The coalescence of adjacent turbulent plumes in a stratified and unstratified environment

Abstract

Plume merger has been the subject of a number of recent theoretical studies that employ turbulent plume theory (including Taylor’s entrainment hypothesis) with ambient fluid external to the plume described using potential flow theory. Notwithstanding this effort, important questions remain as to the rate at which a merged plume formed from two adjacent sources relaxes in the far-field. In this theoretical work, the process of relaxation is examined with respect to the shape of the plume perimeter (as compared against a circle) and the plume volume flux, ${\bar{Q}}$ (as compared against the volume flux predicted from self-similarity). Using these metrics, we find that merging plumes assume a nearly circular cross-sectional shape before ${\bar{Q}}$ realizes its corresponding self-similar value. The discrepancy is especially large when the ambient is stratified. Herein, we consider three different stratification scenarios: uniform ambient, linear stratification and two-layer stratification featuring a thick interface. In each case, the vertical evolution of the plume is modeled using as additional benchmarks the height of first contact between the merging plumes, the height of full merger and, where applicable, the height of neutral buoyancy. Our theoretical model thereby allows us to track the dynamical evolution of the plume and shows, for example, that a merging plume rising through a linearly stratified ambient will almost always spread laterally before completing the process of relaxation. Finally, we highlight some of the open mathematical challenges in extrapolating our two-source model to a situation involving three or more (colinear) sources.

Article highlights

• The merger of adjacent plumes impacts plume growth by the entrainment of external ambient fluid.

• The process of plume merger can be tracked by measuring or calculating the plume shape and its variation with height.

• Plume merger is impacted by ambient density stratification; strong density variations may arrest merging.

Appendices

Appendix 1. Three colinear plumes

In this appendix, we briefly highlight the challenges of modeling plume rise and merger for the unsymmetric case of three plumes arranged in a line. For simplicity of exposition, we restrict attention to the small source limit. On the other hand, it should be clear that the challenges described, and open questions identified, in the following paragraph apply whatever the dimension of the plume sources.

When $\rho _0 \rightarrow 0$ and $n=2$, (2) indicates that

$$\begin{aligned} |Z'^2-1| = k . \end{aligned}$$

(34)

Extending the above equation to a case of two or more plumes yields

$$\begin{aligned} \left| \prod _{j=1}^{\frac{n}{2}}\left[ Z'^2 - (2j-1)^2\right] \right| = k , \qquad {\text{ n even, }} n \ge 2 , \end{aligned}$$

(35)

$$\begin{aligned} \left| Z'\prod _{j=1}^{\frac{n}{2}-\frac{1}{2}}\left[ Z'^2 - (2j)^2\right] \right| = k , \qquad {\text{ n odd, }} n \ge 3 . \end{aligned}$$

(36)

In both of (35) and (36), we assume that plume sources are separated by a horizontal distance 2R. When $n=3$, therefore, plumes originate from sources situated at $(-2R,0)$, (0, 0) and (2R, 0) such that (36) reads

$$\begin{aligned} |Z'(Z'-2)(Z'+2)| = k . \end{aligned}$$

(37)

However,

$$\begin{aligned} Z'(Z'-2)(Z'+2) = \rho ^3\mathrm{cos}\,3\theta - 4\rho \mathrm{cos}\,\theta +\mathrm{i}(\rho ^3\mathrm{sin}\,3\theta -4\rho \mathrm{sin}\theta ) , \end{aligned}$$

(38)

and (37) can therefore be rewritten as

$$\begin{aligned} \rho ^6-8\rho ^4\mathrm{cos}\,2\theta +16\rho ^2-k^2 = 0 , \end{aligned}$$

(39)

which is the analogue of (3). Solutions to (39) are exhibited in Fig. 19, which shows contours in the right-half plane for various k. Unfortunately, the physical interpretation of these contours is not entirely clear. Following the example of Fig. 4, it is necessary to assign to the contours of Fig. 19 an associated value of z or ${\bar{z}}$. Doing so requires first defining the entrainment velocity as in (6), but this entrainment velocity cannot be assumed a-priori to be the same for the middle (plume 2) versus the end member plumes (plumes 1 and 3). However, if $q_e$ is different for different plumes, this suggests that the mapping from k to z is not unique. In other words, a separate mapping is required for plume 2 versus plumes 1 and 3. Once different mappings are permitted, even previously straightforward tasks such as defining the point, $z_{fc}$, of first contact become fraught with ambiguity. Thus, additional progress for the colinear cases $n = 3,\,4,\, 5$, etc. awaits further developments in the field.

Fig. 19

Solutions to (39) for $k = 0,\,0.5,\,1.0,\,1.5,\,2.0,\,2.5,\,3.0,\,3.079,\,3.5,\,4.0, \ldots$. When $k = 3.079$, the contours appear to “kiss” at $(x/R,\,y/R) \simeq (1.155,\,0)$

Appendix 2. Single isolated plume (no merging)

In the case of a single isolated plume, the geometric complications of Sect. 2.1 are thankfully avoided. Rather, we solve (9–11) where the rate of entrainment of external ambient fluid is specified by (13). Simplifying matters still further, we assume a uniform ambient such that the plume buoyancy flux, $F_0$, is constant and is equal to the buoyancy flux associated with either of the plume sources studied in Sect. 3. We therefore seek solutions to

$$\begin{aligned} A\frac{{\mathrm{d}}}{{\mathrm{d}}z}\left( \frac{1}{2}w_s^2\right) =A_sg'_s - 2 \alpha w_s^2 \sqrt{\pi A_s} , \end{aligned}$$

(40)

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}z}(A_s w_s) = 2 \alpha w_s \sqrt{\pi A_s} . \end{aligned}$$

(41)

Here, a subscript ‘s’ has been included to indicate ‘single plume.’ Setting $A_s = Q_s/w_s$ and $g'_s=F_0/Q_s$ then non-dimensionalizing (40) and (41) using (20), it can be shown that

$$\begin{aligned} \frac{{\mathrm{d}}{\overline{w}}_s}{{\mathrm{d}}{\overline{z}}}= & {} \frac{1}{{\overline{w}}_s{\overline{Q}}_s}-2\sqrt{\frac{\pi {\overline{w}}_s^3}{{\overline{Q}}_s}} , \end{aligned}$$

(42)

$$\begin{aligned} \frac{{\mathrm{d}}{\overline{Q}}_s}{{\mathrm{d}}{\overline{z}}}= & {} 2\sqrt{\pi {\overline{w}}_s {\overline{Q}}_s} . \end{aligned}$$

(43)

Solutions to (42) and (43) are considered in Fig. 7.