3.1 Analysis of the laminar ESBL

The laminar boundary layer flow over the librating frustum is realized for small libration amplitudes, that is, we may consider the linearized Navier–Stokes equations (2.3) by letting $Ro\rightarrow 0$ . The problem of a libration-induced boundary layer flow along the slope has only been rarely considered in the literature (most recently by Swart et al. Reference Swart, Manders, Harlander and Maas2010). We therefore include a laminar boundary layer analysis for the frustum in the following. This material comes for a large part from Klein (Reference Klein2016).

Libration of the frustum induces wall-parallel shear of alternating sign. The laminar boundary layer flow is therefore expected to vary only in wall-normal direction for a snapshot at a given time instance. This suggests to decompose the vector of the wall rotation rate into a wall-parallel and a wall-normal component, namely $\boldsymbol{e}_{z}=\boldsymbol{e}_{\bot }\cos \unicode[STIX]{x1D6FC}+\boldsymbol{e}_{\Vert }\sin \unicode[STIX]{x1D6FC}$ . Inserting this into the Coriolis acceleration term in (2.3) and using rigidity of the wall permit to cancel out the geostrophic balance. Hence only the unbalanced wall-parallel projection of the Coriolis force affects the flow evolution near the wall. This is known as the $f$ -plane approximation (e.g. Pedlosky Reference Pedlosky1987). For the frustum with a wall inclination $\unicode[STIX]{x1D6FC}$ (taken with respect to the rotation axis), the strength of the effective Coriolis force is proportional to the dimensionless effective Coriolis parameter

(3.1) $$\begin{eqnarray}\displaystyle f_{\ast }\equiv f\sin \unicode[STIX]{x1D6FC}\quad \text{with }f=2 & & \displaystyle\end{eqnarray}$$

relative to the mean rotation rate $\unicode[STIX]{x1D6FA}_{0}$ . In analogy to the sphere, this yields $f_{\ast }\in [-2,2]$ such that $f_{\ast }=2$ applies to Earth’s North Pole, $f_{\ast }=-2$ applies to the South Pole and $f_{\ast }=0$ applies at the equator or the inner cylinder in geometry  $G1$ . Here, $f_{\ast }$ is constant for the entire frustum of the selected geometry but increases with the apex half-angle $\unicode[STIX]{x1D6FC}$ for the geometries $G1$ – $G7$ (see table 2). We use an asterisk $(\ast )$ to distinguish the local value $f_{\ast }$ from the maximum possible value $f=2$ and retain $f$ to avoid confusion with a numerical factor $2$ in the laminar solution presented below. To simplify the equations, but without loss of generality, we assume $f_{\ast }\geqslant 0$ from here on.

Table 2.

Simulated maximum values and ratios of the spatially averaged wall shear stress components $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x1D70F}_{\Vert }$ obtained for intermittently unstable and stable flow conditions. The subscripts $+$ and $-$ refer to the prograde and retrograde libration half-phases, respectively. The maximum values were divided by $\unicode[STIX]{x1D700}$ and multiplied with $10^{3}$ . The values for the unstable ESBL correspond to figure 7. The deflection angle $\unicode[STIX]{x1D6FD}$ of the Görtler vortices was measured in an axial–azimuthal section for unstable flow conditions; the stress angle $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}$ is given for both stable and unstable conditions.

In order to simplify the analytical treatment, we consider a tangential plane with a local coordinate origin at some radius $R$ on the frustum wall. A local Cartesian treatment is permissible since the frustum radius is much larger than the boundary layer thickness of the ESBL ( $R\gg \unicode[STIX]{x1D6FF}$ ), but requires some care to incorporate curvature effects as we shall see shortly. Let the local Cartesian axes be oriented such that $x^{\prime }$ is the local azimuthal (zonal) coordinate, $z^{\prime }$ is the local wall-normal coordinate and $y^{\prime }$ is the second wall-tangential coordinate (lying in a radial–axial section). The local velocity components are then given by $u^{\prime }$ , $w^{\prime }$ and $v^{\prime }$ corresponding to the local azimuthal, wall-normal and wall-tangential component. This ensures that the local axes are right-handed and that for $\unicode[STIX]{x1D6FC}=0^{\circ }$ the velocities correspond to $u^{\prime }=u_{\unicode[STIX]{x1D703}}$ , $w^{\prime }=u_{r}$ and $v^{\prime }=u_{z}$ .

The most general property of the frustum (or spherical wall) libration is the generation of positive and negative vorticity. This suggests to use the vorticity-divergence form of the $f$ -plane equations as starting point, that is,

(3.2a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}t}+f_{\ast }\unicode[STIX]{x1D6E5}=E\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D701}}{\unicode[STIX]{x2202}z^{\prime 2}},\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}t}-f_{\ast }\unicode[STIX]{x1D701}=E\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x2202}z^{\prime 2}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}=\unicode[STIX]{x2202}v^{\prime }/\unicode[STIX]{x2202}x^{\prime }-\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}y^{\prime }$ is the wall-normal vorticity and $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}x^{\prime }+\unicode[STIX]{x2202}v^{\prime }/\unicode[STIX]{x2202}y^{\prime }$ is the divergence formed by the wall-tangential velocity components. Both $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D6E5}$ can be simplified further for an axisymmetric flow since $\unicode[STIX]{x2202}(\cdot )/\unicode[STIX]{x2202}x^{\prime }=0$ . Note that the unbalanced pressure forces cancel exactly in the equation for $\unicode[STIX]{x1D701}$ , whereas in the equation for $\unicode[STIX]{x1D6E5}$ the pressure gradient terms have been dropped by taking advantage of the boundary layer approximation (i.e. the wall-normal gradients are much larger than the wall-tangential gradients).

The boundary conditions for a single librating wall are given by

(3.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}ccl@{}}\displaystyle u^{\prime }=\frac{Rf}{2}\sin (\unicode[STIX]{x1D714}t), & v^{\prime }=w^{\prime }=0 & \text{at }z^{\prime }=0,\\ \unicode[STIX]{x1D701}=f_{\ast }\sin (\unicode[STIX]{x1D714}t), & \unicode[STIX]{x1D6E5}=0 & \text{at }z^{\prime }=0,\\ u^{\prime },v^{\prime },w^{\prime }\rightarrow 0, & \unicode[STIX]{x1D701},\unicode[STIX]{x1D6E5}\rightarrow 0 & \text{at }z^{\prime }\rightarrow +\infty .\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Equations (3.2) subject to the boundary conditions (3.3) have the solution

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(z^{\prime },t) & = & \displaystyle \frac{f_{\ast }}{2}\sin (\unicode[STIX]{x1D714}t)\left[\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)+\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{f_{\ast }}{2}\cos (\unicode[STIX]{x1D714}t)\left[\unicode[STIX]{x1D70E}\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)-\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right],\quad\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}(z^{\prime },t) & = & \displaystyle \frac{f_{\ast }}{2}\sin (\unicode[STIX]{x1D714}t)\left[\unicode[STIX]{x1D70E}\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)+\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{f_{\ast }}{2}\cos (\unicode[STIX]{x1D714}t)\left[\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)-\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right],\end{eqnarray}$$

in which the following parameters were defined,

(3.6a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}\equiv \text{sgn}(f_{\ast }-\unicode[STIX]{x1D714})\quad \text{and}\quad \unicode[STIX]{x1D6FF}_{\pm }\equiv \sqrt{\frac{2E}{|f_{\ast }\pm \unicode[STIX]{x1D714}|}}. & & \displaystyle\end{eqnarray}$$

The sign $\unicode[STIX]{x1D70E}$ is positive $(+1)$ for libration frequencies $\unicode[STIX]{x1D714}<f_{\ast }$ , whereas it is negative $(-1)$ for libration frequencies $\unicode[STIX]{x1D714}>f_{\ast }$ . The vorticity-divergence solution reveals that there are two competing boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{+}$ and $\unicode[STIX]{x1D6FF}_{-}$ which define the ESBL. We will discuss the boundary layer structure shortly in more detail after we have computed the velocity profiles.

The wall-normal velocity profile $w^{\prime }(z^{\prime },t)$ follows from a wall-normal integration of the continuity equation $\unicode[STIX]{x2202}w^{\prime }/\unicode[STIX]{x2202}z^{\prime }+\unicode[STIX]{x1D6E5}=0$ . This shows that the wall-normal outflow $w^{\prime }$ is induced by the wall-tangential divergence $\unicode[STIX]{x1D6E5}$ which is a result of libration, viscous diffusion and Coriolis force. The wall-tangential velocity profiles $u^{\prime }(z^{\prime },t)$ and $v^{\prime }(z^{\prime },t)$ of the librating frustum are obtained by integrating $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D6E5}$ along the frustum wall (along $y^{\prime }$ ). A consistent wall velocity is obtained by letting $u^{\prime }\propto Rf/2$ , which crosses zero at the apex of the cone that describes the frustum, and yields $y^{\prime }=-R/\sin \unicode[STIX]{x1D6FC}$ . The integration along $y^{\prime }$ must also account for the azimuthal curvature of the frustum. This effect can be accounted for in a laminar boundary layer flow by considering the circulation around the frustum. Doing so simply yields the additional factor $1/2$ in the otherwise planar treatment of the local dynamics. The local 1-D velocity solution of the ESBL at the librating frustum is thus given by

(3.7a-c ) $$\begin{eqnarray}\displaystyle u^{\prime }(z^{\prime },t)=\frac{R}{2\sin \unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D701}(z^{\prime },t),\quad v^{\prime }(z^{\prime },t)=-\frac{R}{2\sin \unicode[STIX]{x1D6FC}}\unicode[STIX]{x0394}(z^{\prime },t),\quad w^{\prime }(z^{\prime },t)=-\int _{0}^{z^{\prime }}\unicode[STIX]{x0394}(\unicode[STIX]{x1D709},t)\,\text{d}\unicode[STIX]{x1D709}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

By expanding (3.7) we can finally obtain the time-dependent velocity profiles of the laminar ESBL. These are given by

