### Initial dilution of pollutant that emerges from submarine orifice into a stratified sea

Project leader: dr. Vlado Malačič

Duration: continuously from 1998

Data for the MATLAB script:

- winter

#### Matlab script description:

In coastal areas sewage or stormwater frequently ends in a submarine outfall, where the latter possibly ends with a diffuser. In any case, with or without the diffuser the (polluted) fluid enters the sea (or the lake) from an orifice with a known diameter, known inclination of its normal to the horizontal plane, and with a ‘known’ flowrate (or outflow mean speed). The ambient fluid (the sea) is vertically stratified. The emergent fluid rises as a buoyant jet into the stratified sea. The ambient seawater entrains in it during its rise. This process dilutes the emergent buoyant jet. The stratification may limit the rise so that the fluid reaches the ambient density and moves further upwards for about a meter at most, solely due to inertia. However, the stratification of ambient fluid may not be sufficient (during winter period) to overcome inertia and the polluted fluid rises to the surface in this case. The height that the polluted fluid reaches is of engineering interest, together with the factor of initial dilution. These two quantities are calculated by the *splinmat* Matlab script.

*Splinmat.m* script is a child of the Delphi (Pascal) program *Splinrun*, developed in 1998 and published firstly in 2001 (Malačič, 2001). It calculates the buoyant jet that emerges from an orifice with the adaptive step Runge-Kutta solver. This fourth order method solves four non-linearly coupled explicit differential equations of the first order. The four quantities of interest are: the core velocity along the curved jet’s axis (*u*), the radius at which the supposed Gaussian profile falls to 1/e value (*b*), the inclination of the slice of a buoyant jet (*θ*), tangential to jet’s curved axis, and the density difference *Δρ *between the ambient density *ρ _{a}* and the density of the core of the jet’s slice

*ρ*, both at the height

_{0}*z*. The latter (the height of the slice) is calculated together with the horizontal distance

*x*of the slice’s core from the orifice from consecutive values of

*s*– distance of the slice along the curved axis of the jet and from the inclinations of the slice normal.

The adaptive step method (Fourth order Runge-Kutta) keeps the linear combination of the absolute and relative error of the method below a chosen limit (by using also the fifth order correction). Therefore, the step ds of the independent coordinate (the length s along the jets axis) varies from one step to another and the method may also ‘make a ‘tango dance’, moving forward and backward along the jet’s axis. Steps are particularly small when gradients of quantities to be solved are large, and this usually happens when the vertical density gradient of ambient fluid is large (the pycnocline). However, when the neutral (*Δρ* = 0), or the sea surface is reached the horizontal spread becomes huge (*b → ∞*) and this singularity is also taken into account. Nonetheless, due to the ‘nature’ of the chosen method one cannot predict in advance the exact step length *ds*, or the height and distance the buoyant jet can reach. At that particular height z the ambient density *ρ _{a}* is simultaneously calculated by the method of (cubic) splines that takes into account known density values at some known heights, that do not coincide with the heights the buoyant jet reaches during its rise. This is where the name of the script comes from. However, the stratification is the quantity that matters, not just the ambient density. Therefore, the derivative

*dρ*is also calculated (analytically) with the spline method.

_{a}/dz

The run for the complete calculation of a buoant jet takes a few seconds at most on a usual portable computer. Much less than the time needed to read this text…

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