Near Wake of a Horizontal Circular Cylinder in Stably Stratified Flows

The near wake of a circular cylinder in linearly stratified flows of finite depth was experimentally investigated by means of flow visualization and measurements of vortex sheddiñg frequencies, at Reynolds numbers 3.5 x l&'1 .2 xl 4 and stratification parameters kd 0-2.0. The non-dimensional parameter kd is defined as kNd/U, where N is the Brunt Vaisala frequency d, the diameter of the cylinder and U the approaching flow velocity The study demonstrates that as kd increases from zero, the vortex shedding from a circular cylinder progressively strengthens, while the Strouhal nümber gradually becomes lower than that for homogeneous flow. This phenomenon can be explained by the effect of the increasingly stable stratification which enhances the two-dimensionality of the near wake flow of the circular cylinder the enhanced two dimensionality of the flow strengthens the roll up of the separated shear layer. Above a certain value of k,.,, however, vortex fòrmation and shedding are strongly suppressed and the Strouhal number rises sharply This observation is attributable to the development of stationary lee waves downstream of the circular cylinder because the lee waves strongly suppress vertical fluid motions.


Introduction
In stratified fluids with vertical gradients of density and/or temperature, buoyancy effects become significant.
For instance, the strength of strátification is a predominant factor for fluid motion, turbulence transport, and dispersion in the atmosphere and ocean. Accordingly, a large number of investigations have been conducted on stratified flow around terrain or objects on the Earth's surface (e.g. Blumen'). Some of these investigations were conducted to improve understanding of meteorological phenomena such as lee waves and local winds (e.g. Snyder2). Other investigations have centered around issues of the atmospheric environment that require understanding of advection and dispersión of air pollutants. In contrast to the investigations on atmospheric flows, there exist few investigations on stratified flow around basic objects such as cylinders or splferes. In stratified flow, the flow around an object is generally governed by the Reynolds number and other non-dimensional stratification numbers (e.g. the Froudé number). Honji35 investigated two-dimensional flows around a circular cylinder placed horizontally in stratified flow for a range of Reynolds numbers less than Near Wake of a Horizontal Circular Cylinder in Stably Stratified Flows wake-flow patterns are observed behind a horizontal circular -cylinder placed perpendicular to the flow direction. However, with increasing stratification, the wake-flow pattern makes a gradual transition to that dóminated by internal waves. When the stratification becomes sufficiently high, blocking occurs and upstream wakes emerge6'7. Furthermore, in a highly stratified fluid of finite depth, columnar disturbances emerge and propagate upstream of an object8'9. Boyer et al.7 investigated the near-wake flows of a circular cylinder for Reynolds numbers less than 4000 and over a wide range of stratification. With a flow visualization experiment, the wake patterns of these investigated flows were classified in detail. Hwang and Lin'0 numerically computed stratified flows around a circular cylinder for Reynolds numbers less than 500. The results of their numerical computation showed suppressed vortex shedding with increasing density stratification, consistent with the expérimental results above, In homogeneous flows, the effect of the Reynolds number is known to be large on the near wake flow of an object, particularly on the wake of a circUlar cylinder and a sphere. In contrast, the influence Of; stratification on the wake of anobject is not yet well understood over a wide range of Reynolds numbers, particularly for high Reynolds numbers. Given this background, the present study investigates the influence of stratification on two-dimensional near wake flows around a circular cylinder with the use of a density-stratified wind tunnel. The density-stratified flow is generated with a Reynolds number in the range of3.5x103_l.2x104. The near wake flow around a circular cylinder is investigated with a flow visualization technique and measurements of wake velocity fluctuations Some of the phenomena observed in the current paper were previously reported as a letter'1. The present paper will investigate the mechanisms of these phenomena.

