2.1 Wind tunnel

Systematic drag force measurements were performed in a wind tunnel of Goettingen type for different cylinder configurations. The working cross-section downstream of the nozzle was $75~\text{cm}\times 75~\text{cm}$ . For the diameter of the cylinder ( $D_{cyl}=7$  cm), Reynolds numbers ranged from $3.5\times 10^{4}$ to $1.4\times 10^{5}$ . This corresponded to a velocity range from $8~\text{m}~\text{s}^{-1}$ to $28~\text{m}~\text{s}^{-1}$ . During the drag measurements, the velocity was controlled by a Prandtl tube installed upstream of the test section. Precise laser-Doppler anemometry (LDA) measurements revealed that the approaching flow was uniform in vertical and horizontal direction (top-hat profile) and turbulence intensity was less than 1 %. Figure 1 shows the horizontal velocity profiles in the wind tunnel cross-section for four mean velocity variations (from $8.45~\text{m}~\text{s}^{-1}$ to $11.45~\text{m}~\text{s}^{-1}$ ), measured at a distance of three cylinder diameters upstream of the cylinder’s test position.

Figure 1.

Horizontal velocity profile $3D_{cyl}$ upstream of cylinder position, measured by LDA without test cylinder.

2.2 Cylinders with integrated porous layer

The cylinders consisted of balsa wood and had a precise and constant outer diameter of $D_{cyl}=7$  cm, a length of $L_{cyl}=70$  cm and a smooth surface. The cylinders were placed horizontally in the test section, see figure 1. A porous layer of polyester foam material was inserted on the leeward side of the cylinders. It was symmetrical to the horizontal stagnation line. Four different cylinder configurations with coating angles of $\unicode[STIX]{x1D6FD}=40^{\circ }$ , $\unicode[STIX]{x1D6FD}=70^{\circ }$ , $\unicode[STIX]{x1D6FD}=100^{\circ }$ , $\unicode[STIX]{x1D6FD}=160^{\circ }$ were used for the force measurements, see table 1. For each angle configuration, three different layer thicknesses ( $d_{M}=3$ , 5, 10 mm) and three different porosities of 10, 20, 30 PPI (pores per inch) were investigated, see figure 2. In addition, a non-coated smooth balsa wood cylinder was built and its drag was measured for comparison with results of cylinders with a porous coating on the leeward side.

The fluid mechanical properties of the porous foam material can be defined by the pressure loss coefficient $\unicode[STIX]{x1D706}$ ( $\text{m}^{-1}$ ):

It describes the difference between the windward and the leeward pressure of a block of porous foam material filling the cross-section of a test channel divided by the dynamic pressure and the characteristic thickness $d$ of the foam material. For the different foam materials of PPI 10, 20 and 30, $\unicode[STIX]{x1D706}$ can be specified as $\unicode[STIX]{x1D706}_{PPI10}\approx 250$ ( $\text{m}^{-1}$ ), $\unicode[STIX]{x1D706}_{PPI20}\approx 500$ ( $\text{m}^{-1}$ ) and $\unicode[STIX]{x1D706}_{PPI30}\approx 1000$ ( $\text{m}^{-1}$ ), respectively.

Figure 2.

Sketch of cylinder configuration with inserted porous layer on the leeward side, which is symmetrical to the horizontal stagnation line (a); porous foam material of the three different pore sizes (b).

Table 1.

Configurations of cylinders with partial porous coating (force measurements). Pressure measurements were performed for underlined configurations. For the asterisked configurations particle image velocimetry (PIV) measurements were performed.

