General solutions of the plume equations: towards synthetic plumes and fountains

Previous mathematical models of quasi-steady turbulent plumes and fountains have described the flow that results from a prescribed input of buoyancy. We offer a new perspective by asking, what input of buoyancy would give rise to a plume, or fountain, with given properties? Addressing this question by means of an analytical framework, we take a first step toward enabling a plume with specific characteristics, i.e. a synthetic plume, to be designed. We develop analytical solutions to the conservation equations that describe four kinds of turbulent flow: axisymmetric plume, starting fountain, line plume and wall plume. Crucially, our solutions do not require the buoyancy distribution to be specified, whether this be the source or off-source distribution. Key to our approach, we specify a function for the volume flux, $Q=f(z)$, and take advantage of the weak coupling between the conservation equations to uniquely express general solutions in terms of $f$. We show that any analytic function $f$ can form the basis for a set of solutions for the fluxes, local variables and local Richardson number, though $f/({\rm d}f/{\rm d}z)>0$ is a necessary condition for physically realistic solutions. As an example of plume synthesis, we show that an axisymmetric plume can have an invariant radius if there is an exponentially increasing input of buoyancy to the plume centreline. We also consider how plume synthesis could be achieved practically.