Experimental investigation of the wake behind a rotating sphere

The wake behind a sphere, rotating about an axis aligned with the streamwise direction, has been experimentally investigated in a low-velocity water tunnel using laser-induced fluorescence visualizations and particle image velocimetry measurements. The measurements focused on the evolution of the flow regimes that appear depending on two control parameters: the Reynolds number Re and the dimensionless rotation or swirl rate $\mathrm{\Omega }$, which is the ratio of the maximum azimuthal velocity of the body to the free-stream velocity. In the present investigation, we cover the range of Re smaller than 400 and $\mathrm{\Omega }$ from 0 and 4. Different wakes regimes such as an axisymmetric flow, a low helical state, and a high helical mode are represented in the (Re, $\mathrm{\Omega }$) parameter plane.

Figure 1

Schematic view of the experimental setup. The water channel and the detailed view of the test section showing the dye injection system and support. The sphere axis is supported by a group of $74\mu \mathrm{m}$ nylon wires with the closest wire located $5\mathrm{cm}$ upstream from the sphere.

Figure 2

PIV laser plane at a given $x$ position behind the sphere.

Figure 3

Side view visualization for $\text{Re}=250$. From top to bottom: $\mathrm{\Omega }=0$ (2CRV regime), 0.2, 0.5 (LH), 0.8, 1.2 (HH).

Figure 4

Side view visualization for $\text{Re}=300$. From top to bottom: $\mathrm{\Omega }=0$ (HS regime), 0.2, 0.5 (LH), 0.8, 1.2 (HH).

Figure 5

Full bifurcation map in the $(\text{Re},\mathrm{\Omega })$ parameter plane. The boundary between regions has been estimated by modal analysis and frequency analysis of the PIV measurements. The different predominant modes correspond as follows: $m=0$ axisymmetric flow, $m=1$ LH, and $m=2$ HH, and HS stands for HS instability. Each point represents a single PIV measurement.

Figure 6

Instantaneous velocity and vorticity fields from PIV measurements for $\text{Re}=250$. In each row the background color represents velocity components $v_y$ (m/s), $v_z$ (m/s), and the longitudinal vorticity ${\omega }_x$ (1/s) (from left to right). The consecutive rows correspond to a different rotation rate: (a) $\mathrm{\Omega }=0$, 2CRV, (b) $\mathrm{\Omega }=0.2$, axisymmetric, (c) $\mathrm{\Omega }=0.5$, LH, (d) $\mathrm{\Omega }=1.6$, HH.

Figure 7

Spatio-temporal representation of the measured streamwise vorticity at $x=2D$ behind the sphere. $\text{Re}=250, \mathrm{\Omega }=0$ (2CRV), 0.2 (axisymmetric), 0.5 (LH), 1.6 (HH) from top to bottom. The dark and light (red and blue in color version) isosurfaces correspond to ${\omega }_x=\pm 0.3{\omega }_{max}$ respectively, where ${\omega }_{max}$ denotes the maximum value of vorticity for each case. Horizontal axis corresponds to the elapsed time.

Figure 8

Fourier azimuthal decomposition patterns for instantaneous field of $\text{Re}=250$ and $\mathrm{\Omega }=0.5$ (each mode has its own color scale).

Figure 9

Spatial-temporal representation of the filtered streamwise vorticity. We present the sum of the modes $m=1$ and $m=2$ as a filtered vorticity field without $m=0$ (b) with a comparison with the full vorticity field (a). $\text{Re}=250, \mathrm{\Omega }=0.5$ (top), $\mathrm{\Omega }=1.6$ (bottom). (The dark and light (red and blue in color version) isosurface corresponds to ${\omega }_x=\pm 0.3{\omega }_{max}$ respectively, where ${\omega }_{max}$ denotes the maximum value of vorticity for a given case).

Figure 10

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=75$. (a) Mode $m=0$ of the average field grows linearly up to $\mathrm{\Omega }=2.7$, axisymmetric flow; (b) modes $m=1$, 2, 3, 4: at 2.7 modes $m=2$ and $m=4$ appear along with HH.

Figure 11

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=100$: mode $m=0$ of the average field (a) and modes $m=1$, 2, 3, 4 (b). The axisymmetric to HH transition at $\mathrm{\Omega }=2.4$.

Figure 12

Spectra for $\text{Re}=75$ (a) and 100 (b). The sphere rotation frequency (dashed line) was detected for axisymmetric flow up to value of $\mathrm{\Omega }$ (a) 2.6 and (b) 2.4. For higher $\mathrm{\Omega }$ the frequency at ${\text{St}}_{75}^2\approx {\text{St}}_{100}^2=0.48$ was observed (independent from the rotation). Red and white denote the highest and zero amplitude, respectively.