(3.8) $$\begin{eqnarray}\displaystyle u^{\prime }(z^{\prime },t) & = & \displaystyle \frac{R}{2}\sin (\unicode[STIX]{x1D714}t)\left[\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)+\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{R}{2}\cos (\unicode[STIX]{x1D714}t)\left[\unicode[STIX]{x1D70E}\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)-\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right],\quad\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle v^{\prime }(z^{\prime },t) & = & \displaystyle \frac{R}{2}\sin (\unicode[STIX]{x1D714}t)\left[-\unicode[STIX]{x1D70E}\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)-\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\sin \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{R}{2}\cos (\unicode[STIX]{x1D714}t)\left[\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{-}}\right)-\exp \left(-\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\cos \left(\frac{z^{\prime }}{\unicode[STIX]{x1D6FF}_{+}}\right)\right],\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle w^{\prime }(z^{\prime },t) & = & \displaystyle -\frac{f_{\ast }}{2f}\sin (\unicode[STIX]{x1D714}t)[\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}_{-}(1+2I_{sin}(z^{\prime }/\unicode[STIX]{x1D6FF}_{-}))+\unicode[STIX]{x1D6FF}_{+}(1+2I_{sin}(z^{\prime }/\unicode[STIX]{x1D6FF}_{+}))]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{f_{\ast }}{2f}\cos (\unicode[STIX]{x1D714}t)[\unicode[STIX]{x1D6FF}_{-}(1+2I_{cos}(z^{\prime }/\unicode[STIX]{x1D6FF}_{-}))-\unicode[STIX]{x1D6FF}_{+}(1+2I_{cos}(z^{\prime }/\unicode[STIX]{x1D6FF}_{+}))],\end{eqnarray}$$

where we have used the following shorthand for some integrals

(3.11a,b ) $$\begin{eqnarray}\displaystyle I_{sin}(\unicode[STIX]{x1D709})=-\frac{\text{e}^{-\unicode[STIX]{x1D709}}}{2}(\cos \unicode[STIX]{x1D709}+\sin \unicode[STIX]{x1D709})\quad \text{and}\quad I_{cos}(\unicode[STIX]{x1D709})=-\frac{\text{e}^{-\unicode[STIX]{x1D709}}}{2}(\cos \unicode[STIX]{x1D709}-\sin \unicode[STIX]{x1D709}). & & \displaystyle\end{eqnarray}$$

Equations (3.8)–(3.10) constitute the 1-D laminar boundary layer solution of the time-dependent $f$ -plane equations valid locally for a given radial position $R$ on the librating frustum. All velocities have been scaled with the libration amplitude $\unicode[STIX]{x1D700}$ . For the solution presented it is instructive to consider the limiting cases of very low and high libration frequencies discussed in the following.

For very low libration frequencies, $\unicode[STIX]{x1D714}\ll f_{\ast }$ , one has $\unicode[STIX]{x1D70E}=1$ and $\unicode[STIX]{x1D6FF}_{\pm }\rightarrow \unicode[STIX]{x1D6FF}_{E}=\sqrt{2E/f_{\ast }}$ , which is the thickness of the classical Ekman boundary layer. All terms in the second lines of equations (3.8)–(3.10) vanish, which results in a quasi-steady Ekman layer that oscillates in phase with the wall libration. In addition, this Ekman layer has non-zero wall-normal velocity $w^{\prime }$ (Ekman pumping/suction) due to non-zero vorticity of the boundary condition (e.g. Greenspan Reference Greenspan1969; Busse et al. Reference Busse, Dormy, Simitev and Soward2007). With increasing $\unicode[STIX]{x1D714}$ , the difference between $\unicode[STIX]{x1D6FF}_{-}$ and $\unicode[STIX]{x1D6FF}_{+}$ increases and leads to a larger phase lag, which reaches its constant maximum value for $\unicode[STIX]{x1D714}\geqslant f_{\ast }$ .

For very high libration frequencies, $\unicode[STIX]{x1D714}\gg f_{\ast }$ , one has $\unicode[STIX]{x1D70E}=-1$ and $\unicode[STIX]{x1D6FF}_{\pm }\rightarrow \unicode[STIX]{x1D6FF}_{S}=\sqrt{2E/\unicode[STIX]{x1D714}}$ , which is the thickness of the classical Stokes boundary layer as discussed by Sauret et al. (Reference Sauret, Cébron, Le Bars and Le Dizès2012). In this case, the right-hand sides of (3.9) and (3.10) vanish and there is only an azimuthal velocity $u^{\prime }$ in the boundary layer. The fluid near the frustum is dragged along with the wall due to viscosity, but the time scale of libration is too short for significant momentum diffusion to occur before the wall velocity reverses sign. With decreasing libration frequency, momentum diffusion can affect fluid further away from the wall so that the Coriolis force becomes notable at some point, which finally giving rise to non-zero $v^{\prime }$ and $w^{\prime }$ components once $\unicode[STIX]{x1D6FF}_{-}$ and $\unicode[STIX]{x1D6FF}_{+}$ differ notably as $\unicode[STIX]{x1D714}\rightarrow f_{\ast }$ .

Figure 2.

Similarity curve of the ESBL in terms of the theoretical boundary layer thicknesses $\unicode[STIX]{x1D6FF}$ as a function of the time scale ratio $\unicode[STIX]{x1D6FE}$ . All boundary layer thicknesses have been normalized by the steady EBL thickness $\unicode[STIX]{x1D6FF}_{E}$ . The ESBL thickness $\unicode[STIX]{x1D6FF}_{-}$ erupts at $\unicode[STIX]{x1D6FE}=1$ (solid line), whereas the ESBL thickness $\unicode[STIX]{x1D6FF}_{+}$ remains finite (thick dashes). The limiting cases are the steady EBL for $\unicode[STIX]{x1D6FE}\ll 1$ (thin dashes) and the SBL for $\unicode[STIX]{x1D6FE}\gg 1$ (dotted). Note that the geometries investigated ( $G2$ – $G7$ ) are located to the right of the boundary layer eruption.

At intermediate frequencies ( $\unicode[STIX]{x1D714}$ close to $f_{\ast }$ ) the ESBL is not so easily described. Figure 2 therefore shows how the ESBL structure changes with $\unicode[STIX]{x1D714}$ or $f_{\ast }$ when varied over orders of magnitude. More precisely, the normalized boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{\pm }/\unicode[STIX]{x1D6FF}_{E}$ is shown as function of the normalized libration frequency $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D714}/f_{\ast }$ . Due to this scaling, the two graphs shown describe every possible ESBL. The asymptotic behaviour for $\unicode[STIX]{x1D6FE}\ll 1$ (EBL) and $\unicode[STIX]{x1D6FE}\gg 1$ (SBL) can be observed by means of $\unicode[STIX]{x1D6FF}_{\pm }$ converging to a constant value (EBL) or an inclined line (SBL). Only for $\unicode[STIX]{x1D714}$ close to $f_{\ast }$ ( $\unicode[STIX]{x1D6FE}\sim O(1)$ ) there is a notable difference between $\unicode[STIX]{x1D6FF}_{+}$ and $\unicode[STIX]{x1D6FF}_{-}$ , and $\unicode[STIX]{x1D6FF}_{-}$ diverges at $\unicode[STIX]{x1D714}=f_{\ast }$ ( $\unicode[STIX]{x1D6FE}=1$ ). The configurations $G2$ – $G7$ studied below fulfil $\unicode[STIX]{x1D6FE}>1$ and are given by vertical lines. ( $G1$ is not shown since it has a straight wall with a pure SBL.)

Let us come back to the velocity solution given by (3.8)–(3.10). It is worth noting that the wall-normal velocity $w^{\prime }$ does not depend on the frustum radius $R$ , but on the local wall-normal vorticity (say $f_{\ast }$ ) in contrast to the wall-tangential velocities $u^{\prime }$ and $v^{\prime }$ . The Ekman pumping/suction associated with the induced velocity $w^{\prime }$ is maximal for a wall perpendicular to the rotation axis (EBL case $|f_{\ast }|=f$ ) and reaches zero for an axial wall (SBL case $f_{\ast }=0$ ). Equations (3.8)–(3.10) show that $w^{\prime }$ becomes negligibly small compared to $u^{\prime }$ and $v^{\prime }$ in case of a very large curvature radius ( $R\gg 1$ ). That is, the kinetic energy density at the wall ( $u^{\prime 2}\propto R^{2}f^{2}$ ) increases with the radius, but the vorticity remains constant ( $\unicode[STIX]{x1D701}^{\prime }\propto f_{\ast }$ ). So, for $R\gg 1$ , the solution developed above approaches the planar ESBL studied by Salon & Armenio (Reference Salon and Armenio2011), who considered a pressure-driven (tidally forced) flow on the $f$ -plane. This suggests that the boundary layer over the librating frustum is always an ESBL, but it depends on the local wall curvature how strong the Ekman pumping/suction is in comparison to the wall-tangential flow trapped in the boundary layer.

Figure 3.

ESBL profiles over the librating frustum at two different libration phases $\unicode[STIX]{x1D711}=0$ and $\unicode[STIX]{x03C0}/2$ for the geometry  $G3$ . The analytical solution is compared to the DNS for a wall inclination of $\unicode[STIX]{x1D6FC}=6^{\circ }$ towards the axial direction and the libration amplitude $\unicode[STIX]{x1D700}=0.2$ (Rossby number $Ro=0.2$ ) in the stable regime. The origin of the wall-normal coordinate $z^{\prime }=0$ corresponds to the point $(r,z)\approx (0.905,1.0)$ and all velocity components have been rescaled with the local frustum radius $R\approx 0.905$ . The Ekman layer thicknesses $\unicode[STIX]{x1D6FF}_{-}$ is given by a dotted vertical line for orientation. The temporal variability is indicated by grey shading between the envelope of the 1-D analytical solutions.

Now that we have the analytical expressions for the laminar velocity profiles we can move on by computing the wall shear stress and the Ekman flux in analogy to (2.8) and (2.9) used for the DNS. This provides further insight into the properties of the ESBL and will proof useful in the analysis of the stable and unstable ESBL addressed later.

