3 The river flow models

The water currents are computed on the base of the number of formulations as follows :

• St. Venan equation based model for large long rivers;
• "Shallow water" model with influence of bed shape, bottom friction, water surface elevation and wind stresses;
• Algebraic drift model to account the simultaneous effect of surface currents, waves, ice-covering conditions and the wind velocities on the mean advection of the surface oil and
• Three-dimensional flow with surface wave emulations.

3.1 Flows in large long rivers

If the river in question is very long and large the only practicable approach is currently to treat it as one dimensional object.

To predict water currents distributions over time and space for such rivers and their networks one has to consider one-dimensional mathematical formulation.

ROSA can use the St. Venan equation based model to calculate the distribution of river flow discharge, q in m^3 per sec, and water level, z in m, over time, t in sec, and river course distance, s in m.

The governig equations are as follows :

dw/dt + dq/ds = d

dq/dt + d(q.q/w)/ds = -g.w.dz/ds - g.w.q.|q|/k^2

where d is river course influx; w stands for river cross sectional area; g is gravity acceleration; z stands for water level and k is flow discharge modulus.

The value of the latter is empirical coefficient. By default, ROSA treats it as being dependent on the cross sectional area, w, Chezy's coefficient, c, and hydraulic radius, R, as follows:

k = w.c.sqrt(R)

The Chezy's coefficient is related to bottom friction coefficient, cf.

The outcome of one-dimensional calculations can be seen here on the upper left hand corner of the picture.

3.2 Shallow-water flows

In many river streams, the mean-flow quantities vary but little in the vertical direction so that mean-flow equations in two-dimensions can be used with big advantages ( Spalding, 1975 ).

By default, ROSA uses conventional, small Froude number, variant of depth averaged governing equations.

The more general equations are obtained by integrating three-dimensional equations of water momentum conservation over water depth, assuming pressure distribution to be hydrostatic.

The set of equations which is used generally for both supercritical ( large Froude numbers ) and subcritical flow regions will follow next.

d()/dt + div[.U] = 0

d(.ui)/dt + div[ui.U - enul..grad(ui)] = -gradi-.gradiZb + windstr - botstr + momsor

d(.f)/dt + div[.U.f - enul..grad(f)/prndtlf] = sourf

wherein:

= rho.h
= rho.g.h^2/2 and

• rho stands for water density;
• h stands for water depth;
• U stands for water velocity vector;
• ui stands for water velocity component;
• Zb stands for vertical coordinate of the bottom;
• f stands for scalar variable;

3.3 Algebraic surface drift

For advection or drifting of the fluid particles and/or parcels on the water surface, ROSA employs the basic concept of vector addition of the wind and current forces to drive the surface water ( Yapp et al., 1991 ). In addition to the simplicity of the resulting mathematics, there are advantages in that the wind and water current factors can be quickly adjusted to fit actual field observation and experimental data.

In its ROSA embodiement weighted vector summation of water currents and wind velocity is extended for proper account of ice cover effects: the mean drift velocity under ice cover is now believed to depend upon oil/water properties and ice cover friction amplification quite realistically.

More details will be given in Section 5.1 below.

3.4 Three-dimensional flow with surface wave emulations

This formulation should be used when in-depth distribution of water currents and pollutant concentration is important.

For this case, ROSA uses, for the description of the river flow, the Eulerian equations built in PHOENICS for single and/or two-phase flows, which have the generic form as :

d(rho.r.f)/dt + div[r.((rho.u.f) - Gf.grad(f))] = Sf + Sp

where Sp is the source representing interaction with the dispersed phases, if any.

To simulate the surface waves the compressibility is introduced for the near open surface layer of cells to emulate the true shallow water assumption restricted to that part of water body.

The variables, f, for which ROSA will expect PHOENICS to solve governing equations and source terms, Sf, are as follows:

• Velocity components in each of the coordinate directions are to be considered;
• When transport turbulence models are used, the appropriate governing equations must be solved;
• Momentum source/sink terms accounting for bottom, wind and ice cover stresses must be set up.

The solution of all equations listed in this chapter can be replaced by the simple variable storage ( without solution ) of either values when ROSA is to track oil slick and calculate oil pollutant concentration in a " frozen " river flow-field.