Exp Fluids (2010) 49:341–353 DOI 10.1007/s00348-010-0885-1

RESEARCH ARTICLE

Experimental investigation of a free-surface turbulent jet with Coanda effect M. Miozzi • F. Lalli • G. P. Romano

Received: 7 July 2009 / Revised: 19 February 2010 / Accepted: 13 April 2010 / Published online: 28 May 2010 Ó Springer-Verlag 2010

Abstract The deviation of a jet from the straight direction due to the presence of a lateral wall is investigated from the experimental point of view. This flow condition is known as Coanda jet (from the Romanian aerodynamicist Henry Marie Coanda who discovered and applied it at the beginning of XXth century) or offset jet. The objective of the work is to detail the underlying mechanisms of such a phenomenon aiming to use it as a flow control method at polluted river flows mouth. To do this, a large laboratory free-surface tank with an incoming channel has been set up and velocity field measurements are performed by Optical Flow methods (namely Feature Tracking). Preliminary tests on the well-known free jet configuration without any marine structure (i.e. lateral wall) are performed to allow comparison with free jet scaling and self-similar solutions. The presence of the free-surface gives rise to centerline velocity decay which is lower than in free unbounded plane or circular jets due to the vertically limited ambient fluid entrainment. In the second part of the paper, the effect of a lateral wall on the jet configuration is examined by placing it at different lateral distances from the jet outlet. The resulting velocity fields clearly show an inclined Coanda jet with details which seems to depend on the lateral wall

M. Miozzi INSEAN, Via di Vallerano 139, 00128 Rome, Italy e-mail: [email protected] F. Lalli ISPRA, Via Vitaliano Brancati 48, 00144 Rome, Italy e-mail: [email protected] G. P. Romano (&) University of Rome ‘‘La Sapienza’’, via Eudossiana 18, 00184 Rome, Italy e-mail: [email protected]

distance itself. The analysis of self-similarity along the inclined jet direction reveals that for wall distances larger than 5 jet widths this dependence almost disappears.

1 Introduction The name ‘‘Coanda effect’’ (Henry Marie Coanda) is referred to the phenomenon in which a jet is deflected from the straight direction by an adjacent surface (wall), being inclined just toward the surface. This effect, although not completely established especially in unsteady conditions (how the initial deflection of the jet starts?), is related to pressure gradients normal to the jet induced by the presence of the lateral wall (Newman 1961). A review by Wille and Fernholz (Wille and Fernholtz 1965) considered the phenomenon as inviscid (thus theoretically independent on Reynolds number), since equilibrium is established between pressure pushing the jet toward the surface and centrifugal force. In the past, the Coanda effect has been employed to investigate and test flow control devices for increasing lifting and optimizing propulsion systems in aeronautical applications. In the present work, the Coanda effect is experimentally investigated in presence of a free surface in the framework of coastal engineering applications. In particular, for fully immersed jets there are suggestions toward the possibility of modifying the jet ejecting directions immediately after the outlet by placing a wall sideways (Nozaki et al. 1979). Here, the outlet of a water jet in free-surface conditions into a wide ambient resembles a river discharging into the sea. So far, the question to be solved is: how does the river water (presumably polluted) spread into the open field and how such spreading can be modified, to redirect the flow along given directions (for

