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# HorizontalBuoyantJet

For the numerical simulation of turbulent buoyant jets, normally there are two methods available: integral methods and simulation by CFD.

Integral methods are based on the basic laws of conservation of mass, momentum, energy, and species concentration. To integrate the partial differential equations, empirical profile shapes (i.e. Gaussian distribution) for velocity, temperature and concentration are assumed. The entrainment of ambient gas into the turbulent buoyant jet is also needed to be assumed to close the equation system [5-8]. Pantokratoras[9] modified the integral Fan-Brooks model[10] to calculate the horizontal penetration of inclined thermal buoyant water jets, and the modified model predictions are in good agreement with the trajectory measurements.

Jirka[11-12] formulated an integral model, named CorJet, for turbulent buoyant jets (round and planar) in unbounded stratified flows, i.e., the pure jet, the pure plume, the pure wake. Based on the Boussinesq approximation, CorJet integral model appears to provide a sound, accurate and reliable representation of complex buoyant jet physics.

However integral methods are difficult to extend to more complex turbulent buoyant flows which are essentially three dimensional and often obstacles and walls interact with the flow. It is very difficult to prescribe the profile shapes and to relate the entrainment rate to all of the local parameters.

Therefore CFD methods become more popular which do not make assumptions for the profile and entrainment but yield them as a part of the solution. However, if not applying DNS, the turbulence model in the engineering CFD code is one of the most significant factors to affect the simulations of turbulent buoyant jets and plumes. Most of the turbulence models were developed and tested for non-buoyant flows. Turbulence models including buoyant effects [13] were reviewed by Hossain and Rodi[14]. In recent studies models with buoyancy related modification were evaluated, assessing in particular buoyancy effect on the production and dissipation of the turbulent kinetic energy in buoyant plumes[15-17]. The standard models were suspected to seriously underpredict the spreading rate of vertical buoyant jets and to overpredict the entrainment of horizontal flows.

Comparing to the vertical buoyant jet, much fewer data and calculations for the study of horizontal turbulent buoyant jets can be found in the open literatures[18-23]. Almost all the models and calculations available are based on Boussinesq approximation, which the environmental and atmospheric engineers are more interested in. What we are concerning is hydrogen, helium, or steam released into air, which involves a large density difference between the jet and ambient, making the Boussinesq approximation invalid.

In the table below, integral models of horizontal buoyant round and plane jets are introduced. The system of ordinary differential equations can be solved by 4th order Runga-Kutta method to obtain the horizontal buoyant jet trajectory, the velocity, the density, and the tracer concentration. For small density difference cases, the Boussinesq integral model can obtain reasonable results as the CorJet model shows in the table. For hydrogen safety analysis, the large density difference will violate the Boussinesq approximation.

 Horizontal buoyant plane jet Horizontal buoyant round jet Diagram of horizontal buoyant jet into uniform ambient Volume flux $$q_0 = U_0 b_0$$(specific flux) $$Q_0 = U_0 A_0 {\rm{ = }}U_0 \pi r_0^2$$ Momentum flux (in kinematic units) $$m_0 = U_0^2 b_0$$ (specific flux) $$M_0 = U_0^2 A_0 = U_0^2 \pi r_0^2$$ Buoyancy flux(in kinematic units) $$j_{\rm{0}} = U_0 \left( {{{(\rho _a - \rho _0 )g} \over {\rho _{\rm{0}} }}} \right)b_0$$ (specific flux) $$J_0 = U_0 \left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)\pi r_0^2$$ Jet/plume transition length scale $$L_M = m_0 /j_0^{2/3}$$ $$L_M =M_0^{3/4}/J_0^{1/2}$$ Froude number $$Fs = {{U_0 } \over {\sqrt {g'b_0 } }} = {{U_0 } \over {\sqrt {\left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)b_0 } }}$$ $$Fs = {{U_0 } \over {\sqrt {g'r_0 } }} = {{U_0 } \over {\sqrt {\left( {{{(\rho _a - \rho _0 )g} \over {\rho _0 }}} \right)r_0 } }}$$ Assumptions The fluids are incompressible; The flow is fully turbulent. Molecular transport can be neglected in comparison with turbulent transport which means there is no Reynolds number dependence; The profiles of velocity, buoyancy, and concentration are similar at all cross sections normal to the jet trajectory; longitudinal turbulent transport is small compared with latitudinal convective transport. Velocity profile $$u = u_s e^{ - n^2 /b^2 }$$ $$u = u_s e^{ - r^2 /b^2 }$$ Density deficiency profile $${{\rho _a - \rho } \over {\rho _a }} = \left( {{{\rho _a - \rho _s } \over {\rho _a }}} \right)e^{ - n^2 /(\lambda b)^2 }$$ $${{\rho _a - \rho } \over {\rho _a }} = \left( {{{\rho _a - \rho _s } \over {\rho _a }}} \right)e^{ - r^2 /(\lambda b)^2 }$$ Tracer concentration profile $$c = c_s e^{ - n^2 /(\lambda b)^2 }$$ $$c = c_s e^{ - r^2 /(\lambda b)^2 }$$ entrainment $$E = 2\alpha \rho _a u_s$$ $$E = 2\pi \alpha b\rho _a u_s$$ Entrainment coefficient $$\alpha = \alpha _j \exp \left[ {\ln \left( {{{\alpha _p } \over {\alpha _j }}} \right)\left( {{{Ri_{j - p} } \over {Ri_p }}} \right)^{3/2} } \right]$$ $$\alpha = \alpha _j \exp \left[ {\ln \left( {{{\alpha _p } \over {\alpha _j }}} \right)\left( {{{Ri_{j - p} } \over {Ri_p }}} \right)^2 } \right]$$ Jet entrainment coefficient $$\alpha _j = 0.052 \pm 0.003$$ $$\alpha _j = 0.0535 \pm 0.0025$$ Plume entrainment coefficient $$\alpha _p = 0.102$$ $$\alpha _{\rm{p}} = 0.0833 \pm 0.0042$$ Width ratio $$\lambda = 1.35$$ $$\lambda = 1.19$$ conservation equations $$\int_{ - \infty }^\infty {{{\partial (\rho u)} \over {\partial s}}dn} = 2\alpha \rho _a u_s$$ $$\int_{ - \infty }^\infty {{{\partial (\rho uu\cos \theta )} \over {\partial s}}dn} = 0$$ $$\int_{ - \infty }^\infty {{{\partial (\rho uu\sin \theta )} \over {\partial s}}dn} = g \int_{ - \infty }^\infty {(\rho _a - \rho )dn}$$ $$\int_{ - \infty }^\infty {{{\partial (cu)} \over {\partial s}}} dn = 0$$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho u)} \over {\partial s}}r} drd\varphi } = 2\pi \alpha b\rho _a u_s$$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho uu\cos \theta )} \over {\partial s}}r} drd\varphi } = 0$$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (\rho uu\sin \theta )} \over {\partial s}}r} drd\varphi } = \int_0^\infty {\int_0^{2\pi } {(\rho _a - \rho )grdr} } d\varphi$$ $$\int_0^\infty {\int_0^{2\pi } {{{\partial (cu)} \over {\partial s}}r} drd\varphi } = 0$$

Boussinesq integral model for horizontal buoyant plane jet.