Asymptotic approximations for swirling turbulent plume rising from circular sources

Governing equations of swirling turbulent buoyant plumes rising from horizontal circular sources into a stationary surrounding are established with the plume function considered. In an attempt to find out the analytical solutions for both lazy and forced plumes, we derive the asymptotic approximations with first-order expansions for all swirling plume variables, including the radius, swirl ratio, axial velocity, and temperature, by applying regular perturbation methods with the swirl term being the perturbative part. Finally, the asymptotic solutions are compared with the numerical evaluations conducted through the fourth-order Runge-Kutta method. The results show that, for lazy plumes, the zeroth-order expansions are good enough to approximate the solutions for each variable, while the first-order expansions are found to match the numerical solution much better for forced plumes, indicating that swirling motions slightly influence lazy plumes but largely affect forced ones. It is also found that, in the presence of swirls, the plume radius slightly increases, while the centerline axial velocity decreases and the temperature barely changes, in both lazy and forced plumes. Additionally, as the input plume function value increases, the swirl ratio decays faster and further decreases the impact on other variables. Especially, a swirl can even turn a moderate forced plume into a lazy plume due to the dominated perturbative part in the near field, which might cause the method for categorizing plumes to be called into question.

Figure 1

Schematic diagram of a steady swirling plume rising from a circular source.

Figure 2

Vertical evolution of swirling lazy plumes ($G_i=0.6$). The solid lines are numerical solutions, while the dashed lines and dotted lines represent the zeroth- and first-order expansions of the asymptotic approximations.

Figure 3

Vertical evolution of swirling forced plumes ($G_i=0.6$). The solid lines are numerical solutions, while the dashed lines and dotted lines represent the zeroth- and first-order expansions of the asymptotic approximations.