslash mu^2)(1+\backslash mu)^\backslash beta\}\{(1-\backslash mu)^\backslash alpha\}\backslash left[c-\; \backslash int\_1^\backslash mu\; \backslash frac\{(1+\backslash mu)^\backslash beta\}\{(1-\backslash mu)^\backslash alpha\}\backslash right]^\{-1\},$

where $\backslash alpha,\backslash \; \backslash beta,\backslash \; c$ are constants. The physically relevant solution to the jet corresponds to the case $\backslash alpha=\backslash beta=0$ (Equivalently, we say that $c\_1=c\_2=c\_3=0$, so that the solution is free from singularities on the axis of symmetry, except at the origin).Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge university press. Therefore,

:$f\; =\; \backslash frac\{2(1-\backslash mu^2)\}\{c+1-\backslash mu\}\; =\; \backslash frac\{2\backslash sin^2\backslash theta\}\{c+1-\backslash cos\backslash theta\}.$

The function $f$ is related to the stream function as $\backslash psi\; =\; \backslash nu\; r\; f$, thus contours of $f$ for different values of $c$ provides the streamlines. The constant $c$ describes the force at the origin acting in the direction of the jet (this force is equal to the rate of momentum transfer across any sphere around the origin plus the force in the jet direction exerted by the sphere due to pressure and viscous forces), the exact relation between the force and the constant is given by

:$\backslash frac\{F\}\{2\backslash pi\backslash rho\; \backslash nu^2\}\; =\; \backslash frac\{32(c+1)\}\{3c(c+2)\}\; +\; 8(c+1)\; -\; 4(c+1)^2\backslash ln\; \backslash frac\{c+2\}\{c\}\; .$

The solution describes a jet of fluid moving away from the origin rapidly and entraining the slowly moving fluid outside of the jet. The edge of the jet can be defined as the location where the streamlines are at minimum distance from the axis, i.e., the edge is given by

:$\backslash theta\_o\; =\; \backslash cos^\{-1\}\; \backslash left(\backslash frac\{1\}\{1+c\}\backslash right).$

Therefore, the force can be expressed alternatively using this semi-angle of the conical-boundary of the jet,

:$\backslash frac\{F\}\{2\backslash pi\; \backslash rho\; \backslash nu^2\}\; =\; \backslash frac\{32\}\{3\}\backslash frac\{\backslash cos\backslash theta\_o\}\{\backslash sin^2\backslash theta\_o\}\; +\backslash frac\{4\}\{\backslash cos\backslash theta\_o\}\; \backslash ln\; \backslash frac\{1-\backslash cos\backslash theta\_o\}\{1+\backslash cos\backslash theta\_o\}\; +\; \backslash frac\{8\}\{\backslash cos\backslash theta\_o\}.$

When the force becomes large, the semi-angle of the jet becomes small, in which case,

:$\backslash frac\{F\}\{2\backslash pi\; \backslash rho\; \backslash nu^2\}\; \backslash sim\; \backslash frac\{32\}\{3\backslash theta\_o^2\}\; \backslash ll\; 1$

and the solution inside and outside of the jet become

:$\backslash begin\{align\}$

f(\theta)&\sim \frac{4\theta^2}{\theta^2+\theta_o^2},\quad \theta<\theta_o,\\

f(\theta)&\sim 2(1+\cos\theta),\quad \theta>\theta_o.

\end{align}

The jet in this limiting case is called the Schlichting jet. On the other extreme, when the force is small,

:$\backslash frac\{F\}\{2\backslash pi\; \backslash rho\; \backslash nu^2\}\; \backslash sim\; \backslash frac\{8\}\{c\}\backslash gg\; 1$

the semi-angle approaches 90 degree (no inside and outside region, the whole domain is considered as single region), the solution itself goes to

:$f(\backslash theta)\backslash sim\backslash frac\{2\}\{c\}\backslash sin^2\backslash theta.$

• Schlichting jet
• Schneider flow

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Category:Flow regimes

Category:Fluid dynamics

Category:Lev Landau