2.3 Cylinder with porous layer applied to the surface in the lee

A second type of cylinder model was made. It consisted of a smooth Plexiglas cylinder with the same diameter of $D_{cyl}=7$  cm. On its lee side, a thin porous layer of 3 mm thickness and PPI 20 was not inserted into the cylinder’s shape as before, but applied onto the surface. The coating angle was $\unicode[STIX]{x1D6FD}=100^{\circ }$ , and the coating was symmetrical to the horizontal stagnation line, see figure 3. To prevent three-dimensional flow conditions at the ends of the cylinders, all configurations were mounted between circular end plates (splitter plates) and fixed to a steel frame outside of the wind tunnel, see figures 4 and 5. The splitter plates were decoupled from the tested cylinders by fitting the models in circular holes of the splitter plates. The diameter of these holes slightly exceeded that of the models. The outer diameter of the splitter plates was $D_{plate}=3.5\cdot D_{cyl}$ . The splitter plates were designed according to the findings of Stansby (Reference Stansby1974) and Szepessy (Reference Szepessy1993), corresponding to a distance $x_{l}$ from the leading edge to the cylinder axis of $x_{l}/D=1.75$ and a distance $x_{t}$ from the cylinder axis to the trailing edge $x_{t}/D=1.75$ . On the wall sides the plates were provided with PVC-caps to prevent gap flow between cylinder model and splitter plates.

Figure 3.

Plexiglas cylinder with additional porous layer applied on the leeward side.

Figure 4.

Cylinder configuration in the wind tunnel; view in upstream direction.

Figure 5.

Splitter plate with PVC-cap and strain gauge sensor.

2.4 Drag force measurement

Total drag was measured for different flow velocities by a force balance which consisted of two strain gauge sensors. The sensors had a nominal force of 50 N each and an accuracy of 0.04 %. The cylinder configurations were connected directly to the sensors, which were fixed to the steel frame installed outside, see figures 5 and 6. 20 000 voltage values were recorded at a sampling frequency of 500 Hz for each tested velocity.

The dynamic characteristics of the force balance were determined by measuring the system response over a frequency range of $5~\text{Hz}\leqslant f\leqslant 100~\text{Hz}$ . This was realized by a calibration apparatus which consisted of a mini shaker, a frequency generator and an amplifier. Via the frequency generator sinusoidal signals of different frequencies but with constant amplitude were fed to the shaker as system input. The volt values of the force sensors were taken as output signals of the system. The frequency response of the system with peaking at $f=28~\text{Hz}$ is shown in figure 13(c). In addition, a second method – the impulse response method – was also applied and the impulse signal also indicated a peak at about $f=28~\text{Hz}$ in the frequency domain. This corresponds to the resonance frequency of the system.

Figure 6.

Force balance with cylinder configuration installed in the wind tunnel.

The mean drag force $\overline{F_{D}}$ can be expressed as the sum of form drag force $\overline{F_{p}}$ and friction drag force $\overline{F_{f}}$ :

Thus, the total drag force can be obtained by summing up all pressure and friction forces acting on the surface elements around the bluff body, which can be expressed by:

Dividing (2.3) by $1/2\cdot \unicode[STIX]{x1D70C}\cdot U^{2}\cdot A$ , with $A=L_{cyl}\cdot D_{cyl}$ , defines the drag coefficient $c_{d}$ :

Especially with respect to similarity, another interesting point is the question regarding on which dimensionless numbers the drag coefficient depends in this particular case. This can be answered by dimensional analysis, where the governing equation is appropriately normalised in order to form dimensionless groups that appear in the resulting dimensionless equation. The mean drag force per unit length of the cylinder $\overline{F_{D}^{\prime }}$ obviously depends on the approaching flow velocity $U$ , density of the fluid $\unicode[STIX]{x1D70C}$ , dynamic viscosity of the fluid $\unicode[STIX]{x1D707}$ , diameter of the cylinder $D_{cyl}$ , pressure loss coefficient of the porous material characterised by the pressure drop per unit length $\unicode[STIX]{x1D706}$ and thickness of the porous layer $d_{M}$ . Thus, the equation can be written as:

Expressing the dimension of each physical quantity in the brackets as a product of the basic physical dimensions and normalizing the quantities $n$ times yield a dimensionless equation, which reads in the present case:

As can be inferred from (2.6), the first quantity on the right-hand side is the Reynolds number $Re$ and the second is similar to an Euler number $Eu$ describing the ratio of pressure loss in the porous medium to inertia force of the flow. The third quantity reflects the ratio of porous layer thickness to cylinder diameter and accounts for geometrical similarity.

Figure 7.

Sketch of cylinder model with two lines of pressure tappings in the middle segment (a) and photo (b); here: smooth cylinder without porous coating.

