Positively buoyant jets: Semiturbulent to fully turbulent regimes

We experimentally study a buoyant jet flow where a heavy fluid is injected into a rectangular reservoir filled with a light ambient fluid. The reservoir dimensions are sufficiently large so that the ambient fluid can be considered as unbounded, over the time scale of our interest. The jet and ambient fluids are miscible and the jet direction is downward. We use nonintrusive experimental methods, such as high-speed camera, laser imaging, and ultrasound velocimetry techniques, to analyze the flow versus the Reynolds (Re), Froude (Fr), and Archimedes (Ar) numbers. These dimensionless parameters cover a wide range, i.e., $3\times {10}^2\le \text{Re}\le 15\times {10}^3$, $2\le \text{Fr}<2\times {10}^2$, and $0\le \text{Ar}\le 25\times {10}^4$, governing the flow via an interplay between inertia, viscosity, and buoyancy. We analyze the jet quasisteady characteristics, in particular the laminar length, which enables us to classify the flow into fully turbulent and semiturbulent regimes. Via evaluating the characteristic momentum-buoyant parameters, we also succeed in dividing the semiturbulent regime into momentum and buoyancy dominated subregimes. We quantify the regime transition boundaries and develop a correlation to predict the laminar length for each (sub-)regime. To have a global view of the buoyant jet flow, we also analyze other jet quasisteady characteristics, including the jet radius, spread angle, virtual origin, velocity profiles, and energy dissipation. Finally, we quantify the starting jet characteristics, including the penetration length and the tip velocity, by developing correlations for the short and long time behaviors, as well as the corresponding transition time.

Figure 1

Schematic representation of the experimental setup. The main measurement technique is image analysis of gray scale images but complementary approaches are also used (i.e., PLIF and UDV) to quantify certain jet characteristics.

Figure 2

Sequence of experimental snapshots, showing a typical flow development in (a) the fully turbulent regime and (b) the semiturbulent regime. The parameters in subfigure (a) are $\text{Re}=6857$ and $\text{Ar}=100$ and the parameters in subfigure (b) are $\text{Re}=1368$ and $\text{Ar}=100$. The dimensionless experimental times are displayed below each snapshot. The arrow along with symbol $\widehat{g}$ represent the gravity direction, here and throughout the paper. The dimensionless field of view for each snapshot is $87\times 231$. (c) A typical positively buoyant miscible jet. A heavy fluid is injected through a nozzle with a circular cross-section area into a reservoir containing a light fluid. The dark thick line roughly marks the boundary of the jet. The injection forms a jet with different parts: the jet penetration length, $L_p$, represents the length of the jet at a specific time; $L_m$ denotes the extent of the jet stable region, without any mixing, i.e., the laminar length; $r_j$ is the jet radius. The white dashed line shows the centerline passing through the nozzle center. The dimensionless field of view is $93\times 256$.

Figure 3

(a) Qualitative spatiotemporal concentration diagram in the plane of $x$ and $t$. The colorbar represents the normalized concentration field averaged over the widths of the reservoir (in the $z$ and $y$ directions). (b) Variation of the time-averaged jet radius, vs the axial position. In both subfigures (a) and (b), the flow parameters are $\text{Re}\approx 1480$ and $\text{Ar}\approx 35000$, and the vertical dashed line represents the laminar length, marked at $x=17.6$. (c) Velocity contours obtained from the UDV technique in the laminar length region at $x=8.5$, where $L_m\approx 14$ (i.e., $x<L_m$). The dash-dot lines mark a virtual boundary equal to the nozzle diameter. The flow parameters are $\text{Re}=1370$ and $\text{Ar}=3248$.

Figure 4

(a) Variation of the dimensionless laminar length vs the dimensionless time, for two experimental conditions. The flow parameters are $\text{Ar}\approx 100$ at $\text{Re}\approx 1400$ (green) and $\text{Re}\approx 2700$ (blue). The amplitude of the laminar length oscillations is indicated by $L_A$. The red solid lines show the mean laminar length for both cases. (b) Variation of $L_A$ vs Re and Ar. The results correspond to $\text{Ar}\approx 42000$ (blue) and $\text{Re}\approx 2500$ (red).

Figure 5

Experimental snapshots at different injection velocities. The horizontal dashed lines represent the laminar length, indicating where the structured laminar jet starts to break down. The flow parameters are $\text{Ar}\approx 8\times {10}^4$ and $\text{Re}\approx$ 371, 1459, 5130, 6766, and 13 478, from left to right. The dimensionless field of view for each snapshot is $50\times 106$.

