Dean instability in double-curved channels

We study the Dean instability in curved channels using the lattice Boltzmann model for generalized metrics. For this purpose, we first improve and validate the method by measuring the critical Dean number at the transition from laminar to vortex flow for a streamwise curved rectangular channel, obtaining very good agreement with the literature values. Taking advantage of the easy implementation of arbitrary metrics within our model, we study the fluid flow through a double-curved channel, using ellipsoidal coordinates, and study the transition to vortex flow in dependence of the two perpendicular curvature radii of the channel. We observe transitions not only to two-cell vortex flow but also to four-cell and even six-cell vortex flow, and we find that the critical Dean number at the transition to two-cell vortex flow exhibits a minimum when the two curvature radii are approximately equal.

Figure 1

Geometry of the curved channel in cylindrical coordinates $(r,\theta ,z)$. The colored cross sections depict the vorticity of the axisymmetric flow, where the blue (lower) and red (upper) spots correspond to low and high values of the vorticity, respectively. The dashed lines indicate the periodicity of the channel in the $\theta$ and $z$ directions.

Figure 2

Average vorticity $\langle {\omega }^{\theta }\rangle$ depending on the Dean number $\mathrm{De}$ for a cylindrical channel with aspect ratio $\eta =r_i/r_o=0.80$. As can be seen, the bifurcation from Poiseuille flow to two-cell vortex flow occurs at ${\mathrm{De}}_c\approx 42$, followed by a second bifurcation to four vortices at ${\mathrm{De}}_{c2}\approx 54$. The colored pictures depict the velocity streamlines on a channel cross section perpendicular to the stream direction. The colors represent the strength of the vorticity, where blue and red correspond to clockwise and counterclockwise rotating vortices, respectively.

Figure 3

Radial, streamwise, and axial velocity profiles for the four-cell vortex flow for a cylindrical channel with aspect ratio $\eta =0.80$ at Dean number $\mathrm{De}=56$.

Figure 4

Critical Dean number ${\mathrm{De}}_c$ at the bifurcation from Poiseuille flow to two-cell vortex flow as function of the aspect ratio $\eta =r_i/r_o$ for a cylindrical channel. The results obtained by our improved method agree very well with the literature values of Finlay et al.

Figure 5

Second critical Dean number ${\mathrm{De}}_{c2}$ at the bifurcation from two-cell vortex flow to four-cell vortex flow as function of the aspect ratio $\eta =r_i/r_o$ for a cylindrical channel.

Figure 6

Relative error of the critical Dean number as function of the number of grid points for a cylindrical channel with aspect ratio $\eta =r_i/r_o=0.9$.

Figure 7

Geometry of the double-curved channel. The colored spots on the channel cross section at the inlet illustrate the strength of the flow vorticity. The dashed lines indicate the periodicity of the channel in the $\varphi$ direction.

Figure 8

Average vorticity $\langle {\omega }^{\theta }\rangle$ as function of the Dean number $\mathrm{De}$ for a double-curved channel with inner radius $r_i=1$, aspect ratio $\eta =r_i/r_o=0.9$, and $b=0.9$. The bifurcation from Poiseuille flow to vortex flow occurs at $\mathrm{De}\approx 30.5$. The colored pictures depict the vorticity on a channel cross section, where the blue (right) and red (left) spots correspond to clockwise and counterclockwise rotating vortices, respectively.

Figure 9

Critical Dean number at the bifurcation points versus the streamwise curvature radius ${\mathcal{R}}_{\theta }=r_i{b}^2$ for a double-curved channel with aspect ratio $\eta =r_i/r_o=0.9$. The colored pictures depict the streamwise vorticity of the flow on a channel cross section.

Figure 10

Critical Dean number at the bifurcation points as function of the cross-sectional curvature radius ${\mathcal{R}}_{\varphi }=r_i$. The colored pictures depict the streamwise vorticity of the flow on a channel cross section.

Figure 11

Velocity profile of six-cell vortex flow in a double-curved channel with aspect ratio $\eta =0.9$, streamwise curvature ${\mathcal{R}}_{\theta }=r_i{b}^2=1$, and cross-sectional curvature ${\mathcal{R}}_{\varphi }=r_i=1.8$. The corresponding Dean number is $\mathrm{De}=44$.