Tags: category:benchmark feature:2d feature:cartesian feature:analytical-solution

Layered flow with viscosity contrast

This section was contributed by Cedric Thieulot.

The idea behind this benchmark is to construct an analytical solution to the incompressible Stokes equation in the case where the viscosity field showcases a viscosity contrast at location \(y=y_0\) whose amplitude and width can be controlled. The viscosity is defined as

\[\eta(y)=\frac{1}{\frac{1}{\pi} \tan^{-1} (\frac{y-y_0}{\beta} ) + 1/2 + \epsilon}\]

where \(\beta\) and \(\epsilon\) are chosen by the user. Viscosity profiles for different values of \(\beta\) and \(\epsilon\) are shown in Fig. 265 and Fig. 266. The set up of this benchmark allows testing how discretizations deal with abrupt changes in the viscosity (if \(\beta\) is small) as well as large changes in the viscosity (if \(\epsilon\) is small).

Fig. 265 Layered flow benchmark: Viscosity profiles for various \(\beta\) values, using \(y_0 = 1/3\) and \(\epsilon = 0.0\).

Fig. 266 Layered flow benchmark: Viscosity profiles for various \(\beta\) values, using \(y_0 = 1/3\) and \(\epsilon = 0.2\).

The flow is assumed to take place in an infinitely long domain (in the horizontal direction) and bounded by \(y=-1\) and \(y+1\). At the bottom we impose \(v_x(y=-1)=0\), while we impose \(v_x(y=+1)=1\) at the top. The density is set to 1 while the gravity is set to zero. Under these assumptions, the flow velocity and pressure fields are given by:

\[\begin{split}\begin{aligned} v_x(x,y)&=&\frac{1}{2\pi} \left( -\beta C_1 \log [\beta^2 + (z-z_0)^2] + 2 (z-z_0) C_1 \tan^{-1} \frac{z-z_0}{\beta} + \pi (1+2\epsilon) z C_1 + C_2 \right), \nonumber\\ v_y(x,y) &=& 0, \nonumber\\ p(x,y) &=& 0, \end{aligned}\end{split}\]

where \(C_1\) and \(C_2\) are integration constants:

\[\begin{split}\begin{aligned} C_1 &=& 2\pi \Bigl[ \beta \log [\beta^2 + (1+z_0)^2] - 2(1+z_0) \tan^{-1} \frac{1+z_0}{\beta} \\ &&\qquad\qquad -\beta \log [\beta^2 + (1-z_0)^2] + 2(1-z_0) \tan^{-1} \frac{1-z_0}{\beta} + 2\pi (1+2\epsilon) \Bigr]^{-1},\\ C_2 &=& \left[ \beta \log [\beta^2 + (1+z_0)^2] - 2(1+z_0) \tan^{-1} \frac{1+z_0}{\beta} + \pi(1+2\epsilon) \right]C_1. \end{aligned}\end{split}\]

The viscosity and velocity fields are shown in Fig. 267 and Fig. 268 for \(\beta=0.01\) and \(\epsilon=0.05\).

Fig. 267 Velocity fields for \(\beta=0.01\) and \(\epsilon=0.05\) at uniform level 8 resolution, using \(y_0=1/3\).

Fig. 268 Viscosity fields for \(\beta=0.01\) and \(\epsilon=0.05\) at uniform level 8 resolution, using \(y_0=1/3\).