The wall shear stress components due to the analytical velocity profiles read

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}\simeq E\frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}z^{\prime }}=E\left(-\frac{R}{2}\frac{\unicode[STIX]{x1D6FF}_{-}+\unicode[STIX]{x1D6FF}_{+}}{\unicode[STIX]{x1D6FF}_{-}\unicode[STIX]{x1D6FF}_{+}}\sin (\unicode[STIX]{x1D714}t)-\frac{R}{2}\frac{\unicode[STIX]{x1D6FF}_{-}-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}_{+}}{\unicode[STIX]{x1D6FF}_{-}\unicode[STIX]{x1D6FF}_{+}}\cos (\unicode[STIX]{x1D714}t)\right), & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{\Vert }\simeq E\frac{\unicode[STIX]{x2202}v^{\prime }}{\unicode[STIX]{x2202}z^{\prime }}=E\left(-\frac{R}{2}\frac{\unicode[STIX]{x1D6FF}_{-}+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}_{+}}{\unicode[STIX]{x1D6FF}_{-}\unicode[STIX]{x1D6FF}_{+}}\sin (\unicode[STIX]{x1D714}t)+\frac{R}{2}\frac{\unicode[STIX]{x1D6FF}_{-}-\unicode[STIX]{x1D6FF}_{+}}{\unicode[STIX]{x1D6FF}_{-}\unicode[STIX]{x1D6FF}_{+}}\cos (\unicode[STIX]{x1D714}t)\right), & \displaystyle\end{eqnarray}$$

where the boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{\pm }$ and the sign $\unicode[STIX]{x1D70E}$ have been defined in (3.6).

The Ekman flux is an oscillating mass flux confined to the ESBL. This flux has an azimuthal and a wall-parallel component. The azimuthal component is due to the Stokes property (i.e. momentum diffusion), whereas the wall-parallel component is due to the Ekman property (i.e. Coriolis force). Here, we want to focus on the Ekman property by looking into the wall-parallel component $Q$ . Over the frustum, $Q$ has radial and axial contributions. Computing $Q$ in analogy to (2.9) but here for the analytical velocity profiles yields

(3.14) $$\begin{eqnarray}\displaystyle Q=-\frac{\unicode[STIX]{x1D6FF}_{-}+\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}_{+}}{4\unicode[STIX]{x1D70E}}\sin (\unicode[STIX]{x1D714}t)+\frac{\unicode[STIX]{x1D6FF}_{-}-\unicode[STIX]{x1D6FF}_{+}}{4}\cos (\unicode[STIX]{x1D714}t)=A\sin (\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D712}), & & \displaystyle\end{eqnarray}$$

where the amplitude $A$ and the phase shift $\unicode[STIX]{x1D712}$ are given by

(3.15a,b ) $$\begin{eqnarray}\displaystyle A=\sqrt{\frac{E}{2}}\sqrt{\frac{f_{\ast }}{f_{\ast }^{2}-\unicode[STIX]{x1D714}^{2}}},\quad \tan \unicode[STIX]{x1D712}=-\frac{\sqrt{f_{\ast }+\unicode[STIX]{x1D714}}-\sqrt{f_{\ast }-\unicode[STIX]{x1D714}}}{\sqrt{f_{\ast }+\unicode[STIX]{x1D714}}+\sqrt{f_{\ast }-\unicode[STIX]{x1D714}}}\quad \text{for }\unicode[STIX]{x1D714}<f_{\ast }, & & \displaystyle\end{eqnarray}$$

(3.16a,b ) $$\begin{eqnarray}\displaystyle A=\sqrt{\frac{E}{2}}\sqrt{\frac{\unicode[STIX]{x1D714}-\sqrt{\unicode[STIX]{x1D714}^{2}-f_{\ast }^{2}}}{\unicode[STIX]{x1D714}^{2}-f_{\ast }^{2}}},\quad \tan \unicode[STIX]{x1D712}=1\quad \Rightarrow \quad \unicode[STIX]{x1D712}=\unicode[STIX]{x03C0}/4\quad \text{for }\unicode[STIX]{x1D714}>f_{\ast },\quad & & \displaystyle\end{eqnarray}$$

where we assume positive values for $f_{\ast }$ and $\unicode[STIX]{x1D714}$ . The geometries $G2$ – $G7$ correspond to the second case where the phase shift is fixed but the amplitude of the induced mass flux depends on the forcing frequency ( $\unicode[STIX]{x1D714}$ ) and the wall inclination ( $f_{\ast }$ ). There are two interesting properties specific to the ESBL: (i) the fixed phase shift (3.16) is a result of the Ekman property by means of a fixed intrinsic time scale $f_{\ast }^{-1}$ of the Coriolis force, and (ii) the mass flux amplitude tends to zero for $\unicode[STIX]{x1D714}\gg f_{\ast }$ but ‘erupts’ for $\unicode[STIX]{x1D714}\rightarrow f_{\ast }$ (Klein et al. Reference Klein, Seelig, Kurgansky, Ghasemi, Borcia, Will, Schaller, Egbers and Harlander2014). It turns out that small (large) libration frequencies in comparison to the effective Coriolis parameter emphasize the Ekman (Stokes) property of the ESBL. So, in order to have a significant Ekman property in the ESBL, we consider a regime with $\unicode[STIX]{x1D714}$ and $f_{\ast }$ not too far apart from each other.

3.2 Comparison of the local numerical and analytical ESBL solutions

The DNS requires a finite libration amplitude, which means finite Rossby number and, thus, per se weakly nonlinear conditions. Nevertheless, as long as the flow remains stable, a fair comparison between weakly nonlinear DNS and the linear theory can be performed in the vicinity of the wall. For the case shown in figure 3 a Rossby number of $Ro=0.2$ and an Ekman number of $E=3\times 10^{-5}$ has been selected such that the flow remained axisymmetric and stable over the frustum. The simulations were conducted as described in § 2. For better comparison with the $f$ -plane solution, the velocities $(u_{\unicode[STIX]{x1D703}},u_{r},u_{z})$ were transformed to a local reference frame such that $(u^{\prime },v^{\prime },w^{\prime })$ denote the local azimuthal, axial (wall parallel in a radial–axial slice), and wall-normal velocity components. Wall-normal velocity profiles (coordinate $z^{\prime }$ ) have been recorded over a selected location on the frustum.

Figure 3 shows various local wall-normal velocity profiles $(u^{\prime },v^{\prime },w^{\prime })$ over the librating frustum comparing the 1-D ESBL solution to the 3-D DNS for the two libration phases $\unicode[STIX]{x1D711}=0$ and $\unicode[STIX]{x03C0}/2$ . The inclination of the frustum wall is $\unicode[STIX]{x1D6FC}=6^{\circ }$ (geometry  $G3$ ), but the flow is representative for all geometries $G2$ – $G7$ with an inner frustum. Any of the velocity components has been normalized with the local libration velocity $Rf/2$ . The temporal variability of the flow is illustrated by a shading of the area between the minimum and maximum values of the 1-D solution.

It is quite remarkable how well the 1-D theory captures the stable 3-D DNS near the wall. The agreement between 1-D theory and DNS is virtually perfect for the wall-tangential components $u^{\prime }$ and $v^{\prime }$ in figure 3(a,b) up to the five theoretical boundary layer thicknesses ( $z^{\prime }\leqslant 5\unicode[STIX]{x1D6FF}_{-}$ ) shown. By contrast, the wall-normal velocity profile $w^{\prime }$ in figure 3(c) deviates notably from the DNS, but especially for $z^{\prime }>2\unicode[STIX]{x1D6FF}_{-}$ . This can be explained by curvature and geometry effects. On the one hand, we accounted for curvature only by formulating the boundary conditions on the $f$ -plane and then continued with a Cartesian formulation. On the other hand, the 1-D solution does not account for adverse pressure gradients, mean bulk flow (e.g. Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012), oscillatory bulk flow due to inertial waves (e.g. Klein et al. Reference Klein, Seelig, Kurgansky, Ghasemi, Borcia, Will, Schaller, Egbers and Harlander2014) or secondary circulations due to the lids (e.g. Wang Reference Wang1970; Hollerbach & Fournier Reference Hollerbach, Fournier, Bonanno, Rüdiger and Rosner2004). These additional flows are likely to modify the boundary layer structure, in particular the wall-tangential divergence, and are thus likely to have an effect on the wall-normal velocity profile. In any case, the wall-normal velocity is very small compared to the wall-tangential velocity components ( $u^{\prime }:v^{\prime }:w^{\prime }\approx 1:10^{-1}:10^{-4}$ ) and deviations might be less critical.

Note that the larger one ( $\unicode[STIX]{x1D6FF}_{-}$ ) of the two theoretical boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{\pm }$ is taken as the overall thickness of the ESBL. This boundary layer thickness is resolved roughly by the distance between the markers shown in figure 3, hence, the mesh size near the wall resolves $\unicode[STIX]{x1D6FF}_{-}$ with $N_{\unicode[STIX]{x1D6FF}}\approx 4$ grid boxes. This is just good enough to capture the boundary layer structure given the ratio of $\unicode[STIX]{x1D6FF}_{-}/\unicode[STIX]{x1D6FF}_{+}\lesssim 1.6$ for the geometries $G2$ – $G7$ investigated. It is worth to point out that the level of agreement between DNS and the 1-D theory is comparable to the reference results by Lopez & Marques (Reference Lopez and Marques2011) and Salon & Armenio (Reference Salon and Armenio2011), which indicates that the near-wall dynamics is sufficiently resolved in the present DNS and that the 1-D theory is acceptable for the near-wall laminar flow.

In order to study further the effects of nonlinearity, let us come back to the scaling of the governing equations. We can now make use of the laminar boundary layer structure to yield appropriate length scales by means of $\unicode[STIX]{x1D6FF}_{\pm }$ (3.6) as well as the well-known boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{S}$ of the Stokes layer (e.g. Batchelor Reference Batchelor1967, pp. 353–355) and the thickness $\unicode[STIX]{x1D6FF}_{E}$ of the Ekman layer over an inclined wall. Wall libration of the frustum suggests to use the libration frequency to obtain a time scale $\unicode[STIX]{x1D714}^{-1}$ and to keep the velocity scale $U=\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}r_{1}$ , but to express spatial gradients more carefully with an appropriate length scale. A term-by-term analysis of the momentum balance yields

(3.17) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \left|\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}\right|\sim \unicode[STIX]{x1D714},\quad |2\boldsymbol{e}_{z}\times \boldsymbol{u}|\sim f_{\ast },\quad |E\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D735}\times \boldsymbol{u})|\sim \frac{E}{\unicode[STIX]{x1D6FF}^{2}},\\ \displaystyle |Ro(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}|\sim \frac{Ro}{\unicode[STIX]{x1D6FF}},\quad |\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|\sim \frac{Ro}{\unicode[STIX]{x1D6FF}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where each term was normalized by $\unicode[STIX]{x1D700}^{-1}U^{2}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}^{2}r_{1}^{2}$ . Hence, $\unicode[STIX]{x1D714}$ and $f_{\ast }$ are taken dimensionless relative to the mean rotation rate $\unicode[STIX]{x1D6FA}_{0}$ , and the length scale $\unicode[STIX]{x1D6FF}$ is dimensionless due to division by the maximum frustum radius $r_{1}$ . The balance of forces governs the type of the boundary layer. The viscous term is of the order of $E/\unicode[STIX]{x1D6FF}_{S}^{2}=\unicode[STIX]{x1D714}$ for the SBL ( $\unicode[STIX]{x1D6FF}_{S}=\sqrt{2E/\unicode[STIX]{x1D714}}$ ), whereas it is of the order of $E/\unicode[STIX]{x1D6FF}_{E}^{2}=f_{\ast }$ for the EBL ( $\unicode[STIX]{x1D6FF}_{E}=\sqrt{2E/f_{\ast }}$ ) and of the order of $E/\unicode[STIX]{x1D6FF}_{\pm }^{2}=|\unicode[STIX]{x1D714}\pm f_{\ast }|$ for the ESBL.

