# Population Distribution in the Wake of a Sphere

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## Abstract

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## 1. Introduction

## 2. Physical Model, Numerical Method and Boundary Conditions

## 3. Results on Spatial Structure of Steady Wake

## 4. Discussions and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the computational domain used in the simulations. The centre of the sphere is in the origin of the coordinate system. The flow moves from the left to the right in the picture. Two streamlines at Re = 200 are shown as an example.

**Figure 3.**Spatial distribution of the dimensionless stream-wise component of fluid velocity u in color and the contour lines of a scalar ${\theta}_{1}$ in black ($Sc=0.71$) and another scalar ${\theta}_{2}$ in white ($Sc=0.61$) for various steady axisymmetric and oblique $Re$. The visualization is across two central orthogonal planes ($z,x$) and ($y,x$) passing through the center of the sphere with an extent of [−1.5,1.5] along the horizontal axes and [−1.5,7.5] along the vertical x axis. Contour lines for ${\theta}_{1}$ and ${\theta}_{2}$ are plotted at magnitudes of $0.2,0.35,0.45,0.6,0.7,0.8$ and $0.9$, ascending from the ambient towards the sphere.

**Figure 4.**Distribution of the dimensionless stream-wise velocity component u for various $Re$. $u=0.95$ contours are drawn in solid lines along with $p=0$ pressure contours in dashed thin lines along the orthogonal (

**a**) ($z,x$) and (

**b**) ($y,x$) planes. A horizontal dotted line at $x=-0.325$ is drawn to divide the upstream spatial structure of u from the downstream one. Normalized population density function ${N}^{*}\left(u\right)=N\left(u\right)/A$ (A is the area of the orthogonal plane) for the u sample population across the orthogonal ($z,x$) and ($y,x$) planes are plotted respectively in (

**c**,

**d**). ${N}^{*}\left(u\right)$ for the upstream, downstream and the entire planes are respectively plotted as the bottom, middle and the top sets of curves. A scale difference is created by amplifying the ${N}^{*}\left(u\right)$ of the downstream and the entire domain 30 and 900 times respectively. Sample extent in (

**a**,

**b**) is $[-3,3]$ along the horizontal and $[-3,11]$ along the vertical axes, whereas, in (

**c**,

**d**) it is $[-3.5,3.5]$ along the horizontal and $[-5,20]$ along the vertical axes.

**Figure 5.**Distribution of the pressure p and velocity component v for various $Re$. The spatial distribution of v in color along with the contour lines of p at $0.1,0.05,0.0,-0.05,-0.1$ magnitudes respectively in red, orange, white, cyan, and blue solid lines along the orthogonal ($y,x$) plane for the axisymmetric $Re=175$ in (

**a**) and for the oblique $Re=275$ in (

**b**). Normalized population density of pressure ${N}^{*}\left(p\right)$ across the entire orthogonal ($y,x$) central plane is plotted in (

**c**), whereas ${N}^{*}\left(v\right)$ is plotted in (

**d**). The sample extent is similar to Figure 4.

**Figure 6.**Three dimensional spatial structure of velocity components, v and w. The surface contours of $w=-0.06$ and $0.06$ are plotted respectively in cyan and yellow in (

**a**), and $v=-0.06$ and $0.06$ contours are plotted respectively in blue and red in (

**b**). (

**c**,

**d**) present both the v and w contours for the oblique $Re=275$ and axisymmetric $Re=175$ flow fields respectively.

**Figure 7.**Spatial evolution of the normalized population density of scalar ${N}^{*}\left({\theta}_{1}\right)$ along the ($z,x$) plane is presented in (

**a**) and along the ($y,x$) plane in (

**b**). Evolution ${N}^{*}\left({\theta}_{2}\right)$ along the ($z,x$) plane is plotted in (

**c**) and along the ($y,x$) plane in (

**d**). These orthogonal planes pass through the center of the sphere and extends to the entire simulated domain of [-3.5:3.5] in the horizontal $y,z$, and [-5:20] in the stream-wise x directions.

**Figure 8.**Spatial distribution of convective scalar flux $\dot{Q}=u\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\theta}_{1}$ for various $Re$. Spatial distribution of $\dot{Q}$ in color along the orthogonal ($y,x$) plane for the axisymmetric $Re=175$ in (

**a**) and for the oblique $Re=275$ in (

**b**). The white contour lines represent $\dot{Q}=0.069$ in (

**a**) and $\dot{Q}=0.077$ in (

**b**), while the pink contour lines are at $\dot{Q}=0.11$ in (

**a**) and $\dot{Q}=0.096$ in (

**b**) respectively. Normalized population density of convective scalar flux ${N}^{*}\left(\dot{Q}\right)$ across the entire orthogonal ($z,x$) and ($y,x$) central planes are plotted respectively in (

**c**,

**d**). The sample extent is similar to Figure 4.

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**MDPI and ACS Style**

Bhowmick, T.; Wang, Y.; Iovieno, M.; Bagheri, G.; Bodenschatz, E.
Population Distribution in the Wake of a Sphere. *Symmetry* **2020**, *12*, 1498.
https://doi.org/10.3390/sym12091498

**AMA Style**

Bhowmick T, Wang Y, Iovieno M, Bagheri G, Bodenschatz E.
Population Distribution in the Wake of a Sphere. *Symmetry*. 2020; 12(9):1498.
https://doi.org/10.3390/sym12091498

**Chicago/Turabian Style**

Bhowmick, Taraprasad, Yong Wang, Michele Iovieno, Gholamhossein Bagheri, and Eberhard Bodenschatz.
2020. "Population Distribution in the Wake of a Sphere" *Symmetry* 12, no. 9: 1498.
https://doi.org/10.3390/sym12091498