Flow in a weakly curved square duct: Assessment and extension of Dean's model

The simplified model by W. R. Dean, based on a low-curvature assumption, provided an early understanding of the laminar flow in curved ducts. However, most of the following studies relied on computer simulations of the complete curvilinear Navier-Stokes equations controlled by two nondimensional parameters, the curvature ratio and the Reynolds number. In the present article an extended version of Dean's model is used and compared to existing results. The set of equations is unsteady, parabolic in a streamwise direction, expressed in Cartesian coordinates, and contains a single control parameter, namely, the Dean number. The equations are identical to those used for the Görtler instability in boundary layer flows. Nonetheless, their extension to duct flows remains to be validated, which is the main purpose of the present article. The model satisfactorily reproduces benchmark results in the literature. In particular we retrieve the successive bifurcations between steady, unsteady, and chaotic regimes for 2D flows. The model also reproduces the development of three-dimensional flow in an elbow with a curvature radius equal to 15.1 times the square duct width. In addition, the present results confirm the Dean number as the single control parameter for laminar flows in a weakly curved ducts.

Figure 1

Representation of the Cartesian coordinates system in a weakly curved duct.

Figure 2

Filled circles: values of the squared 2-norm of the velocity vector at the instants when the Poincaré section is crossed. Open circles: values of the square of the 2-norm of the velocity vector in the stationary cases. The first bifurcation from a steady regime to an unsteady, periodic regime occurs at $\text{De}=128.32$. Starting from $\text{De}\approx 193$ we observe two periods which are incommensurable, characteristic of an aperiodic regime. Then the system quickly evolves into chaos from $\text{De}\approx 195$. For $\text{De}\in [227,318]$ the system is again stationary, and after a narrow periodicity interval the system becomes again chaotic at $\text{De}=324$. A magnification of the bifurcation diagram for $\text{De}\in [190,230]$ is shown in the inset

Figure 3

Injection-velocity norm phase diagram at $\text{De}=130$. The system follows a periodic orbit passing close to the two stationary solutions, indicated by the solid circle ($I\approx {41.5,\left|\right|\mathit{u}\left|\right|}_2^2\approx 0.655$) and the solid square ($I\approx {43.5,\left|\right|\mathit{u}\left|\right|}_2^2\approx 0.647$). The small dots along the curve are recorded for a constant $\mathrm{\Delta }t$. The higher density of points near the the two stationary solutions indicates the attractive nature of these fixed points. The long arrows indicate the instantaneous states of the system. The flow evolves following the small arrow: from the weak four-cells, symmetric state to the symmetric strong four-cells, when symmetry is broken a kinetic energy burst is observed. The system then returns to a weak four-cells state regaining its symmetry.

Figure 4

Streamwise velocity at $y=0.74,z=0.5$ for $\text{De}=125,127,150$, 3D flow development problem with straight, laminar inflow. Computed with the present parabolic model (lines), compared with the experiments by Bara et al. [23] (circles). Cross-flow velocity vectors are shown at $\theta ={12}^{\circ },{80}^{\circ }$, and ${300}^{\circ }$.