Analytical solutions for turbulent non-Boussinesq plumes

Abstract

Analytical solutions are developed for non-Boussinesq turbulent plumes rising from horizontal area sources in unconfined quiescent environments of uniform density. The approach adopted follows and extends an earlier approach for Boussinesq plumes and replaces the non-Boussinesq area source of interest and located at $z\,{=}\,0$ with an idealized point source located at a virtual origin $z\,{=}\,z_v$ such that the flow above the idealized source approximates that from the actual source. Asymptotic analytical expressions are developed for the location of the virtual source that are valid for large vertical distances above the non-Boussinesq source. The non-Boussinesq source is characterized by a non-dimensional parameter $\Gamma_{\hbox{\scriptsize{\it nb}}}$ which is a measure of the relative strengths of the mass, momentum and density deficit fluxes at, or at a specified height above, the source. The vertical distance between the actual and virtual sources scales on the length scale $\ell$ that characterizes the height over which the flow is non-Boussinesq and expressions for $z_v/\ell$ are developed for lazy ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{>}\,1$) and forced plume ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,1$) sources. For pure-plume source conditions $\Gamma_{\hbox{\scriptsize{\it nb}}}\,{=}\,1$, and the virtual source provides an exact representation of the actual plume above $z\,{=}\,0$. The limiting cases of a nearly pure lazy plume and of a highly lazy plume are also explored analytically. For fire plumes, $\Gamma_{\hbox{\scriptsize{\it nb}}}$ is determined from the balance of fluxes immediately above the combustion region and a procedure for estimating these fluxes is given. Solutions expressing the dependence of the mass flux with height are also developed for the near-field flow regions and thereafter an approximation for the mass and momentum fluxes valid for all heights and for source conditions yielding $0\,{<}\,\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,\infty$ is deduced. Applications of the model may include plumes above fires and forced releases of highly buoyant gas into the atmosphere, for example, following the rupturing of a pressurized container vessel.