Figure 13

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=250$. (a) $m=0$ of the average field grows linearly up to $\mathrm{\Omega }=0.6$ and stagnates at $\mathrm{\Omega }=1.5$ (HH); (b) modes $m=1$, 2, 3, 4: an isolated point of $m=1$ of 2CRV is seen at $\mathrm{\Omega }=0$, decays immediately as $\mathrm{\Omega }$ increases and reappears at 0.5 (LH).

Figure 14

Dimensionless enstrophy of mode $m=0$ as a function of $\mathrm{\Omega }$ and $\text{Re}$. In panel (b), where the enstrophy is nondimensionalized with the vorticity induced by the rotation ($\sim v_{\theta }/D$), the initial mode $m=0$ is nearly constant and equal to 20. In panel (a) the points of $\text{Re}=75$ are shifted, to maintain linearity, according to ${\epsilon }_{\mathrm{plot}}={\epsilon }_{\mathrm{real}}+{\epsilon }_{\mathrm{sh}}$, where ${\epsilon }_{\mathrm{sh}}=7.54$.

Figure 15

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=150$. (a) Mode $m=0$ of the average field grows linearly up to $\mathrm{\Omega }=1$, (axisymmetric flow), followed by small deviation due to LH and stagnation at $\mathrm{\Omega }=2$ (HH); (b) modes $m=1$, 2, 3, 4: first detection of mode $m=1$ (LH) at 1.2. The substantial increase of mode $m=2$ is at $\mathrm{\Omega }=2$.

Figure 16

Spectra for $\text{Re}=150$ (a) and 200 (b). Red and white denote the highest and zero amplitude, respectively. As $\mathrm{\Omega }$ increases, three main groups was observed: linear growth with rotation frequency (dashed line), LH frequency ${\text{St}}_{150}^1=0.14$ and ${\text{St}}_{200}^1=0.12$, HH frequency ${\text{St}}_{150}^2=0.45$ and ${\text{St}}_{200}^2=0.41$.

Figure 17

Spectra for $\text{Re}=250$ and the superimposed results of Ref. [13] related to the frequency of the leading eigenmode for axisymmetric basic wake. Red and white denote the highest and zero amplitude, respectively.

Figure 18

Dimensionless enstrophy of mode $m=1,2$ as a function of $\mathrm{\Omega }$ for all investigated $\text{Re}$. (a) Mode $m=1$: bifurcation points for LH regime: starts with 0.25 for $\text{Re}=350$ up to 1.2 for $\text{Re}=150$; (b) Mode $m=2$: bifurcation points for HH regime: 1.1 to 2.7 for $\text{Re}=350$ to $\text{Re}=75$, respectively.

Figure 19

Main frequency as a function of $\text{Re}$. The frequency of HH is nearly two times higher than the frequency of LH.

Figure 20

$\delta {\epsilon }^{1/2}(m=0)$ as a function of enstrophy of mode $m=2$ and $m=4$ for different $\text{Re}$ in log-log scale. (a) The mode $m=2$ is proportional with coefficient 1 to the deviation from linearity of $m=0$; (b) the analogous proportionality coefficient for $m=4$ is nearly equal to 0.5.

Figure 21

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=300$. (a) $m=0$ of the average field grows linearly up to $\mathrm{\Omega }=0.6$ and stagnates at $\mathrm{\Omega }=1.3$ (HH); (b) modes $m=1$, 2, 3, 4: an isolated point of $m=1$ of HS is seen at $\mathrm{\Omega }=0$, drops immediately almost to 0 at $\mathrm{\Omega }=0.1-0.2$, subsequently LH is observed and $m=1$ continues to grow up to $\mathrm{\Omega }=1.2$, when HH appears.

Figure 22

Spectra for $\text{Re}=300$ (a) and 350 (b). Numerical results of Refs. [12, 13] were superimposed for $\text{Re}=300$. Strong correlation between numerics and experiment can be seen. Red and white denote the highest and the zero amplitude, respectively.

Figure 23

Dimensionless enstrophy of modes as a function of $\mathrm{\Omega }, \text{Re}=350$. (a) $m=0$ of the average field grows linearly up to $\mathrm{\Omega }=0.5$ and stagnates at $\mathrm{\Omega }=1.2$ (HH); (b) $m=1$, 2, 3, 4: $m=1$ of HS is seen at $\mathrm{\Omega }=0$, drops down and continues to grow while LH is observed.

Figure 24

Spatio-temporal representation of the streamwise vorticity measured at $x=2D$ behind the sphere. $\text{Re}=350, \mathrm{\Omega }=0$ (HS), 0.2, 0.5 (LH), 1.2 (HH) from top to bottom. The dark and light (red and blue in color version) isosurface corresponds to ${\omega }_x=\pm 0.3{\omega }_{max}$ respectively, where ${\omega }_{max}$ denotes the maximum value of vorticity for a given case.

Figure 25

DMD modes related to the predominant frequency, $\text{Re}=300$ (top) and $\text{Re}=350$ (bottom), $\mathrm{\Omega }=0,0.05,0.1,0.15,...,0.4$ (left to right).