Figure 6

(a) Variation of the dimensionless laminar length, $L_m$, vs the Reynolds number, Re, at different values of the Archimedes number, Ar. (b) Variation of $L_m^{*}$ vs Re at different values of Ar, for the same data as in subfigure (a). In both subfigures (a) and (b), the data correspond to experiments with $\text{Ar}\approx 200$ ($•$), $\text{Ar}\approx 1800$ ($▴$), $\text{Ar}\approx 39200$ ($\blacksquare$). In subfigure (a), the solid lines correspond to the literature data of pure jets (e.g., with $\text{Ar}\approx 0$): according to Table 2, the literature data are shown using solid lines with colors of black [57], cyan [56], green [48], purple [50], and pink [55].

Figure 7

Experimental snapshots for two experimental conditions: (a) $L_m^{*}={\widehat{L}}_m/{\widehat{L}}_s<1$ ($\text{Re}\approx 3900$ and $\text{Ar}\approx 41700$) and (b) $L_m^{*}={\widehat{L}}_m/{\widehat{L}}_s>1$ ($\text{Re}=2300$ and $\text{Ar}=41700$). The dimensionless field of view for each snapshot is $43\times 93$.

Figure 8

(a) Comparison between the experimental $L_m$ and the predicted $L_m$ using Eq. (13), i.e., $L_m^{\text{correlation}}$. The dotted line is $L_m=L_m^{\text{correlation}}$. The colors and symbol sizes represent the magnitude of Fr. (b) Variation of $L_m^{\text{correlation}}$ and the experimental $L_m$ vs Re and Ar. Regarding the variation vs Re, the results correspond to $\text{Ar}=0$ (solid line), $\text{Ar}=50$ (dashed line), $\text{Ar}=4\times {10}^3$ (dotted line), and $\text{Ar}=18\times {10}^4$ (dash-dot line). The experimental data for $\text{Ar}\approx 4\times {10}^3$ ($▵$) are also superimposed. Regarding the variation vs Ar, the results correspond to $\text{Re}=5000$ (dotted line), $\text{Re}=3000$ (dashed line), and $\text{Re}=2000$ (dash-dot line). The experimental data for $\text{Re}\approx 3000$ ($\square$) are also superimposed. (c) $L_m^{*}$ vs Fr, along with the fitted dashed line of Eq. (15) valid for $L_m^{*}\gtrsim 1$ (marked by the short horizontal solid line). (d) $L_m^{*}$ vs Eq. (16), i.e., $L_m^{*,\text{correlation}}$.

Figure 9

Regime classification in the plane of Re and Ar. (a) The symbols represent experiments of semiturbulent ($•$) and fully turbulent ($\circ$) flows. (b) The colors and symbol sizes represent the magnitude of $L_m^{*}$. In both subfigures, the solid and dashed lines represent Eqs. (18) and (19), respectively. The flow regimes are marked as follows: I, fully turbulent; ${\mathrm{II}}_1$, semiturbulent (momentum dominated); ${\mathrm{II}}_2$, semiturbulent (buoyancy dominated). The prediction to $L_m^{*}$ in each regime is given in Eq. (16).

Figure 10

Experimental images using PLIF, for $\text{Re}\approx 4550$ and $\text{Ar}\approx 41703$. (a) Side-view results in the $x\text{−}y$ plane. (b)–(d) Bottom-view cross-section results in the $z\text{−}y$ plane, at $x=11.1$, 58.3, and 75.0, with respect to the nozzle exit. In panel (a), the PLIF image corresponds to the middle of the jet in the $x\text{−}y$ plane. The equivalent jet radius ($r_j$) and times are mentioned in each subfigure. Note that $r_j$ is calculated using $r_j=\sqrt{\frac{A}{\pi }}$, where $A$ is the jet cross-section area. In subfigures (b4), (c4), and (d4), the jet boundary for ten specific times and the equivalent jet radius averaged over these times (${\overline{r}}_j$) are shown by the black lines and the red dashed line, respectively.

