**5.3. Numerical results**

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


**Table 1.** Computation settings.

The recirculation zones that occur at the corners of the channel and beyond the contraction section have the lengths S1, S2 , and S3l , S3u, respectively (see Figure 15).

Mathematical Modelling and Numerical Investigations on the Coanda Effect 123

**Figure 16.** The computed separation (S1) and reattachment lengths (S2) for computed symmetric

**Figure 17.** Pathlines for *k* = 2 and Re = 3600: (a) half-channel, (b) full-channel symmetric mesh and (c)

solution in a half-channel and full channel (lower and upper corner).

full-channel with asymmetric orthogonal mesh.

**Figure 15.** The geometry and the reference lenghts that characterize the recirculating zones (near corners and in downstream channel).

For each contraction coefficient *k*, two computation cases occur: a first case in which we have used half-channel (*y* = 0 is a symmetry boundary) and a second case in which the flow through the whole channel has been studied. The computational domain allows for generating orthogonal grids.

For a given value of *k*, without changing the value of the Reynolds number, the numerical simulations lead to similar values of the length(s) of the recirculation zone (S1 and S2) in the corner(s) of the upstream channel, in accordance with the results from the works of Hawken [30]. The relative error between the simulated values S1 and S2 (whole channel, upper and lower corner) and the values obtained from the half-channel simulations is under 1% (see Figure 16).

Figure 17 shows that, for the half-channel symmetric solution, the reattachment length S3 is linear and monotonously increasing.

Concerning the full channel solution we make the following remarks:


section have the lengths S1, S2 , and S3l

corners and in downstream channel).

generating orthogonal grids.

linear and monotonously increasing.

symmetric solution (see Figure 17b);

*y* and keeping the grid orthogonal).

Figure 16).

wall ( 3 3

The recirculation zones that occur at the corners of the channel and beyond the contraction

**Figure 15.** The geometry and the reference lenghts that characterize the recirculating zones (near

For each contraction coefficient *k*, two computation cases occur: a first case in which we have used half-channel (*y* = 0 is a symmetry boundary) and a second case in which the flow through the whole channel has been studied. The computational domain allows for

For a given value of *k*, without changing the value of the Reynolds number, the numerical simulations lead to similar values of the length(s) of the recirculation zone (S1 and S2) in the corner(s) of the upstream channel, in accordance with the results from the works of Hawken [30]. The relative error between the simulated values S1 and S2 (whole channel, upper and lower corner) and the values obtained from the half-channel simulations is under 1% (see

Figure 17 shows that, for the half-channel symmetric solution, the reattachment length S3 is



*l u S S* ) (see Figure 17c). The results hereby have been obtained using four layers

of cells, in the neighborhood of the lower wall, each cell splitted into two parts (keeping

Concerning the full channel solution we make the following remarks:

, S3u, respectively (see Figure 15).

**Figure 16.** The computed separation (S1) and reattachment lengths (S2) for computed symmetric solution in a half-channel and full channel (lower and upper corner).

**Figure 17.** Pathlines for *k* = 2 and Re = 3600: (a) half-channel, (b) full-channel symmetric mesh and (c) full-channel with asymmetric orthogonal mesh.

Figure 18 shows the evolution of the reattachment length S3 for both half-channel and full channel solutions, on the lower wall of downstream channel ( S3l ) and on the upper wall (S3u) respectively. Note that up to a certain value of the Reynolds number the full channel numerical solution is identical to the half-channel numerical solution (the relative error is under 1%). For values greater than the aforementioned Reynolds number, a longer recirculation zone occurs on one of the walls of the channel, e.g., the lower wall, in our case. The critical value of the Reynolds number that leads to the bifurcation of the solution lies in the range: (1) 3050 Re 3100 *cr* for *k* = 2, (2) 1350 Re 1400 *cr* for *k* = 4 and (3) 1050 Re 1100 1150 *cr* for *k* = 8.

Mathematical Modelling and Numerical Investigations on the Coanda Effect 125

) is larger than the critical value Recr the symmetry of

The numerical simulations confirm the pitchfork bifurcation of the solution. The asymmetric solution given by the two different recirculation zones that occur on the upper and lower wall, respectively, can be stabilised up to a value of approximately 3800 of the Reynolds

Another application where the pitchfork bifurcation occurs is the study of two flows that go through a channel [34]. We consider the case when the velocity profile is described by equation (42), and assume that the flows have identical velocity profiles (see Figure 19). The flowing regime is characterized by small Reynolds numbers ( < 25-30) such that the flows are laminar and stationary. The domain is discretized in 400 x 600 nodes [35]. When the

the flow is lost, hence the flow becomes asymetric. Figure 20 shows the flow patterns for S/D = 10, S/H = 0.4 and L/H = 15, for Re = 15, 19 and 24, respectively. In Figure 20.b the jets unite into a single jet deflected towards one side-wall, which is then redirected to the opposite side-wall downstream. According to Figure 20.c the number of separation bubbles increase