In a wind tunnel having a cross-section of height $H_{tunnel}=75$  cm, a cylinder with a diameter of $D_{cyl}=7$  cm mounted perpendicular to the flow direction leads to a blockage of 9.3 %. As a consequence of the blockage effect, flow velocity around the cylinder and, thus, drag measurement must be corrected e.g. according to the image concept of Allen & Vincenti (Reference Allen and Vincenti1944) (used in this study), see also Dalton (Reference Dalton1971), or according to the momentum concept of Maskell (Reference Maskell1963). Such correction methods consider the change in dynamic pressure caused by the interaction of fixed walls and blocking cylinder, which leads to a wall–wake interference.

2.5 Pressure measurement

For the pressure measurements, cylinder configurations with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , $d_{M}=10$  mm and foam material of PPI 10, 20 and 30 were investigated, see table 1 underlined. The cylinders were made of smooth Plexiglas and consisted of three segments. The smaller middle segment included the pressure tappings and was fixed between two smooth segments, see figure 7. The pressure tappings were grouped in two lines. The increment between the pressure tappings for each line was $10^{\circ }$ . The first pressure tapping of line 1 was placed at the stagnation point ( $\unicode[STIX]{x1D719}=0^{\circ }$ ), the last one was at $\unicode[STIX]{x1D719}=180^{\circ }$ . Line 2 was displaced by $5^{\circ }$ and the first tap of this line started at $\unicode[STIX]{x1D719}=5^{\circ }$ and ended at $\unicode[STIX]{x1D719}=175^{\circ }$ . This resulted in a maximum number of 37 pressure tappings on half of the cylinder circumference. In the leeward region, the pressure taps were installed below the porous foam material, see figure 8. To exclude that pressure measurements performed at taps below the porous foam material leads to wrong results, pressure was also measured using tubes that flush with the porous surface, see § 3.5. In order to measure accurately the pressure fluctuations, corrections of distortion effects caused by the tubing system have been performed. This was realized by a calibration apparatus. With a frequency generator and a loudspeaker an acoustical sinusoidal signal was applied simultaneously to port 1 and port 2 of a multiport pressure sensor. The connection from loudspeaker to port 1 was realized by a short tube with a length of $L_{tube}=2$  cm. This served as corresponding reference pressure of an undistorted signal. The tubing system of the cylinder configuration was interconnected between loudspeaker and port 2. The connection of the corresponding pressure tap to the loudspeaker was also realized by a short tube length ( $L_{tube}=2$  cm). The frequency response was measured within the range of 30 Hz ${\leqslant}f\leqslant 75$  Hz and is shown in figure 9. The pressure measurements were corrected respectively.

Figure 8.

Layout of pressure tappings for both lines: configuration PPI 10, $d_{M}=10$  mm,  $\unicode[STIX]{x1D6FD}=100^{\circ }$ .

Figure 9.

Amplitude ratio of the tubing system.

2.6 PIV flow field measurements

The flow field was measured using a DANTEC 2C-2D TR-PIV system for the configurations asterisked, see table 1. The system consisted of double-pulsed Nd:YAG lasers (532 nm) and a high-speed camera with a complementary metal-oxide-semiconductor (CMOS) sensor of $1280\times 800$ pixel resolution. The light sheet was aligned in main stream direction, in the middle of the cylinder configuration. To provide the two-dimensional (2-D) flow and optical accessibility, the cylinder was mounted between the Plexiglas wall on the one side and the splitter plate on the other, see figure 10. PIV data were obtained for the field of view I (FOV I), see figure 11, corresponding to the shear layer and wake regions. The size of the measurement area was $300\times 185$  mm. The images were evaluated using an adaptive correlation algorithm with a final interrogation size of $32\times 32$ pixel by 50 % overlap. The universal outlier detection technique was used to eliminate spurious vectors (DANTEC Software v3.41) based on the work of Westerweel & Scarano (Reference Westerweel and Scarano2005). The sampling frequency was $f_{sample}=1000$  Hz and the total number of images was $N_{images}=5000$ .

Figure 10.

PIV plane.

Figure 11.

Field of view I.

