Direct numerical simulation of turbulent elliptical pipe flow under system rotation about the major axis

The effect of Coriolis forces on the turbulent flow in an elliptical pipe subjected to spanwise rotation about the major axis has been studied using direct numerical simulations (DNS). In response to the system rotation, large-scale secondary flows appear in the cross-stream plane as a pair of counterrotating vortices, which significantly impact the turbulence statistics and structures of the flow. To capture the most energetic turbulent eddy motions in the streamwise direction, the pipe length has been extended to $L_z=20\pi b$ (here, $b$ is the minor semiaxis of the elliptical pipe), which is the longest in the current literature for the study of elliptical pipe flows. Laminarization occurs on the suction side of the flow and propagates toward the pressure side as the speed of the system rotation increases. The system rotation imposed radically alters the budget balance of Reynolds stresses through its effects on the mean and turbulent flow fields and through the Coriolis term. At a moderate rotation number, the Coriolis term starts to dominate the energy transfer from $\langle w^{\prime }{w}^{\prime }\rangle$ to $\langle v^{\prime }{v}^{\prime }\rangle$, and also acts on $\langle v^{\prime }{w}^{\prime }\rangle$ as an additional source term. This mechanism far surpasses the role of the pressure-strain term, which undergoes a significant reduction in response to the system rotation. The characteristics of the turbulence field is investigated in both physical and spectral spaces through analyses of the first- and second-order statistical moments, as well as the budget balance of the Reynolds stress transport equation and coherent flow structures.

Figure 1

Computational domain of a spanwise-rotating elliptical pipe in the Cartesian coordinate system. The major and minor semiaxes are denoted as $a$ and $b$, respectively. The two Coriolis force components $2\mathrm{\Omega }w$ and $-2\mathrm{\Omega }v$ are shown in the $y$ and $z$ directions, respectively, resulting from the spanwise rotation about the major ($x$) axis.

Figure 2

Cross-sectional view of the mesh. The domain is discretized into 1024 quadrilateral elements using a spectral-element method. Each element is constructed using a eighth-order Gauss-Lobatto-Legendre-Lagrange polynomial. The mesh is then expanded in the streamwise direction through 960 modes (not shown).

Figure 3

Profiles of the Reynolds shear stress ${\langle v^{\prime }{w}^{\prime }\rangle }^{+}$ along the pipe minor axis (for $\theta ={90}^{\circ }$ or ${270}^{\circ }$) at ${\text{Ro}}_{\tau }=1.0$, averaged over different time durations ($t_{\text{avg}}$). The profiles have been nondimensionalized using $u_{\tau 0S}^2$.

Figure 4

Contours of the instantaneous axial velocity $w^{+}$ nondimensionalized using $u_{\tau 0S}$ at four different rotation numbers.

Figure 5

Contours of the mean axial velocity ${\langle w\rangle }^{+}$ nondimensionalized using $u_{\tau 0S}$ with superimposed streamlines of the mean secondary flows in the cross-stream plane at ${\text{Ro}}_{\tau }=0.0$, 0.25, 4.0, and 24.0. The averaging time is over 40 LETOTs for rotating flow cases and over 80 LETOTs for the nonrotating flow case.

Figure 6

Bulk mean velocity $U_b^{+}$ nondimensionalized using $u_{\tau 0S}$ and nondimensionalized wall friction velocities $u_{\tau }/u_{\tau 0}$ at different rotation numbers. Superscripts $t, s$, and $b$ correspond to azimuthal locations $\theta ={90}^{\circ }, {180}^{\circ }$ (or ${360}^{\circ }$), and ${270}^{\circ }$, respectively.

Figure 7

Profiles of the mean axial velocity ${\langle w\rangle }^{+}$ in the vertical plane ($\theta ={90}^{\circ }$ or ${270}^{\circ }$) and horizontal plane ($\theta =0^{\circ }$ or ${180}^{\circ }$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, 0.5, 1.0, 4.0, 8.0, 16.0, and 24.0. The profiles are nondimensionalized using $u_{\tau 0S}$. The arrow indicates a monotonic trend with an increasing ${\text{Ro}}_{\tau }$.

Figure 8

Profiles of the mean axial velocity ${\langle w\rangle }^{+}$ displayed in the semilogarithmic wall coordinate system in the vertical plane ($\theta ={90}^{\circ }$ or ${270}^{\circ }$) from (a) the pressure side and (b) the suction side at different rotation numbers. The profiles are nondimensionalized using the time- and axially averaged local friction velocity $u_{\tau }$. Specifically, the value of $\langle w\rangle$ is nondimensionalized by $u_{\tau }^t$ and $u_{\tau }^b$ on the pressure and suction sides of the elliptical pipe, respectively. DNS data of Nikitin and Yakhot [19] (denoted as “N&Y (2005)”) are also presented for the purpose of comparison. The thin black lines represent the classical law-of-the-wall based on the two-layer boundary-layer model of von Kármán for a circular pipe flow. The arrow indicates a monotonic trend with an increasing ${\text{Ro}}_{\tau }$.