Figure 21 shows the attachements point locations *xatt* for flows at various Reynolds numbers, for fixed ratio S/D = 10. This point aparently remains unchanged for S/H < 0.5, since the core region between jets is distant from side-walls and the walls do not influence it. For S/H 0.5, the walls are relatively closer to the jets, and the Coanda effects lead to the "attraction" of jets towards the walls with the merging point suddenly jumping to a further downstream

The transonic airfoil buffet [36, 37] is a stability issue that leads to shock oscillations and large variations of the lift coefficient. The practical problem of the airplane buffet is given by

The prediction of the onset and character of the unsteady transonic flow field is a great challenge. The transonic flow around an airfoil has been used as a model problem for

Many researchers analyze the problem using the Reynolds-averaged Navier–Stokes equations with adequate turbulence closure, which are a necessary approximation to cover

understanding the unsteady forcing, phenomenon similar to airplane buffeting [39].

the dynamic response of the elastic structure at the flow field [38].

the high Reynolds numbers at which transonic buffet occurs.

number for *k* = 2, 2000 for *k* = 4 and 1700 for *k* = 8, respectively.

Reynolds number ( Re / *V D med*

**Figure 19.** Twin-jet flow configuration.

location (observed for Re > 15).

with the Re number, and the flow becomes unsteady.

**Figure 18.** The plot of reattachment length S3 after contracted section (i.e. downstream channel) as a function of Reynolds number .

The numerical simulations confirm the pitchfork bifurcation of the solution. The asymmetric solution given by the two different recirculation zones that occur on the upper and lower wall, respectively, can be stabilised up to a value of approximately 3800 of the Reynolds number for *k* = 2, 2000 for *k* = 4 and 1700 for *k* = 8, respectively.

Another application where the pitchfork bifurcation occurs is the study of two flows that go through a channel [34]. We consider the case when the velocity profile is described by equation (42), and assume that the flows have identical velocity profiles (see Figure 19). The flowing regime is characterized by small Reynolds numbers ( < 25-30) such that the flows are laminar and stationary. The domain is discretized in 400 x 600 nodes [35]. When the Reynolds number ( Re / *V D med* ) is larger than the critical value Recr the symmetry of the flow is lost, hence the flow becomes asymetric. Figure 20 shows the flow patterns for S/D = 10, S/H = 0.4 and L/H = 15, for Re = 15, 19 and 24, respectively. In Figure 20.b the jets unite into a single jet deflected towards one side-wall, which is then redirected to the opposite side-wall downstream. According to Figure 20.c the number of separation bubbles increase with the Re number, and the flow becomes unsteady.

**Figure 19.** Twin-jet flow configuration.

124 Nonlinearity, Bifurcation and Chaos – Theory and Applications

1050 Re 1100 1150 *cr* for *k* = 8.

function of Reynolds number .

channel solutions, on the lower wall of downstream channel ( S3l

Figure 18 shows the evolution of the reattachment length S3 for both half-channel and full

(S3u) respectively. Note that up to a certain value of the Reynolds number the full channel numerical solution is identical to the half-channel numerical solution (the relative error is under 1%). For values greater than the aforementioned Reynolds number, a longer recirculation zone occurs on one of the walls of the channel, e.g., the lower wall, in our case. The critical value of the Reynolds number that leads to the bifurcation of the solution lies in the range: (1) 3050 Re 3100 *cr* for *k* = 2, (2) 1350 Re 1400 *cr* for *k* = 4 and (3)

**Figure 18.** The plot of reattachment length S3 after contracted section (i.e. downstream channel) as a

) and on the upper wall

Figure 21 shows the attachements point locations *xatt* for flows at various Reynolds numbers, for fixed ratio S/D = 10. This point aparently remains unchanged for S/H < 0.5, since the core region between jets is distant from side-walls and the walls do not influence it. For S/H 0.5, the walls are relatively closer to the jets, and the Coanda effects lead to the "attraction" of jets towards the walls with the merging point suddenly jumping to a further downstream location (observed for Re > 15).

The transonic airfoil buffet [36, 37] is a stability issue that leads to shock oscillations and large variations of the lift coefficient. The practical problem of the airplane buffet is given by the dynamic response of the elastic structure at the flow field [38].

The prediction of the onset and character of the unsteady transonic flow field is a great challenge. The transonic flow around an airfoil has been used as a model problem for understanding the unsteady forcing, phenomenon similar to airplane buffeting [39].

Many researchers analyze the problem using the Reynolds-averaged Navier–Stokes equations with adequate turbulence closure, which are a necessary approximation to cover the high Reynolds numbers at which transonic buffet occurs.

