Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 5

Use the flux normalization of the supplied equations, with the common factor $\pi$ suppressed. Put $q=w{b}^2$, $m=w^2{b}^2$ and $B_0=g^{\prime }q$. The top-hat plume model reduces toEliminate height to obtain $dm/dq=B_{0}q/(2\alpha m^{3/2})$. Integrating with zero source volume and momentum flux gives the pure plume balanceEquivalently, seek $b=dz$ and $w=s{z}^{-1/3}$. The volume balance gives $d=6\alpha /5$, and the momentum and buoyancy balances give $g^{\prime }=(4s^2/3)z^{-5/3}$ and $B_0=4d^2{s}^3/3$. ThusA finite source replaces the singular point origin by a plume virtual origin and source conditions. The Boussinesq approximation requires the fractional density difference to be small, so density can be replaced by ambient density in inertia and entrainment while retaining the small difference in buoyancy. In this solution it requires $g^{\prime }/g\ll 1$. The approximation necessarily fails sufficiently near the ideal point source, where $g^{\prime }$ diverges, and can also fail for strongly heated or otherwise large-density-contrast sources. The similarity solution is appropriate farther away once entrainment has diluted the anomaly, assuming a uniform ambient, constant entrainment coefficient and negligible confinement, heat loss and stratification.

The mass and heat entrainment equations immediately giveUse the specified relation $\rho T={\rho }_0{T}_0=\beta$. ThenMeanwhile the kinematic buoyancy flux isSince $T_0$, ${\rho }_0$ and $c_p$ are constant,Thus conservation is a consequence of conserved excess heat in this ideal-gas, uniform-ambient model, not merely an assumption that arbitrary non-Boussinesq buoyancy fluxes are conserved in every environment.

From the definitions of mass and momentum fluxes,The density relation follows by solving the conserved-buoyancy identity in the preceding part. Hence $wb\sqrt{\rho }=\sqrt{M}$ andThe second equality uses $B=g({\rho }_0-\rho )Q/({\rho }_0\rho )$ and $b^2=Q^2/(\rho M)$.

Set $q=Q/{\rho }_0$ and $m=M/{\rho }_0$. The equations become $q^{\prime }=2\alpha \sqrt{m}$, $m^{\prime }=Bq/m$, exactly the same flux system as for the pi-suppressed Boussinesq pure plume. For the heated point source take zero initial mass and momentum flux and the finite conserved heat/buoyancy supply. DefineThe resulting mass and momentum fluxes areSubstituting $Q$ into the density relation gives the non-Boussinesq pure-plume density transitionThe two expressions for $z_b$ agree because $B=4d^2{s}^3/3$. Density increases from the very dilute near-source regime toward the ambient value as entrainment increases the mass flux.

The flux solution gives $w=M/Q=s{z}^{-1/3}$, while $b^2=Q^2/(\rho M)$. ThusAt $z=z_b$, $\rho ={\rho }_0/2$, $T=2T_0$ and $g^{\prime }=g/2$. This height marks the transition between a strongly non-Boussinesq near-source plume and the dilute far-field plume. For $z\gg z_b$, $b\sim dz$ and $g^{\prime }\sim g(z_b/z{)}^{5/3}=(4s^2/3)z^{-5/3}$, recovering the Boussinesq solution. For $z\ll z_b$, $b\sim d{z}_b^{5/6}{z}^{1/6}$, $g^{\prime }\to g$ and $\rho \propto z^{5/3}$.

The ideal point-source limit still has a divergent velocity and temperature; it is not a literal resolved finite burner. In particular, zero source mass flux does not imply zero physical hot-gas volume flux: $w{b}^2$ tends to $B/g$ in this limiting solution. Source size, compressibility and heat transfer impose additional near-source limits on the model.