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example far away from the shoreline) using Coanda effect? The problem is complex, due to the interactions of the river flow and related large-scale vortices with pre-existent marine structures (laterally and in front of the jet outlet) and with those eventually added to activate the effect itself. Some specific investigations have been performed to determine the effect of the distance from the jet outlet to the wall (here indicated as H, where H = 0 is equivalent to the so-called wall jet configuration) (Nozaki et al. 1979; Hoch and Jiji 1981; Lund 1986; Nars and Lai 1997, 1998). Moreover, the inclination of the wall (b) in respect to the streamwise direction (Lai and Lu 2000) has been also considered. The main results of these studies are that the reattachment length of the jet onto the wall scales almost linearly with the distance H and increases non-linearly with the angle b. In addition, the evolution of the velocity field downstream the outlet is strongly modified by the presence of the wall (for example the decrease of streamwise velocity along the jet axis) in a way which is interesting to relate to other well-known fluid-mechanics problems as free and wall jets (Neuendorf and Wygnanski 1999). In particular, the deviation of the jet from the streamwise direction gives rise to centrifugal instabilities which generate asymmetric vortices on the two side of the jet in comparison with free jet conditions (Han et al. 2005). For the present investigation, the problem is complicated by the presence of the free-surface (which is necessarily considered in coastal engineering applications) and by the possible effects of Reynolds and Froude numbers involved. Therefore, the experiments are performed on a large tank which can simulate the main features of coastal river flows. At first, the well-known free jet configuration without any marine structure (i.e. wall) is considered and the results are compared with self-similarity solutions. Here, deviations due to the presence of the free surface are outlined. Secondly, the effect of a lateral wall on the jet configuration is examined by placing it at different lateral distances from the jet outlet and by looking at the resulting velocity field. The self-similarity along the inclined jet direction is again considered and compared to the free and wall jet conditions. Measurements are performed on a horizontal plane by optical flow methods (namely Feature Tracking) which demonstrated to attain accuracies similar to Particle Image Velocimetry (PIV) (Stanislas et al. 2008) with the advantage of a lower dependence on seeding conditions. The effect of different Froude numbers is also considered.

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(about 2 m 9 2 m) with transparent lateral walls to allow optical access. A long free-surface channel (about 1 m) with an horizontal section W = 5 cm is connected to the middle of the tank to mimic the effect of a plane jet (i.e. a river flow) as depicted in Fig. 1. The water jet flows into the basin as feeded by an hydraulic pump after passing into a small tank to damp pump instabilities. Honeycombs ensure smoothing large-scale vortical structures and a set of trimmers forces transition to turbulence into the injecting channel. As a result, a rectangular free-surface turbulent jet is obtained. To allow flow discharge from the basin into the recirculating circuit, a sharp crested weir with Vshaped notches is placed on the basin wall at the opposite side of the input channel. The height of the weir can be adjusted, thus allowing an accurate control of the freesurface level inside the basin. The discharged water falls inside a vessel and is reintroduced into the hydraulic circuit. Sideways to the channel, a lateral wall can be positioned at different distances from the jet outlet (which is varied from H = 3W to H = 7W) in order to generate the Coanda effect (the measurements without any lateral wall are referred as free jet measurements). The wall lengths range from 80 to 110 cm, equivalent to more than 15 jet widths W. The water flow from the pump is accurately controlled by means of two magnetic induction fluximeters, placed just upstream of the channel. The flow rate fluctuations are lower than the resolution of the fluximeters (0.01 l/s) that, for the Reynolds number of the experiment, corresponds to an error of less than 2%. The Reynolds number Re, evaluated from the average velocity at the channel inlet (U0), the water height at the outlet (h, which is the same in the input channel and in the basin) and the

2 Experimental setup To investigate the discharge of a jet in free-surface water, a large basin has been set up on a aluminum plate basement

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Fig. 1 The free-surface basin setup and reference system

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kinematic viscosity of water (m), is between 5 9 103 and 104. In those conditions, the water depth has been kept constant at h = 0.05 m. Simultaneously, the Froude number is changed from 0.05 to 0.45 (based on the same reference quantities). To attain an almost constant Reynolds number and variable Froude number, both U0 and h are changed (being W a constant). In Table 1, the different experimental conditions used for the free jet and the Coanda jet are summarized (note that hereafter * indicates normalization by the jet width W). A set of six lamps (maximum power equal to 200W for each one) is placed above the free surface to illuminate the tracer motion. The free surface is seeded by floating woodpowder (size dp ^ 200 lm) up to a few seconds before starting image acquisition, thus minimizing the disturbances on the fluid motion. Even if the seeding particle size seems quite large, it must be considered that the imaged field is also large (about 70 9 50 cm2) and that the particle relaxation time (t ¼ qp dp2 =18qf mf ’ 103 s, where q is the density and subscripts p and f refer to particle and fluid respectively) is much smaller than the large structure turnover time (W/U0 ^ 0.1 s), so that large structures are very well resolved. As test case, the free jet configuration is considered just to point out the error level attained by present facility and setup (optics, seeding, camera and image processing) by comparison with well-known referenced results available in the literature for that case. Images are acquired by means of a commercial video camera (standard PAL, 720 9 576 px, 25 frames/s, 24 bits per Table 1 Experimental conditions for free and Coanda jet

