**3.1. Coanda effect. Computational analysis**

In this section the effect of the surface curvature (Coanda effect) on the development of a two-dimensional wall jet is numerically investigated. The main goals are providing a systematic survey of the performance of selected eddy-viscosity models in a range of curved flows and establishing more clearly their potential and limitations.

Reynolds averaged Navier-Stokes simulations (RANS) with different turbulence models have been employed in order to compute the two-dimensional turbulent wall jet flowing around a circular cylinder: (1) Spalart and Allmaras (SA - one turbulence model equation) [13], (2) Launder and Spalding *k* model [14], (3) Wilcox *k* model [15] and (4) Menter *k* SST model [16]. The predictions yielded by the simulations were compared to available experimental measurements from the literature. The surface curvature enhances the near-wall shear production of turbulent stresses and is responsible for the entrainment of the ambient fluid which causes the jet to adhere to the curved surface.

112 Nonlinearity, Bifurcation and Chaos – Theory and Applications


controls - magnetodynamics).

surface, see [11].

frequencies on Coanda surfaces [12].

**3.1. Coanda effect. Computational analysis** 

flows and establishing more clearly their potential and limitations.

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18.5 ; *<sup>m</sup>*

In many applications that use boundary layer control by tangential blowing, the solid surface downstream of the blowing slot is strongly curved and, in this case, the prediction of the jet involves separation and a more accurate knowledge of the flow (radial and tangential

**3. Control of the two-dimensional turbulent wall jet on a Coanda surface** 

Flow control refers to the ability to alter flows with the aim of achieving a desired effect. Examples include the delay of boundary layer separation and drag reduction, noise attenuation, improved mixing or increased combustion efficiency, among many other industrial applications. There are two possibilities to approach the problem of flow separation control: (1) passive control (vortex generators, flaps/slats, slots, absorbant surfaces and riblets) and (2) active control (mobile surface, planform control, jets, advanced

The active control without additional net mass flow can be achieved by synthetic jets or small vibrating flap. A synthetic jet is a concept that consists of an orifice or neck driven by an acoustic source in a cavity, as in [10]. At sufficiently high levels of excitation by the acoustic source, a mean stream of flow has been observed to emanate from the neck. The

Another technique of flow control on the convex surfaces is to use passive devices, one of these being the slot mounted between lower-pressure and high-pressure points (near the separation point) on the upper surface. The tendency of equalization of the pressure will produce blowing-suction jets which maintain the boundary layer attached to the upper

We investigate three issues related to flow control with applications to aerospace and wind energy: finding the appropriate turbulence model for the study of jets on convex surfaces, the passive control using a slot and the active control using a synthetic jet at medium

In this section the effect of the surface curvature (Coanda effect) on the development of a two-dimensional wall jet is numerically investigated. The main goals are providing a systematic survey of the performance of selected eddy-viscosity models in a range of curved

excitation cycle increases the ability of the boundary layer to resist separation.

*V b U R*

\* \* 2/3 1/2 ~, ~ , . *<sup>m</sup> y xV x x R* 

*j*

pressure - velocity profiles) which can be done with CFD methods.

The particular configuration shown in Figure 5 is considered cylindrical. The wall jet properties have been reported by Neuendorf and Wygnanski [17] and provide the means to evaluate the simulation results (diameter *d* = *2R =* 0.2032 m, nozzle height *b* = 2.34 mm and jet-exit velocity *U*j = 48 m/s).

The computational grid used for these investigations consists of 900 x 220 nodes. For the turbulence models used in these calculations the laminar sublayer needed to be resolved. The *y* values of the wall-next grid points were between 0.4 and 1, and the *x* values were between 50 and 300. The grid resolution in the jet was between 40 and 180 times the local Kolmogorov length scale. A fully developed channel velocity profile was prescribed at the nozzle inflow (no near field), with a medium turbulence. The ambient was quiescent.

For some of these turbulence models the jet-velocity decay and jet-half-thickness versus the streamwise angle are plotted in Figure 6. The jet-half-thickness (*y*1/2) represents the thickness where the jet velocity (*Uj*) is half of the maximum jet velocity (*Um*) through the same section.

When the *k* model was used in combination with the *k* SST model, a close match of the jet-velocity decay with the measured data was achieved. However, even with this model, the downstream development of the jet-half-thickness was poorly predicted.

The shape of the normalized velocity profiles is best predicted by the *k* model (see Figure 7). Since the predicted half-thickness (*y*1/2) is small for all models, the normalized velocity profiles do not match the experimental velocity profiles neither in the mild pressure region, nor in the adverse pressure region.

For the *k* SST model, the separation location was slightly closer to the experimental data. When the *k* and Spalart-Allmaras models were used, the jet remained attached to the cylinder for more than 260 degrees (see Figure 8).

One weakness of the eddy-viscosity models is that these models are insensitive to streamline curvature and system rotation. Based on the work of Spalart and Shur [18] a modification of the production term has been derived, which allows the *k* SST model to sensitize to the curvature effect.

The results obtained with the corrected (curvature correction – *c.c.*) *k* SST turbulence model were presented in Figures 6, 7 and 8, respectively. The results are close to the experimental data up to about 120 degrees. For larger values, the development of the jet was poorly predicted.

Mathematical Modelling and Numerical Investigations on the Coanda Effect 115

**Figure 7.** The shape of normalized velocity profiles at 900 and 1800 [*y/y*1/2=*f*(*U/Um*) ].

**Figure 8.** Streamline function: (a) Spalart-Allmaras turbulence model, (b) k- model (enhanced wall

option), (c) k- SST model and (d) k- SST c.c. model.

**Figure 5.** Configuration used in analysis.

**Figure 6.** Jet velocity decay and jet-half-thickness (Exp.-Ref.[17]).

Mathematical Modelling and Numerical Investigations on the Coanda Effect 115

**Figure 7.** The shape of normalized velocity profiles at 900 and 1800 [*y/y*1/2=*f*(*U/Um*) ].

114 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 5.** Configuration used in analysis.

**Figure 6.** Jet velocity decay and jet-half-thickness (Exp.-Ref.[17]).

**Figure 8.** Streamline function: (a) Spalart-Allmaras turbulence model, (b) k- model (enhanced wall option), (c) k- SST model and (d) k- SST c.c. model.