Figure 11

(a) Jet radius development (obtained from the bottom-view PLIF results) vs time, at three different axial distances from the nozzle exit, i.e., $x=11.1$ (green), $x=58.3$ (blue), and $x=75.0$ (red), for the same experiments as shown in Fig. 10. The jet radii reach the plateau (nearly steady) values at long times, marked by the black dashed lines. (b), (c) Variation of the jet radius at long times (nearly steady) vs the axial position. (b) Results for $\text{Ar}\approx 15700$ and with $\text{Re}\approx 1100$ ($▴$), $\text{Re}\approx 3100$ ($\blacksquare$), and $\text{Re}\approx 12200$ ($•$). (c) Results for $\text{Re}\approx 1300$ with $\text{Ar}\approx 0$ ($▴$), $\text{Ar}\approx 450$ ($\blacksquare$), and $\text{Ar}\approx 35006$ ($•$). The vertical solid lines show the laminar length for each case. The dashed lines are added as eye guides in both subfigures.

Figure 12

(a) Jet geometrical parameters, including the jet spread angle, marked on a typical experimental result of the radius vs the axial position. Note that here $r_{j,\text{max}}$ and $L_{j,\text{max}}$ are the maximum jet radius and the distance between the laminar length and the maximum jet radius, respectively. (b) Variation of the spread angle, $\theta$, as a function of the Reynolds number, Re, with $\text{Ar}=35006$ ($•$), $\text{Ar}=450$ ($▴$), and $\text{Ar}=0$ ($\blacksquare$). The data points from the literature are also superposed: $\text{Ar}=7200$ ($\circ$) [68], $\text{Ar}=296450$ ($\mathbf{\nabla }$) [69], and $\text{Ar}=583200$ ($\square$) [69]. (c) Variation of $\theta$ vs Re in the fully turbulent regime (i.e., where $L_m$ disappears). The values of Ar are indicated by the colors and symbol sizes. The horizontal dashed line and the shaded area represent $\theta ={25}^{\circ }$ and ${25}^{\circ }\pm 3^{\circ }$, respectively.

Figure 13

(a) Schematic views of the virtual origin position (red point) in two cases: (1) fully turbulent flows (i.e., without a laminar length, usually at high Re) and (2) semiturbulent flows (i.e., with a laminar length, usually at low Re). (b) Variation of the virtual origin, $X_v$, vs the Reynolds number, Re. The values of Ar are indicated by the colors and symbol sizes. The horizontal dash-dot line marks $X_v=0$.

Figure 14

(a)–(d) Time-averaged axial velocity profiles (solid line) obtained from the UDV technique measurements, downstream of the nozzle exit, superimposed on experimental snapshots for $\text{Ar}=3248$: (a) $x=8.5$ and $\text{Re}=1370$; (b) $x=17$ and $\text{Re}=1370$; (c) $x=8.5$ and $\text{Re}=11531$; (d) $x=17$ and $\text{Re}=11531$. The dotted line presents the time-averaged velocity profiles for high Re, calculated based on Eq. (25). The dimensionless field of view in each subfigure is $18\times 39$. (e) Variation of the dimensionless centerline velocity vs $x$, based on $V_c=(c/x)$ (solid line) [79, 84] and $V_c={(c/x)}^2$ (dashed line), with $c$ being a constant (here $c=6$). Our experimental data for $\text{Re}=11531$ and $\text{Ar}=3248$ ($•$) are also superimposed. (f) Variation of the jet energy flux, based on Eq. (27) and using the UDV results, vs time, at $x=8.5$ (green) and $x=17$ (red), for the same experiments of subfigures (c) and (d). The mean energies of the jets are marked by the blue solid lines.

Figure 15

(a) Variation of the dimensionless jet penetration length, $L_p$, vs the dimensionless time, $t$, at fixed $\mathrm{Ar}\approx 97000$ with $\mathrm{Re}\approx 6800$ ($•$), $\mathrm{Re}\approx 6100$ ($▴$), $\mathrm{Re}\approx 2600$ ($\blacksquare$). (b) Variation of the dimensionless jet penetration length, $L_p$, vs the dimensionless time, $t$, at fixed $\mathrm{Re}\approx 5300$ with $\mathrm{Ar}\approx 106000$ ($•$), $\mathrm{Ar}\approx 97000$ ($▴$), $\mathrm{Ar}\approx 47000$ ($\blacksquare$). The insets show the logarithmic scale of the same data as in the main plots. In all the subfigures, the dash-dot lines represent $L_p=t$.

