Jets and Plumes. Agenda. Definition of jets, plumes, and others. Engineering applications of Jets/Plumes. Basic properties of Jets/Plumes

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1 Dr. (IfH) Environmental Fluid Mechanics II Stratified Flow and Buoyant Mixing Jets and Plumes Dong-Guan Seol INSTITUTE FOR HYDROMECHANICS National Research Center of the Helmholtz Association Agenda Definition of jets, plumes, and others Engineering applications of Jets/Plumes Basic properties of Jets/Plumes Modeling issues o Basic assumption o Jets o Plumes

2 Dr. (IfH) Definition of jets and plumes Jets: driven by momentum (continuous injection) o Puffs: Intermittent injection of same fluid with momentum Plumes: driven by buoyancy (continuous injection) o Thermals: Intermittent injection of same fluid with different temperature Buoyant jet: Plumes with the added propulsion of momentum Why we care? Basic mixing process in nature Environmental engineering applications o Disposal of waste and toxic matters Engineering applications Jets: Waste water discharge system, marine outfall

3 Dr. (IfH) Plumes Volcano: Iceland Oil well blowout: IXTOC I Oil well Smokestack Oil spill: Deepwater Horizon, GoM, USA

4 Dr. (IfH) Weathering of spilled oil waves current w/o emulsion Oil slick oil Dispersed oil droplets water o/w dispersion Convergence of oil slick: Langmuir cell

5 Dr. (IfH) Basic properties of Jets/Plumes Flow properties Basic properties of Jets/Plumes Mixing properties o Jet: directly related to the inertia of the turbulent eddies o Plume: buoyant force produces inertia to lead mixing

6 Dr. (IfH) Modeling issues Self-similarity Momentum (buoyancy) conservation Entrainment hypothesis Modeling issues Self-similarity o Velocity profile : velocity profile : centerline (max.) velocity : characteristic width of jet/plume If we know u max and R, we can predict the flow field

7 Dr. (IfH) Velocity profile Velocity profiles (normalized with initial velocity and width) Velocity profiles (normalized with centerline velocity and characteristic width) Usually, we take Gaussian velocity profile as where : standard deviation If we define radius as R=2 for jet for plume Modeling of Jet (neutrally buoyant or pure jet) Momentum conservation Initial momentum Momentum at control section To get momentum over the cross section, integrate over the radial distance

8 Dr. (IfH) Modeling of Jet (neutrally buoyant or pure jet) Integral equation for momentum Introduce Gaussian velocity profile From the momentum conservation and Boussinesq approximation Modeling of Jet (neutrally buoyant or pure jet) Entrainment hypothesis: draws the ambient irrotational fluid into the jet Volume flux over the cross section Volume flux increment over height Entrained volume over unit height From the above, entrainment velocity is given by

9 Dr. (IfH) Modeling of Plume (homogeneous ambient) Continuous rise of lighter fluid thru the ambient dense fluid o local density difference : temperature anomaly : ambient temperature where : fluid s thermal expansion coefficient o local buoyancy or reduced gravity o Driving force of a plume: heat flux (buoyancy flux) where : fluid s heat capacity : characteristic plume rise velocity : characteristic plume radius Modeling of Plume (homogeneous ambient) Mass conservation o [Mass flux exiting control volume] = [Mass entering control volume] + [Mass entraining from side by entrainment] After introducing Buossinesq approximation, express in differential form

10 Dr. (IfH) Modeling of Plume (homogeneous ambient) Momentum conservation o [Momentum flux exiting control volume] = [Momentum entering control volume] + [Momentum entrained from side] + [Upward buoyancy] [Upward buoyancy] =[weight of displaced water]-[actual weight of plume segment] Modeling of Plume (homogeneous ambient) Momentum conservation conservation (continued) o [Momentum flux exiting control volume] =[Momentum entering control volume] + [Momentum entraining from side] + [Upward buoyancy] Introducing Buossinesq approximation

11 Dr. (IfH) Modeling of Plume (homogeneous ambient) So far, we have 3 equations But, 4 unknowns: To close this problem, we introduce the entrainment coefficient as where : entrainment coefficient ( for pure plume)