Dual-plane turbulent jets and their non-Gaussian velocity fluctuations

Direct numerical simulations are performed to investigate the spatial evolution of dual-plane jet flows with different separation lengths between the two jets. Based on the scaling law and the probability density function of the turbulent/nonturbulent interface of a single plane jet, the jet-interaction length scale $X_{*}$ is introduced. It is shown that for different separation lengths, the streamwise evolutions of various statistics along the centerline all scale with $X_{*}$. This finding may explain the linear relationship between the location of the merge point and the separation length. Of particular interest is the evolution of the probability distributions and energy spectra of the streamwise velocity fluctuations in the developing region. Similar to the case of grid-generated turbulence, the probability distribution of the velocity fluctuations can also be non-Gaussian in a dual-plane jet flow. For all flow configurations considered, close to the inlet (e.g., $X/X_{*}\simeq 1.0)$ where the two jets have not yet joined together, the skewness of the streamwise velocity fluctuations is negative. In contrast, at a further downstream location (e.g., $X/X_{*}\simeq 2.0)$, where the turbulence intensity and mean pressure reach their maximum values, the skewness takes a positive value instead. Our study suggests that there are two different physical mechanisms responsible for the formation of the intense oscillations of the velocity fluctuations. The negative value of the skewness in the upstream region is caused by the large-scale movement of the contrarotary vortices, whereas the streamwise position of the positive skewness appears to be correlated to the location of peak intensity near (or after) the merging of the jets.

Figure 1

(a) Schematic front view and (b) schematic side view of the computational domain. The two drawings are not to scale.

Figure 2

Streamwise evolution of the spatial resolution ${\left(\mathrm{\Delta }X\mathrm{\Delta }Y\mathrm{\Delta }Z\right)}^{1/3}/\eta$ along the centerline.

Figure 3

Comparison of the vertical distribution of the normalized mean streamwise velocity $(U-U_A)/(U_C-U_A)$ at three different downstream locations (i.e., $X/L_0=24$, 26, and 28) for the case with $L_d/L_0=4$ with previous studies (a) experiments by Gutmark and Wygnansky [46], Ramparian and Chandrasekhara [47], and Watanabe et al. [31]. (b) DNSs by Stanley et al. [32], da Silva and Pereira [48], and Watanabe et al. [31].

Figure 4

Vertical distribution of the normalized mean streamwise velocity $(U-U_A)/(U_C-U_{max})$ at the far downstream location (i.e., $X/L_0=28)$ for all three cases (i.e., $L_d/L_0=4$, 6, and 8). To investigate the influence of the lateral domain, the vertical distribution of $U$ within the range $-L_Y/2\le Y/L_0\le L_Y/2$ is plotted.

Figure 5

Streamwise evolution of the streamwise turbulence rms value $U_{rms}/U_J$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline.

Figure 6

Streamwise evolution of the mean streamwise velocity $U_C/U_J$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The two dashed lines in the three subfigures indicate $U_C/U_J=0.1$ and 0.0.

Figure 7

Vertical distribution of $U/U_J$ at $X/X_{*}=3$ for all three cases.

Figure 8

Evolution of the small-scale anisotropy $K=\langle {(\partial v^{\prime }/\partial x)}^2\rangle /\langle {(\partial u^{\prime }/\partial x)}^2\rangle$ with $\langle \rangle$ denoted as the ensemble average over time and $Z$. $K$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The horizontal dashed lines indicate $K=2$.

Figure 9

Streamwise evolution of the large-scale anisotropy $U_{rms}/V_{rms}$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The horizontal dashed lines indicate $U_{rms}/V_{rms}=1$.

Figure 10

Streamwise evolution of the skewness of the streamwise velocity fluctuations $S_u=\langle u^{\prime 3}\rangle /{\langle u^{\prime 2}\rangle }^{3/2}$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The horizontal dashed lines indicate $S_u=0$.

Figure 11

Streamwise evolution of the kurtosis of the streamwise velocity fluctuations $F_u=\langle u^{\prime 4}\rangle /{\langle u^{\prime 2}\rangle }^2$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The horizontal dashed lines indicate $F_u=3$.

Figure 12

Streamwise evolution of the normalized mean pressure $P/\left(\rho U_J^2\right)$ vs (a) $X/L_0$, (b) $X/X_{\mathrm{peak}}$, and (c) $X/X_{*}$ along the centerline. The horizontal dashed lines indicate $P=0$.

Figure 13

Magnitude of instantaneous vorticity field $\mathrm{\Omega }/(U_J/L_0)$ in the $X\text{−}Y$ plane for the case with $L_d/L_0=6$. This visualization is in the region $-10\le Y/L_0\le 10$ at $Z/L_0=0$ covering only $50\%$ of the simulation domain. The two horizontal lines indicate the centerlines of the jet exits.

Figure 14

Mean streamlines in the vicinity of the jet exits for the case with $L_d/L_0=6$. The color contours correspond to the normalized mean streamwise velocity $U/U_J$. This visualization is in the region $2\le X/L_0\le 16$ and $-6\le Y/L_0\le 6$.

Figure 15

Streamwise evolution of the local Reynolds number based on the Taylor microscale along the centerline for the case with $L_d/L_0=6$.

