the velocity profiles in (a).](Fig1.pdf){width="80.00000%"}

![\[fig:localisation\_factor\] The localisation factor, i.e., the part in the parentheses in Eqs. \[equ:localised\_forcing\], as a function of the distance to the centre of the forcing area, $\sqrt{(x-x_c)^2+(z-z_c)^2}$, given $R=7$ and $B=0.2$, 0.5 and 1.0.](Fig2.pdf){width="70.00000%"}

Besides, to localise the force, we multiply the force by a tanh-function as the following $$\label{equ:localised_forcing}
\boldsymbol F=(F_x, 0, F_z)=\left(0.5-0.5{\text {tanh}} \frac{\sqrt{(x-x_c)^2+(z-z_c)^2}-R}{\sqrt{2} B}\right)(A_xf_x, 0, A_zf_z),$$ where $B$ determines the sharpness of the localisation, $A_x$ and $A_z$ are the nominal amplitude of the force component $f_x$ and $f_z$, respectively, and are adjustable independently. $R$ is the nominal radius and $x_c$ and $z_c$ are the streamwise and spanwise coordinates of the center of the localised forcing area. As an example, figure \[fig:localisation\_factor\] shows the localisation factor as a function of the distance to the center of the forcing area in (\[equ:localised\_forcing\]) (the part in the parentheses) for $R=7$ and $B=0.2$, 0.5 and 1.0. The smaller $B$, the steeper the force curve at the boundary. The localisation factor falls nearly to zero at distances considerably larger than $R$ (e.g. larger than approximately 8 for $B=0.5$).

It should be noted that, in the proximity of the border of the forcing area, this localised force does 