Interaction of vertical round turbulent buoyant jets—Part II: Superposition method

Yannopoulos, Panayotis C.; Noutsopoulos, George C.

doi:10.1080/00221686.2006.9521678

Cited by 27 publications

(21 citation statements)

References 14 publications

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“…For the most interesting region concerning initial dilution, which is roughly in the range 0 < z < 100t, practically T values greater than 0.12 indicate that buoyancy governs the merging process and when T is lower than 0.012 inertia becomes the dominant force. The same behaviour identification in either linear diffusers or rosette-type risers has been found by Yannopoulos and Noutsopoulos [29], and Bloutsos and Yannopoulos [4], respectively. These studies indicated the existence of five flow regimes of interacting buoyant jets, namely: (a) Zone of original three-dimensional (3D) behaviour, prior to merging; (b) Lower transition zone of merging, developing from 3D to two-dimensional (2D); (c) Zone of 2D behaviour, where the merger effect is integrated and the flow and mixing fields resemble those of an infinite slot diffuser; (d) Upper transition zone, where flow and mixing fields are tending to the finite diffuser character as their behaviour is gradually turning to 3D, and (e) Final zone, in which a 3D behaviour is again reached.…”

Section: Introductionsupporting

confidence: 52%

“…In the following, the steady-state PDEs of continuity, momentum, tracer conservation and kinetic energy for the mean motion in three dimensions are formulated in both the Cartesian O(x, y, z) and cylindrical K(r, ϕ, z) coordinate system, where z is vertical. Their derivation is based on the usual assumptions-approximations [4,13,28,29], with the difference that the second-order terms of the turbulence contribution to momentum and buoyancy fluxes have been included [25]. Momentum…”

Section: Basic Assumptions and Governing Equationsmentioning

confidence: 99%

“…It is reminded that SM solution for interacting jets is unique, as it has been proved utilising the Reichardts inductive theory for turbulent fluctuations. Author [29] has extended SM applicability to interacting plumes, but without insuring the uniqueness of SM solutions. Since SM solutions satisfy linear differential equations, they provide that the complicated field (general solution) may be obtained by the superposition (addition) of all particular single fields (particular solutions).…”

Section: Introductionmentioning

confidence: 98%

“…It is based on ERA procedure incorporating SOA and advanced composite profiles produced by the Superposition Method (SM) [29]. Note that SM is based on the superposition of pertinent quantities of the mean flow and mixing fields of single vertical buoyant jets in order to yield the complicated fields of interacting jets or plumes of any number N from 2 to infinity.…”

Section: Introductionmentioning

confidence: 99%

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Advanced integral model for groups of interacting round turbulent buoyant jets

Yannopoulos

2010

Environ Fluid Mech

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An integral model that combines all advantages of Superposition Method (SM), Entrainment Restriction Approach (ERA) and Second Order Approach (SOA) is proposed to predict the mean axial velocity and concentration fields of a group of N interacting vertical round turbulent buoyant jets. SM is successful in predicting the fields of mean axial velocity and mean concentration for a group of N interacting jets or plumes and ERA is advantageous in predicting the above fields for either two or large number (N → ∞) of interacting buoyant jets in the whole range of buoyancy. SOA takes into consideration in a dynamic way the turbulent contribution to the momentum and buoyancy fluxes and provides better accuracy than the usual procedures. A novelty of the proposed model is the production and utilisation of advanced profile distributions, convenient for the mean axial velocities and concentrations in a cross-section of the entire group of buoyant jets. These profiles are developed on the basis of flux conservation of momentum, buoyancy and kinetic energy for the mean motion. They enhance dynamic adaptation of the individual buoyant jet axes to the group centreline. Due to these profile distributions, the present model owns generality of application and better accuracy of predictions compared to usual integral models using simple Gaussian or top-hat profiles; thus it conferred the name Advanced Integral Model (AIM). AIM is herein applied to predict the mean flow properties of two different arrangement types of any number of buoyant jets: (a) linear diffusers and (b) rosette-type risers. Present results are compared to available experimental data and traditional solutions based on Gaussian profiles. Findings may be useful for design purposes and environmental impact assessment.

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Section: Introductionsupporting

confidence: 52%

Section: Basic Assumptions and Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Advanced integral model for groups of interacting round turbulent buoyant jets

Yannopoulos

2010

Environ Fluid Mech

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“…Thus, the local fluid density is considered nearly constant and equal to a reference density r , where it is taken as either r = 0 or r = a . The usual assumptions and approximations are described in more detail elsewhere [17][18][19], while [5,9,20] have included the second-order terms of the turbulence contribution to momentum and buoyancy fluxes. Following Batchelor [14] and considering (3), (4), and (10), the term k ⋅ ∇k of (13) can be written with respect to the local coordinates as…”

Section: Momentum Equationmentioning

confidence: 99%

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

Bloutsos

Yannopoulos

2018

Mathematical Problems in Engineering

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The development of a local system of orthogonal curvilinear coordinates, which is appropriate to monitor the flow of an inclined buoyant jet with reference to the basic Cartesian coordinate system is presented. Such a system is necessary for the correct application of the integral method, since the well-known Gaussian profiles should be integrated on the cross-sectional area of inclined buoyant jet, where they are valid. This is the major advantage of the present work compared to all other integral methods using Cartesian coordinate systems. Consequently, the flow and mixing governing partial differential equations (PDE), i.e., continuity, momentum, buoyancy, and/or tracer conservation, are written in the local orthogonal curvilinear coordinate system and, then, the Reynolds substitution regarding mean and fluctuating components of all dependent variables is applied. After averaging with respect to time, the mean flow PDEs are taken, omitting second-order terms, as the dynamic pressure and molecular viscosity, compared to the mean flow and mixing contributions of turbulent terms. The latter are introduced through empirical coefficients. The Boussinesq's approximation regarding small density differences is taken into consideration. The system of PDEs is closed by assuming known spreading coefficients along with Gaussian similarity profiles. The methodology is applied in the inclined two-dimensional buoyant jet; thus, PDEs are integrated on the jet cross-sectional area resulting in ordinary differential equations (ODE), which are appropriate to be solved by applying the 4th order Runge-Kutta algorithm coded in either FORTRAN or EXCEL. The numerical solution of ODEs, concerning trajectory of the inclined two-dimensional buoyant jet, as well as longitudinal variations of the mean axial velocity, mean concentration, minimum dilution, and entrainment velocity or entrainment coefficient, occurs quickly, saving computer memory and effort. The satisfactory agreement of results with experimental data available in the literature empowers the usefulness of the proposed methodology in inclined buoyant jets.

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2007

Hydrological Processes

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Interaction of vertical round turbulent buoyant jets—Part II: Superposition method

Cited by 27 publications

References 14 publications

Advanced integral model for groups of interacting round turbulent buoyant jets

Advanced integral model for groups of interacting round turbulent buoyant jets

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

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