**5.1. Physical model**

The flow equations through the channel are characterized by the Navier-Stokes equations for a laminar, incompressible, stationary flow, given by

$$\begin{aligned} \nabla \vec{V} &= 0\\ \rho (\vec{V} \cdot \nabla) \vec{V} &= -\nabla p + \mu \nabla^2 \vec{V} \end{aligned} \tag{41}$$

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 velocity profile

$$\mu(y) = 6 \text{ V}\_{med} \left[ 0.25 - \left( y / D \right)^2 \right] \tag{42}$$

where /2 /2 <sup>1</sup> ( ) *D med D <sup>V</sup> u y dy <sup>D</sup>* .

120 Nonlinearity, Bifurcation and Chaos – Theory and Applications

the mixing area.

**contraction** 

with contraction.

**5.1. Physical model** 

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.

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

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

An important application is the study of the incompressible flow in a symmetric 2D channel

Experimental and numerical research (see the works [26 – 32]) were performed in order to evaluate the flow through the 2D channel, especially after contraction occurs. The experiments done by Cherdron and Sobey (see [27], [28]) show the preferential formation of a recirculating zone on one of the channel walls, at a given Reynolds number. For values larger than the critical value Re*cr* , the flow through the channel loses its symmetry with respect to the channel summetry axis. This phenomenon is known as pitchfork bifurcation. Physically, in the fluid, a momentum transfer process occurs, causing the appearance of a pressure gradient across the channel. Such a pressure gradient may lead to an asymmetric

The flow equations through the channel are characterized by the Navier-Stokes equations

*VV p V*

0 ( ) *V* 

2

(41)

 

**5. The pitchfork bifurcation flow in a symmetric 2D channel with** 

strong influence on the behavior and performance of the Coanda ejector.

flow. We refer to this phenomenon as the Coanda effect.

for a laminar, incompressible, stationary flow, given by

The Reynolds number is defined by the mean value of the velocity *V*med and the maximum value of the height of the channel *D*, i.e., Re / *V D med* , where is the fluid (air) density and is the dynamical viscosity.

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