The 2D annulus benchmark#

This section was contributed by C. Thieulot and E. G. Puckett.

This benchmark is based on a manufactured solution in which an analytical solution to the isoviscous incompressible Stokes equations is derived in an annulus geometry. The velocity and pressure fields are as follows:

\[\begin{split}\begin{aligned} v_r(r,\theta) &=& g(r) k \sin(k\theta), \\ v_\theta(r,\theta)&=& f(r) \cos(k \theta), \\ p(r,\theta) &=& k h(r) \sin(k \theta), \\ \rho (r,\theta) &=& \aleph(r) k \sin (k \theta),\end{aligned}$$ with $$\begin{aligned} f(r)&=&Ar+B/r, \\ g(r) &=& \frac{A}{2}r + \frac{B}{r} \ln r + \frac{C}{r}, \\ h(r)&=& \frac{2g(r)-f(r)}{r}, \\ \aleph(r) &=& g'' - \frac{g'}{r} - \frac{g}{r^2} (k^2 - 1) + \frac{f}{r^2} + \frac{f'}{r}, \\ A &=& -C\frac{2(\ln R_1 - \ln R_2)} { R_2^2 \ln R_1 - R_1^2 \ln R_2}, \\ B &=& -C \frac{R_2^2-R_1^2}{R_2^2 \ln R_1 - R_1^2 \ln R_2}.\end{aligned}\end{split}\]

The parameters \(A\) and \(B\) are chosen so that \(v_r(R_1)=v_r(R_2)=0\), i.e. the velocity is tangential to both inner and outer surfaces. The gravity vector is radial and of unit length.

The parameter \(k\) controls the number of convection cells present in the domain, as shown in Fig. 3.

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Fig. 158 Pressure, density and velocity fields for k=0,1,2,3 for the 2D annulus benchmark.#

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Fig. 159 Pressure, density and velocity fields for k=0,1,2,3 for the 2D annulus benchmark.#

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Fig. 160 Pressure, density and velocity fields for k=0,1,2,3 for the 2D annulus benchmark.#

In the present case, we set \(R_1=1\), \(R_2=2\) and \(C=-1\). Fig. 4 shows the velocity and pressure errors in the \(L_2\)-norm as a function of the mesh size \(h\) (taken in this case as the radial extent of the elements). As expected we recover a third-order convergence rate for the velocity and a second-order convergence rate for the pressure.

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Fig. 161 Velocity and pressure errors in the L2-norm as a function of the mesh size for the 2D annulus benchmark.#