3.1 Influence of a porous layer applied to the leeward side of a smooth cylinder

At first, drag measurements were carried out with a fully smooth cylinder made of Plexiglas. Subsequently, a thin porous layer was applied to the leeward side, see figure 3. As can be inferred from figure 12, the applied thin layer causes the drag to be reduced almost constantly in the Reynolds number range. When averaged over the tested Reynolds number range, drag reduction is approximately 9.8 %. This demonstrates the effectiveness of leeward coating as a passive device reducing the drag and influencing the flow field around the coated structure.

In figure 13, a part of the time series of the voltage signal recorded by the force sensors is shown for both configurations, the reference cylinder (figure 13 a) and the identical cylinder with the porous layer coating in the lee (figure 13 b), respectively. The mean velocity of this experiment was $U=24.2~\text{m}~\text{s}^{-1}$ . The voltage curve can be taken as a measure of the curve of the resulting horizontal wind force acting on the cylinder. The instantaneous force measurements were performed in a velocity/frequency range far away from the system resonance frequency as indicated in the diagram of figure 13(c).

As can be seen, the signal curve of the cylinder with the porous layer coating shows qualitatively a significant reduction in amplitude oscillation and is much smoother and more regular. This difference is recorded during the whole sampling time for all velocity settings.

Figure 12.

Drag coefficient of the configuration with an additional thin porous layer on the leeward side.

Figure 13.

(a,b) Time series of voltage data of the force sensors for a smooth Plexiglas cylinder of 7 cm in diameter and $U=24.2~\text{m}~\text{s}^{-1}$ in mean velocity: without any porous layer (a), with porous layer coating on the leeward side (b). (c) Dynamic characteristics of the force balance with Plexiglas cylinder model, measured over a frequency range of $5~\text{Hz}\leqslant f\leqslant 100~\text{Hz}$ .

The results in figures 12 and 13 originate from measurements carried out for verification purposes rather than for obtaining systematic data. Systematic data were obtained using modified cylinder models with shape-integrated porous layers, see figures 2 and 8. As a result, the outer contour of the cylinders was no longer dependent on the layer thickness and remained always constant with a diameter of 7 cm.

Figure 14.

Drag coefficient of the cylinder coated on the leeward side with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , a constant layer thickness of $d_{M}=10$  mm and varying porosity.

Table 2.

Varying parameters.

3.2 Drag coefficient of cylinders with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$

Figures 14–17 present the measurement results of the drag coefficient $c_{d}$ for a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ and compare them to those of the reference cylinder (smooth cylinder with no porous layer in the lee). The curves of the $c_{d}$ coefficient are fairly constant over the measured range of Reynolds numbers. The following subsections present the data for varying key parameters, i.e. porosity (pore sizes), porous layer thickness and coating angle. Table 2 gives an overview of the varying parameters.

Figure 15.

Drag coefficient of the cylinder coated on the leeward side with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , constant PPI 10 foam material and varying layer thickness.

Figure 16.

Drag coefficient of the cylinder coated on the leeward side with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , constant PPI 20 foam material and varying layer thickness.

Figure 17.

Drag coefficient of the cylinder coated on the leeward side with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , constant PPI 30 foam material and varying layer thickness.

3.3 Drag coefficient averaged over the measured Reynolds number range for varying coating angles $\unicode[STIX]{x1D6FD}$

As can be seen from figures 14–17, the drag reduction reached for a cylinder/porous layer combination is almost constant in the measured Reynolds number range (horizontal $c_{d}$ -curve before the drag crisis). The constancy is found similar to the curve of the reference (smooth) cylinder and is due to the fact that this subcritical Reynolds number range characterized by a laminar boundary layer, periodic vortex shedding and a turbulent wake shows rather stable separation conditions. This allows for the averaging of the drag coefficients in that range and the determination of its deviation from the reference cylinder data, see table 3. Thus, the measured curve of drag coefficients shrinks to a single value. The $Re$ -averaged drag $\overline{c_{d}}$ over the measured velocity range ( $U_{i}$ , with $i=1{-}17$ velocities) is determined by:

Deviation from the $Re$ -averaged reference cylinder value can be written as:

