**2. Similar solution for a Coanda flow**

Jets are frequently observed to adhere to and flow round nearby solid boundaries. This general class of phenomena, which may be observed in both liquid and gaseous jets, is known as the Coanda effect. In recent years, great interest has been taken in flows deflected by a curved surface. Studying this phenomenon is very important due to the possibility of using the Coanda effect to aircrafts with short takeoff and landing, for fluidic vectoring.

Mathematical Modelling and Numerical Investigations on the Coanda Effect 103

the turbulent viscosity, assumed constant in a cross-section like shear

*U Rj <sup>y</sup> y r*

 

*<sup>R</sup>* \*

. Since \* *y* is much smaller than the unity, it may be

(4)

*<sup>f</sup>* (5)

 / , 1, *y*

(assumed constant in cross-

section), Re is the Reynolds number based on the cylinder radius and *y* is the radial

The dimensionless continuity equation (1) is satisfied by a stream function, chosen such that

neglected compared with the unity in the dimensionless equation (2). Introducing a

\* <sup>1</sup> 1 1 Re *<sup>c</sup> c a*

 *y* 

<sup>1</sup> , *a c*

 

and the turbulent shear stress, where the

*r* 

), has the form

*r* 

 

contribution of the laminar sublayer is neglected (omit the term *<sup>V</sup>*

The variables in equations (1)-(3) can be made dimensionless, as follows

where *Uj* is the velocity of the jet at the exit of nozzle <sup>0</sup>

*r*

 

\* \* Re , 1,

where the laminar shear stress is *<sup>V</sup>*

, *<sup>t</sup> V r* 

layer, i.e., / . *<sup>c</sup> t* 

 with *<sup>t</sup>* 

**Figure 1.** Coordinate system and notation.

 \* <sup>1</sup> / *Vr <sup>r</sup>* 

 

modelling variable of the form

and with the stream function chosen as

\* \* \*\* <sup>2</sup> , ,, , *<sup>r</sup> <sup>r</sup> j j j V V r P V V rp U UR <sup>U</sup>* 

distance from the cylinder surface, i.e.,*r R* .

and \* *V*

 

This section deals with the steady two-dimensional, laminar and turbulent flow of an incompressible fluid that develops like a jet-sheet on a cylinder surface, i.e., a Coanda flow [5]. We show that this flow can be approximated well enough by similar solutions for both the laminar and the turbulent regime. Basically we use Falkner-Skan transformations of the momentum equations that can be reduced to one ordinary differential equation (ODE). These solutions are presented in this section for both the laminar and the turbulent flow. The results are given in the form of analytical expressions for the mass flow, thrust and jetsheet thickness depending on the angle of deviation.

We also consider the possibility of the thrust augmentation yielded by the fluid entrainment of the jet flow. Thrust vectoring of aircraft which is the key technology for current and future air vehicles, can be achieved by utilizing the Coanda effect to alter the angle of the primary jet from an engine exhaust nozzle. Furthermore, the increased entrainment by the Coanda surface coupled with the primary jet fluid can augment the thrust, see e.g., [6].

The problem considered here is only a crude approximation of the physical phenomenon. However, we believe that the singular solutions that we develop pave the way towards a further, more accurate approach of the problem.
