1 Introduction

Junction flows occur in many aerodynamic and hydrodynamic situations in practice, e.g., external aerodynamics, turbomachinery, underwater vehicle, electronic component cooling, and river/bridge flows. In these cases, the complex interference flow fields and three-dimensional (3D) separations are produced by an upstream boundary layer on a surface that encounters an obstacle attached to the surface, e.g., a wing or cylinder, turbine blade, sail or conning tower, electronic chip, and bridge pier^[1]. Past experimental and numerical studies^[1-2] on junction flows are extensively reviewed, and the underlying physics are discussed. The most striking finding of previous experiments is that horseshoe vortices are dominated by coherent low-frequency unsteadiness and characterized by the bimodal histograms of velocity probability density functions (PDFs)^[3]. Especially, in turbulent junction flows at high Reynolds numbers, the vortices are highly unsteady and responsible for high turbulence intensities^[4], high surface pressure fluctuations^[5], high heat transfer rates^[6], and erosion scour in the nose region of the obstacle^[1]. In the vicinity of the junction vortex, though the turbulence stresses are much greater and reach values many times larger than those normally observed in turbulent flows, they are associated with the bimodal (double-peaked) histograms of velocity fluctuations produced by a bistable velocity variation. The observations are consistent with the large-scale low-frequency unsteadiness of instantaneous flow structures associated with the junction vortices.

According to the work quoted above, the intensive pressure fluctuations are associated with the unsteady motions of junction vortices. While the unsteady motions of a turbulent junction flow would be caused by a variety of reasons, e.g., unsteady movements of horseshoe vortices, instability of incoming flow or flow outside the boundary layer, and the interactions of the above. Therefore, it is too complex to reveal the underlying physics. This paper focuses on the unsteady motions of horseshoe vortices and their influence on pressure fluctuations. To get rid of other disturbances, it is going to discuss how the pressure fluctuations change when the unsteadiness of the horseshoe vortices in a laminar junction flow is attenuated. Whether the pressure fluctuations are reduced or not once the unsteadiness of junction vortices is weakened is crucial for the reduction of the intensity of the flow acoustic noise source of a junction flow. Accordingly, a series of experimental research on this topic is conducted in a laminar junction flow in a low speed wind tunnel by smoke wire visualization and two-dimensional (2D) time-resolved particle image velocimetry (PIV) measurements.

2 Experimental setup and mean velocity profile 2.1 Wind tunnel and test model

Experiments are carried out in a low-speed open-return wind tunnel with the test section dimensions 0.3 m× 0.3 m× 1.9 m, which is a through type tunnel with an exhaust recirculated back into the room via a series of filters. A 10 mm-thick clear acrylic flat-plate with a 2:1 elliptical leading edge and a sharp trailing edge is mounted horizontally in the test section and 0.15 m after a 7:1 area ratio contraction, with a 200 mm-distance left to install the circular cylinder vertically at the centreline of the flat-plate and below the top wall. The cylinder has the diameter D =31 mm and height 195 mm, and there is a 5 mm-gap intentionally above the top wall to prevent junction vortices and their impact on the junction flow interested in Refs. [7] and [8]. A sketch for the model arrangement and coordinates is shown in Fig. 1, where the flow is from the left to the right. The plane O-x'y is a vertical plane in an angle of 30° to the O-xy plane. The plane illuminated by the laser sheet for flow visualization and PIV measurements is in an angle of 30° to the symmetric plane (see the flow visualization and PIV meaurement area in Fig. 1), and the end-plate in yellow is located vertically in the symmetric plane upstream the nose of the cylinder. The two smoke-wires in blue solid lines are crossing the mean flow 4 mm or 9 mm above the flat-plate, respectively.

2.2 Mean velocity profile

The flow field around the circular cylinder-plate junction is studied with a Dantec dynamics 2D time-resolved PIV system in the O-x'y plane (see Fig. 1). Olive oil seeding particles with the diameter of nominally 1 µm are introduced upstream the wind tunnel contraction section from a TSI 9307-6 oil droplet generator via a seeding rake^[9]. Image pairs are typically taken at a frame rate of 1 000 Hz with a 100 µs interval. The mean streamwise velocity profile of the boundary layer above the flat-plate is detected by the time-resolved PIV. The particle image pairs are captured at a rate of 1 000 Hz for 4 s. When the freestream velocity is U_∞=1.6 m/s (correspondingly, Re_D=U_∞ D/ν), the thickness of the boundary layer δ ranges from 8.18 mm to 9.22 mm, herein U =0.99U_∞ at y=δ. The value is within 5% of the flat-plate solution δ/x_l = 5.0Re_xl^-1/2. The shape factor is H =2.58, and the mean streamwise velocity profile closely matches the Blasius solution for a laminar flat-plate boundary layer (see Fig. 2).