Experimental set-up and methodology 2.1 Generation and evaluation of density stratification
Experiments were performed in a density-stratified wind tunnel of a closed-circuit type'2. The test section of the wind tunnel was O.4m wide, O.6m high and l.7m lông. Designed to generate predefined density-stratified flows, the wind tunnel, except for the contractiOn and test sectiOns, was divided horizontally into six stories (Figure 1). A mixture of air and sulfur hexafluoride, SF6, was injected into each story from an external source to directly produce density stratificatioñ. Sulfur hexafluoride is a high-density gas with a molecular weight approximately 5 times that of air. Various types and strengths of vertical density gradients were produced in the test section by appropriately adjusting the volume of SF6 injected into each story. The density of the gas mixture in the test section was indirectly measured with the use of an oxygen meter. To evaluate the volume fraction of oxygen, the gas mixture was sampled through a gas sampling tube placed upstream of the test section and sent to the oxygen meter. The volume fraction of oxygeñ is approximately 1/5 of a given volume of air. Thus, the volume fraction of oxygen evaluated by the oxygen meter allowed for the calculation of the volume fraction of the air, thus that of SF6 in the gas mixture. Finally, the molecular ratio of air to SF6 was used to evaluate the specific density of the gas mixture with respect to the air, P'Pa' at the locatiOn of the gas sampling.

Wind velocity control and measurements
The wind in the test sectioñ was generated by 18 microfans, eäch driven by a micrömotor; three microfans were mounted side by side on each story. The generated wind velocity, U, ranged from 0 to 1.2 m/s. In homogeneous flow (air only), añ ltype hot-wire anemometer wasP deployed to measure low wind velocities with high accuracy. As for the.wind velocity measurements in the stratified flow of the air-SF6 mixture, the smoke-tracer technique was adopted; that is, the visualized flow was video recorded and the transit time required for markers to pass between two reference points was analyzed to determine the wind velocity.

Test models and measUrement Of velocity fluctuations in the wake
The models used in the present study were cirôuÌar cylinders with diameters, d=5, 7.5, 10 and 13 cm and two rectangular cylinders. The spanwise length of each of the cylinders was 30 cm. For both of the rectangular cylinders, the long side of the rectangular cross-section, d, was 10 cm. The symbol d is chosen for both the circular and rectangular cylinders because it represents the vertical height in both cases and will simplif' the notation used in later analysis (see below and Section 5.1). The ratiO of the short side to the long side Of eaòh rectangular cylinder1 bld, was 0.1 and 0.6, where b is the short side of the rectangular crOss-section. All the cylinders were mounted in the test section so that their axes were horizontal and perpendicular to the flow direction. The rectangular cylinders were mounted with the long side of the rectangular cross-section facing the on-coming flow. To ensure the two-dimensionality of the flow, a 3Øx5Q cm plate was .attached to each end of the cylinders (models), and the plates were placed 5 cm away from the side walls (Figurel).
The velocity fluctuations in the wake of a cylinder were measured by an I-type hot-wire anemometer at the oùter edge of the wake behind the cylinder. Spectral analyses were performed on these data with an FFT analyzer. Einally, spanwise correlations of velocity fluctuations in the wake of the cylinder were evaluated using multiple I-type hot-wires deployedsiniultaneously in the spanwise direction (y-direction).
The smoke-wire technique was adopted to visualize the flow field around a cylinder ( Figure 1). A mixture of liquid paraffin and iron powder was used to prolong the duration of the smoke release. To illuminate the visualized flows, three 1kW projectors were installed above the test section of the wind tunnel. The visualized flow was photographed by a Nikon F3 Camera (lens: 55 mm; exposure time: 1/125 sec; f-stop: 1.2; ASA400 black and white film) and video-taped simultaneously.
In the present study, the non-dimensional parameter, kd, is adopted to quanti1' the density stratification. The parameter kd is the iñverse of the Froude number and defined as kd=Nd/U where N is the Brunt-Valsälä frequency and N2=-(g/po)/(dpdz). The variables g and Po indicate the acceleration of gravity and the reference density, respectively. The reference density, Po. was set to the density value measured at the installation position of the circular cylinder, z30 cm above the floor of the Airflow with various vertical gradients of density was generated in the wind tunnel ( Figure 2). Over this range of the vertical gradient of density, the generated density profiles were approximately linear. Figure