For low Rossby numbers and moderate Ekman numbers (not too small boundary layer thickness) equation (3.17) reveals that the local acceleration prescribed by the wall libration and the Coriolis acceleration imposes two time scales. In analogy to Noir et al. (Reference Noir, Calkins, Lasbleis, Cantwell and Aurnou2010), we define the parameter

(3.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}\equiv \frac{\unicode[STIX]{x1D714}}{f_{\ast }}, & & \displaystyle\end{eqnarray}$$

which expresses the strength of the librational acceleration relative to the (local) Coriolis acceleration in terms of a time scale ratio. Three different flow regimes of the ESBL can be distinguished with the aid of $\unicode[STIX]{x1D6FE}$ . These are given by (i)  $\unicode[STIX]{x1D6FE}\ll 1$ , the rotation-dominated regime with the EBL limit; (ii)  $\unicode[STIX]{x1D6FE}\sim 1$ , the intermediate or librational–rotational regime with a ‘true’ ESBL; and (iii)  $\unicode[STIX]{x1D6FE}\gg 1$ , the libration-dominated or viscous regime with the SBL limit. These regimes have been discussed above for the laminar ESBL in terms of $\unicode[STIX]{x1D714}$ and $f_{\ast }$ instead of $\unicode[STIX]{x1D6FE}$ (see pp. 13, below equation (3.11)).

It is illustrative to express the boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{\pm }$ in terms of the time scale ratio $\unicode[STIX]{x1D6FE}$ to yield

(3.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{\pm }(\unicode[STIX]{x1D6FE})=\frac{\unicode[STIX]{x1D6FF}_{E}}{\sqrt{1\pm \unicode[STIX]{x1D6FE}}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{E}=\sqrt{2E/f_{\ast }}$ denotes the thickness of the steady EBL over the inclined (frustum) wall. The functional dependency $\unicode[STIX]{x1D6FF}_{\pm }(\unicode[STIX]{x1D6FE})$ is shown in figure 2, in which one can distinguish the three flow regimes mentioned. Regime (ii) is located in the central part covering a $\unicode[STIX]{x1D6FE}$ -interval of approximately three orders of magnitude centred at $\unicode[STIX]{x1D6FE}=1$ . In this regime the two boundary layer thicknesses $\unicode[STIX]{x1D6FF}_{\pm }$ are notably different from each other indicating strong competition between the libration and Coriolis time scales. In the limits, regime (i) is approached on the left side for $\unicode[STIX]{x1D6FE}<5\times 10^{-2}$ , whereas regime (iii) is approached on the right side for $\unicode[STIX]{x1D6FE}>5\times 10^{1}$ . These regimes exhibit a collapse of the two ESBL thicknesses to either the Ekman boundary layer in regime (i) or the Stokes layer in regime (iii). The values of $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}_{\pm }$ corresponding to geometries $G2$ – $G7$ are given for orientation and found exclusively to the right of the boundary layer eruption $\unicode[STIX]{x1D6FE}=1$ (see table 2 for the exact values). For the geometry $G2$ with $\unicode[STIX]{x1D6FE}\approx 10$ , still the librational forcing gives the main contribution to the boundary layer dynamics, thus it is categorized as the regime (iii). Increasing $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FE}$ gradually decreases and thus the effect of the rotation on the flow dynamics increases in such a way that for the geometry $G7$ with $\unicode[STIX]{x1D6FE}\approx 1.5$ both rotation and libration give almost the same contribution to the boundary layer dynamics.

The parameter $\unicode[STIX]{x1D6FE}$ allows us to estimate fairly easily which of the forces dominates the boundary layer dynamics and, thus, to infer which boundary layer structure is present. This can be relevant for applications, like librating planets or moons, which exhibit all possible wall inclinations thanks to their spherical symmetry, and thus one is left with determining their libration frequency (e.g. Comstock & Bills Reference Comstock and Bills2003). It is worth mentioning that the values of $\unicode[STIX]{x1D6FE}$ in table 2 can also be recovered using the maximum ratio of the wall shear stress components as will be discussed in § 3.5 (see table 2, $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}/|\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}$ ).

Until now we have considered only the stable laminar ESBL and used 1-D theory to obtain the axisymmetric near-wall velocity solution over a curved and inclined (conical) wall. In practical applications, however, one is more frequently confronted with (locally) unstable or turbulent flows, which lack specific symmetries. In the following, we will therefore put the focus on nonlinearity, instability and curvature effects by studying the unstable ESBL with DNS for a large libration amplitude ( $\unicode[STIX]{x1D700}=0.8$ ) but moderate Ekman number ( $E=4\times 10^{-5}$ ). We will discuss further the instability mechanism and how it affects the mean flow excited by comparing DNS for the unstable flow regime to those of the stable flow regime.

3.3 Görtler instability of the ESBL

In order to steer the discussion in the direction of the unstable flow conditions, let us begin with a brief review of the laminar flow and the instability mechanism at work in the case of a librating coaxial inner cylinder (geometry $G1$ ). In this case the induced flow is notably curved, but when the cylinder radius is large compared to the boundary layer thickness, it is well known that an SBL is established over the librating wall and curvature is barely notable for the laminar flow (e.g. Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012). The SBL is basically a damped wave, which follows the boundary oscillation and thus yields a strong velocity shear in the vicinity of the librating wall. The absence of the Ekman property is reflected in $\unicode[STIX]{x1D6FE}$ tending to infinity, which means that librational forcing and molecular momentum diffusion dominate the flow dynamics exclusively. Non-zero radial and axial velocity components in the DNS are the result of secondary flows that may be due to geometry, nonlinearity and lid effects under stable conditions (e.g. Wang Reference Wang1970; Busse Reference Busse2011) or due to instability (e.g. Noir et al. Reference Noir, Hemmerlin, Wicht, Baca and Aurnou2009; Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012). Focusing on the instability mechanism, it is worth pointing out that the SBL over the librating inner cylinder becomes centrifugally unstable once the libration amplitude exceeds a critical value. This means that curvature effects are vital for the instability. The instability is intermittent and confined to the vicinity of the librating wall. It was reported that structures very much alike Görtler vortices develop. These vortices accelerate the flow locally due to nonlinear advection, which results in a local enhancement of the Coriolis force and leads to a rapid propagation of the Görtler vortices into the fluid bulk (Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016).

The aim of the following paragraphs is to clarify how much of the instability mechanism of the SBL remains in an ESBL. We tackle this question by varying the wall slope of the inner cylinder (frustum).

Figure 4.

Contour plots of the instantaneous azimuthal $u_{\unicode[STIX]{x1D703}}$ (a–d) and radial velocity component $u_{r}$ (e,f) visualizing the Görtler instability of the ESBL for intermittently unstable flows driven by frustum libration with libration amplitude $\unicode[STIX]{x1D700}=0.8$ and frequency $\unicode[STIX]{x1D714}=1$ using an Ekman number of $E=4\times 10^{-5}$ . Panels (a,b,e) on the left correspond to the geometry $G2$ , panels (c,d,f) on the right correspond to the geometry $G6$ . The contours are shown in axial–radial (a–d) and wall-parallel–azimuthal sections (e,f) of the annular domain. The latter case, the wall-parallel surface is extracted at about $2\unicode[STIX]{x1D6FF}_{-}$ . The contours have been recorded at the end of the prograde libration half-period when the instability is fully developed: libration phase $\unicode[STIX]{x1D711}=7\unicode[STIX]{x03C0}/8$ (a,c,e,f), and $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$ (b,d).

For the selected Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{0}r_{1}^{2})=4\times 10^{-5}$ all the geometries $G1$ – $G7$ were found to be unstable for a dimensionless libration amplitude of $\unicode[STIX]{x1D700}=0.8$ , and remained stable for $\unicode[STIX]{x1D700}=0.2$ . In that respect the ESBL studied here behaves very similar to the SBL investigated by Sauret et al. (Reference Sauret, Cébron, Le Bars and Le Dizès2012) and Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016), i.e. the ESBL becomes centrifugally unstable in the prograde half-phase of a libration cycle. The DNS results shown in figure 4 reveal the generation of Görtler vortices over the librating frustum. A few representative examples of the developed instability are given, whose details and consequences will be discussed further in the following.

Figure 4 shows various contour plots of instantaneous velocity components in different sections of the annular domain. Figure 4(a–d) shows $u_{\unicode[STIX]{x1D703}}$ in axial–radial slices for two different configurations $G2$ (a,b) and $G6$ (c,d), and for two different libration phases $\unicode[STIX]{x1D711}=7\unicode[STIX]{x03C0}/8$ and $\unicode[STIX]{x03C0}$ at the end of the prograde libration half-period. Figure 4(e,f) shows a wall-parallel–azimuthal section of $u_{r}$ for the libration phase $\unicode[STIX]{x1D711}=7\unicode[STIX]{x03C0}/8$ at the end of the prograde libration cycle. Geometry $G2$ exhibits a small effective Coriolis parameter $f_{\ast }$ and represents the transitional regime (iii) where the viscous diffusion is much larger than the Coriolis force ( $\unicode[STIX]{x1D6FE}\sim O(10)$ ). Geometry $G6$ exhibits the second largest value of $f_{\ast }$ investigated and has $\unicode[STIX]{x1D6FE}\sim O(1)$ .