The curves of measured drag coefficients displayed above refer to a fixed coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ . In fact, not only this coating angle was investigated but also other coating angles in the range of $\unicode[STIX]{x1D6FD}=40^{\circ }{-}160^{\circ }$ . Using the $Re$ -averaged drag coefficients makes it easy to display the measured curves as points in a plot and to accommodate many configurations therein. In figure 18(a–c) the $Re$ -averaged drag coefficient is plotted over the coating angles of $\unicode[STIX]{x1D6FD}=40^{\circ },70^{\circ },100^{\circ }$ and $160^{\circ }$ for a constant porosity. Figure 19(a–c) shows the drag coefficient over the same coating angles, but for a constant porous layer thickness. The drag coefficient for the coating angle $\unicode[STIX]{x1D6FD}=0^{\circ }$ corresponds to that of the smooth reference cylinder without any coating.

Table 3.

Drag coefficient averaged over the measured range of Reynolds numbers and deviation from the reference cylinder drag value for the $\unicode[STIX]{x1D6FD}=100^{\circ }$ configuration.

Figure 18.

Drag coefficient $\overline{c_{d}}$ with constant porosity over varying coating angle (a) PPI 10, (b) PPI 20, (c) PPI 30.

3.4 Reproducibility of the results

In some cases, the measured forces in the wind tunnel differed slightly only. To check whether these small differences can be measured unambiguously and are not subject to a limited measurement accuracy, the reproducibility of the measurements has been investigated. Force measurements for each configuration were repeated on different days during the measurement period. Each time, re-installation in the measuring section was required. Figure 20 shows two examples of drag coefficient measurements performed on different days with newly installed test cylinders of coating angle $\unicode[STIX]{x1D6FD}=100^{\circ }$ , porosity PPI 10, and layer thickness $d_{M}=10$  mm. The reproducibility of the results was very good and corresponded to a deviation of drag coefficient of less than 0.15 %.

3.5 Influence of leeward porous coating on base pressure

Figure 21 compares the pressure coefficient $c_{p}$ results for half the circumference of the cylinder with a configuration of $\unicode[STIX]{x1D6FD}=100^{\circ }$ , $d_{M}=10$  mm and PPI 10 to the results of the reference cylinder. The pressure distributions for both cylinders are almost congruent on the windward side up to a circumferential angle of about $\unicode[STIX]{x1D719}=60^{\circ }$ . From that point on, pressure around the cylinder with an inserted porous layer increases when compared to the reference cylinder. In the region of the porous layer ( $\unicode[STIX]{x1D719}\geqslant 130^{\circ }$ ) an increase of base pressure of up to 29 % is recorded. Apparently, the porous layer on the leeward side of the cylinder also influences the pressure behaviour in the region $\unicode[STIX]{x1D719}\leqslant 130^{\circ }$ , i.e. upstream of the porous layer. This changed pressure distribution is similar to the upstream pressure change due to a splitter plate in the wake of a cylinder, see Roshko (Reference Roshko1955, Reference Roshko1961). Vortex shedding exists also for the configuration $\unicode[STIX]{x1D6FD}=100^{\circ }$ , $d_{M}=10$  mm. Flow separation occurs for both cylinders with and without porous layers at a circumferential angle of approximately $\unicode[STIX]{x1D719}=75^{\circ }$ . The separation angle was determined at the location where the root-mean-square value $c_{prms}$ has its maximum, see West & Apelt (Reference West and Apelt1993). The shedding frequency can be seen as a clear peak in the spectrum in figure 22 for the reference cylinder (a) and for the cylinder with a porous layer in the lee (b). Comparing both spectra reveals an increase in shedding frequency from 39.3 Hz to 43.1 Hz, a narrowing of the spectrum and a decrease of fluctuations for the cylinder with a porous layer in the lee. The decrease of standard deviation of pressure fluctuations for the cylinder with a porous layer in the lee is shown in detail in figure 23. Due to the significant reduction of pressure fluctuations, the porous layer in the lee of the cylinder obviously leads to a regularization of the flow, which is in accordance with the findings of Price (Reference Price1956) and Wong (Reference Wong1979).

Figure 21.