Re

Fr

H~

Free jet 10000

0.29

Coanda jet 4000

0.05

5

0.07 0.09 0.10 0.11

Fig. 2 Negative of a typical acquired image with subtracted background

pixel) and long time histories of the flow evolution (about 40 s) have been stored on a PC. This corresponds to a total of 1000 samples for each data set, 400 of them statistically independent as resulting from the ratio between acquisition time (40 s) and large structure turnover time just introduced (0.1 s). The resulting statistical error on mean velocity is in the order of 2%. All images in each time history are used to produce a mean background, that is subtracted to each original image thus minimizing background noise effects (Fig. 2). The almost continuous distribution of the seeding emphasizes the capabilities of the Feature Tracking, because of the benefic effect of the continuous spatial distribution of image intensity gradients. The imaged field is extended over a region equal to 70 9 50 cm2 corresponding to 15 W 9 10 W. Considering that the maximum jet velocity, equal to 20 cm/s, corresponds to a displacement equal to about 10 px between each frame couple, it is possible to state that the displacements are well resolved in time in all tested conditions. The measurements are also sufficiently resolved in space (1 pixel corresponds to about 1 mm) to allow a correct investigations on the behavior of large intermediate scales in a free-surface flow with Coanda effect.

0.16 0.45 5000

0.14

3 4 5 6 7

10000

0.29

3 4 5 6 7

3 Feature tracking Images of the fluid motion have been analyzed by means of a Feature Tracking algorithm (Miozzi 2004). This method consists of an optical flow–based correlation technique, that defines its best correlation distance as minimum of the Sum of Squared Differences between the interrogation windows intensity in two successive images. The linearized system is solved, with an iterative procedure, only around points (features) where the solution of the system is guaranteed to exist and to be numerically stable. Interrogation windows

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are centered on those features, thus the signal-to-noise ratio is maximized. The solution of the problem is gained in two successive steps: first, the interrogation window is allowed to rigidly translate (thus determining a good set for the second step which is less stable numerically), secondly, a full affine deformation determines translation, rotation, scale and shear of the investigated area. The final set of deformation parameters includes both velocity data (displacements of the centroid of the interrogation window) and velocity gradients in a Lagrangian framework. It is to note that one main advantage of the employed algorithm is that these velocity gradients have been obtained without using any finite difference scheme and that grid resolution effects are avoided (Miozzi 2005a). Subpixel accuracy requires an interpolation step in order to resample the interrogation window at subpixel locations. The use of a deformed interrogation window introduces a further critical factor in the subpixel interpolation scheme, due to the varying distance between original image pixels and deformed interrogation window nodes. The selection of an appropriate interpolation method requires particular attention, especially when higher-order moments of the observable quantity are involved in the analysis. It is known from literature that interpolation methods based on B-spline basis functions (Unser et al. 1993) offers great accuracy in image interpolation, with the penalty of a high computational time cost. The 5th-order B-Spline method implementation offers a good compromise between accuracy and computational time and for this reason it has been adopted in the present work. The Lagrangian velocity and velocity gradients have been obtained by using a 15 9 15 px interrogation window, with an initial feature minimum distance of 4 px that produces an overlap degree of about 65%. Points near the lateral walls are tracked by using an adaptive logic mask scheme, that excludes the portion of the interrogation window that falls outside the velocity field (Miozzi et al. 2007). In order to reduce data on a regular grid, a secondorder Delaunay-triangulation-based natural neighbors interpolation scheme has been adopted both for the velocity and for the velocity gradients (Miozzi 2005b).

