**4.1. Mathematical model**

**Dimensionless Forms of Fluid Transport Equations**. The fluid transport equations such as the mass (continuity), momentum, and energy conservation equations are used. We define: *c* , the characteristic (inlet) density of the fluid ( <sup>3</sup> kg/m ), *U*, the characteristic (inlet) velocity of the fluid (m/s), *ct* , the characteristic time (s), and *L*, the characteristic length, which is equal to the inlet diameter of ejector (m) [23, 24].

Mathematical Modelling and Numerical Investigations on the Coanda Effect 119

**Figure 12.** a) Geometry of the Coanda ejector 3D view ; b) detail of the throat gap (primary nozzle).

In Figures 14a and 14b the flow velocities at *x* = 0, and *x* = 550 are plotted versus the diameters of the Coanda ejector for various values of *e*. Note that the graph can be split into two parts: the first part characterized by a large velocity gradient with high velocities (the primary flow) and a second part (the induced flow) where the velocity gradient is small. The

Also the flow velocities for two diameters of the Coanda ejector are analysed. Although cross sectional area increases when the diameter increases, the increment in mass flow rate

**Figure 13.** a) Velocity vectors; b) Mach number contour.

is quite small.

flat portion of the velocity profile indicates a mixed flow.

**Figure 14.** a) Velocity profiles at x = 0, and b) at x = 0.55 m - b

Then each term is converted to its dimensionless form by multiplying and dividing each term by their characteristic parameters, and then rearranging the equation to the dimensionless parameters. Since the geometrical configuration of the ejector is axisymmetric, the continuity equation and the momentum conservation equation have been used in axisymmetric coordinates.

In compressible fluids, the energy equation is used together with the transport equations in order to calculate fluid properties.

The equations can be spatially averaged to decrease computational cost, yet the averaging process yields a system with more unknowns than equations. Hence, the unclosed system requires a model (e.g., turbulence, or subgrid scale) to make the problem well posed.

**Turbulence Closure Equations.** The basic idea behind the SST model (see [16]) is to retain the robust and accurate formulation of the Wilcox model in the near wall region, and to take advantage of the free stream independence of the model in the outer part of the boundary layer. In order to achieve this, the *k* model is transformed into a *k* − ω formulation by means of a function that has the value one in the near wall region and zero away from the surface. The final form, the model parameters and the implementation are presented in detail in paper [16].

All the equations stated above are used to calculate fluid properties in a CFD code.
