**4.2. Numerical model and results**

A numerical model of axisymmetric Coanda ejectors (Fig. 12a) have been built using the CFD software Fluent with a preprocessor, Gambit. The grid size was optimized to be small enough to ensure that the CFD flow results are virtually independent of the size, see [25]. The used grid is divided in a structured grid near the wall and an unstructured grid otherwise. The numerical results have been obtained for a total pressure value of 5 bar, imposed at the reservoir inlet. The computational domain includes the adjacent regions of the ejector with the physical opening boundaries condition. The flow is considered to be steady. We have used the following geometrical configurations (Fig. 12b): 1*e* 0.25 mm, *R*<sup>1</sup> 7.5 mm; 2*e* 0.4 mm, *R*<sup>2</sup> 37.5 mm .

Figures 13a and 13b show the velocity vectors and the Mach number contours for the investigated axisymmetric Coanda ejector. The induced flow does not follow the path defined by the primary jet. The Mach contours clearly show the flow patterns of the primary and the induced flows and how they mix in the divergent portion of the ejector.

### Mathematical Modelling and Numerical Investigations on the Coanda Effect 119

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

118 Nonlinearity, Bifurcation and Chaos – Theory and Applications

equal to the inlet diameter of ejector (m) [23, 24].

**Dimensionless Forms of Fluid Transport Equations**. The fluid transport equations such as the mass (continuity), momentum, and energy conservation equations are used. We define:

 , 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

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

In compressible fluids, the energy equation is used together with the transport equations in

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

**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.

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].

A numerical model of axisymmetric Coanda ejectors (Fig. 12a) have been built using the CFD software Fluent with a preprocessor, Gambit. The grid size was optimized to be small enough to ensure that the CFD flow results are virtually independent of the size, see [25]. The used grid is divided in a structured grid near the wall and an unstructured grid otherwise. The numerical results have been obtained for a total pressure value of 5 bar, imposed at the reservoir inlet. The computational domain includes the adjacent regions of the ejector with the physical opening boundaries condition. The flow is considered to be steady. We have used the following geometrical configurations (Fig. 12b): 1*e* 0.25 mm,

Figures 13a and 13b show the velocity vectors and the Mach number contours for the investigated axisymmetric Coanda ejector. The induced flow does not follow the path defined by the primary jet. The Mach contours clearly show the flow patterns of the primary

and the induced flows and how they mix in the divergent portion of the ejector.

model is transformed into a *k* − ω formulation by means of a

requires a model (e.g., turbulence, or subgrid scale) to make the problem well posed.

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

**4.1. Mathematical model** 

used in axisymmetric coordinates.

order to calculate fluid properties.

In order to achieve this, the *k*

**4.2. Numerical model and results** 

*R*<sup>1</sup> 7.5 mm; 2*e* 0.4 mm, *R*<sup>2</sup> 37.5 mm .

*c* 

> 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 flat portion of the velocity profile indicates a mixed flow.

> 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 is quite small.

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

The optimization study of the Coanda ejector is attempted mainly based on the primary nozzle throat and the stagnation pressure ratio. Based on the computational results, it is seen that the throat gap and the stagnation pressure ratio are the two critical parameters which have great influence on the flow characteristics through the ejector and then on the performance of the Coanda ejector, see [24]. Based on these studies, the optimal configuration of a Coanda ejector might be obtained, in order to maximize the ratio of the mass flow rates.

Mathematical Modelling and Numerical Investigations on the Coanda Effect 121

(42)

is the fluid (air) density

The geometry of the channel is shown in Figure 15 and it has a symmetry axis (y = 0). The contraction of the channel is given by the contraction ratio *k* = D/d. The inflow of the channel is at a coordinate 1 *x L* with respect to the contraction section, and has the following

<sup>2</sup> ( ) 6 0.25 / *med uy V y D*

The Reynolds number is defined by the mean value of the velocity *V*med and the maximum

The outflow ( <sup>2</sup> *x L* ) is chosen sufficiently far from the contraction suction, such that the velocity gradient associated with the velocity profile is zero ("outflow" output condition).

On the solid boundaries (the walls of the channel) we impose a "no slip" condition. If, for numerical simulations, a half-channel is used, then the presence of the symmetry axis is

The "pressure-based" solver that is used has a SIMPLE-C algorithm implemented (see [33]), together with a multi grid technique for increasing the rate of convergence of stationary flow problems (Ansys Fluent). The spatial discretization is of second order accuracy, with a under-relaxation coefficient of 0.5 for both the pressure and the momentum. The solution has converged when the global <sup>2</sup>*L* - norm of the pressure and the velocity residuals is lower

The Reynolds number sets were selected such that the flow in channel is laminar and

In the numerical simulations we use three channels with the contraction coefficients *k* = 2.4 and 8, repectively (see Table 1, and Figure 15). The fixed dimensions of the channel are D =

*k d* = D/*k* (m) Re *s* = min(*x*,*y*)/D Grid nodes (full channel)

2 0.1 500…3600 2.50e-4 413,400 4 0.05 500…2000 1.25e-4 415,990 8 0.025 400…1700 1.00e-5 433,800

imposed assuming a zero flux of all quantities across a symmetry boundary.

 , where

velocity profile

*med*

**5.2. Numerical model** 

**5.3. Numerical results** 

0.2 m, *L*1 = 0.5 m and *L*2 = 1 m.

**Table 1.** Computation settings.

than 10-8.

stationary.

/2

/2 <sup>1</sup> ( ) *D*

is the dynamical viscosity.

value of the height of the channel *D*, i.e., Re / *V D med*

*D <sup>V</sup> u y dy <sup>D</sup>* .

where

and 

By performing a computational study the effect of various geometric parameters on the performance of the Coanda ejector has been analyzed. The throat gap of the primary nozzle (*e*) has a strong influence on the ratio of mass flow rates of the induced flow and the primary flow and a critical control over the mixing length as well. For reduced throat gaps, the mixing length decreased, and this possibly indicates the rapid mixing layer growth in the ejector. The mixing layer was more developed for higher values of the diameters of the ejector throat. Validity limits of the calculation laws used in the numerical code have been confirmed by comparisons between numerical and experimental data. The present computational study has allowed us to identify the important parameters which have a strong influence on the behavior and performance of the Coanda ejector.

Further investigations are needed on the primary jet stability and its influence on the flow in the mixing area.