3 Unsteadiness of a laminar junction flow 3.1 Flow visualization

Smoke-wire flow visualization is conducted to figure out what is happening at the base of the circular cylinder (for the smoke-wire visualization of the whole field, please refer to Ref. [10]). Two wires with the diameter of 100 µm are horizontally placed upstream the cylinder by 55 mm within the incoming boundary layer, where the heights are 4 mm and 9 mm above the flat-plate, respectively. The O-x'y plane is illuminated by a laser sheet, and the images are captured at 1 kHz. Therefore, the time interval between the two adjacent images is 1 ms. A process of the vortex motions in a cycle is presented by six images in Figs. 3(a_i) (i=1, 2, ..., 6).

As seen from Fig. 3(a₁), the corner vortex (CV) goes downstream together with the primary vortex (PV) until when it is extremely close to the cylinder (see Fig. 3(a₂)), and changes its direction to go upstream, pointing to the PV in Fig. 3(a₃). Then, the CV runs into the PV, and starts turning around as a part of the PV, leading to deformations and oscillations (see Fig. 3(a₄)). Finally, the CV is merged into the primary vortex, and becomes a new CV of the next cycle (see Figs. 3(a₅) and 3(a₆)). In a cycle, the vortex pattern is alternating from six-vortex into four-vortex, which takes on a highly unsteady process. It is the first time to experimentally visualize the impact and merge into the horseshoe vortices in a laminar junction flow (see Fig. 3(a₄)). It is believed to be responsible for the unsteadiness of the laminar junction vortex system and highly oscillation of wake flows. The horseshoe vortices around the laminar junction mentioned above form upstream the cylinder, where the flow separates, rolls up, and stretches downstream and sideward. The developing vortex (DV) and PV are connecting to the incoming boundary layer indicated by the streamlines illuminated in Figs. 3(a_i) (i=1, 2, ..., 6) and keep themselves energetic by the energy transferring from the incoming flow. Moreover, the CV separation and energy transfer from the incoming flow are few. Therefore, it is the most isolated and weakest one of the three clockwise rotating vortices of the laminar junction flow, and goes upstream when it is not strong enough to stand the large adverse pressure gradient as going downstream up to around 9 mm upstream the cylinder in the 30° plane. The thickness Reynolds number of the vertical end-plate in the symmetric plane (see Fig. 1) is far less than the critical Reynolds number of separation, and the weakest CV is cut off. The vortices of the laminar junction flow are seemingly steady, as indicated by the smoke wire flow visualization in Figs. 3(b_i) (i=1, 2, ..., 6).

3.2 2D time-resolved PIV measurements

The 2D velocity fields are acquired by the planar time-resolved PIV, and the unsteadiness of the laminar junction flow is discussed with respect to the histogram for the fluctuating velocity (see Fig. 4) and the trajectory of the vortex core in the phase-averaged flow field (see Fig. 5) in both cases with and without end-plate control.

The PDFs of the longitudinal velocity fluctuations are at eight vertical locations above the flat-plate at x'/D=-0.47, i.e., x'=14.57 mm, marked by " ▲" in Fig. 3. They are within the reversal zone according to the smoke wire flow visualization (see Fig. 3). The histogram shape of the natural (see Fig. 4(a)) laminar junction flow varies as away from the flat-plate, i.e., bimodal (double-peaked) at y/D=0.014, trimodal (treble-peaked) at the next two locations, bimodal again from y/D=0.047 to y/D=0.058, and has a quasi-Gaussian distribution from y/D=0.091 and further except for the weakly bimodal one in the middle, which denotes the highly unsteadiness of a laminar junction flow. The histograms of the case with end-plate control (see Fig. 4(b)) have always a quasi-Gaussian distribution from the first vertical location y/D=0.015 up to y/D=0.091, which covers the reversal zone of a laminar junction flow (see Fig. 4(a)).

Besides, the trajectories of the vortex core of the phase-averaged flow field of a laminar junction give a direct view of the unsteady vortex motions of laminar junction flows (see Fig. 5). The trajectories of the vortex core of the CV, PV, and DV during a phase-averaged cycle of a natural laminar junction flow draw a curve in red at different levels. During a phase-averaged cycle, the three clockwise vortices are oscillating in a similar streamwise range from 6.9 mm to 7.8 mm with a 1 mm difference, which is tolerated by the spatial resolution of the PIV system and vortex core identification method. The compact and merging occur at a streamwise location, where the PV stops and the new CV starts (see Figs. 3(a_i) (i=1, 2, ..., 6) and Fig. 5), corresponding to the maximum height of the vortex core's trajectory of a cycle. Under the end-plate control, the trajectories of both CV and PV are oscillating within 1 mm, which might be due to the spatial resolution of the PIV system and vortex core identification method, and the DV oscillates in the same streamwise range as that of the case without control (6.9 mm), because the DV is mostly dominated by the incoming boundary layer, which is not changed apparently by putting a vertical end-plate in the symmetric plane. Therefore, the flow field becomes steady with respect to the fluctuating velocity histogram (see Fig. 4) and the trajectory of the vortex core of phase-averaged flow fields (see Fig. 5) by putting an end-pate in the symmetric plane of a laminar junction.