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numbers are equal to Re = 10000 and Fr = 0.29 respectively. This choice allows to perform comparisons with numerical computations at subcritical Froude numbers which are those typical of practical applications to river flows outlets in shallow waters. Results are presented by using dimensionless distances, i.e. as x~ ¼ WX and y~ ¼ WY with the origin of the axis at the center of the outlet jet, as reported in Fig. 1. Due to the upstream boundary conditions, the present free-surface jet discharging into the water basin follows a meandering motion, by swaying around the mean position along the jet axis. When averaging in time several instantaneous fields, the horizontal velocity (Fig. 3) presents a quite symmetrical shape, with the classical jet development both along the streamwise and transverse directions. The average vorticity map given in Fig. 4 confirms this overall behavior and shows a flow field which is symmetric within 10%. From these plots it could be stated that the present setup is suitable to reproduce and measure the global features of jet flows. To further stress this point, the well-known jet self-similarity is examined in detail. 4.2 Scaling and Self-similarity The self-similarity of a turbulent jet during decay along the propagation direction is related to the identification of one (or more) length and velocity scales which allow to derive almost identical transverse profiles of scaled turbulent quantities. Usually, the proposed velocity and length scales are the local maximum velocity, U x , and the local jet half width, y~1=2 , which is the transverse coordinate y at which Uðx; yÞ ¼ U2x (Pope 2000). For turbulent circular jets, the scaling behaviors along the axis x as derived from simplified conservation laws in integral form and from entrainment hypothesis (Rajaratman 1976; Pope 2000) is U0 ’ x~ dðxÞ ¼ y~1=2 ’ x~ Ux

ð1Þ

Therefore, for a circular jet, a linear spread must be expected together with a velocity decay as the inverse of

4 Free jet 4.1 Averaged fields As a preliminary test for setup and algorithms related to flow measurement, image acquisition and data processing as well as for a reference condition to be compared to, the free-surface jet configuration without any lateral wall has been investigated. For these measurements, being the input velocity of the jet about U0 = 0.2 m/s and the level of free surface about h = 5 cm, the resulting Reynolds and Froude

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Fig. 3 Free jet: time average of horizontal velocity:

U U0

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Fig. 4 Free jet: time average of vorticity:

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wz wz0

the axial distance. Similarly, for plane jets the following scaling behaviors are obtained (Gutmark and Wygnanski 1976; Kuang et al. 2001) U0 ’ x~1=2 Ux

dðxÞ ¼ y~1=2 ’ x~

Fig. 5 Free jet: scaling of UU0 for the present free-surface jet (red x symbols), for the plane jets by Kuang et al. (2001) (K) and Gutmark and Wygnanski (1976) (G) and for the circular jet by Djeridane et al. (1993) (D)

ð2Þ

which show that while the spreading is still linear, the velocity decay is slower in comparison with the circular jet due to reduced entrainment. The multiplicative coefficients appearing in previous equations could depend on geometrical and dynamical boundary conditions. Using the above reported scales, it is possible to derive self-similar solutions for the jet transverse profiles of turbulent quantities. Among the others, for comparing data on average streamwise velocity, the following empirical selfsimilar law for plane jets is considered (Kuang et al. 2001):   Uðx; yÞ ab g ¼ ab g sech2 : ð3Þ 2b U0 y and a, b are empirical constants to be where b g ¼ dðxÞ determined by data fit. The data measured for the present free-surface jet allow to test the scaling of centerline velocity (normalized by jet outlet velocity) at each downstream distance and to compare it with results from Djeridane et al. (1993) for a circular jet at a Reynolds number equal to 20.000 and from Kuang et al. (2001), Gutmark and Wygnanski (1976) for the plane jet at Reynolds numbers larger than 3000 as reported in Fig. 5. While close to the jet outlet, x~ \ 5, the present free-surface jet centerline velocity decays as for the circular jet data, for x~ [ 7 the decay slope is very close to that of a plane jet (note that the inverse of the normalized local axial velocity is plotted in the figure). However, in comparison with the plane jet data, there is a relevant difference in the extension of the core region (or development zone). This could be partially ascribed to the different upstream and boundary conditions (Romano 2002; Kim and Choi 2009), but also to the different geometry of the present free-surface jet (bounded at the bottom by the tank wall, at the top by the free surface and