Pressure coefficient $c_{p}$ of the reference cylinder and the cylinder with $\unicode[STIX]{x1D6FD}=100^{\circ }$ , PPI 10, $d_{M}=10$  mm;  $Re=6.4\times 10^{4}$ .

Figure 22.

Amplitude spectrum of the pressure fluctuations at $\unicode[STIX]{x1D719}=80^{\circ }$ , (a) smooth cylinder, (b) smooth cylinder with porous layer in the lee ( $\unicode[STIX]{x1D6FD}=100^{\circ }$ , PPI 10, $d_{M}=10$  mm),  $Re=6.4\times 10^{4}$ .

Figure 23.

Pressure coefficient $c_{p}$ with standard deviation (bars) for the smooth cylinder and the smooth cylinder with porous layer in the lee ( $\unicode[STIX]{x1D6FD}=100^{\circ }$ , PPI 10, $d_{M}=10$  mm);  $Re=6.4\times 10^{4}$ .

Figure 24.

Comparison of the signals of pressure taps beneath the porous coating with those which are flush with the coating surface. Mean pressure coefficient and standard deviation (bars) for line 2 of the pressure measurements.

Figure 24 shows the comparison of the pressure measurements for the taps beneath the porous coating and for the taps that are flush with the porous surface. The results show no difference for the mean pressure coefficient $\overline{c_{p}}$ . Only very small differences are visible for the standard deviation. The results show that the pressure measurements below the highly porous material used in our experiments are not subject to additional distortion.

3.6 Surface roughness of the porous material

To consider the effect of surface roughness of the foam material and, therefore, to investigate how much the roughness of the porous material contributes to the obtained results of this study additional experiments were carried out with foam material PPI 10, which is the material with the highest surface roughness. The mean size of the pore diameter of PPI 10 foam material is ${d_{p}}_{PPI10}=2.54$  mm. In comparison to a rough surface and therefore to impermeable rough layers, the characteristic property of a porous layer is that fluid can flow through it. If it would be possible to produce the thinnest porous layer of PPI 10 foam material, the pore arrangement must be ideal with one pore next to each other. For that case, the minimum height $d_{M}$ of the layer would be the pore diameter ${d_{p}}_{PPI10}=2.54$  mm itself. To satisfy the condition of a flow inside the layer, the assumption of the roughness height $h_{r}$ must be: $h_{r}<{d_{p}}_{PPI10}$ . Kuznetsov & Becker (Reference Kuznetsov and Becker2004) recommend a roughness height of a porous medium of half of the pore diameter. This corresponds to $h_{r}={d_{p}}_{PPI10}/2$ and is equal to $h_{r}=1.27$  mm.

Figure 25.

Mean pressure distribution around the cylinder. (a) Comparison of the two different emery papers. (b) Comparison of reference cylinder; emery paper $h_{r}<1.5$  mm;  $\unicode[STIX]{x1D6FD}=100^{\circ }$ , PPI 10, $d_{M}=10$  mm.

The surface of the PPI 10 material with a roughness height of $h_{r}=1.27$  mm is categorized as a very rough surface. The implementation of the rough surface on the leeward side of the cylinder was realized by emery paper which was fixed with a coating angle of $\unicode[STIX]{x1D6FD}=100^{\circ }$ on the reference cylinder. A roughness height of $h_{r}=1.1$  mm ( $h_{r}/D_{cyl}=1571\times 10^{-5}$ ) was used. This is the roughest emery paper available in the market. For even rougher conditions, a second self-made emery paper of equal distributed gravel with densest packing was fixed on the cylinder. This corresponds to a roughness height of $1.3~\text{mm}<h_{r}\leqslant 1.5$  mm ( $h_{r}/D_{cyl}=2142\times 10^{-5}$ ).

The two different emery papers deliver only small differences in the mean pressure coefficient, see figure 25(a). Compared to the reference cylinder and to a configuration with an inserted porous layer on the leeward side the pressure plot of the rough cylinder is very similar in shape to the plot of the smooth cylinder with a slightly increased pressure coefficient in the region of the rough surface. Regarding the pressure plot of the configuration with the porous layer, there is an increase even further upstream of the inserted region and an increased $c_{p}$ in the porous region can be detected, see figure 25(b). The fraction of streamwise pressure force within the modified region (for $\unicode[STIX]{x1D719}>130^{\circ }$ ) is presented in table 4. This shows that pure surface roughness delivers a pressure force reduction of 6.7 %, whereas the porous coating leads to a much higher reduction of 18.9 %. Thus, increasing surface roughness on the leeward side of a cylinder exerts a certain effect on the drag, but cannot explain alone the measured drag reduction by leeward porous coating.