Flow Field Visualization
Flow around a circular cylinder (d10 cm) was observed in a stratified wind field that had a nearly uniform vertical profile of wind velocity and a linear vertical profile of density as in Figure 2. With the flow visualization technique, change in the near-wake flow pattern was examined with respect to the stratification parameter, kd. Figure  a large density stratificatiòn (large kd) was initially generated, and the density stratification was progressively reduced (smaller kd). Specificälly, during an experiment with a constant wind speed, the value of kd was nudged gradually by adjusting the volume ratio of the two, gasses iii the gas mixture. Figure 5a shows the temporal change of the velocity fluctuations in the wake of a circular cylinder with d13 cm, and is characterized by a sudden increase of the magnitude of the velocity fluctuatión. The accompanyiñg change in the predominant frequency of the vortex shedding is shown in Figure Sb, that is, the wake-flow pattern suddenly switches to that of low-frequency vortex shedding. The changes observed in this example are consistent with changes that would be expected in the transition between two flow regimes.
3.4 Summary of experimental results and discussion Figure 3 shows the transition of the wake pattern with increasing stability: (b,c)(d,e)--(í). Initially, the near-wake pattern is characterized by highly-turbulent vortices (b and c). With increasing stratification, the turbulent near-wake becomes narrow (d and e) and finally, turbulent mixing occurs in isolated regions downstream of the cylinder (f). A similar three-regime transition of the ,cyliìider wake pattern was òbserved in the salt water experiment of Boyer et aL1 Boyer et al. refer to the 'isolated regions of turbulent mixing similar to those in (f) as "isolated mixed regions." In their experiment, wakes were generated behind cylinders in stably stratified water-tank flow at a Reynolds number up to 4000, which was smaller than the range of Reynolds numbers from our wind tunnel experiment A fürther examination of the changes in the wake pattern in the present experiment (Section 3.3) reveals two flow regimes. First, with increasing kd (increasing stability) up to a critical value, the vortex fOrmation behind the circular cyliñder is enhanced and the Strouhal number, St, decreases gradually. The corresponding critical value of kd will be referred to as (kd)C, hereafter.
Second, the strength of vortex formation behind the circular cylinder reaches its maximum at kd (kd).
With further increases in kd, vortex formation is sharply suppressed, and vortex formation becomes weak. Furthermore, this transition is accompanied by a sharp increase in St. The rapid change in the flow pattern at kd ()cr will be referred as the critical phenomenon below. In the follówing sectiOns, the generation mechanisms of the phenomena in these two flow regimes will be discussed. In the present experimental set-up, stably stratified flows were produced in the flow channel bounded by solid' horizontal top' and bottom walls. Thus, characteristics of stratified flows of finite depth are first summarized in the following section.