Fig. 6 Free jet: lateral spreading of the horizontal velocity (red circles) for the free-surface jet with related linear fits (black lines)

unbounded laterally) in comparison with common plane jets (unbounded in all directions even if with a different geometry in comparison with circular jets). Thus, a slower decay of centreline velocity in comparison with the plane jet is to be expected as a consequence of the limited vertical extension. In particular, the centerline velocity decay coefficient of the present jet is equal to 2.35 which is very close to the value 2.4 reported by Gutmark and Wygnanski (1976). The spread of the free-surface jet is shown in Fig. 6, reporting the streamwise positions of dð~ xÞ. The spread shows that the free-surface jet expands by following the linear law (after the end of the potential core) as expected from previous consideration: dð~ xÞ / 0:10~ x

ð4Þ

This spreading corresponds to literature data for turbulent jets (Chu and Lee 1999). The velocity decay and lateral spreading are used in the following analysis of self-similarity. Using scaled quantities, the experimental results have been reported as transverse profiles of streamwise velocity in Fig. 7. A well-defined self-similar behavior is

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empirical result Eq. (3) from x~ ¼ 4:5 to the end of the measurement region. Thus, the preliminary analysis of the present free-surface jet allows to state that it well reproduces the characteristic behaviors of a plane jet with some differences in the entrainment of ambient fluid due to boundaries.

5 Coanda Jet 5.1 Description of the global configuration

Fig. 7 Free jet: self similarity of the mean horizontal excess of velocity U x . Velocity is scaled with its centerline value U x ð~ xÞ. y , where d(x) is the jet spread. The Lengths are scaled with b g ¼ dðxÞ coefficients are: a = 0.12 (see Fig. 6), and b = 0.037 is an universal constant (Schlichting 1979)

observable, whose extension ranges from -2 d up to 2 d, with some weak deviation far from the centerline. All profiles show a good agreement with the provided Fig. 8 Coanda jet: timeaveraged vector field overlapped to the vorticity field at Re = 10000, with H~ ¼ 3. Red lines are the loci of maximum of horizontal velocity (outwards from the jet origin and inwards in the lower region), green lines are the position of U max2 ðxÞ (outwards only) and blue lines are the positions of the zeros of UðxÞ

Fig. 9 Coanda jet: timeaveraged vector field overlapped to the vorticity field at Re = 10000, with H~ ¼ 5: Lines color as in Fig. 8

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The presence of a lateral wall strongly modifies the corresponding shear layer in comparison with the free jet condition. Due to the reduced entrainment on the side where the lateral wall is placed (in comparison with the other side), asymmetric pressure gradients take place between the jet and the external flows thus deflecting the jet toward the wall itself. Examples of the overall average fields for distances of the lateral wall from the jet H~ ¼ 3; 5 and 7 are presented in Figs. 8, 9 and 10 (similar figures are