Table 4.

Fraction of streamwise pressure force within the modified region for $\unicode[STIX]{x1D719}>130^{\circ }$ (for the region of a rough surface and the region of a porous layer) in comparison with the smooth reference cylinder.

3.7 Flow field analysis

For a detailed analysis of the flow field around the leeward porous coated cylinder, 2C-2D TR-PIV measurements were performed in the wind tunnel. The turbulence intensity of the flow was less than 1 %. The measurements were carried out at a Reynolds number of $Re=3\times 10^{4}$ characterizing the subcritical range in which a laminar boundary layer and a turbulent wake occur. The measurements were performed with a completely smooth cylinder and a smooth but leeward porous coated cylinder with coating angles of $\unicode[STIX]{x1D6FD}=40^{\circ }$ and $\unicode[STIX]{x1D6FD}=100^{\circ }$ . The coating material with porosity PPI 10 and thickness $d_{M}=10$  mm was chosen.

Pressure and flow field on the windward side of the cylinders are almost identical. The boundary layers of the cylinders are in laminar state and separation occurs at the same region and seems not to be affected by the leeward coating.

As stated by Zdravkovich (Reference Zdravkovich1997), the transition can occur in three different regions around the cylinder (near wake, shear layer, boundary layer) depending on the Reynolds numbers. Increasing the Reynolds number will firstly turn the near wake into turbulent state and, with further increase, the shear layer and, then, after another increase also the boundary layer of the cylinder. Thus, the transition moves in counter-streamwise direction with increasing Reynolds number. In the subcritical case, the near wake is turbulent and the transition takes place in the shear layer. The transition in the shear layer ends when the shear layer is fully turbulent and the transition finally shifts to the boundary layer. Concerning the present study, the region where the leeward porous layer changes the flow conditions compared to the reference cylinder is the shear layer and the near wake.

In figures 26–32 (except figures 29 and 30) the time-averaged data of the PIV measurements for a Reynolds number of $Re=3\times 10^{4}$ are presented. The data are scaled with the approaching flow velocity  $U$ . The presented contour plots show the upper half of the field of view due to the symmetric appearance of the flow structure in the near wake.

Figure 26.

Mean velocity components.

Figure 27.

Variances of the velocity components.

Considering first the mean velocity components, $\overline{u}$ and $\overline{v}$ , behind the completely smooth and leeward porous coated cylinders, see figure 26, it can be inferred that with increasing coating angle the area of small streamwise and lateral velocity is extended. It can also be seen from figure 26(a) that with increasing coating angle the horizontal free stream velocity outside of the free shear layer protrudes longer into the wake region. The isopatterns of the $\overline{u}$ and $\overline{v}$ component are shifted downstream. This is supported by figure 27, showing a shift of the normal stress pattern in downstream direction and a decrease in strength. At a position $x/D_{cyl}=0.52$ and $y/D_{cyl}=0.58$ the streamwise normal stress within the shear layer decreases significantly, from the value 0.33 to 0.19, with increasing coating angle. Figure 28 presents the maximum root-mean-square value of the streamwise component $\sqrt{\overline{u^{\prime 2}}}$ within the shear layer for the smooth cylinder and the configuration $\unicode[STIX]{x1D6FD}=100^{\circ }$ , $d_{M}=10$  mm, PPI 10. Compared to the smooth cylinder it can be seen that the maximum root-mean-square value of the configuration with a porous layer on the leeward side is reduced in the shear layer at all locations. For $0<x/D_{cyl}\leqslant 0.29$ the development is very similar with the same slope for both configurations. The maximum value is reached at $x/D_{cyl}=0.29$ . After that both curves drop until they reach a minimum level. After the minimum the curve slightly rises for the smooth cylinder whereas the curve of the configuration with the porous layer stays more constant on a minimum plateau over several distances $x/D_{cyl}$ before it rises. The porous layer is inserted in the cylinder shape at a location $x/D_{cyl_{porous}}=0.32$ , which is immediately after the maximum value at the location $x/D_{cyl}=0.29$ . This shows that the porous layer is responsible for a steeper fall of  $\sqrt{\overline{u^{\prime 2}}}$ .