Critical phenomenon around a non-circular cylindrical object -
To investigate i the critical phenomenon at the critical stratification, kd(kd)cr, is unique to flOw around a circular cyliñder, additional experiments were performed with two rectangular cylinders (Section 2.3). The long side of the rectangular-cross-section of both cylinders (d=lOcm) was facing the on-coming flOw, and this length was identical to the -diameter of the circular cylinder (d=lOcm) for which the near wake-flow was investigated in Section 3. As with the experiment with the circular cylinder, the flow around the square cylinders was investigated for various degrees of stratification, kd, and flo' visualization experiments *ere conducted. In addition, the fluctuating velocity in the near-wake was measured to evaluate the peak frequency, fu.-As in the flow around the circular cylInder -of d=lOcrn (Figure 4), a sudden change in the flow pattern and a sharp increase in the value of St are observed -in the proximity of kd=O.52 or K=l (Figures 6   and 7). When kd exceeds (kd)cr, the pattern of the near-wäke flow behind the rectangular cylinder changes and vortex formation is suddenly suppressed. (change from Figure 6a to 6b). The present investigation suggests that the critical phenomenon in the proximtty of (k(l)cr is nOt unique to the near-wake flow behind a circular cylinder. The critical phenomenon occurs at a similar value of (kd)cr behind a rectangular cylinder that has the same vertical dimension as the circular cylinder.  The critical stratification, for near-wake flows is investigated behind circular cylinders of three additional diameters, d5, 7.5, 13cm, in the test section of height H=6Ocm. The results are summarized together with those from the experiment with the two rectangular cylinders (Section 5.1) in Figure 8. For an object with a certain diameter d (or a height d facing the on-coming flow), the critical phenomenon occurs at a similar value of kd, independent of the Reynolds number. Furthermore, the emergence of the critical phenomenon coincides with the lower limit of kd at which stationary lee waves are predicted to occur by the linear theory ( Figure 9).
To further investigate the dependency of (kd) on d/R (or K), the height of the text section, H, was modified to H=46.3cm, and near-wakes behind a circular cylinder of d7.Scm (d/H=0.162) were studied. The value of (kd)c. observed for this case is approximately 0.52, which falls on the predicted lower limit for the occurrence of stationary lee waves ( Figure   9). The present result suggests 1) the value of (  6. Intensification of vortex formation 6.1 Non-circular cylindrical bluff bodies As discussed in Section 3, the vortex formation behind a circular cylinder is intensified with increasing stratification between k0 (homogeneous flow) and (lSd)cr. The intensification of vortex formation with increasing, stability was also examined for the near-wake flows of non-circular cylindrical bluff bodies. Generally, the Strouhal number, St, is small for the wake behind a bluff body that is conducive to vigorous vortex formation. Therefore,, the value of St is negatively correlated to the formation of strong vortices. The decreasing trends of St between ,kd=0 and (kd)cr for the wakes behind the rectangular cylinders of b/d=0.l, 0.6 ( Figure 7) are smaller than those for the wakes behind the circular cylinders ( Figure 4). In particular, the corresponding slope for the wake behind the rectangular cylinder of b/d=0.6 is small. The flow visualization experiments also confirmed that enhancement of vortex formation behind the rectangular cylinder of bIdO.6 was not as evident as that behind the circular cylinders.
62 Spanwise correlation of fluctuating u-velocities in near-wake flows Behaviors of the spanwise correlation of lhe fluctuating u-velocities were investigated iñ the near-wakes of circular (d=5, 10 cm) and rectangular (b/d0.6) cylinders. The investigations were made in homogeneous flow and stratified flow at kd slightly less than (k)cr. For the near-wakes behind the circular cylinders, the correlation of the fluctuating u-velocities is significantly higherin stratified flow at kd(kd)cr than that in homogeneous flow ( Figure 11). in contrast, with T Reports of Research Institute for Applied Mechanics, Kyushu Uñiversity No.138 March 2010 21 kd approaching (kd)cr, the correlation for the wake of the rectangular cylinder is nearly the same as that for homogenous flow. This result is attributable to the highly two dimensional nature of the near wake flow around the rectangular cylinder as will be discussed in the next subsection.
6.3 Dependency of vortex formation enhancement on the shape of the bluff body Flow visualization experiments and measurements of the velocity fluctuatioñ frequencies in the near-Wake led to the following conclusions: With increasing values of kd(NdIIJ), i.e., increasing stratification, vortex formation and shedding behind a circular cylinder are gradually enhanced with respect to those in homogeneous flow, and the Strouhal number decreases These observations can likely be explained by the effect of increasing stratification which enhances the two-dimensionality of the near wake flow behind a circular cylinder. The enhanced two-dimensionality of the flow strengthens the roll-up of the separated shear layer.
The strength of vortex. formatiòn and shedding reaches its maximum at kd (kd), . and is suddenly suppressed when kd exceeds (kd)cr; This critical phenomenon is accompanied by a suddèn change in the flow pattern, that is, oñly weak vortex formation occurs downstream of the obstacle and the Strouhâl number increases sharply. In terms of the stratification parameter K(N1-VitU) which includes the depth of stratified flow, H, the critical phenomenon occurs close to K1 for all sizes of circular and rectangular cylinders investigated. The linear theory predicts that lee waves will develop for K>!. Thus, in this critical phenomenon, the emergence of a stationary lee wave suppresses the vertical fluid motions of the vortices formed behind an object and causes a Sùdden change iñ the flow pattern.
With further increase of k.3, the wake flow pattern transitions to one dminated by internal waves.