Mushroom-like structures in the axial–radial plane and azimuthally elongated structures along the wall are clearly visible in figures 4(a–d) and 4(e,f), respectively. The later have been obtained by taking a wall-parallel–azimuthal section approximately $2\unicode[STIX]{x1D6FF}_{-}$ above the frustum. Streaks of positive $u_{r}$ are surrounded by those of negative $u_{r}$ which suggests that the propagation of the Görtler vortices out of the ESBL (and into the bulk) is compensated by a local flow entrainment into the ESBL. The close visual correspondence of the elongated vortices in the ESBL and the Görtler vortices observed in an SBL (see Lopez & Marques Reference Lopez and Marques2011; Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012; Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016) suggest that the structures in the unstable ESBL are also Görtler vortices. The time sequence of events during the instability development favour this interpretation. Assuming that the Stokes property of the ESBL is the cause of the instability, the analysis of Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016) for a pure SBL can be applied straightforwardly to the ESBL: First, the Görtler vortices are generated by a centrifugal instability mechanism after the frustum started to decelerate. Second, the Görtler vortices propagate radially into the fluid bulk by forming mushroom-like structures that decay as time progresses. The axial–radial sections in figures 4(a,b) and 4(c,d), respectively, reveal the temporal evolution of the Görtler vortices in the ESBL.

An interesting feature of the Görtler vortices in the ESBL is their tilt with respect to the azimuthal coordinate. This tilt is apparent in figure 4(f) and can be quantified by defining a deflection angle $\unicode[STIX]{x1D6FD}$ . One could argue that the Coriolis force induces the tilt of the Görtler vortices so that $\unicode[STIX]{x1D6FD}$ might not depend on $\unicode[STIX]{x1D6FC}$ but rather on the time scale ratio $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D714}/f_{\ast }$ . The values of $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FD}$ for the geometries $G1$ – $G7$ are given in table 2.

Figure 5.

Schematic drawing of the laminar boundary layer flow in terms of the boundary layer mass flux  $\boldsymbol{Q}$ in the unstable phase where Görtler vortices develop (a,b), and the stress angle $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}$ normalized by the vortex deflection angle $\unicode[STIX]{x1D6FD}$ as function of the wall inclination angle $\unicode[STIX]{x1D6FC}$ (c). Stress angles of the stable and unstable flow have been normalized using $\unicode[STIX]{x1D6FD}$ from the $\unicode[STIX]{x1D700}=0.8$ case. The streamlines are aligned with the azimuthal coordinate for the inner cylinder (a), but oblique over the frustum (b). The tilt of the streamlines over the frustum tends to increase the effective curvature radius of the local flow ( $r^{\prime \prime }>r^{\prime }$ for locally comparable wall radii).

The tilt of the Görtler vortices can be related to the direction of the flow within the boundary layer, that is, the direction of the wall shear stress that will be discussed in detail in § 3.5. Figure 5(a,b) shows a sketch of the flow along the inner cylinder (a) and the inner frustum (b) during the unstable phase of the libration cycle. The flow is not just azimuthal in the case of the frustum like for the pure Stokes boundary layer and neither just down slope like for the pure Ekman flux. The sum of both effects leads to a spiral-shaped downward flow and the Görtler vortices will be mainly oriented parallel to this flow. This can be analysed quantitatively by comparing the deflection angle $\unicode[STIX]{x1D6FD}$ to the components of the wall shear stress components $\tan \unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}=|\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}/|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}$ (see table 2). Figure 5(c) shows the normalized stress angle $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6FD}$ as a function of the wall slope $\unicode[STIX]{x1D6FC}$ . This normalization is arbitrary for $\unicode[STIX]{x1D700}=0.2$ since there are no Görtler vortices for this case. Under these stable conditions, $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6FD}$ can be interpreted as the tilt angle that could be expected if the flow would be unstable for $\unicode[STIX]{x1D700}=0.2$ . As can be seen in figure 5(c), for both stable and unstable conditions, with increasing cone angle  $\unicode[STIX]{x1D6FC}$ the stress angle $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}$ becomes slightly smaller than the Görtler vortex deflection angle $\unicode[STIX]{x1D6FD}$ but the dependency is rather weak. Linear regression $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D6FD}=b\unicode[STIX]{x1D6FC}+c$ yields a slope of $b=-0.034$ for $\unicode[STIX]{x1D700}=0.8$ and $b=-0.015$ for $\unicode[STIX]{x1D700}=0.2$ , respectively. This has an interesting consequence. The effective curvature radius $r^{\prime \prime }$ increases with $\unicode[STIX]{x1D6FC}$ as sketched in figure 5(b) so that a critical value of $\unicode[STIX]{x1D6FC}$ can be expected above which the Görtler instability will vanish. We will see later that the Görtler instability is the main driver for a bulk mean flow, which means we can expect that for large cone angle $\unicode[STIX]{x1D6FC}$ this mean flow will be significantly reduced.

Finally we give the power scaling of the deflection angle $\unicode[STIX]{x1D6FD}$ with respect to $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D714}/(2\unicode[STIX]{x1D6FA}_{0}\sin \unicode[STIX]{x1D6FC})$ . For the values listed in table 2 we find

(3.20a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})\approx 1.48\unicode[STIX]{x1D6FC}^{0.89}\quad \text{and}\quad \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FE})\approx 29.4\unicode[STIX]{x1D6FE}^{-0.89}, & & \displaystyle\end{eqnarray}$$

which yields a close-to-linear dependency of the deflection angle on any of the librating wall’s inclination parameters. This is because the values of $\unicode[STIX]{x1D6FC}$ are small ( $\unicode[STIX]{x1D6FC}\leqslant 18^{\circ }$ ), so that a dependency on $\sin \unicode[STIX]{x1D6FC}$ is not notable because of the uncertainty in $\unicode[STIX]{x1D6FD}$ . The almost linear dependencies confirm that the cause of the Görtler vortex deflection is the Ekman flux $Q$ along the inner cone, which is given by (3.14) and (3.16) for the stable boundary layer here.

In order to clarify what determines the deflection of the Görtler instability, the near-wall flow structure has to be addressed more carefully. The laminar ESBL discussed in the previous section provides the environment in which the instability develops. We will therefore discuss the relevance of the Ekman flux and the wall shear stress in the following two sections.

3.4 Ekman flux of the ESBL under stable and unstable flow conditions

The axial displacement of the Görtler vortices can be related to the oscillating Ekman flux by considering the time scales of the instability and the forcing. The Görtler vortices develop only during a fraction of the prograde libration cycle, that is, between libration phase $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x03C0}$ . In the remaining $3/4$ of a libration period the Görtler vortices merely decay while propagating towards the fluid bulk. This suggests that the Ekman flux properties are only important in the fraction of the libration period during which the Görtler vortices develop.

To assess this further, we assume that the Ekman flux $Q$ is dominated by linear dynamics with a laminar near-wall flow, which is described by the analytical solution given by (3.14) and (3.16). For the geometries studied here we have to select the case with constant phase shift ( $\unicode[STIX]{x1D712}=\unicode[STIX]{x03C0}/4$ ). The amplitude is positive ( $A>0$ ) so that the Ekman flux becomes negative,

(3.21) $$\begin{eqnarray}\displaystyle Q=A\sin (\unicode[STIX]{x1D711}+\unicode[STIX]{x1D712})<0\quad \text{for }3\unicode[STIX]{x03C0}/4<\unicode[STIX]{x1D711}<7\unicode[STIX]{x03C0}/4. & & \displaystyle\end{eqnarray}$$

Remember that the local axes used for the analytical computation is oriented such that $Q>0$ corresponds to axially upward flow. Consequently, the Görtler vortices start to develop at $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$ in the ESBL with some upward directed Ekman flux, which decreases gradually while the Görtler vortices intensify. (This is supported also by the DNS results in figure 6 to be discussed shortly.) From there on, the Görtler vortices are immersed in a downward directed Ekman flux until they have decayed or left the Ekman flux influence region of the ESBL.

The evolution of the wall-parallel Ekman flux $Q$ over one libration period for statistically stationary flow conditions is shown in figure 6 for all the geometries investigated. Figure 6(a) shows the DNS results for the stable ESBL excited by wall libration with the dimensionless amplitude $\unicode[STIX]{x1D700}=0.2$ , whereas figure 6(b) shows those for the intermittently unstable ESBL excited with $\unicode[STIX]{x1D700}=0.8$ . The dimensionless libration frequency $\unicode[STIX]{x1D714}=1$ and the Ekman number $E=4\times 10^{-5}$ were kept constant.

The simulated Ekman fluxes $Q$ in the stable ESBLs shown in figure 6(a) are sinusoidal in time as expected from the analytical analysis. All time series pass through the same zero crossings at $\unicode[STIX]{x1D711}\approx 5\unicode[STIX]{x03C0}/8$ and $13\unicode[STIX]{x03C0}/8$ and the local extrema can be found half-way in between. This is interesting, since the phase shift is larger in comparison to the analytical solution given by (3.14) and (3.16). The origin of the remaining phase shift is not clear yet but potentially related to an interaction of the ESBL with the excited bulk flow. (We will discuss the mean bulk flow further in § 3.6.) For the rest, the Ekman flux of the stable ESBL agrees with the analytical solution, in particular with respect to the increase in amplitude together with cone angle due to an increase of the effective Coriolis force.

Figure 6.

Evolution of the spatially averaged wall-parallel mass flux $Q$ (Ekman flux) scaled with the dimensionless libration amplitude $\unicode[STIX]{x1D700}$ for statistically stationary flow conditions. $Q$ is a property of the ESBL driven up and down the librating frustum during one libration cycle. DNS results are shown for stable flow conditions excited with $\unicode[STIX]{x1D700}=0.2$  (a) and intermittently unstable flow conditions excited with $\unicode[STIX]{x1D700}=0.8$  (b). Time series are shown for all geometries $G1$ – $G7$ and exhibit an increase of $Q/\unicode[STIX]{x1D700}$ with the cone angle $\unicode[STIX]{x1D6FC}$ of the frustum. The grey region indicates the time interval of the development of the Görtler instability.