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Fig. 10 Coanda jet: timeaveraged vector field overlapped to the vorticity field at Re = 10000, with H~ ¼ 7: Lines color as in Fig. 8

obtained for the other distances). In all conditions the jet is clearly moving toward the lateral wall generating upstream of the reattachment point a large recirculation region between the jet and the wall, whereas the flow reattaches to the lateral wall downstream that point. This behavior resembles that observed in the wellknown backward facing step case with curved streamlines directed toward the lateral wall, a successive separation in correspondence of a saddle point with zero horizontal velocity, an upstream recirculating region and a developing wall jet downstream. On the average, the recirculation region consists of a big vortex adjacent to the main jet and a counter-rotating smaller one, confined in the lower left corner of the field. However, in comparison with the backward facing step case, where the step height is usually limited to a fraction of the jet width, here the distance of the lateral wall from the jet is up to 7 jet widths. Therefore, the details of recirculation and reattachment regions are dependent on such a parameter (which is also of fundamental importance in practical applications). To determine the reattachment point and the global scaling features of the flow field as a function of the lateral wall distance from the jet, it is necessary to identify some special loci, e.g. the line of maximum horizontal velocity (upper red lines in former figures) and the lines where onehalf of this value is attained (green lines). These are the counterparts of the classical free jet scaling analysis for the Coanda jet condition. In addition to the free jet case, due to the presence of the large recirculation region, a new set of lines can be considered: the one corresponding to zero horizontal velocity between the jet and the recirculation (upper blue lines in former figures), the one related to the minimum velocity in the recirculation (lower red lines) and that connected to the secondary recirculation at the corner

between the horizontal and lateral walls (lower blue lines). All those lines have been derived from the data set presented in this work by using an ad hoc procedure. The eduction of these lines starts with the identification of the maximum horizontal velocity position with subgrid accuracy. The main problem relies on the discrete nature of the Eulerian grid. The subgrid position of the maximum has been detected by fitting a set of grid points (4 before and 4 after) around the maximum (initially evaluated at integer position) with a 9th-degree polynomium. Thus, analytical derivatives allow to extract the maximum position with subgrid accuracy. Similarly, the lines of one-half velocity are obtained by a 2nd-order fit around the first value (at integer position) as also both two zero velocity lines inside recirculation. Lastly, the line of minimum velocity is derived by adopting a 5th order fit. The choice of the order of previous polynomials is driven by considering the specific velocity profiles, with the aim of using as less points as possible. In particular, when dealing with the maximum velocity line, a high-degree polynomium (9th) is adopted due to the large zone with the same curvature sign on both sides of the line itself (as reported in previous figures), and also to the change in magnitude of the line curvature from one side to the other. On the other hand, around the line of minimum velocity, a smaller polynomial degree (5th) is used because the velocity profile has the same curvature sign on a smaller extension and a reduced magnitude. Lines of one-half velocities are positioned in places where the velocity profile is almost rectilinear and the 2nd-degree polynomium is accurate enough. The obtained global description of each field will be used in next sections to determine Coanda jet (or offset jet) both global topology features as the reattachment length and velocity-spreading scaling behaviors. As for the free jet

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case, instantaneous fields show series of traveling developing vortices along the shear layers with a swaying motion. Consequently, at this Reynolds number, the reattaching point periodically moves backward and forward along the lateral wall. Thus, the analysis will be usually performed by considering time-averaged quantities as in the previous figures. An investigation of the swaying unsteady behavior of the Coanda jet is reported in the Sect. 5.3. 5.2 Reattachment length The global mean flow configuration is governed mainly by the offset distance, whose extension defines the recirculating zone dimension. For small H~ this is quite small and confined near the reattachment (see Fig. 8) but, when the ratio is improved, it expands over the entire inner region thus minimizing the small counter-rotating structure in the lower left corner (Fig. 10). The position at which the offset jet reaches the lateral wall (i.e. the average reattachment point) has been evaluated and compared with results obtained by other authors. A comparative sketch of these and present results is reported in Fig. 11, with data for both Re = 5000 and Re = 10000. There is a quite good agreement between present data and those by other authors thus confirming that the evolution of the reattachment distance is a slightly non-linear ~ The following empirical law has function of the ratio H. been proposed as a fit to data (Nars and Lai 1998) ~ 0:851 ð~ xÞ ¼ 2:632ðHÞ

ð5Þ

and the present data points seems to follow reasonably this behavior. It is of a certain interest to observe that the