Mathematical Modelling and Numerical Investigations on the Coanda Effect 127

**Figure 22.**Extreme values of the lift coefficient *CL* as function of the Mach number M*∞* for self-sustained flow oscillations about the particular symmetric airfoil (relative airfoil thickness *h* = 0*.*09) at 00 incidence and Re = 1.1 e+7: domains A, B,C and D describes the flow regimes with different location of supersonic regions [40].

**Figure 23.** (A., B.,C., D.) - The flow regimes with different supersonic regions computational domains has 1.1 million cells (hybrid mesh with quadrilateral cells near the airfoil). Solver used: unsteady

Reynolds-averaged Navier-Stokes implicit solver with SST k- turbulence model.

**Figure 20.** Streamlines for case S/D = 10, S/H = 0.4: (a) Re=15, (b) Re=19 and (c) Re=24.

**Figure 21.** The attachment length for various Re and S/H for S/D = 10.

A simple exemplification for **bifurcation in transonic flow** over an particular airfoil is presented in the following section. The reference model can be found in ref . [40].

In figure 22, for the set of Mach incidence numbers 0.852 0.868 *M* , one may notice the appearance of the solution bifurcation. The bifurcation is given by the relation *CL* = f(M), resulting in four domains with different supersonic flow profiles:


**Figure 20.** Streamlines for case S/D = 10, S/H = 0.4: (a) Re=15, (b) Re=19 and (c) Re=24.

**Figure 21.** The attachment length for various Re and S/H for S/D = 10.

resulting in four domains with different supersonic flow profiles:

uniform flow at an angle of 00;

solution at 10 and M=0.86.

A simple exemplification for **bifurcation in transonic flow** over an particular airfoil is

In figure 22, for the set of Mach incidence numbers 0.852 0.868 *M* , one may notice the appearance of the solution bifurcation. The bifurcation is given by the relation *CL* = f(M),



presented in the following section. The reference model can be found in ref . [40].

**Figure 22.**Extreme values of the lift coefficient *CL* as function of the Mach number M*∞* for self-sustained flow oscillations about the particular symmetric airfoil (relative airfoil thickness *h* = 0*.*09) at 00 incidence and Re = 1.1 e+7: domains A, B,C and D describes the flow regimes with different location of supersonic regions [40].

**Figure 23.** (A., B.,C., D.) - The flow regimes with different supersonic regions computational domains has 1.1 million cells (hybrid mesh with quadrilateral cells near the airfoil). Solver used: unsteady Reynolds-averaged Navier-Stokes implicit solver with SST k- turbulence model.

The nonlinear flow equations, the initial solution used for the numeric computations and the length of the airfoil midpart with a small or zero curvature are the principal factors for the onset of flow bifurcations [40, 41].

Mathematical Modelling and Numerical Investigations on the Coanda Effect 129

performance parameters. The mixing layer growth plays a major role in optimizing the performance of the Coanda ejector as it decides the ratio of secondary mass flow rate to

Because single jet flows or multi-jet flows are extensively applied in conjunction with the Coanda surface, as confined or free jet flows, in the last part of the chapter we have provided further insight into complexities involving issues such as the variety of flow

We have considered two cases: i) the flow bifurcation in the symmetric planar contraction channel for different contraction ratio and Reynolds number (single jet) and ii) the flow structure as bifurcation phenomena involved in the confined twin-jet flow field, related to the parameters of jet momentum (Re), side-wall confinement and jet proximity effects.

Also was presented a simple exemplification for bifurcation in transonic flow over an

Thus, we have determined the conditions and the limits within which one can benefit from

*Institute of Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania* 

*Institute of Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania "POLITEHNICA" University of Bucharest, Faculty of Aerospace Engineering, Bucharest,* 

*Imperial College London, Dept. Electrical and Electronic Eng., Control & Power Group, London, UK* 

[1] Bourque C., Newman BG. Reattachment of a two – dimensional - incompressible jet of

[2] Newman BG. The deflection of plane jets adjacent boundaries – Coanda effect. In: Lachmann GV. (ed.) Boundary Layer and Flow Control Principles and Applications,

[3] Kruka V., Eskinazi S. The wall – jet in a moving stream. Journal of Fluid Mechanics

[4] Williams JC., Cheng EH., Kim KH. Curvature effects in a laminar and turbulent free jet

an adjacent flat plate. The Aeronautical Quarterly 1960; XI part 3 201-232.

vol. I, New York: Pergamon Press Inc.; 1961. p232-264.

boundary*.* AIAA Journal 1971; 4 733-736.

primary mass flow rate and the mixing length.

particular airfoil.

**Author details** 

A. Dumitrache

F. Frunzulica

*Romania* 

T.C. Ionescu

**7. References** 

1964; 20(4) 555-579.

the advantages of Coanda-type flows.

structure and the related bifurcation and flow instabilities.
