Integral model for multiple forced plumes arranged around a circle in a linearly stratified environment

An integral model is developed to describe the merging of multiple forced plumes released from sources arranged evenly around a circle against a linearly stratified background. The velocity potential of an array of line sinks is introduced to identify the distorted boundaries of interacting plumes. The variation of the entrainment coefficient for forced plumes is modeled as a function of the local Richardson number in the model and an improved entrainment assumption based on the boundary-curvature analysis is proposed to regulate the merging. The present model is shown to be feasible to provide a self-consistent prediction for the dynamics of multiple coalescing forced plumes. The amplification of the maximum rise height of two plumes due to superposition is faithfully reproduced and its scaling predicted by the model is reasonably consistent with the observations of available experiments. The model results further suggest that the mean entrainment efficiency per unit length of the plume boundary attains a local minimum at the touching height, implying significant mutual entrainment in that region.

Figure 1

Solutions for the values of $k$ for cases 6–10. The red line indicates the value of 1, implying the touching height of two plumes.

Figure 2

Illustration of the boundaries of (a) the two plumes for case 7 and (b) the three-plume counterpart. Several typical contour lines of $k$ are plotted and projected onto the plane $\overline{z}=0$.

Figure 3

Solutions for dimensionless buoyancy flux for cases 6–10. The blue dashed line represents the mean of those for the single-plume counterparts and the red line indicates the value of 0, implying the level of neutral buoyancy.

Figure 4

Solutions for dimensionless (a) vertical velocity and (b) momentum for cases 6–10. The blue dashed lines represent the mean of those for the single-plume counterparts and the horizontal red lines indicate the neutrally buoyant plume heights for each case of two plumes.

Figure 5

Solutions for dimensionless (a) volume flux and (b) horizontal plume length scale for cases 6–10. The blue dashed lines represent the mean of those for the single-plume counterparts and the red dashed lines indicate the fitted lines of the horizontal plume length scale in the near vent region and the merging region, respectively.

Figure 6

Comparison between experimental measurements, numerical results, and model predictions for the maximum plume rise height scalings of two plumes with different dimensionless source separations, where the dashed line indicates the tendency of all these data. The outcomes of the present model without any entrainment enhancements, i.e., $f_r=1$, are also shown for comparison.

Figure 7

Vertical profiles of (a) the local Richardson number and (b) the corresponding entrainment coefficient for cases 6–10. The blue dashed lines represent the mean of those for the single-plume counterparts and the red dashed line indicates the value of the entrainment coefficient for pure jets.

Figure 8

Vertical profiles of the mean entrainment coefficient for (a) cases 1–5, (b) cases 6–10, and (c) cases 11–15. The blue dashed lines represent the mean of those for the single-plume counterparts.

Figure 9

Dependence of the effective entrainment ${\alpha }_{\text{eff}}/{\alpha }_e$ for cases 6–10 with height. The red and blue dashed lines represent the values of $2^{-1/2}$ and 1, respectively.