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# Buoyant Jets and Plumes

After the hydrogen-rich cloud losses its inertia, buoyant forces become more and more dominant. Usually, this cloud is less dense than the surrounding air and then the buoyancy force is directed upwards. When the hydrogen released is very cold, buoyant forces could point downwards. However, the heat and mass transfers during the mixing process reduce the mixture density and invert the buoyant force direction. The fluid structure established is a plume, where two regions are distinguished: forced plumes and buoyant plumes. At the forced plume both forces (inertia and buoyancy) are of similar magnitude and separate pure inertia region (jet) apart from the pure buoyant region (plume).

The buoyancy to inertia ratio is expressed by the densimetric Froude number:

 Fr = {{\rho_0 U_0^2} \over {(\rho_\infty - \rho_0 )gd_0}} (1)
 {\omega} = {{\rho _\infty} \over {\rho _0}} (2)
 {L_{Mo}} = Fr^{1/2}{\omega}^{-1/4}d_0 (3)

where U0, ρ0 and d0 are the fluid velocity, the fluid density and the diameter at the break point, respectively, g is the gravity acceleration and ρ is the bulk density. Using this dimensionless number, (GebhartB:1988) recommend the following expressions (Tab. 1) for the three regions of a vertical upward structure. The x-direction is along the centreline of the structure, uc, ρc and cc the fluid velocity, the fluid density, and concentration at the centreline. Notice that velocity (u) and concentration (c) profiles at any transversal plane are expressed by Gaussian functions.

When the structure shape is very different from an upward vertical one, these regions need to be established by numerical simulations.

Table 1: Recommended expressions for the three regions of a vertical upward structure from (GebhartB:1988).

 Regions INERTIA INTERMEDIUM BOUYANCY STRUCTURE JET FORCED PLUME (OR BUOYANT JET) BUOYANT PLUME BOUNDS $${x \over L_{Mo}} < 0.5$$ $$0.5 \le {x \over L_{Mo}} \le 5$$ $$5 < {x \over L_{Mo}}$$ $${{u_c } \over {U_0 }}$$ $$6.2 {\omega}^{1/2}\left({x \over {d_0 }}\right)^{- 1}$$ $$7.26Fr^{ -1/10} {\omega}^{-9/20} \left( {{x \over {d_0 }}} \right)^{ - 4/5}$$ $$3.5Fr^{ -1/3} {\omega}^{-1/3} \left( {{x \over {d_0 }}} \right)^{ - 1/3}$$ $${{(\rho _\infty - \rho _c )} \over {(\rho _\infty - \rho _0 )}}$$ $$5{\omega}^{1/2} \left( {{x \over {d_0 }}} \right)^{ - 1}$$ $$4.4Fr^{1/8} {\omega}^{7/16} \left( {{x \over {d_0 }}} \right)^{-5/4}$$ $$9.35Fr^{1/3} {\omega}^{1/3} \left( {{x \over {d_0 }}} \right)^{-5/3}$$ $${{c_c } \over {C_0 }}$$ $$6.34\left( {{x \over {d_0 }}} \right)^{ - 1}$$ -- $$12.16Fr^{1/3} {\omega}^{-1/3} \left( {{x \over {d_0 }}} \right)^{ - 5/3}$$ $${u \over {u_c }}$$ $$\exp \left[ { - 92\left( {{r \over x}} \right)^2 } \right]$$ -- $$\exp \left[ { - 90\left( {{r \over x}} \right)^2 } \right]$$ $${c \over {c_c }}$$ $$\exp \left[ { - 52\left( {{r \over x}} \right)^2 } \right]$$ -- $$\exp \left[ { - 80\left( {{r \over x}} \right)^2 } \right]$$

## References

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