Theoretical and numerical investigation of flow transition in rotating curved annular pipes (Q1405763)

The authors investigate flow transition phenomena in rotating curved annular pipes theoretically and numerically. The simplified equations are solved by employing the perturbation method, and a perturbation solution is presented up to the second order. The fully nonlinear equations for axial velocity and axial vorticity are solved by the finite volume method. Analysis indicates that this flow is governed by four parameters: the Dean number \(De\), the force ratio \(F\), the curvature \(\kappa\) and the radius ratio \(\delta\). However, for simplified problems with small curvature, the governing parameters are \(De\), \(F\), \(\delta\) and Rossby number \(Ro\). Comparisons are made between the numerical results and the perturbation solutions. They are in close agreement for small curvature and small Dean number. However, the numerical solution can be used to estimate the range of validity of the perturbation solution. The secondary flow direction and number of cells are discussed with varying \(F\), and the complete reversal secondary flow and the four-cell, six-cell and eight-cell structures are also reported. Whether the maximum axial velocity is near the inner bend or the outer bend, depends on the value of \(F\), but is also affected by curvature. Increasing curvature significantly affects the flow structure which indicates that the simplified problem for a small curvature really loses some flow properties. Increasing the inner circle radius enlarges the cell near the inner wall. The maximum stream function \(\psi_{\text{max}}\), which represents the secondary flux, is used to quantify the intensity of the secondary flow. The behaviour of \(\psi_{\text{max}}\) with varying \(F\) is similar for different curvatures. When \(F\) decreases from \(1.0\) to \(-3.0\), \(\psi_{\text{max}}\) first decreases and reaches its minimum value at \(F\approx-1.0\), then it increases. Obviously, it should be of interest in engineering applications, because the secondary flow is closely related to the heat and mass transfer of the flow in a curved system. The Taylor-Proudman phenomenon is obtained when \(Ro\) is small. Two procedures are used to approach this flow structure. Although the two procedures reach the same flow structure, their effects on the secondary flow intensity are completely different. The wall shear stresses \(\sigma_s\) and \(\sigma_\theta\) are discussed for different \(F\) and \(\kappa\). The distribution of \(\sigma_s\) along the wall is similar to that of axial velocity, while \(\sigma_\theta\) is similar to that of a secondary flow. The friction factor ratio \(\lambda_c/\lambda_\theta\) increases for co-rotation with \(F\) increasing, but for counter-rotation, \(\lambda_c/\lambda_\theta\) first decreases and reaches its minimum about 1 at \(F\approx -1\), then it increases with \(| F| \) increasing. The larger the curvature, the larger \(\lambda_c/\lambda_\theta\) will be. \(\lambda_c/\lambda_\theta\) decreases with \(\delta\) increasing. Although the secondary structure is more complicated in a rotating curved annular pipe than that in a circular pipe or a square duct, multiple solutions are not detected in this work. The authors assume that the existing inner circle wall stabilizes the flow and makes the critical Dean number larger than that for the circular and rectangular cross-section.