Fig. 11 Comparison of reattachment point position vs. H~ from various authors. Results from the present work are reported for both Re = 5000 and Re = 10000. Other results from: BN (Bourque and Newmann 1960), HJ (Hoch and Jiji 1981), N (Nozaki et al. 1979), P (Perry 1967), L (Lund 1986), B (Bourque 1967)

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reattachment distance does not depend on the Reynolds number, at least for the tested range, as observed in Fig. 11. A more complex behavior is observed when the Froude number influence on the reattachment point position is considered. The previous results have been obtained with fixed Froude and Reynolds numbers (equal to 0.14 and 5000 for the first series and to 0.29 and 10000 for the second, compare Table 1) while changing the lateral wall ~ To investigate the effect of Froude number, the distance H. lateral wall distance is now kept constant (H~ ¼ 5) as also the Reynolds number (Re = 4000) while Froude number is varied between 0.05 and 0.45. In Fig. 12, the horizontal profiles of average horizontal velocity, U, very close to the lateral wall (~ y ¼ 4:8) are presented. It can be observed that for almost all data at different Froude numbers the reattachment point, corresponding to the position where ~ ¼ 10 except for the data at U ¼ 0 and oU o~ x [ 0, is around x Fr = 0.45. In this case, the reattachment point moves to x~ ¼ 13, revealing a strong modification of the flow configuration. Some elements to clarify this point can be deduced from the velocity field at Fr = 0.45 (Fig. 13), in which the flow shows a topology which differs from that observed for Fr = 0.05 at the same value of H~ ¼ 5 (Fig. 14). The set of lines which have been superimposed to the velocity field illustrates that the main recirculating structure is confined near the reattachment point and that a new elongated vortex (not observed at smaller Froude numbers) takes place in the region near the jet outlet. The presence of this secondary vortex pushes the main recirculating region farther from the jet outlet, thus determining an increment of the reattachment distance. This behavior confirms finding of the free jet case in which a limited vertical extension for jet development is the main consequence of the simultaneous presence of bottom wall and free surface, considering that a high Froude number is attained by reducing the water depth in the tank.

Fig. 12 Coanda jet: variable Froude series, with fixed H~ ¼ 5: Evolution of the horizontal velocity along the line at y~ ¼ 4:8

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Fig. 13 Coanda jet: timeaveraged vector field overlapped to the vorticity field at Fr = 0.45, Re = 4000 and H~ ¼ 5: Lines color as in Fig. 8

Fig. 14 Coanda jet: timeaveraged vector field overlapped to the vorticity field at Fr = 0.05, Re = 4000 and H~ ¼ 5: Lines color as in Fig. 8

5.3 Some characters of the swaying motion The time evolution of the Coanda jet configuration exhibits a more or less regular swaying motion, which remarks the vortex shedding at the jet outlet. The complex vortex dynamics determines the presence of counter-rotating structures at both sides of the instantaneous maximum velocity line as shown at a specific time instant in Fig. 15 for the Coanda configuration at Fr = 0.45 and Re = 4000. In this type of pictures the rotating zones have been emphasized by adopting the Jeong and Hussain method for the identification of the vortical structures (Jeong and Hussain 1995).The identified areas have been multiplied by the vorticity sign, in order to retain the rotational sign. From the observation of the instantaneous fields it emerges that the jet maximum velocity line splits the whole field in two regions and that the greater and more intense is the vortex, the stronger is the deflection of the jet toward its

side. A slight tendency of the vortical structures to dispose themselves in a varicose distribution has been observed, i.e. structures with opposite sign alternates along the jet path. A deeper analysis has been developed to determine the evolution of the jet swaying frequency, f, when the Froude number varies. The frequencies of the peaks in the spectrum of the vertical velocity are presented in Fig. 16 for the data series at Re = 4000 and Froude number ranging from 0.05 to 0.45, together an expected trend for the frequency evolution, derived from the expression of the Strouhal number (St) as a function of frequency, Reynolds and Froude numbers: St ¼ f