Figure 16

Streamwise evolution of the normalized mean pressure $P/\left(\rho U_J^2\right)$ along the centerline for the case with $L_d/L_0=6$.

Figure 17

PDFs of the normalized streamwise velocity fluctuation $u^{\prime }/u_{rms}$ at $X/L_0=7$, 12, and 25. The Gaussian fit of $P\left(u^{\prime }\right)$ at $X/L_0=25$ is also shown for comparison.

Figure 18

Normalized fluctuating streamwise velocity component at $X/L_0=7$, 12, and 25 for the case with $L_d/L_0=6$. The filled circle symbols represent the selected three times at the streamwise location $X/L_0=7$ with $t_1/(L_0/U_J)=50.7,t_2/(L_0/U_J)=61.3$, and $t_3/(L_0/U_J)=83.5$. The horizontal dashed line corresponds to $u^{\prime }/U_{rms}=0$.

Figure 19

(Top) Normalized fluctuating streamwise velocity component $u^{\prime }$ and the magnitude of vorticity $\mathrm{\Omega }$ at $X/L_0=7$ and for the case with $L_d/L_0=6$. The horizontal dashed line represents $u^{\prime }=0$ and $\mathrm{\Omega }=0$. The same filled circle symbols plotted in Fig. 18 are also included. (Bottom) Instantaneous streamwise velocity field and the streamlines in the reverse flow region in the $X\text{−}Y$ plane for the case with $L_d/L_0=6$. From left to right, subfigures (a), (b), and (c) are plotted in the sequence of time $t_1,t_2$, and $t_3$. To highlight the existence of recirculation vortices, only the streamlines corresponding to large negative $u$ are drawn.

Figure 20

(a) Time-averaged area fraction of turbulent region at $X/L_0=7$, 12, and 25. The two vertical dashed lines (i.e., ${\mathrm{\Omega }}_{tr}/{\mathrm{\Omega }}_{max}=0.001$ and 0.003) show the range within which for all three cases plateaus of $A_t/A$ exist. (b) Streamwise evolution of $\gamma$ along the centerline under three different thresholds. The horizontal dashed line represents fully turbulent state (i.e., $\gamma =1.0)$. The two vertical dashed lines (i.e., $X/L_0=6$ and 9) define the intermittent region.

Figure 21

Vertical distribution of the normalized mean streamwise velocity $U/U_J$ at three different streamwise locations, (a) $X/L_0=7$, (b) $X/L_0=12$, and (c) $X/L_0=25$, for the case with $L_d/L_0=6$.

Figure 22

Vertical distribution of the normalized mean vertical velocity $V/U_J$ at three different streamwise locations, (a) $X/L_0=7$, (b) $X/L_0=12$, and (c) $X/L_0=25$, for the case with $L_d/L_0=6$.

Figure 23

Joint PDFs of $u^{\prime }/U_{rms}$ and $v^{\prime }/V_{rms}$ at three different streamwise locations (a) $X/L_0=7$, (b) $X/L_0=12$, and (c) $X/L_0=25$ for the case with $L_d/L_0=6$. The contour levels are 0.1, 0.05, ${10}^{-2},{10}^{-3}$, and ${10}^{-4}$.

Figure 24

Time traces of the normalized fluctuating streamwise and vertical velocity components at $X/L_0=12$ for the case with $L_d/L_0=6$. The two ranges defined by vertical dashed lines clearly imply that $u^{\prime }$ and $v^{\prime }$ can be either in phase or out of phase with each other.

Figure 25

(a) Energy spectra of the streamwise velocity fluctuations at $X/L_0=7$, 12, and 25 for the case with $L_d/L_0=6$. The transparent profiles correspond to the original spectra and the opaque profiles correspond to the smoothed spectra and (b) the smoothed compensated energy spectra. The vertical lines describe the ranges of the $-5/3$ scaling. The two horizontal dashed lines represent deviations of $-7.5\%$ and $7.5\%$, respectively, from the mean value.

Figure 26

Spectra of $u_{>}^{\prime }$ and $u_{<}^{\prime }$ at two streamwise locations, (a) $X/L_0=12$ and (b) $X/L_0=25$, for the case with $L_d/L_0=6$. For comparison, the corresponding unfiltered spectra (represented by the red dotted lines) are also plotted. The vertical dashed lines indicate the cutoff frequency $f{L}_0/U_J=0.3$.

Figure 27

PDFs of the increments of the velocity fluctuations at $X/L_0=12$ and 25 for the case with $L_d/L_0=6$. The separation distance $r$ is using the Kolmogorov scale $\eta$ with respect to the downstream coordinate, that is, $\eta =\eta (X/L_0).$

Figure 28

Streamwise evolution of the centerline statistics (a) $U_C/U_J$ and (b) $U_{rms}/U_J$ corresponding to different time steps $N_s$ and averaged over $Z$ for the case with $L_d/L_0=6$.

Figure 29

Streamwise evolution of the centerline statistics (a) $S_u$ and (b) $F_u$, corresponding to different time step $N_s$ and averaged over $Z$ for the case with $L_d/L_0=6$. The two dashes indicate, respectively, $S_u=0$ and $F_u=3$.