The simulated Ekman fluxes $Q$ in the intermittently unstable ESBL shown in figure 6(b) are neither sinusoidal nor symmetric even though the sinusoidal forcing is still dominating the time series. For this unstable flow regime, the negative part of the time series of $Q$ between $\unicode[STIX]{x1D711}=5\unicode[STIX]{x03C0}/8$ and $13\unicode[STIX]{x03C0}/8$ is notably modified, which can be attributed to the development of Görtler vortices for libration phases $\unicode[STIX]{x1D711}\geqslant \unicode[STIX]{x03C0}/2$ . However, once the Görtler vortices have decayed the laminar solution starts to dominate again for $\unicode[STIX]{x1D711}\gtrsim 7\unicode[STIX]{x03C0}/4$ . Consequently, the positive part of the time series of $Q$ is comparable to the stable regime (compare with figure 6 a).

The Ekman flux is affected more by the Görtler vortices when the cone angle $\unicode[STIX]{x1D6FC}$ and the effective Coriolis parameter $f_{\ast }$ are small and the time scale ratio $\unicode[STIX]{x1D6FE}$ is large. In the case of geometry $G2$ in figure 6(a,b), for example, the minimum value of $Q$ reached at $\unicode[STIX]{x1D711}\approx 9\unicode[STIX]{x03C0}/8$ in the stable regime is about eight times larger in magnitude than the corresponding value of $Q$ in the intermittently unstable flow regime. With increasing $\unicode[STIX]{x1D6FC}$ or $f_{\ast }$ (decreasing $\unicode[STIX]{x1D6FE}$ ) the negative values of $Q$ in the unstable flow regime are much closer to the values obtained in the stable flow regime. For the geometry $G7$ in figure 6(a,b), for example, the minimum value of $Q$ reached at $\unicode[STIX]{x1D711}\approx 9\unicode[STIX]{x03C0}/8$ in the stable flow regime is only approximately $1.16$ times larger in magnitude than the corresponding value in the unstable flow regime. Altogether, the Coriolis force (as property of the ESBL) appears to have a stabilizing effect on the boundary layer flow. We therefore conjecture that there is a critical cone angle $\unicode[STIX]{x1D6FC}_{c}$ (wall slope) above (below) which the Görtler instability ceases to exist for given values of the similarity parameters $\unicode[STIX]{x1D6FE}$ and $E$ of the laminar ESBL. The proof of this conjecture is beyond the scope of the present study but would be worth addressing elsewhere.

With close with noting that the development of the Görtler vortices causes near-wall mixing of angular momentum. The negative part of the time series of $Q$ is notably flattened (figure 6 b), which indicates that the structure of the ESBL is partly destroyed, but only temporarily. Therefore, not only the Görtler vortices are affected by the Ekman flux, but also the Ekman flux is modified by the Görtler vortices.

3.5 Wall shear stress components under stable and unstable flow conditions

For the geometry $G2$ with cone angle $\unicode[STIX]{x1D6FC}=3^{\circ }$ and effective dimensionless Coriolis parameter $f_{\ast }\approx 0.1$ it follows that the time scale ratio is $\unicode[STIX]{x1D6FE}\approx 10$ . Thus, according to the classification above, librational forcing dominates the boundary layer flow and the boundary layer is almost a Stokes layer. With $\unicode[STIX]{x1D6FC}$ increasing from $G2$ to $G7$ , $\unicode[STIX]{x1D6FE}$ decreases gradually to about $1.6$ so that the Coriolis force influences more and more the boundary layer dynamics. This change in quality is exhibited not only by the boundary layer thickness (see figure 2) but also by the wall shear stress, which is very sensitive to the flow structure above the wall. The wall shear stress components $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x1D70F}_{\Vert }$ (as defined by (2.8)) contain information about the ESBL and can therefore be used to characterize the boundary layer in addition to or as an alternative to the time scale ratio $\unicode[STIX]{x1D6FE}$ . The stable ESBL excited with the dimensionless libration amplitude $\unicode[STIX]{x1D700}=0.2$ is accurately described by the analytical solution (see (3.8)–(3.10) figure 3) so that we focus on the unstable ESBL in the following paragraphs.

Figure 7.

Evolution of the azimuthal wall shear stress component $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ (a) and the almost axial along-wall component $\unicode[STIX]{x1D70F}_{\Vert }$ (b) during a libration cycle in the case of statistically stationary flow conditions in the intermittently unstable flow regime (dimensionless libration amplitude $\unicode[STIX]{x1D700}=0.8$ ). Time series are shown for various geometries ( $G1$ – $G7$ ) with different wall inclinations (cone angles $\unicode[STIX]{x1D6FC}=0^{\circ }$ – $16^{\circ }$ ). The Ekman number and dimensionless libration frequency were fixed at $E=4\times 10^{-5}$ and $\unicode[STIX]{x1D714}=1$ , respectively. The grey region indicates the time interval of the developing Görtler instability.

Figure 7 shows various simulated time series of the scaled wall shear stress components $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D70F}_{\Vert }/\unicode[STIX]{x1D700}$ covering a complete libration period for the flow in the statistically stationary state under intermittently unstable flow conditions ( $\unicode[STIX]{x1D700}=0.8$ ). Each profile corresponds to one of the geometries $G1$ – $G7$ . The wall shear stress is scaled with $\unicode[STIX]{x1D700}$ in analogy to the 1-D boundary layer solution given by (3.12) and (3.13). For all geometries investigated, the boundary layer over the inner librating wall becomes unstable in the prograde half-phase of libration just after the wall libration has passed its maximum amplitude at $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$ , which is precisely the onset of the wall deceleration.

The azimuthal component of the wall shear stress $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ is shown in figure 7(a) and reaches its maximum value at $\unicode[STIX]{x1D711}\approx \unicode[STIX]{x03C0}/4$ during the prograde half-phase of libration. The amplitude decreases with increasing cone angle $\unicode[STIX]{x1D6FC}$ from $G1$ to $G7$ due to stronger Coriolis force effects. The time series of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ are almost symmetric in the prograde and retrograde libration half-phases. This indicates that the Görtler instability has only a weak effect along the azimuth, which is therefore the direction with merely linear dynamics.

The axial component of the wall shear stress $\unicode[STIX]{x1D70F}_{\Vert }$ is shown in figure 7(b) and reaches its maximum value at $\unicode[STIX]{x1D711}\approx 3\unicode[STIX]{x03C0}/4$ , which is also in the prograde half-phase of libration but phase shifted by $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2$ with respect to $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ . The maximum value increases with the cone angle $\unicode[STIX]{x1D6FC}$ or the wall-parallel component of the Coriolis force, respectively. This suggests that the straight cylinder case $G1$ has $\unicode[STIX]{x1D70F}_{\Vert }\neq 0$ only because the developing Görtler instability (compare with Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012; Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016) since the SBL lacks a boundary layer mass flux in the axial direction. The time series of $\unicode[STIX]{x1D70F}_{\Vert }$ are not symmetric with respect to the prograde and retrograde libration half-periods in contrast to $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ . This indicates a strong effect of the Görtler instability (i.e. nonlinear dynamics) in the along-wall direction. In fact, the Görtler vortices yield the largest gradients (shear) approximately in axial direction (see figure 4 e,f).

It is worth to note that the magnitudes of $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x1D70F}_{\Vert }$ are comparable in the DNS results. Moreover, the phase shifts of $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/4$ to the libration exhibited by $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x1D70F}_{\Vert }$ , respectively, are present in the DNS results and the 1-D laminar boundary layer solution, ((3.12) and (3.13), case with $\unicode[STIX]{x1D714}>f_{\ast }$ and $\unicode[STIX]{x1D70E}=-1$ ). This suggests that there is a trace of both the linear and nonlinear dynamics. The azimuthal flow is dominated by the viscous linear dynamics due to the forcing, whereas the axially directed flow is dominated by the nonlinear dynamics of the Görtler vortices.

Furthermore, note that the evolution of the wall shear stress components for stable flow conditions is perfectly sinusoidal and qualitatively similar to the Ekman flux shown in figure 6(a). The time series are therefore not shown here, but the corresponding values of the wall shear stress components are given in table 2. It is quite interesting that the ratio of the wall shear stress components $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}/|\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}$ during prograde libration is comparable for the stable and intermittently unstable ESBL (see table 2). This ratio apparently scales with $\unicode[STIX]{x1D6FE}$ so that its maximum value may be used for the flow regime identification of the ESBL as an alternative to $\unicode[STIX]{x1D6FE}$ . The ratio $|\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}/|\unicode[STIX]{x1D70F}_{\Vert }|_{-}^{max}$ reflects the asymmetry of the Ekman fluxes observed in the prograde and retrograde libration half-periods. For the stable ESBL, this ratio shows an asymmetry of approximately $15\,\%$ . For the intermittently unstable ESBL, the asymmetry increases with increasing cone angle $\unicode[STIX]{x1D6FC}$ so that it is approximately $65\,\%$ for the geometry $G7$ .

The maximum axial wall shear stress $|\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}$ is notably different for the stable and unstable flow conditions (see table 2). It exhibits an approximately linear dependency on the effective Coriolis parameter $f_{\ast }$ corresponding to an approximately inverse proportionality to $\unicode[STIX]{x1D6FE}$ (provided the libration frequency is constant), that is,

(3.22) $$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D70F}_{\Vert }|_{+}^{max}\approx \left\{\begin{array}{@{}l@{}}0.0028\unicode[STIX]{x1D6FE}^{-0.90}\quad \text{for }\unicode[STIX]{x1D700}=0.2,\\ 0.0034\unicode[STIX]{x1D6FE}^{-1.02}\quad \text{for }\unicode[STIX]{x1D700}=0.8.\end{array}\right. & & \displaystyle\end{eqnarray}$$

A similar analysis for the maximum azimuthal wall shear stress $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}$ (see table 2) reveals only a weak dependency on $\unicode[STIX]{x1D6FE}$ , namely $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}\approx 0.27\unicode[STIX]{x1D6FE}^{0.13}$ , for both the intermittently unstable ( $\unicode[STIX]{x1D700}=0.8$ ) and stable ( $\unicode[STIX]{x1D700}=0.2$ ) ESBLs. From geometry $G2$ to $G7$ , $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}$ changes by almost $20\,\%$ . But at the same time also the mean frustum radius $r_{m}=(r_{1}+r_{3})/2$ changes from $r_{m}=1$ ( $G1$ ) to $r_{m}=0.8$ ( $G7$ ) which is precisely a $20\,\%$ difference. This suggests that the variation of $|\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D703}}|_{+}^{max}$ is merely due to geometry and a result of the change of the wall velocity due to smaller mean frustum radius at larger cone angle $\unicode[STIX]{x1D6FC}$ .

