Abstract
Buoyant turbulent plumes are often categorized by their geometry and described as either round plumes, issuing from a point source, or line/planar plumes, issuing from an elongated source. As line plumes rise above their source they get thicker (normal to the source axis) and, far from the source, they will no longer be planar but more resemble a round plume. However, the vast majority of experimental measurements of line plumes focus on the near source region, where they are still planar and the flow is two-dimensional. Further, these experiments constrain the ends of the plume with barriers to prevent entrainment through the ends of the plume and maintain a two-dimensional flow. Herein, results are presented from a series of experiments that were conducted to measure the transition of an unconstrained line plume into a round plume. A model is presented that allows the calculation of the entrainment into a plume of arbitrary cross sectional shape in terms of the hydraulic radius of the plume defined as the cross-sectional area divided by the perimeter over which entrainment is occurring. This formulation, along with a smooth transition function that changes both the geometry and entrainment coefficient, is used to make predictions of the front position over time for a line plume in a filling box. The model was run for different values of the nozzle width to box height ratio. Results of the model were compared to the experimental front position measurements and show that an unconstrained line plume will transition to a round plume at a height equal to approximately three times the source width. This is consistent with the idea that the line plume will transition when its thickness is similar in magnitude to its nozzle width.
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Abbreviations
- \(\alpha \) :
-
Entrainment coefficient (-)
- \(\alpha _L\) :
-
Entrainment coefficient for a line plume (-)
- \(\alpha _{L.G}\) :
-
Entrainment coefficient for a line plume with Gaussian velocity profile (-)
- \(\alpha _R \) :
-
Entrainment coefficient for a round plume (-)
- \(\varDelta \rho \) :
-
Density difference between the plume and ambient fluid (\(kg/m^3\))
- \(\eta \) :
-
Non-dimensional first front height (-)
- \(\varGamma_{T} \) :
-
Flux balance parameter at the transition height (-)
- \(\lambda \) :
-
Non-dimensional plume transition height (-)
- \(\phi \) :
-
Non-dimensional transition distance (-)
- \(\psi \) :
-
Ratio of the round plume to line plume filling box time (-)
- \(\rho \) :
-
Plume fluid density (\(kg/m^3\))
- \(\rho _0 \) :
-
Ambient density (\(kg/m^3\))
- \(\tau \) :
-
Non-dimensional time (-)
- \(\forall \) :
-
Tank volume (\(m^3\))
- \(\zeta _T \) :
-
Plume transition height scaled on the tank height (-)
- b:
-
Plume radius or half width (m)
- f:
-
Buoyancy flux per unit width (\(m^3/s^3\))
- g:
-
Gravitational acceleration (\(m/s^2\))
- g’:
-
Reduced gravity \(m/s^2\))
- h:
-
Height of the first front (m)
- m:
-
Momentum flux per unit width (\(m^3/s^2\))
- q:
-
Flow rate per unit width (\(m^2/s\))
- u:
-
Top hat vertical velocity (m/s)
- z:
-
Vertical coordinate measured from the plume source (m)
- \(z^*\) :
-
Non-dimensional height (-)
- \(z_T\) :
-
Plume transition height (m)
- \(z_v\) :
-
Virtual origin height (m)
- A:
-
Plume cross sectional area (\(m^2\))
- \(A_T\) :
-
Tank cross sectional area (\(m^2\))
- \(C_L\) :
-
Line plume flow rate coefficient (-)
- \(C_R\) :
-
Round plume flow rate coefficient (-)
- F:
-
Buoyancy flux (\(M^4/s^3\))
- \(F_B\) :
-
Buoyancy force (N)
- H:
-
Tank height (m)
- M:
-
Momentum flux (\(m^4/s^3\))
- P:
-
Entraining perimeter (m)
- Q:
-
Volume flux (\(m^3/s\))
- \(Q^*\) :
-
Non-dimensional volume flux (-)
- R:
-
Hydraulic Radius (m)
- \(R_T\) :
-
Equivalent radius of tank (m)
- \(R_L\) :
-
Line plume hydraulic radius (m)
- \(R_R\) :
-
Round plume hydraulic radius (m)
- S.G.:
-
Plume fluid specific gravity (-)
- \(T_{fill}\) :
-
Filling box time for a line plume (s)
- W:
-
Line plume source width (m)
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Acknowledgements
The authors would like to thank Dr. Abdul Khan for the use of the visualization tank for the experiments and Scott Black for assistance in the design and construction of the line plume nozzle. This material is based upon work supported by the National Science Foundation under Grant No.1703548. Any opinions, findings, and conclusions or recommendations expressed in the material are those of the author and do not necessarily reflect the views of the NSF.
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Kaye, N.B., Robinson, D.M., Akhter, R. et al. The transition of a line plume to round plume. Environ Fluid Mech 22, 763–787 (2022). https://doi.org/10.1007/s10652-022-09852-7
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DOI: https://doi.org/10.1007/s10652-022-09852-7