5.3
Symmetric and asymmetric flow patterns
This section presents the time-averaged velocity profiles for purely laminar, transitional and fully turbulent regimes, respectively. Particular attention is given to the transitional regime to characterize asymmetry in time.
5.3.1
Asymmetry parameter
An asymmetry parameter α is introduced to quantify the degree of azimuthal flow asym- metry and reveal how the asymmetry develops with streamwise distance along the pipe. The parameter is defined as:
α = RR 0 R2π 0 |u(r, θ ) − uM(r)| · r · dθ · dr RR 0 uM(r) · 2πr · dr (5.5) where uM is the arithmetic mean of the streamwise velocity u at the same specified radial loca-
tion r along the circumference direction defined as below:
uM(r) =
R2π
0 u(r, θ ) · dθ · r
2πr (5.6)
The velocity integral is weighted by 2πr such that the flow rate can be considered from the physical perspective. The uM in Equation 5.6 is actually linked with volumetric flow rate if the
axial velocity profile is perfectly axisymmetric.
5.3.2
Time-averaged velocity profiles
Previous studies regarding the asymmetry mainly focussed on the description of the time- averaged streamwise velocity profile by means of LDV. The LDV data was generally collected in hours to produce the time-averaged velocity profile (it took 3 to 4 hours to generate a velocity profile consisting of 50 probing points at a certain Reynolds number using LDV in experimental work of Escudier et al. [54]). In the current SPIV study, a frame rate of 5Hz is used to compare with LDV, and the corresponding dimensionless acquisition time t∗≈1720 (dimensional time
is 200s) when the flow is in the laminar-turbulent transition regime at Rew≈7939. Figure 5.8
shows the time-averaged axial velocity profiles at different Reynolds numbers. The flow pat- terns from the SPIV data present axisymmetric velocity profiles for both laminar (α ≈ 0.021, the definition of asymmetry indicator α is in Equation 5.5, Rew ≈ 990) and turbulent flow
(α ≈ 0.047, Rew ≈ 14720) and a significant departure from axisymmetry is observed when
flow is in the transitional regime (α ≈ 0.157, Rew≈ 7939). This is consistent with observations
from previous LDV data but the novel SPIV data here provides a three-dimensional view of the asymmetrical flow pattern. In a single run of camera recording, 1000 successive pairs of PIV images can be obtained. For a good statistical convergence purpose, 21 repeated runs were implemented at the same flow condition thus a total of 21000 realizations of the instantaneous velocity field were acquired (the according overall experimental duration is approximately 60 minutes). Intriguingly, the asymmetry prefers to stay in a certain azimuthal location in r − θ plane in all of the datasets tested. The ensemble average of all the SPIV experimental datasets validates the existing LDV results and confirms the asymmetry in the transitional regime.
Figure 5.8: Time-averaged streamwise velocity profile normalized by bulk velocity for laminar (a), transitional (b) and turbulent (c) flow regime, respectively.
5.3.3
Time-varying nature of asymmetry
One of the major insights provided by our SPIV measurements in comparison to all other previous work is the time-varying behaviour of the asymmetric flow pattern, which was hitherto considered stationary. The general consensus of previous LDV experiments was that, for the
5.3. SYMMETRIC AND ASYMMETRIC FLOW PATTERNS
asymmetric flow pattern, the location of the peak velocity remained at a fixed point in space. This agrees with our new experimental data to some degree since the time-averaged SPIV ve- locity profile also reveals the asymmetry pattern prefers to stay a certain location in r − θ plane after a long time averaging, however we found this asymmetry is indeed not stationary. This conclusion is supported by a direct numerical simulation of pipe flow using power-law model suggested by Rudman et al. [139]. The authors wrote “active region of the flow continually moves along the pipe and appears to preferentially occur at one azimuthal location for ex- tended times, so that the average velocity profile shows some asymmetry”. However, in their work, due to the limitation of computational effort, the mean asymmetric velocity profile was estimated approximately in the domain of 100 pipe diameters (i.e. t∗=100) and the authors ob-
served the asymmetry by extracting the contours of instantaneous axial velocity but did not give a comprehensive description of the asymmetry.
The following will provide experimental evidence of the time-varying nature of the asym- metry and attempt to give a more clear scenario about the asymmetry in contrast with the DNS conclusion. The experiments were repeatedly concluded in the transitional regime (Rew≈7939).
The dimensionless time t∗ for one single run was roughly 1720 and 1000 successive velocity
vector maps were obtained. The time history of the asymmetry factor and selected instantaneous velocity profiles are presented in Figure 5.9. The variation of asymmetry factor clearly indicates the flow pattern is not stationary, although the asymmetry preferentially appears in certain az- imuthal position. When the asymmetry is very pronounced (indicated by a higher asymmetry factor α ≈ 0.22), the instantaneous flow pattern (Figure 5.9a) is similar to the time-averaged flow pattern (Figure 5.8b). This flow pattern dominates the flow in the vast majority of observa- tion instances and therefore has a significant influence on the asymmetric time-averaged veloc- ity profile. Some small amplitude fluctuation of the asymmetry factor α can be observed, for example, that in time domain t∗= 100 ∼ 300. Physically, the asymmetric flow pattern slightly
fluctuates around the favoured asymmetry position in the circumferential direction. The small amplitude fluctuation is essentially different from the flow event which is shown in Figure 5.9
Figure 5.9: The time-varying nature of the asymmetric flow pattern at Rew≈8000 is shown by
the time history of asymmetry factor, α (bottom) and instantaneous cross-stream snapshots of the streamwise velocity (U /Ub). (a) asymmetric flow with preferred orientation, (b) temporarily
axisymmetric flow induced by a turbulent puff, and (c) a brief visit to asymmetric flow with an alternative orientation. The experimental duration is approximately 200s and the SPIV acquisi- tion rate is 5Hz. The two insets show the location of the peak velocity in the radial-azimuthal plane as the asymmetry returns following the passing of a puff.
(b), where the asymmetry factor abruptly drops to a low value, indicating that the flow is briefly returning to a quasi-axisymmetric state. In Figure 5.9 (b), the instantaneous velocity presents the “breakdown” of this asymmetry and clearly flow is restored to more axisymmetric pattern instantaneously. The time scale of asymmetry to axisymmetry transition is comparatively short with a dimensionless time scale t∗≈10. Applying Taylor hypothesis [36, 111] to transfer the flow event from time domain to space domain, it implies the asymmetry disappears in a spatial scale of ≈10 pipe diameters and in this domain, the flow is characterized by a strong random turbulent velocity fluctuation.
Although the asymmetric flow pattern is most commonly observed in this orientation, there are numerous brief instances when the orientation is different. One example is indicated in