Direct numerical simulation of low Reynolds number turbulent swirling pipe flows

Direct numerical simulation of swirling pipe flows is performed to investigate the effects of swirl on turbulence statistics as well as the inertial regime. The swirling motion is imposed via a constant azimuthal body force coupled with a body force in the axial direction that drives the flow. The friction Reynolds number is ${\text{Re}}_{\tau }\approx 170$ with a pipe length of $8\pi \delta$ (where $\delta$ is the pipe radius). The simulations are performed at various swirl numbers. The swirling motion introduces additional drag into the flow and reduces the bulk flow rate for the swirl strengths investigated in this study. We also show from our data as well as via the mean equations that similar to the total axial stress, the total azimuthal stress when normalized by the azimuthal friction velocity follows a linear decrease from wall to the pipe centerline independent of swirl strength. In the inertial regime, with negligible viscous effects, the linear relationship holds for the turbulent shear stresses. We derive the modified axial mean momentum equation, which when studied alongside the azimuthal momentum equation shows that increasing the swirl strength increases the inertial region in the pipe by pushing the beginning of the inertial region closer to the wall. This effect is governed by the axial viscous forces rather than the azimuthal ones.

Figure 1

Instantaneous axial velocity in the $x\text{−}r\theta$ plane for all cases at wall-normal location $y^{+}\approx 15$. (a) Nonswirling pipe F0, (b) 50% forcing F50, (c) 100% forcing F100, and (d) 200% forcing F200.

Figure 2

Comparison of swirl angle at $y^{+}\approx$ 15 as a function of (a) forcing ratio $f_{\theta }/f_x$, (b) swirl number $S$, and (c) azimuthal Reynolds number ${\text{Re}}_{\tau ,\theta }={\delta }^{\times }$. The symbols are F0 (gray $▵$), F50 (red $◯$), F100 (black $\square$), and F200 (blue $\diamond$).

Figure 3

Instantaneous axial velocity profile. (a) Nonswirling pipe F0, (b) 50% forcing F50, (c) 100% forcing F100, and (d) 200% forcing F200.

Figure 4

(a) Mean streamwise velocity profile, (b) streamwise turbulence intensity, (c) azimuthal turbulence intensity, and (d) radial turbulence intensity. Black dot line (nonswirling pipe F0). Red line (F50), black line (F100), blue line (F200). Arrow indicates increasing $f_{\theta }$. Inset in (a) shows the bulk velocity $U_b$ normalized by the bulk velocity of nonswirl case $U_{b,f0}$.

Figure 5

Mean swirl velocity $U_{\theta }$ profile. (a0 and (b) $U_{\theta }$ normalized by $u_{\tau ,\theta }$, $U_{\theta }^{\times }$, in linear and log scale ($x$ axis), respectively. (c) $U_{\theta }$ normalized by $u_{\tau ,x}$, $U_{\theta }^{+}$. (d) $U_{\theta }$ normalized by $u_{\tau ,x\theta }$. Line symbols are as in Fig. 4. The arrow denotes increasing $f_{\theta }$.

Figure 6

The total stresses Reynolds stress for cases F50 (red), F100 (black), and F200 (blue). (a) ${\tau }_{xr}^{+}=\frac{d{U}_x^{+}}{d{y}^{+}}-{\langle u_r{u}_x\rangle }^{+}$; dashed lines ${\tau }_{xr}^{+}$, dotted line ${\langle u_r{u}_x\rangle }^{+}$, and solid line $\frac{d{U}_x^{+}}{d{y}^{+}}$. (b) ${\tau }_{r\theta }^{\times }=\left(\frac{d{U}_{\theta }^{\times }}{d{y}^{\times }}-\frac{U_{\theta }^{\times }}{y^{\times }}\right)-{\langle u_r{u}_{\theta }\rangle }^{\times }$; dashed lines ${\tau }_{r\theta }^{\times }$, dotted line ${\langle u_r{u}_{\theta }\rangle }^{\times }$, and solid line $\left(\frac{d{U}_{\theta }^{\times }}{d{y}^{\times }}-\frac{U_{\theta }^{\times }}{y^{\times }}\right)$.

Figure 7

(a) Contribution of the pressure gradient (PG), viscous force (VF), and turbulent inertia (TI) to the mean axial momentum equation [Eq. (1)] for all cases. (b) Zoomed-in region of (a) from $20\le y^{+}\le$ 55. Line are dashed line (nonswirling pipe F0), red line (F50), black line (F100), and blue line (F200). The arrow indicates increasing $f_{\theta }$. The dots indicate the intercept of TI and VF. The inset in (b) shows the wall-normal location $y^{+}$ of VF = TI versus ${\delta }^{\times }$ for all swirl cases. The best-fit dash line equation is $y^{+}=-0.106{\delta }^{\times }+45.9$.

Figure 8

(a) Contribution of the pressure gradient (${\mathrm{PG}}^{\times }$), viscous force (${\mathrm{VF}}^{\times }$) and turbulent inertia (${\mathrm{TI}}^{\times }$) to the azimuthal mean momentum equation (9) for all three cases. (b) Zoomed-in region of (a) from $10\le y^{\times }\le$ 30. Line colors are as in Fig. 4. The arrow indicates increasing $f_{\theta }$. The circular symbols in the insert in (b) show the wall-normal location $y^{\times }$ of ${\mathrm{VF}}^{\times }$ = ${\mathrm{TI}}^{\times }$ versus ${\text{Re}}_{\tau ,\theta }$ for all swirl cases. The corresponding best-fit dash line equation is $y^{\times }=0.122\sqrt{{\delta }^{\times }}+4$. The triangular symbols are for data of $y^{+}$ from the inset of Fig. 7 converted to the corresponding $y^{\times }$ values for comparison.