Figure 9

Profiles of the mean vertical velocity ${\langle v\rangle }^{+}$ in the horizontal plane ($\theta =0^{\circ }$ or ${180}^{\circ }$) along the major ($x$) axis. Given the significant differences in the magnitude between low and high rotation numbers, to have a clear view, the profiles of ${\langle v\rangle }^{+}$ are presented in two panels. (a) for ${\text{Ro}}_{\tau }=0.0$, 0.25, 0.5, and 1.0, and (b) for 4.0, 8.0, 16.0, and 24.0. The profiles are nondimensionalized using $u_{\tau 0S}$. Arrows indicate monotonic trends with an increasing rotation number ${\text{Ro}}_{\tau }$.

Figure 10

Contours of turbulent kinetic energy $k^{+}$ nondimensionalized using $u_{\tau 0S}^2$ for cases of ${\text{Ro}}_{\tau }=0.0$, 0.25, 4.0, and 16.0 (left to right) in the cross-stream plane. Contours corresponding to TKE levels below $k^{+}=0.2$ are clipped to clearly show the propagation of laminarization.

Figure 11

Profiles of Reynolds normal and shear stresses nondimensionalized using $u_{\tau 0S}^2$ along the pipe minor axis (for $\theta ={90}^{\circ }$ or ${270}^{\circ }$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, 0.5, 1.0, 4.0, 8.0, 16.0, and 24.0. An arrow indicates a monotonic trend with an increasing ${\text{Ro}}_{\tau }$.

Figure 12

Profiles of Reynolds normal and shear stresses nondimensionalized using $u_{\tau 0S}^2$ along the pipe major axis (for $\theta =0^{\circ }$ or ${180}^{\circ }$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, 0.5, 1.0, 4.0, 8.0, 16.0, and 24.0. An arrow indicates a monotonic trend with an increasing ${\text{Ro}}_{\tau }$.

Figure 13

Profiles of the budget terms of ${\langle w^{\prime }{w}^{\prime }\rangle }^{+}$ along the minor axis (nondimensionalized by $u_{\tau 0S}^3/b$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, and 8.0.

Figure 14

Profiles of the budget terms of ${\langle v^{\prime }{w}^{\prime }\rangle }^{+}$ along the minor axis (nondimensionalized by $u_{\tau 0S}^3/b$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, and 8.0.

Figure 15

Profiles of the budget terms of ${\langle v^{\prime }{v}^{\prime }\rangle }^{+}$ along the minor axis (nondimensionalized by $u_{\tau 0S}^3/b$) at ${\text{Ro}}_{\tau }=0.0$, 0.25, and 8.0.

Figure 16

Two-point autocorrelation coefficients of the axial velocity fluctuations along the axial direction of the pipe at wall-normal position $y^{+}=14.1$ on the pressure side of the pipe in the central vertical plane ($\theta ={90}^{\circ }$ or ${270}^{\circ }$) for nonrotating and rotating elliptical pipe flows of ${\text{Ro}}_{\tau }=0$ and 1.0. The value of $R_{33}$ is obtained by performing independent DNS based on various domain sizes (in the range $2\pi b\le L_z\le 20\pi b$) at both rotation numbers.

Figure 17

Effect of spanwise system rotation on the premultiplied velocity spectra $k_z^{+}{\varphi }_{ii}^{+}$ nondimensionalized by $u_{\tau 0S}^2$ along the axial direction at wall-normal position $y^{+}=14.1$ on the pressure side of the pipe in the central vertical plane ($\theta ={90}^{\circ }$ or ${270}^{\circ }$). The premultiplied spectra at various rotation numbers are all calculated based on a fixed pipe length of $L_z=20\pi b$, and the cutoff wavelengths corresponding to shorter pipe lengths are indicated in panel (a). The arrows indicate monotonic trends as a result of an increasing rotation number.

Figure 18

Contours of instantaneous axial vorticity ${\omega }_z^{+}$ nondimensionalized by $u_{\tau 0S}^2/\nu$ in cross-stream planes at different rotation numbers. All contours are extracted at the same time instant (at the beginning of the 41st LETOT of each simulation).

Figure 19

Isosurfaces of instantaneous vortical structures (plotted by setting the swirling strength to ${\lambda }_{ci}=1.0$) for the nonrotating (${\text{Ro}}_{\tau }=0.0$) and rotating (${\text{Ro}}_{\tau }=0.5$ and 8.0) cases. For clarity, only a quarter of the axial domain and one half of the cross-stream domain are shown. The contour colors indicate the nondimensionalized elliptical radius $r/R$. All instantaneous snapshots are extracted from the same time instant (at the beginning of the 41st LETOT of each simulation).

Figure 20

Near-wall streaks visualized using the contours of instantaneous axial velocity fluctuation $w^{\prime +}$ (nondimensionalized using $u_{\tau 0S}$) at wall-normal distance $d^{+}=14.1$ along the periphery for ${\text{Ro}}_{\tau }=0.0$, 0.5, and 8.0. For clarity, only half of the axial domain size is shown. All instantaneous snapshots are extracted from the same time instant (at the beginning of the 41st LETOT of each simulation).

Figure 21

The JPDF of $w^{\prime }$ and $-v^{\prime }$ at $y^{+}=14.1$ measured from the pressure wall along the minor axis for different rotation numbers. The four quadrants are divided by thin white lines.