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1. Buoyant jet model

A type of mathematical model that has been developed for sumerged round buoyant jets is the length-scale model.  Discharges flows can be divided into different regimes each dominated by particular flow properties. Within each regime, the flow may be approximated with simple mathematical relations describing the simplified problem. A model that uses asymptotic solutions is refered to as length-scale model because of length scales to delineate the extent of the regimes for which the mathematical expressions are valid. The pollutant concentration, in a certain instant, and at a distance x (meters) in the X-Axis and at a distance y(meters) in the Y-Axis will be given by: 

c =c_c exp[-(r/b)²]  (1)

where c is the pollutant concentration, r is the distance from the point (that we are calculating) to the center of the line that forms the polluting plume, c_c is the pollutant concentration in the center of the plume line and b is the plume half-width. We attempt to link the momentum dominated and buoyanvy dominated regimes into one relationship by using proposed relations for the transition where: 

z/L_b =2^4/3[(1/2)(x/L_b)²+(L_m/L_b)(x/L_b)]^1/3  (2)

b/L_b =c_b[(1/2)(x/L_b)²+(L_m/L_b)(x/L_b)]^1/3    (3)

S=c_s(u_o/u_a)[(1/2)(L_b/L_m)(x/L_m)²+(x/L_m)]^1/3 (4)

L_b=plume-to-crossflow length scale

L_m=jet-to-crossflow length scale

x=horizontal downstream coordinate in global coordinate system

y=horizontal coordinate in coordinate system perpendicular to ambient crossflow

z=vertical coordinate

u_a=ambient velocity

u_o=discharge velocity

S=dilution along the plume centerline C/C₀ being C the centerline pollutant concentration and the C₀ initial pollutant concentration at the discharge.

c_b=constant of proportionality that can be modified by the user (can be determined experimentally)

c_s=constant of proportionality that can be modified by the user (can be determined experimentally)

c_xy=constant of proportionality that can be modified by the user (can be determined experimentally)

We obtain solutions for a vertical buoyant jet in a crossflow. And buoyant jets discharged horizontally perpendiculat to crossflow.

z/L_b =c_xy(x/L_m)^1/3    (5) 

This model performs satisfactorily for simple flows with no shoreline interaction or attachment. Strong crosscurrents or limited depths causing attachment with the downstrean bank or strong initial buoyancy render this model invalid. In addition, they are incapable of simulating any far-field processes that occur after a certain distance.

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