Flow measurements in the near wake of a smooth sphere and one mimicking a pine cone

A combination of time-resolved particle image velocimetry (PIV) and tomographic PIV was used to study the instantaneous three-dimensional vortex shedding as well as mean velocities and turbulent stresses in the near wake of a smooth and a rough sphere. Sphere roughness was modeled by spiraling grooves found in nature on pine cones. The measurements extended up to five sphere diameters $D$, downstream of the sphere, and were performed across a Reynolds number range of $226<{\mathrm{Re}}_D<$ 5067, spanning several vortex shedding regimes. Our results showed that the main effect of the sphere roughness on the near-wake flow was to lower ${\mathrm{Re}}_D$ for which the clear, temporally averaged recirculating wake pattern observed at low ${\mathrm{Re}}_D$ disappeared. Furthermore, mean velocities and root-mean-square (rms) values of fluctuating velocities scaled best with the mean recirculating wake length, ${\overline{L}}_w$, instead of $D$ that is commonly used in the far wake. Normalized, maximum rms values of velocity fluctuations measured in a horizontal equatorial plane increased with increasing ${\mathrm{Re}}_D$ and were well fitted by a logarithmic function. The mean velocity defect recovery was well described by a power law with exponents decreasing from $-1$ to $-3$ at ${\mathrm{Re}}_D\approx$ 2000, and subsequently increasing to $-2.3$ at ${\mathrm{Re}}_D\approx$ 5000. Most surprisingly, our results showed that ${\overline{L}}_w/D$ was locally minimum at ${\mathrm{Re}}_D\approx 900$. At these ${\mathrm{Re}}_D$, the wake transitioned from a well-organized laminar wake, characterized by a single vortex shedding frequency, to one exhibiting high and low shedding frequency branches. By analyzing the instantaneous centroid positions of the velocity defect in a transverse plane, we showed that the local minimum value of ${\overline{L}}_w/D$ was the result of increased wake “meandering” that resulted in enhanced mixing at this ${\mathrm{Re}}_D$. Moreover, at the transition, the wake alternated between the organized shedding pattern observed at low ${\mathrm{Re}}_D$ and the one lacking any preferred orientation at higher ${\mathrm{Re}}_D$.

Figure 1

Schematic layout of the experimental setup. The inset shows the employed coordinate system with the origin at the sphere center. (Not to scale.)

Figure 2

Example of the spiraling arrangement of scales according to Fibonacci number ratios in a pine cone. Red and blue curves denote counterclockwise- and clockwise-rotating spirals, respectively. Numbers indicate the first (1) and the last (8 or 13) spiral of each set. Courtesy of Warren Wilson College Physics Department [42].

Figure 3

Images showing the similarity between (a) an actual pine cone, (b) a 3D CAD model of the rough sphere, and (c) the 3D-printed CAD model.

Figure 4

The Strouhal number vs ${\mathrm{Re}}_D$ for the smooth ($\square$) and the rough ($◯$) spheres. The cyan “envelope” represents the published data range from different studies as compiled by Sakamoto and Haniu [1].

Figure 5

Instantaneous snapshots of the 3D vortex shedding structure (tomo-PIV) as visualized by isosurfaces of the $Q$ criterion overlaid by the normalized streamwise vorticity component, ${\omega }_{1}D/U_c$. ${\mathrm{Re}}_D$ = (a) 465, (b) 864, (c) 1208, and (d) 2061. See the Supplemental Material [53].

Figure 6

Contour plots of the normalized time-averaged streamwise velocity, ${\overline{U}}_1/U_c$ (TR-PIV), for the smooth sphere (left column) and the rough sphere (right column). Reynolds numbers are (a) 353, (b) 457, (c) 835, (d) 1580, (e) 2775, (f) 355, (g) 460, (h) 841, (i) 1611, and (j) 2804. Level list $[-0.4$:0.1:1] denoting [minimum: step size: maximum]. An example level is indicated in (a). The mean recirculating wake length is indicated in (c). White dashed curves denote the recirculating wake boundary.

Figure 7

Contour plots of the the normalized time-averaged transverse velocity, ${\overline{U}}_2/U_c$ (TR-PIV), for the smooth sphere (left column) and the rough sphere (right column). Reynolds numbers are (a) 353, (b) 457, (c) 835, (d) 1580, (e) 2775, (f) 355, (g) 460, (h) 841, (i) 1611, and (j) 2804. Level list $\pm$ [0.05 0.1 0.15 0.2 0.3]. Blue dashed curves denote the recirculating wake boundary.