Figure 16

(a) Variation of $L_p$ vs $t$ for different experimental conditions. The dash-dot line represents $L_p=t$, showing each data set initially follows this line. The data correspond to experiments with $\mathrm{Re}\approx 1800\mathrm{and}\mathrm{Ar}\approx 400$ ($\ast$), $\mathrm{Re}\approx 6200\mathrm{and}\mathrm{Ar}\approx 97000$ ($\blacksquare$), $\mathrm{Re}\approx 7100\mathrm{and}\mathrm{Ar}\approx 21000$ ($♦$), $\mathrm{Re}\approx 1300\mathrm{and}\mathrm{Ar}\approx 160000$ ($•$), $\mathrm{Re}\approx 4500\mathrm{and}\mathrm{Ar}\approx 47000$ ($▴$), $\mathrm{Re}\approx 1900\mathrm{and}\mathrm{Ar}\approx 250000$ ($\blacksquare$), $\mathrm{Re}\approx 1400\mathrm{and}\mathrm{Ar}\approx 5600$ ($\ast$), $\mathrm{Re}\approx 1800\mathrm{and}\mathrm{Ar}\approx 106000$ ($▴$), $\mathrm{Re}\approx 1200\mathrm{and}\mathrm{Ar}\approx 8000$ ($•$), $\mathrm{Re}\approx 3800\mathrm{and}\mathrm{Ar}\approx 160000$ ($♦$), $\mathrm{Re}\approx 1700\mathrm{and}\mathrm{Ar}\approx 106000$ ($\ast$), $\mathrm{Re}\approx 2400\mathrm{and}\mathrm{Ar}\approx 93000$ ($\blacksquare$), $\mathrm{Re}\approx 2400\mathrm{and}\mathrm{Ar}\approx 400$ ($▴$), $\mathrm{Re}\approx 1800\mathrm{and}\mathrm{Ar}\approx 5600$ ($\blacksquare$), $\mathrm{Re}\approx 3100\mathrm{and}\mathrm{Ar}\approx 106000$ ($•$), $\mathrm{Re}\approx 1500\mathrm{and}\mathrm{Ar}\approx 160000$ ($▵$), $\mathrm{Re}\approx 6900\mathrm{and}\mathrm{Ar}\approx 43000$ ($\square$), $\mathrm{Re}\approx 1100\mathrm{and}\mathrm{Ar}\approx 160000$ ($▴$), $\mathrm{Re}\approx 1600\mathrm{and}\mathrm{Ar}\approx 97000$ ($•$), $\mathrm{Re}\approx 11900\mathrm{and}\mathrm{Ar}\approx 160000$ ($\circ$), $\mathrm{Re}\approx 1700\mathrm{and}\mathrm{Ar}\approx 11700$ ($•$), $\mathrm{Re}\approx 1400\mathrm{and}\mathrm{Ar}\approx 106000$ ($\ast$), $\mathrm{Re}\approx 2600\mathrm{and}\mathrm{Ar}\approx 21000$ ($\blacksquare$), and $\mathrm{Re}\approx 1100\mathrm{and}\mathrm{Ar}\approx 400$ ($\ast$). (b) Variation of $L_p^{*}$ as a function of $t^{*}$ for the same data as in subfigure (a). The dashed line represents Eq. (28), showing the collapse of data sets onto a single curve at long times. (c) Variation of $L_p^{*}$ vs $t^{*}$ for $\mathrm{Re}\approx 1300\mathrm{and}\mathrm{Ar}\approx 160000$ ($•$), $\mathrm{Re}\approx 1800\mathrm{and}\mathrm{Ar}\approx 106000$ ($▴$), and $\mathrm{Re}\approx 2600\mathrm{and}\mathrm{Ar}\approx 21000$ ($\blacksquare$). The dashed line, the dash-dot lines, and the vertical dotted lines (along with the marker “$+$”) represent the results of Eqs. (28, 29, 30), respectively.

Figure 17

(a) Variation of $V_t^{*}$ vs $x^{*}$ for different experimental conditions. The data correspond to the experiments in Fig. 16. (b) Variation of $V_t^{*}$ vs $x^{*}$ for $\mathrm{Re}\approx 7100\mathrm{and}\mathrm{Ar}\approx 21000$ ($♦$), $\mathrm{Re}\approx 6200\mathrm{and}\mathrm{Ar}\approx 97000$ ($\blacksquare$), and $\mathrm{Re}\approx 1300\mathrm{and}\mathrm{Ar}\approx 160000$ ($•$). The dashed line and the dash-dot lines represent Eqs. (32) and (33), respectively.