# Asymmetry Propagation in a Pipe Flow Downstream of a 90\({}^{\circ}\) Sharp Sarp Elbow Bend
## 3 Results  
The main characteristics of the instantaneous velocity field in a pipe upstream and downstream of the sharp bend is shown in Figure 6. Clearly, the velocity boundary condition at the inlet provided a fully developed flow, which was axisymmetric in time-averaged fields, regardless of the \(Re\). This resulted in a laminar flow upstream of the elbow for the lowest \(Re\), \(Re=1400\), and a turbulent flow for the other two flow conditions, \(Re=5600\) and \(Re=10,000\). For each simulated case, there was a strong flow separation observed in the elbow of the pipe. The boundary layer detached at the sharp internal edge of the elbow, bent and reattached further downstream of the flow. The flow from the inlet pipe section impinged into the outer wall of the pipe in the elbow as well as downstream of it. Further downstream of the elbow, flow again achieved fully developed velocity conditions; however, the exact recovery length was not evident from the instantaneous velocity fields. Figure 4: Time-averaged streamwise velocity profiles for fully developed flow: (**a**) \(Re=5600\); (**b**) \(Re=10,000\). Figure 5: Time-averaged streamwise velocity fluctuations for fully developed flow: (**a**) \(Re=5600\); (**b**) \(Re=10,000\). The influence of the pipe elbow was more clearly depicted in the time-averaged velocity fields, which highlighted the development of Dean vortices. As illustrated in Figure 7, these vortices persisted for up to 25 D downstream of the elbow. At first, the Dean vortices displayed nearly perfect symmetry up to around 7 D. However, between 8 D and 10 D from the elbow, this symmetry broke down, resulting in chaotic behavior of the Dean vortices. In the region from 8 D and 10 D, the vorticity of the Dean vortices reversed direction, with the left vortex switching from clockwise to counterclockwise and the right vortex doing the opposite, as indicated by the coloured fields in Figure 7. Downstream of this region, the Dean vortices maintained their vorticity direction until approximately 26 D, where they began to decay in structure. These results were based on turbulent flow conditions at \(Re=5600\), though similar behavior was observed at \(Re=10,000\), with vorticity reversal occurring between 8 D and 11 D and the Dean vortices persisting up to 20 D. In contrast, under laminar conditions at \(Re=1400\), the development of the Dean vortices differed, with four strong vortices appearing between 8 D and 11 D, which gradually decayed further downstream. In order to estimate the flow recovery length, we calculated time-averaged velocity fields and introduced four different criteria. Namely, the first applied criterium was based on calculation of the time-averaged wall shear stress. This criterium gives useful information about flow anisotropy in the near-wall region; however, it reveals no or minimal insight about the flow anisotropy in the bulk region. For that reason, we also introduced profile asymmetry parameter \(\alpha\), which integrates discrepancies between profile function \(f(r)\) and \(f(-r)\) across the pipe diameter \(r\), i.e.,  
\[\alpha(x,\theta)=\int\limits_{-R}^{R}\frac{\left|f(x,r,\theta)-f(x,-r,\theta) \right|}{N}dr \tag{3}\]  
Figure 6: Instantaneous velocity magnitude field for the three simulated cases: (**a**) \(Re=1400\); (**b**) \(Re=5600\); (**c**) \(Re=10,000\). Blue and red color correspond to small and large velocity magnitude, respectively. where \(x\), \(r\), and \(\theta\) are axial, radial and azimuthal coordinates in the pipe downstream of the pipe elbow and \(R=D/2\) is the pipe radius. Clearly, the flow asymmetry \(\alpha\) is proportional to the area between curve \(f(r)\) and its mirrored counterpart \(f(-r)\), as shown with the purple area in Figure 8. For each axial location, a single value for \(\alpha\) is calculated, offering a concise insight into the flow's asymmetry.