Figure 8

The normalized, mean recirculating wake length as a function of ${\mathrm{Re}}_D$. Present results: $▿$: smooth sphere; $▵$: rough sphere; tomo-PIV: $\blacksquare {\overline{L}}_w^{max}/D, \square {\overline{L}}_w/D$. Literature results (uniform flow): (experimental) $\times$ [43], $▹$ [15]; (numerical) $+$ [4], $⧫$ [25], $*$ [3], $◯$ [26], $•$ [27]. Literature results (turbulent upstream conditions): (experimental) $\circ$ TI $>$ 26% [6], $\diamond$ TI = 4.6% [25]. The solid curve depicts the turbulence intensity (TI) in the water channel at the position of the sphere (secondary $y$ axis).

Figure 9

Contour plots of the normalized, ensemble-averaged streamwise velocity, ${\overline{U}}_1/U_c$, in vertical ($x_1\text{−}{x}_3$, left column) and horizontal ($x_1\text{−}{x}_2$, right column) equatorial planes (tomo-PIV, [45]). The supporting rod is depicted. ${\mathrm{Re}}_D$ = (a),(b) 465, (c),(d) 864, (e),(f) 1206, and (g),(h) 2061. Level list $[-0.4$:0.1:1]; an example level is indicated in (c).

Figure 10

Examples of flow “visualizations” in the wake of the smooth (left column) and the rough (right column) spheres. ${\mathrm{Re}}_D$ = (a) 226, (b) 457, (c) 632, (d) 714, (e) 835, (f) 285, (g) 460, (h) 484, (i) 684, and (j) 841. Note that flow patterns are best observed digitally at high magnification, especially for the rough sphere (right column).

Figure 11

Contour plot of the normalized instantaneous velocity defect, $(U_c-U_1)/U_c$, in a transverse plane at $x_1/D=1$ (tomo-PIV, smooth sphere at ${\mathrm{Re}}_D=864$). The sphere perimeter is indicated as a dashed circle, and its center position by a “$+$”. The centroid position of the velocity defect [Eq. (1)] is denoted by $⨁$. Definitions of the orientation angle $\theta$ (positive in the clockwise direction), and radius $\rho$ are shown.

Figure 12

Normalized radii (left column) and instantaneous orientation angles (right column) of the wake centroid position based on the velocity defect at $x_1/D=1$ (tomo-PIV, smooth sphere). ${\mathrm{Re}}_D$ = (a),(b) 465, (c),(d) 864, (e),(f) 1206, and (g),(h) 2061. Note: The $x$-axes scales are not the same.

Figure 13

Profiles of the normalized, time-averaged streamwise velocity at $x_2/D=0$ as a function of $x_1/D$ for all investigated ${\mathrm{Re}}_D$ (TR-PIV). (a) Smooth sphere and (b) rough sphere. The legend numbers in (a) and (b) correspond to ${\mathrm{Re}}_D$.

Figure 14

Profiles of the velocity defect, $1-{\overline{U}}_1/U_c$ (TR-PIV), at $x_2/D=0$ as a function of $x_1/{\overline{L}}_w$ for all investigated ${\mathrm{Re}}_D$ (log-log plot). (a) Smooth sphere and (b) rough sphere. Legend numbers in (a) and (b) correspond to ${\mathrm{Re}}_D$. Solid lines represent examples of the least-squares-fitted power laws for ${\mathrm{Re}}_D$ equaling (a) 353, 714, and 2174, and (b) 355, 684, and 2804. Note that the $x$- and $y$-axis scales are dissimilar in (a) and (b).

Figure 15

Least-squares-fitted power-law exponent $n$ vs ${\mathrm{Re}}_D$ for the smooth and the rough spheres (TR-PIV). Error bars indicate uncertainties based on 95% confidence intervals.

Figure 16

Contour plots of the normalized rms values of the streamwise velocity fluctuations, $u_1^{\prime }/U_c$ (TR-PIV), for the smooth sphere (left column) and the rough sphere (right column). Reynolds numbers are (a) 353, (b) 457, (c) 835, (d) 1580, (e) 2775, (f) 355, (g) 460, (h) 841, (i) 1611, and (j) 2804. Level list [0.04:0.02:0.1 0.14:0.04:0.22]. One level is given in (b). The blue dashed curve demarcates the mean recirculating wake.