So, what will be the net effect of the intermittently unstable ESBL flow? We address this question in the following by investigating the generation of azimuthal mean flows by distinguishing the stable and unstable ESBL.

3.6 Mean flow generation due to Görtler vortices in the unstable ESBL

In this section we focus on the generation mechanism of the mean flow in the case of an intermittently unstable ESBL (excited with the dimensionless libration amplitude $\unicode[STIX]{x1D700}=0.8$ ). We do not discuss in detail the mean flow generation mechanism in the case of a stable ESBL ( $\unicode[STIX]{x1D700}=0.2$ ), but we will describe important properties of this mean flow for comparison.

Figure 8 shows radial profiles of the scaled time-mean azimuthal velocity $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D700}^{2}$ (see § 2.5 for the definition of the mean) for the geometry $G2$ in the libration-dominated regime (iii) and $G6$ in the intermediate rotation-libration regime (ii), respectively. The mean flow is either due to a stable (figure 8 a,b) or unstable (figure 8 c,d) boundary layer flow. The corresponding radial profiles of the azimuthal mean velocity extracted at the axial position $z=1.2$ are shown in figure 8(e,f). In the following, we describe first the mean flow due to a stable ESBL and afterwards that due to the unstable ESBL.

Figure 8.

Azimuthal mean flow $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D700}^{2}$ for the stable (a,b) and intermittently unstable (c,d) ESBLs for the geometries $G2$ (a,c) and $G6$ (b,d). Radial profile of $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D700}^{2}$ at axial height $z=1.2$ for all the geometries from $G1$ to $G7$ for the stable (e) and intermittently unstable (f) ESBLs.

In the case of the stable ESBL (figure 8 a,b), both geometries $G2$ and $G6$ exhibit a prograde jet-like structure in the azimuthal mean velocity field. This jet originates from the bottom corner, extends to the top lid and aligns then with the rotation axis. For the geometry $G2$ , the cone angle is small ( $\unicode[STIX]{x1D6FC}=3^{\circ }$ ) so that the prograde jet remains attached to the frustum. This is different for the geometry $G6$ , where the wall is more oblique ( $\unicode[STIX]{x1D6FC}=15^{\circ }$ ) and the prograde jet has notably detached from the frustum at heights $z\geqslant 0.4$ . In addition, one can also see a retrograde mean flow between the frustum and the prograde jet. The radial profiles shown in figure 8(e) reveal that the position of the maximum of the prograde jet is quite stable and located at radial position $r=1.02$ for the geometries $G2$ – $G7$ investigated. Geometry $G1$ is somewhat different as it exhibits a SBL over the inner wall and only a weak prograde jet close to the inner cylinder (see also Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016, figure 3).

The prograde jet shown in figure 8(e) exhibits a dependency on the wall inclination in addition to the compensated $\unicode[STIX]{x1D700}^{2}$ -dependency. The magnitude of the jet increases with the cone angle $\unicode[STIX]{x1D6FC}$ or the local Coriolis parameter $f_{\ast }$ , respectively. The maximum jet velocity $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}^{max}/\unicode[STIX]{x1D700}^{2}$ is located at $z=1.2$ and $r=1.02$ and exhibits an almost linear dependency on $f_{\ast }$ . A least-squares fit of the data gives

(3.23) $$\begin{eqnarray}\displaystyle \langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}^{max}/\unicode[STIX]{x1D700}^{2}=0.23f_{\ast }^{1.14}. & & \displaystyle\end{eqnarray}$$

It is interesting that the structure of the prograde jet shown in figure 8(e) is reminiscent of the Stewartson layer in a librating spherical shell geometry, in which it is caused by the boundary layer ‘eruption’ at the equator (see e.g. Sauret & Le Dizès Reference Sauret and Le Dizès2013). For the present geometry, however, the origin of this ‘Stewartson layer’ is not yet fully clear. It seems that the divergence of the wall-tangential Ekman flux near the corners adjacent to librating parts of the wall (here in particular the bottom inner corner) is crucial. This hypothesis is supported by the localized boundary layer eruption of the corner flow suggested by Klein et al. (Reference Klein, Seelig, Kurgansky, Ghasemi, Borcia, Will, Schaller, Egbers and Harlander2014).

It is also worth noting that the retrograde mean flow is absent for the geometries $G1$ and $G2$ (see figure 8 e). The generation mechanism for this retrograde flow is completely unclear at present. We conjecture that it is due to the recurrence of low frequency inertial waves to the frustum. These waves propagate mainly in axial direction so that they can get focused towards the upper inner corner provided their frequency is low enough or the cone angle of the frustum large enough (for details see Borcia & Harlander Reference Borcia and Harlander2012; Klein et al. Reference Klein, Seelig, Kurgansky, Ghasemi, Borcia, Will, Schaller, Egbers and Harlander2014). This scenario yields strong localization of wave energy and momentum in the vicinity of the upper inner corner so that the ‘trapped’ waves can interact nonlinearly with each other and/or the ESBL. The retrograde mean flow follows from mixing of background angular momentum (e.g. Maas Reference Maas2001).

In the case of the intermittently unstable ESBL, the prograde azimuthal mean flow tends to cover a larger radial fraction of the domain (figure 8 c,d) in comparison to the stable ESBL (figure 8 a,b). This is even better visible in figure 8(f), where the radial profiles of the compensated azimuthal mean velocity $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D700}^{2}$ are shown. One can see that the prograde mean flow is typically composed of two peaks for the geometries $G2$ to $G5$ which suggests presence of two different generation mechanisms. The first local maximum is located closer to the librating frustum and the second further away towards the bulk. The first is larger than the second for the geometry $G2$ , while it becomes smaller than the second for $G3$ – $G5$ , and disappears for the geometries $G6$ and $G7$ . The first maximum decreases with increasing cone angle $\unicode[STIX]{x1D6FC}$ or local Coriolis parameter $f_{\ast }$ , respectively.

It is worth noting that the mean flow magnitude in the case of an unstable ESBL approaches $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}^{max}/\unicode[STIX]{x1D700}^{2}\approx 0.09$ whereas it keeps increasing in the case of the stable ESBL (see lines for $G2$ – $G7$ in figure 8 e,f). The results in figure 8(f) reveal that the approximately linear dependency $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}^{max}/\unicode[STIX]{x1D700}^{2}\propto f_{\ast }$ according to (3.23) is not valid for the unstable ESBL.

Geometry $G1$ yields an interesting limit by exhibiting only a single-peaked jet (first maximum). This suggests dominance of one generation mechanism. The maximum value $\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}}^{max}/\unicode[STIX]{x1D700}^{2}$ is located in the vicinity of the frustum wall and exhibits a jump of more than an order of magnitude when the ESBL changes from stable to unstable flow conditions. It is the result of the generation of Görtler vortices in the unstable SBL (Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016) and this behaviour is still visible in the geometry  $G2$ . So, what happens for the geometries $G3$ – $G7$ ?

For larger wall inclination angles $\unicode[STIX]{x1D6FC}$ (measured relative to the axis), not only does the ESBL becomes thicker (see table 1) but also the deflection angle $\unicode[STIX]{x1D6FD}$ of the Görtler vortices increases (see figure 4 and table 2). This suggests that, with increasing $\unicode[STIX]{x1D6FC}$ or $f_{\ast }$ , the Görtler vortices travel further within the ESBL and decay partly there. This leaves less momentum and kinetic energy for the mean flow driven in the bulk (Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016).

The intensity of the Görtler vortices is not only reduced due to inertial (upscale energy transfer to the mean flow) and viscous losses, but also due to the cylindrical geometry. As discussed at the end of the § 3.5, the mean radius of the frustum decreases with increasing $\unicode[STIX]{x1D6FC}$ and so does the resultant prograde mean flow due to weaker background angular momentum near the librating frustum wall. Consequently, for a fixed axial position  $z$ well above the bottom lid, the contribution of the Görtler vortices to the generation of the prograde mean flow becomes smaller from $G1$ to $G7$ due to an increase of $\unicode[STIX]{x1D6FC}$ (see figure 8 f). In the geometry $G6$ (figure 8 d), for example, the dominating source for the mean flow is due to Görtler vortices close to the bottom inner corner (edge) and the prograde jet seems to originate from there precisely. Interestingly, the radial profiles of the azimuthal mean flow shown in figure 8(f) indicate that the bulk part of the prograde jet ( $r\geqslant 0.9$ ) is virtually the same in the two geometries $G6$ and $G7$ . This can be understood as follows. On the one hand, the deflection angle $\unicode[STIX]{x1D6FD}$ of the Görtler vortices increases from $G6$ to $G7$ which causes the vortices to travel further within the ESBL and along the frustum. Viscous losses dominate there so that the vortices contribute less to the generation of the prograde jet outside the ESBL. On the other hand, the Ekman flux of the ESBL increases from $G6$ to $G7$ (see figure 6) which yields a more intense source for the prograde jet near the bottom inner corner. This is backed by the empirical finding that the mean flow magnitude scales almost linearly with the Coriolis parameter $f_{\ast }$ (see (3.23)). This suggests that Görtler vortices contribute less to the prograde mean flow generation with increasing cone angle $\unicode[STIX]{x1D6FC}$ , but this is compensated by a larger amplitude of the Ekman flux in the libration-induced ESBL flow.

Altogether, the discussion of the previous paragraphs suggests that the first (inner) mean flow maximum (figure 8(f), $G2$ – $G5$ ) represents the contribution of the Görtler vortices, whereas the second (outer) maximum is due to the Ekman flux and the corner flow in the bottom inner corner. The source for the mean flow in the geometry $G2$ is therefore mainly due to Görtler vortices since the boundary layer flow is dominated by libration and viscosity (Stokes property of the ESBL). In the geometries $G6$ and $G7$ , the main source for the mean flow is the corner flow since the ESBL flow is strongly affected by rotation effects (Ekman property). The prograde jet in the geometries $G3$ to $G5$ is due to an intermediate situation where both the Görtler vortices and the Ekman flux (plus corner flow) have notable contributions. Note that the Ekman flux is also affected by the Görtler vortices, but this is merely restricted to the unstable phase of libration so that the Ekman flux does not fully vanish (figure 6). The perturbation of the Ekman flux, however, is relatively larger for smaller wall inclinations $\unicode[STIX]{x1D6FC}$ (measured relative to the axis).