4 Statistics of the pressure field

Based on the phase-averaged and instantaneous velocity field acquired by the 2D time-resolved PIV, the fluctuating pressure fields are estimated by the multi-path integral method^[11] to reveal the effect of the end-plate control on the basis of the integration of the Navier-Stokes equation.

4.1 Square-mean-root of the pressure fluctuations

The distributions of the square-mean-root of pressure fluctuations within 4 s around a laminar junction flow with and without end-plate control are given in Fig. 6. Significantly intensive pressure fluctuations occur at a location slightly above the place where the unsteady horseshoe vortices appear (see Fig. 6(a)). The maximum value of this highly fluctuating area is 0.2 Pa at the center, i.e., (x'/D, y/D)=(-0.41, 0.08). Whereas there is no noticeable energy concentrated in the end-plate control case (see Fig. 6(b)). The pressure fluctuation is at a comparative or even higher level on the flat-plate surface. Therefore, the pressure field has been changed from a pressure concentrated pattern to a dispersive pattern by the vertical end-plate placed in the symmetric plane, i.e., the pressure fluctuation gradually increases as away from the solid wall.

4.2 Decomposition of phase-averaged pressure field

The decomposition of the pressure field is conducted to obtain the phase-averaged, none phase-averaged, and total pressure distributions in the streamwise direction, from the surface of the flat-plate up to y/D=0.12, of a laminar junction flow (see the solid curves in Fig. 7). The phase-averaged event contributes nearly all of the total values of the pressure fluctuations, both of which reach their maximum values around x'/D =-0.47. There is a tinny difference caused by the none phase-averaged event, which is related to the contour-rotating secondary vortex. The contour-rotating secondary vortex also leads to the third peak around u = 0 of the PDFs in Fig. 4(a) at x'/D = -0.47. In contrast to the "bell" shape distributions of pressure fluctuations in the natural laminar junction flow, the pressure fluctuates at a certain level, which increases as away from the wall in the end-plate control case. Similar to p_rms, the pressure fluctuation is not apparently concentrated in the reversal area in the end-plate control case. However, the pressure fluctuates at a quite high level in the near wall region, which means intensive excitation on the surface for the secondary noise radiation.

According to the above results, we know that the unsteadiness of a laminar junction is significantly weakened by putting an end-pate in the symmetric plane of a laminar junction, and consequently, the intensive pressure fluctuations in the reversal area will disappear. Meanwhile, the pressure fluctuations close to the surface of the flat-plate are enhanced under end-plate control, which means the intensive excitation on the surface of the secondary noise radiation.

5 Concluding remarks and discussion

Many experiments are conducted in a low-speed wind tunnel to reveal the physics of large-scale low-frequency unsteadiness of the instantaneous flow structure and their influence on the pressure field around a cylinder-plate junction. The experimental data and results of a natural laminar junction flow with and without end-plate control are discussed.

It seems that the flow field is dominated by a six- or four-vortex pattern alternating process in a natural laminar junction flow. It is the first time to experimentally report that the CV of a laminar junction flow stops moving downstream and changes its direction to run into and be merged into the PV, leading to shape changes and oscillations of the PV based on a series of snapshots of smoke-wire flow visualization in a 14 ms interval.

The flow field becomes a steady four-vortex system according to smoke-wire flow visualization and time-resolved PIV measurements by putting a vertical end-plate located in the symmetry plane upstream the nose of the circular cylinder. The unsteadiness of a laminar junction flow is discussed with respect to the smoke-wire flow visualization and PDFs of longitudinal velocity fluctuations. In the end-plate control case, the unsteadiness of a laminar junction is noticeably weakened by cutting off the CV of the laminar junction, and consequently, the intensive pressure fluctuations in the reversal area disappear. Meanwhile, the pressure fluctuations on the surface of the flat-plate are enhanced under the end-plate control. It means that the excitation on the surface for the secondary noise radiation of a laminar junction flow is intensified, even though the unsteadiness is noticeably weakened.

In a word, although the phase-averaged event takes a dominant role of intensive pressure fluctuations in the region of the vortex core in a laminar junction flow and the flow radiation noise source is weakened under end-plate control, pressure fluctuations are intensified on the surface of the flat-plate. This new finding leads to following up the investigation in the relationship between the unsteadiness and the intensity of the flow noise source of a laminar junction flow. In the future, further experimental and numerical works will be carried out, including the 3D velocity field and sound pressure in near field and far field so as to reveal the physics underlying.

Acknowledgements  The authors would like to express her sincerely gratitude to the financial support of China Scholarship Council to conduct the research in Professor K. S. CHOI's group of University of Nottingham (U. K.) and the National Key Laboratory of Science and Technology on Hydrodynamics.