Figure 17

Contour plots of the normalized rms values of the transverse velocity fluctuations, $u_2^{\prime }/U_c$ (TR-PIV), for the smooth sphere (left column) and for the rough sphere (right column). Reynolds numbers are (a) 353, (b) 457, (c) 835, (d) 1580, (e) 2775, (f) 355, (g) 460, (h) 841, (i) 1611, and (j) 2804. Level list [0.04:0.02:0.1 0.14:0.04:0.22]. One level is given in (c). The blue dashed curve demarcates the mean recirculating wake.

Figure 18

Transverse profiles of the normalized rms values of the streamwise velocity fluctuations, $u_1^{\prime }/U_c$, at $x_1={\overline{L}}_w$ (TR-PIV). (a) Smooth sphere and (b) rough sphere. The legend numbers denote ${\mathrm{Re}}_D$.

Figure 19

Streamwise profiles of the normalized rms values of the transverse velocity fluctuations, $u_2^{\prime }/U_c$ (at $x_2/D=0$) vs $x_1/{\overline{L}}_w$ (TR-PIV). (a) Smooth sphere and (b) rough sphere. The legend numbers denote ${\mathrm{Re}}_D$. Note that the $x$-axis scales in (a) and (b) are different.

Figure 20

Streamwise profiles of the normalized rms values of the transverse velocity fluctuations, $u_2^{\prime }/U_c$ (at $x_2/D=0$) vs $x_1/D$ (TR-PIV). (a) Smooth sphere and (b) rough sphere. The legend numbers denote ${\mathrm{Re}}_D$. Note that the $x$-axis scales in (a) and (b) are different.

Figure 21

Maximum values of $u_1^{\prime }/U_c$ and $u_2^{\prime }/U_c$ (TR-PIV) as a function of $x_1/{\overline{L}}_w$ for the smooth (open symbols) and the rough (closed symbols) spheres. (a) ${(u_1^{\prime }/U_c)}_{max}$, (b) ${(u_2^{\prime }/U_c)}_{max}$. Present results: legend numbers denote ${\mathrm{Re}}_D$. Literature results: $\times {\mathrm{Re}}_D=3700$ and ${10}^4$ [22], $+{\mathrm{Re}}_D=5000$ [19], $\diamond {\mathrm{Re}}_D=3700$ [23]. Note that the $x$-axis scales in (a) and (b) are different.

Figure 22

Maximum values of (a) $u_1^{\prime }/U_c$ and (b) $u_2^{\prime }/U_c$ (TR-PIV) as a function of ${\mathrm{Re}}_D$ for the smooth (black open symbols) and the rough (black closed symbols) spheres. $\left(▵\right){(u_1^{\prime }/U_c)}_{max}, (\circ ){(u_2^{\prime }/U_c)}_{max}$. Literature results: $\times {\mathrm{Re}}_D=3700$ and ${10}^4$ [22], $+{\mathrm{Re}}_D=5000$ [19], $\diamond {\mathrm{Re}}_D$ = 3700 [23]. Dash lines in (a) and (b) are least-squares fits to the present data, ${(u_i^{\prime }/U_c)}_{max}=p_1{\text{log}}_{10}\left({\text{Re}}_D\right)-p_2$. (a) $p_1=0.147\pm 0.023, p_2=0.276\pm 0.069$ with coefficient of determination, $R^2=0.86$; (b) $p_1=0.185\pm 0.030, p_2=0.425\pm 0.088$ with $R^2=0.85$.

Figure 23

Contour plots of the normalized Reynolds shear stress, $\overline{u_1{u}_2}/U_c^2$ (TR-PIV), for the smooth sphere (left column) and the rough sphere (right column). Reynolds numbers are (a) 353, (b) 457, (c) 835, (d) 1580, (e) 2775, (f) 355, (g) 460, (h) 841, (i) 1611, and (j) 2804. Level list $\pm$ [0.0005 0.001 0.002:0.002:0.02 0.026]. The black dashed curve demarcates the mean recirculating wake.

Figure 24

Transverse profiles of the normalized Reynolds shear stress values, $\overline{u_1{u}_2}/U_c^2$, at $x_1={\overline{L}}_w$ (TR-PIV). (a) Smooth sphere and (b) rough sphere. The legend numbers denote ${\mathrm{Re}}_D$.