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

Abstract

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.

Room Air-Conditioning Operating as a Filling Box

“…In Appendix A, the 2D mode of the integral model EMA equipped with Equation ( 10) is qualified by comparing its predictions for vertical and inclined plane buoyant jets in linearly stratified environments with experimental measurements available in the literature. Therefore, it can be successfully applied to predict plane buoyant jets of positive or negative buoyancy [47]. In general, the results agree well with available experimental data.…”

“…Therefore, this is a well-known filling box operation in conjunction with the inclined positively buoyant jet. For simplicity reasons, an inclined plane buoyant jet is considered, which has been mathematically described and tested [45,47,48]. This model constitutes the 2D mode of the EMA integral model, which in the context of the present study is supplemented with the additional ability to treat flow and mixing in stratified environments.…”

“…The derivation of the conservation equations governing the flow and mixing of inclined buoyant jets is based on the usual assumptions and approximations described by Yannopoulos & Noutsopoulos [52,53] and Bloutsos & Yannopoulos [54] along with the addition of the second-order terms of the turbulence contribution to momentum and buoyancy fluxes [51,55]. More details are given by Yannopoulos & Bloutsos [45] and Bloutsos & Yannopoulos [47] concerning such flows within uniform environments. The derivation regarding stratified environments obeys the same rules with the difference in the definition of the above local parameter g and its value at the exit, g 0 .…”

An Improved Integral Model for a Non-Buoyant Turbulent Jet in Wave Environment

“…In fact, the integral model can achieve fairly good results, while the computational costs are only with the order of minutes. That is the reason why the integral approach is increasingly popular for research purposes [14][15][16][17][18][19][20][21][22].…”

Revisiting Mean Flow and Mixing Properties of Negatively Round Buoyant Jets Using the Escaping Mass Approach (EMA)

“…The escaping mass approach (EMA) model [74] is an integral model that describes the flow and mixing fields of inclined round or plane buoyant jets issuing into stationary ambient environment of uniform density. For the case of round buoyant jets, the conservation partial differential equations (PDE) of continuity, momentum and tracer have been formulated in a curvilinear cylindrical coordinate system, which is relative to the corresponding curvilinear orthogonal Cartesian coordinate system for plane buoyant jets [84]. EMA simulates the reported phenomenon ( [42,47,[49][50][51][52]54,60,[68][69][70][71][72][73] of fluid portions that detach from the main flow of the buoyant jet, especially from the inner boundary, and move vertically upwards (or downwards for brines), causing gravitational instabilities, and consequently destroying the axial symmetry of transverse concentration and velocity profiles.…”