Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 345 / 3

Take inward fluid entrainment speed to be positive. The non-Boussinesq form of the Batchelor entrainment hypothesis iswhere $\alpha$ is the entrainment coefficient. Define physical, cross-section-integrated fluxes for the top-hat plume model byThus $Q$ is mass flux, $M$ is momentum flux, and $F$ is a density-weighted buoyancy flux. The kinematic buoyancy flux in the Boussinesq approximation is $F/{\rho }_0$. These definitions keep the factor $\pi$; conventions suppressing it simply rescale all three fluxes consistently.

For a homogeneous ambient, the sectional volume, mass and momentum balances areThe momentum source is the net upward buoyancy force; entrained ambient fluid initially supplies no vertical momentum. Subtracting the mass balance from ${\rho }_0$ times the volume balance gives conservation of the density deficit:Now $w=M/Q$, $\pi b^2\rho =Q^2/M$, and $\pi b^{2}g({\rho }_0-\rho )=QF/M$. Substitution produces the non-Boussinesq top-hat plume equationsFor reconstructing the fields without a Boussinesq approximation,The last expression uses the plume density in the acceleration denominator; it agrees with the ambient-density definition of reduced gravity to Boussinesq order.

Use the kinematic fluxes $q=Q/(\pi {\rho }_0)$, $m=M/(\pi {\rho }_0)$ and $f_0=F_0/(\pi {\rho }_0)$. In the steady Boussinesq approximation, the non-Boussinesq top-hat plume equations reduce toEliminating $z$ gives $m^{5/2}=5f_0{q}^2/(8\alpha )$, where the integration constant vanishes for a pure plume from a point source. Write $b=az$ and $w=K{z}^{-1/3}$. The two balances give $a=6\alpha /5$ and $K^3=3f_0/(4a^2)=25f_0/(48{\alpha }^2)$. The Boussinesq point-source plume is thereforeHere $q=a^{2}K{z}^{5/3}$ and $m=a^2{K}^2{z}^{4/3}$ tend to zero at the source, while $f_0$ remains positive. The plume Froude number is independent of height:The ideal point source is a far-field similarity idealization. Since $g^{\prime }\propto z^{-5/3}$, the Boussinesq approximation fails near $z=0$ and is valid only when $g^{\prime }/g\ll 1$. In terms of $K$, this requires $z\gg [4K^2/(3g){]}^{3/5}$; the density formula must not be extrapolated into its unphysical negative-density region.

Let $f_0=F_0/(\pi {\rho }_0)$ as above. A self-similar starting plume has no source length or time scale; dimensional analysis therefore givesThe dimensionless constants depend on the entrainment coefficient and thermal closure. Thus $R/B$ is constant because the underlying Boussinesq point-source plume has $B=b(Z)=(6\alpha /5)Z$. The steadily increasing thermal volume scales as $t^{9/4}$, even though its reduced gravity decreases.

Let $V=4\pi R^3/3$, $w_p=w(Z)$ and ${\rho }_p=\rho (Z)$, with the subscript $p$ here referring to the plume fluid. The section at $Z(t)$ moves upward at $U=\dot{Z}$. Its incoming volume flux relative to that moving section is thereforeThe thermal's sectional model allows no direct ambient fluid entrainment, so its volume and mass balances areSubtracting these balances with $g_t=g({\rho }_0-{\rho }_t)/{\rho }_0$ and $g_p=g({\rho }_0-{\rho }_p)/{\rho }_0$ gives the buoyant thermal mass balanceFor the self-similar starting plume, $V\propto t^{9/4}$ and $g_t\propto t^{-5/4}$, so $g_{t}V\propto t$. HenceUsing $\pi B^2{w}_p$ instead of the relative inflow would miss the volume swept out by the rising matching section. The larger thermal reduced gravity reflects fluid accumulated from earlier, lower, more buoyant parts of the plume.

Put $a=6\alpha /5$ and $\lambda =R/B$, so that $B=aZ$ and $R=\lambda aZ$ with constant $\lambda \ge 1$. The volume balance givesConsequentlyFor the definition of $C_t$ printed in the question, using $g_p$ in its denominator, the starting-plume thermal Froude-number ratio isAll factors are constant in the self-similar starting plume. This expression decreases strictly with $\lambda >0$, givingThe stronger bound requested in the PDF does not follow from its stated definition and $R\ge B$. For example, $\lambda =1$ already contradicts that bound. If instead the thermal Froude number is normalized by its own reduced gravity, ${\tilde{C}}_t=U/\sqrt{g_{t}R}$, the preceding result $g_t=9g_p/4$ supplies the missing factor $2/3$:The inequality is strict for $R>B$, with equality allowed by the stated endpoint $R=B$. This alternative normalization recovers the numerical bound while keeping the distinction between the plume and thermal reduced gravities explicit.