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A Stationary Rotating Vortex

We consider a rotating vortex in gaseous flow in two space dimensions. The governing equations are the Euler equations for one polytropic gas. We construct a non-trivial radially symmetric and stationary solution by balancing hydrodynamic pressure and centripetal force per volume element, i.e.

$\frac{d}{dr}p(r) = \rho(r) \frac{U(r)^2}{r},\qquad\qquad\qquad\qquad \rm (1)$

where U(r) denotes the velocity at distance r away from the origin. For $\rho_0:=1$ and the velocity field

$U(r) = \alpha \cdot \left\{ \begin{array}{ll} 2r/R & {\rm if\;} 0<r<R/2, \\ 2(1-r/R) & {\rm if\;} R/2\le r\le R, \\ 0 & {\rm if\;} r>R, \end{array} \right.$ we find by integrating Eq. (1) with boundary condition p(R)=p_0:=2

the pressure distribution

$p(r) = p_0 + 2 \rho_0\alpha^2 \cdot \left\{ \begin{array}{ll} r^2/R^2 + 1 -2\log 2 & {\rm if\;} 0<r<R/2, \\ r^2/R^2 + 3 -4r/R+ 2\log (r/R) & {\rm if\;} R/2\le r\le R, \\ 0 & {\rm if\;} r>R. \end{array} \right.$ In Cartesian coordinates the entire solution for Euler equations in primitive variables reads

$\rho(x,y,t) = \rho_0,\quad u(x,y,t) = -\sin\varphi\, U(r),\quad v(x,y,t) = \cos\varphi\, U(r),\quad p(x,y,t) = p(r)$

for all $t\ge 0$ with $r=\sqrt{(x-x_c)^2+(y-y_c)^2}$ and $\displaystyle \varphi={\rm arctan}\frac{y-y_c}{x-x_c}$ .

This exact smooth solution is applicable to measure the accuracy of Cartesian finite volume schemes alone, but also to quantify the accuracy of imbedded boundary methods by constructing a radially symmetric no-slip boundary around the origin.

Configuration used in accuracy study for available patch solvers:

$r\in[0,0.5], \quad R=0.4, \quad \alpha=R\pi$

Velocity U(r)

Pressure p(r)

-- RalfDeiterding - 23 Feb 2005

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