Figure 28.

Maximum root-mean-square value of the streamwise velocity component $\sqrt{\overline{u^{\prime 2}}}$ in the cross-section within the shear layer.

Figure 29.

Time traces of the horizontal velocity component $u(t)$ within the shear layer at maximum $\sqrt{\overline{u^{\prime 2}}}$ locations for the smooth cylinder and the configuration $\unicode[STIX]{x1D6FD}=100^{\circ }$ , PPI 10, $d_{M}=10$  mm.

Figure 30.

Instantaneous streamwise velocity component $u(t)$ at centreline and $x/D_{cyl}=0.55$ .

Figure 26(b), as well as figure 27(b), indicate that the region of vortex formation is shifted further downstream with increasing coating angle. It is also evident from figures 27(a) and 31(a), that with increasing coating angle the separating free shear layer becomes thinner and, thus, the velocity gradient in the shear layer steepens.

Figure 31.

Turbulent kinetic energy and vorticity.

Figure 29 presents the time traces of the streamwise component $u(t)$ within the shear layer at different locations $x/D_{cyl}$ where the maximum root-mean-square value appears. At the position $x/D_{cyl}=0.07$ the plot of the reference cylinder is in good agreement with the results of Bloor (Reference Bloor1964). It indicates a regular sinusoidal structure of the time trace before it starts to decompose ( $x/D_{cyl}\geqslant 0.23$ ). This is immediately before the transition in the shear layer begins. The time traces of the configuration with a porous layer on the leeward side shows a more random signal from the beginning and no typical sinusoidal structures can be found. The fluctuations start to grow until $x/D_{cyl}=0.29$ . At that position both signals show the same strength in fluctuations. After that point and when the location of the porous layer starts ( $x/D_{cyl}>0.29$ ) the fluctuations decrease whereas the fluctuations of the reference cylinder remain on the same high level as before.

Figure 30 displays the time traces for the streamwise component $u(t)$ at the centreline and close to the porous layer at $x/D_{cyl}=0.55$ . It can be inferred that a certain backflow into the porous layer is measured, whereas the streamwise component $u(t)$ of the reference cylinder remains zero at the same position. Considering figure 31(a), we can conclude that a porous layer on the leeward side of a cylinder reduces the turbulent kinetic energy in the free shear layer. Turbulent kinetic energy in the free shear layer decreases with increasing coating angle. The level of turbulent kinetic energy in the near wake decreases from the value $k=0.5$ for the smooth cylinder to $k=0.4$ for the configuration with the coating angle $\unicode[STIX]{x1D6FD}=100^{\circ }$ . As can be seen from figure 31(b), the vorticity in the shear layer seems to be maintained longer on a high level as in the case of the completely smooth cylinder. In figure 32, the shear stress $\unicode[STIX]{x1D70F}_{corr}$ and the production term of turbulent kinetic energy $P_{k}$ is shown. Also, these results show a shift of the isopattern in downstream direction and a decrease in value with increasing coating angle. As can be inferred from figure 22, the vortex shedding frequency is increased by leeward porous coating and figures 27(a), 31(a) and 32(a) indicate that the width of the wake is reduced. This is in agreement with results of Roshko (Reference Roshko1954, Reference Roshko1955), who found that in the wake of bluff bodies a universal Strouhal number, $St=0.16$ , occurs when it is formed with the width of the wake instead of the width of the body. In other words, this means for the defined bluff body, that the shedding frequency increases when the width of the wake is reduced as it is the case for leeward coated cylinders.

Figure 32.

Shear stress and production term of turbulent kinetic energy.

The vortex street formation is delayed, i.e. shifted downstream, indicating a reduced mixing in the wake. This can be attributed to the increase in base pressure leading to a decrease in pressure fluctuations and a weakened vortex shedding.