3.7 RANS analysis of the prograde mean flow

In this section we consider in more detail the prograde mean jet flow and focus on the case of an unstable ESBL. The flow behaviour is studied in the statistically stationary state for which we inspect the Reynolds-averaged Navier–Stokes (RANS) equations by means of a combination of long time and spatial averaging of the DNS solution. We follow mainly the strategy used by Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016). That is, we still use the RANS equations in cylindrical coordinates and apply them here to the rotating flow in an annular confinement without axial–radial mirror symmetry. The motivation for a cylindrical reference system is given by the simulated azimuthal mean flow, which has a columnar spatial structure aligned with the axis of rotation (see figure 8). Furthermore, the RANS equations have substantially less terms in the cylindrical basis than in the generalized curvilinear one we used for the numerical solver (as described in § 2.3) and this simplifies our analysis.

Figure 9.

Axial and radial Reynolds stress terms $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}/(\unicode[STIX]{x1D700}^{2}f_{\ast })$ (a) and $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}/(\unicode[STIX]{x1D700}^{2}f_{\ast })$ (b), respectively. A stable $(\unicode[STIX]{x1D700}=0.2)$ and an intermittently unstable ESBL $(\unicode[STIX]{x1D700}=0.8)$ are considered for the geometries $G2$ , $G4$ and $G6$ .

In the statistically stationary state, the azimuthal mean flow is axisymmetric and has reached a balance in the averaged sense. This implies that the terms with the time derivative and those with azimuthal derivatives vanish. The azimuthal ( $\unicode[STIX]{x1D703}$ ) component of the RANS equations in the cylindrical co-rotating frame of reference thus reads

(3.24) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}\langle \widetilde{u}_{r}\rangle _{z\unicode[STIX]{x1D703}}}{r}+f\langle \widetilde{u}_{r}\rangle _{z\unicode[STIX]{x1D703}}+\langle \widetilde{u}_{r}\rangle _{z\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{r}\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}\approx E\left(\unicode[STIX]{x2202}_{r}^{2}\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}+\frac{\unicode[STIX]{x2202}_{r}\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}}{r}-\frac{\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}}{r^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \quad -\,\unicode[STIX]{x2202}_{z}\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}-\unicode[STIX]{x2202}_{r}\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}-2\frac{\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}}{r}.\end{eqnarray}$$

There are two remaining Reynolds stress terms in (3.24). One is $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ , the other one is $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ . Inspecting this equation for the geometry $G1$ , for which the boundary layer is a SBL, numerical results reveal that the radial turbulent transport is much larger than the axial one $(\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}\gg \langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}})$ . This case is investigated in detail by Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016, § D), who suggested neglecting the axial transport term.

Present numerical results for cases with the inner cylinder wall tilted with respect to the rotation axis (geometries $G2$ – $G7$ ), however, exhibit a different behaviour, suggesting that the axial Reynolds stress term $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ cannot be neglected in comparison to the radial one $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ . In these geometries, there is an ESBL over the frustum which causes $u_{\unicode[STIX]{x1D703}}^{\prime }$ to be coupled to $u_{z}^{\prime }$ in the vicinity of the wall (Ekman property of the ESBL). This results in a larger correlation $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ .

Figure 9 shows radial profiles of $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ and $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ for the stable and unstable ESBLs for the geometries $G2$ , $G4$ and $G6$ and for different $\unicode[STIX]{x1D6FC}$ or $f_{\ast }$ . As previously, the simulations are done for two different libration amplitudes $\unicode[STIX]{x1D700}$ . Assuming that each velocity fluctuation scales with $\unicode[STIX]{x1D700}$ and making use of the fact that the Ekman flux depends on $f_{\ast }$ , the Reynolds stress terms are scaled with $\unicode[STIX]{x1D700}^{2}f_{\ast }$ . (This compensates approximately the empirical mean flow scaling given in (3.23).)

In figure 9(a) one can see that the radial profiles of $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}/(\unicode[STIX]{x1D700}^{2}f_{\ast })$ for the intermittently unstable ESBL approach those of the stable ESBL with increasing $f_{\ast }$ (geometries $G2$ – $G6$ from right to left). For the geometry $G2$ , the librational forcing governs the ESBL dynamics (regime (iii)) and there is only a weak Ekman flux. By contrast, there is a considerable Ekman flux parallel to the frustum for the geometries $G4$ and $G6$ (compare with figure 6). In the cylindrical coordinates, the Ekman flux parallel to the frustum has always components in both the radial and axial directions, although with different magnitude. As $f_{\ast }$ increases, the Ekman flux increases, causes $u_{z}^{\prime }$ to be increased and results in a larger correlation $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ ( $\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}^{min}=-0.0024$ for $G2$ , but $-0.010$ for $G6$ ). In figure 9(b) and for the geometry $G2$ (regime (iii)) one can see that $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}/(\unicode[STIX]{x1D700}^{2}f_{\ast })$ is larger for the unstable ESBL compared to the stable ESBL. As $f_{\ast }$ increases ( $G2$ – $G6$ , from right to left), the radial profile of the radial Reynolds stress term of the unstable ESBL approaches that of the stable ESBL. For the geometry $G6$ (regime (ii)) those profiles almost coincide again.

Note that the radial profiles shown in figure 9(a,b) are notably different for the stable and unstable ESBL. This is most notable for the geometry $G2$ with weakest $f_{\ast }$ . As we saw already in figure 6(b), the Ekman flux is weak and therefore substantially perturbed by the Görtler vortices. This seems to explain also the differences seen in the Reynolds stresses. Following Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016, §§ C and D), the Görtler vortices generate a mean flow due to the radial gradient of $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ outside of the boundary layer in the case of an inner straight cylinder (geometry $G1$ ). In the geometry $G2$ the situation does not change much implying that the Görtler vortices have a considerable influence on the mean flow in the vicinity of the ESBL. However, as $f_{\ast }$ increases, the ESBL becomes thicker and the vortices need to travel further within the boundary layer (see § 3.6). In the geometry $G6$ , this effect seems responsible for a smaller contribution of the Görtler vortices to the term $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ outside the boundary layer and, consequently, causes the mean flow to be diminished.

Note further that in the case of the stable ESBL there are no Görtler vortices. Nevertheless, there is considerable overlap of the profiles corresponding to the geometry $G6$ shown in figure 9(b). We attribute this contribution to the mean flow to the Ekman flux through the corner flow at the bottom which is now responsible for generating $\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}$ .

On the basis of numerical data we performed a scale analysis with the cylindrical RANS equation (3.24). We kept only those terms larger than a threshold of $5\,\%$ of the largest term. In addition, the $1/r$ -terms where neglected in favour of the radial derivatives because we are facing an azimuthal jet and this jet is relatively thin compared to the annular gap (this is analogous to Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016, §§ D and E). So, with these simplifications in (3.24) we obtain the empirical balance

(3.25) $$\begin{eqnarray}\displaystyle \underbrace{\unicode[STIX]{x2202}_{r}\langle \widetilde{u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}}_{RT}+\underbrace{\unicode[STIX]{x2202}_{z}\langle \widetilde{u_{z}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }}\rangle _{z\unicode[STIX]{x1D703}}}_{AT}+\underbrace{f\langle \widetilde{u}_{r}\rangle _{z\unicode[STIX]{x1D703}}}_{LA}\approx \underbrace{E\unicode[STIX]{x2202}_{r}^{2}\langle \widetilde{u}_{\unicode[STIX]{x1D703}}\rangle _{z\unicode[STIX]{x1D703}}}_{D}, & & \displaystyle\end{eqnarray}$$

where $RT$ is the radial turbulent transport, $AT$ is the axial turbulent transport, $LA$ is the local Coriolis force affecting the azimuthal balance and $D$ is the viscous diffusion. Equation (3.25) is an approximate balance for which the validity has been proven numerically as will be discussed shortly. In view of this, it is not surprising to see in figure 8(a–d) that the radial derivative dominates the viscous terms and that the Coriolis force has a notable contribution only due to the azimuthal mean flow. Note that the turbulent axial transport term $AT$ was not present in the diagnostic RANS equation for the straight annulus (geometry $G1$ ; see Ghasemi et al. Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016). We therefore conjecture that $AT$ is also related to the Ekman property of the ESBL which manifests itself otherwise mainly by driving the Ekman flux up and down the frustum wall.

Figure 10.

Radial profiles of the terms contributing to the approximate balance (3.25) for the geometries $G2$ in regime (iii) (a) and $G6$ in regime (ii) (b) for the intermittently unstable ESBL. The contributions shown are the turbulent axial and radial transport terms ( $AT$ and $RT$ ), the Coriolis force ( $LA$ ), the viscous diffusion ( $D$ ) and the balance as the difference of the right- minus left-hand side ( $R-L$ ). The ESBL thickness $\unicode[STIX]{x1D6FF}$ is given for orientation and corresponds to the laminar value $\unicode[STIX]{x1D6FF}_{-}$ according to equation (3.6).

Figure 10 shows the radial profiles of the different terms given in (3.25) for the case of an unstable ESBL in the geometries $G2$ and $G6$ corresponding to the flow regimes (iii) and (ii), respectively. In figure 10(a), one can see that $RT$ is almost balanced by $D$ for the geometry $G2$ . As we discussed at the beginning of this section, the situation for the geometry $G2$ is very close to the geometry $G1$ and the term $AT$ is generally weak since it is related to the Ekman flux, which is weak here, too. This is in agreement with the previous discussion of figure 8 where we concluded that, for the geometry $G2$ , the prograde mean flow for the unstable ESBL is mainly due to the Görtler vortices. For the geometry $G6$ the magnitudes of $RT$ and $AT$ are very similar outside of the ESBL, but both terms have opposite signs and their magnitude is much larger than that of $D$ . Therefore, one can argue that in (3.25) $RT$ takes the role of a generating agent, whereas $AT$ takes that of a destructing agent for the prograde mean flow. This argument is supported also by figure 8, in which we saw that the prograde azimuthal mean flow is actually missing over the largest part of the frustum for the geometry $G6$ . Instead, only a prograde jet was originating from the bottom corner.

Note that the positive sign of the mean flow driven by an unstable ESBL can also be understood with the aid of (3.25). The negative sign of $RT$ outside of the boundary layer implies generation of a positive (prograde) mean flow. This is fully analogous to Ghasemi et al. (Reference Ghasemi, Klein, Harlander, Kurgansky, Schaller and Will2016, § D).