ðRe mÞ1=3 Fr4=3 g2=3

ð6Þ

Being the Reynolds number a constant for these experiments, the scaling of the frequency with Froude

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Fig. 15 Coanda jet: instantaneous (t = 2.88 s) vortex eduction at Fr = 0.05, Re = 4000 and H~ ¼ 5: Lines color as in Fig. 8. The jet shows a flexuous pattern. Structures with opposite sign alternates along the jet path

Fig. 16 Coanda jet: evolution of the jet swaying frequency vs Fr. Dashed line indicates Fr4/3

number will be an indication of a constant Strouhal number or not. In particular, in such a condition, the frequency of swaying motion should scale as: f ðFrÞ / Fr4=3

ð7Þ

The reported results confirms the expected trend, showing the increase of the jet swaying frequency when the Froude number increases. This also give the suggestion of a continuous change of the Coanda jet pattern with Froude number. 5.4 Coanda jet scaling The objective of finding scaling laws for the Coanda jet is along the same line of the free jet, i.e. to attain self-similarity of average quantities. The scaling of the velocity at the center of the offset jet is evaluated on a curvilinear coordinate system (xc), having axis respectively tangential and normal to the maximum velocity of the jet. Fig. 17, gives the scaling of UU 0 (note again that this is the inverse of xc

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the normalized centerline velocity evaluated along the offset jet axis). For all tested lateral wall distances, the jet centerline velocity exhibits a linear scaling after a distance ~ This distance is from the input, which is variable with H. usually shorter than in the case of the free jet for H~ \ 5, while it is larger for H~ [ 5. The centerline velocity departs from the ð~ xÞ1 behavior at the reattachment and then remains constant or slightly increase again. Thus, before reattachment, the observed behavior corresponds to the scaling of an inclined jet, i.e. the offset jet deviates toward the lateral wall and then behaves as the free jet (inclined). A linear fit has been derived from the data by selecting only the linear part of the profiles. The fit equation U0 ¼ 1 þ Kd ð~ x  Ck Þ U xc

ð8Þ

identifies two coefficients for the velocity, corresponding to the position of a virtual jet origin Ck and to the virtual decay Kd along the inclined direction. The evolution of the virtual origin position as a function of H~ is shown in Fig. 18. Having in mind the different dimension and shape of the centerline position between the various series, it is quite surprising how strictly those data show a linear ~ with a coefficient slightly greater than dependence on H, one. As noticed before, the virtual origin derived for the present free jet (Ck ^ 5.5) is around the Coanda jet condition with H~ ¼ 5. Thus, for H~ \ 5 the Coanda inclined jet develops before (i.e. closer to the jet outlet section) than the free jet, whereas it develops further than the free jet for H~ [ 5. The decaying coefficient of the linear fit is reported in Fig. 19. Due to the observed peculiar behavior, its

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Fig. 17 Coanda jet: with their linear fits

U0 U xc

351

along the centerline of the jet for various H~

Fig. 18 Coanda jet: virtual origin of the jet centerline horizontal velocity as a function of lateral wall distance

Fig. 20 Coanda Jet: spread of the offset jet as signed distance along the normals to the jet centerline. Squares are for the external (i.e. upper layer) spread, circles for the internal (i.e. lower one)

Kd ^ 0.06 - 0.11), the decaying coefficient is now faster ~ for both H\5 and H~ [ 5 so that the Coanda jet slows down much more than a free jet. This behavior is partially balanced by an equivalent increasing spreading of Coanda jets in comparison with free jets. As reported in Fig. 20, the spread of the Coanda jet is of course non-symmetrical between the upper and lower shear layers and the whole jet width does not increase as linearly as for the free jet (note that the abscissa is non-dimensional by the reattachment length so that only behavior for values less than 1 should be ~ considered). There is some evidence of linearity for H\5: In particular, the upper shear layer spreading is around 0.15–0.20 (i.e. larger than the free jet value which is equal to 0.10), whereas for the lower layer it is around 0.09 (similar to the free jet). For H~ ¼ 5 or larger, there is a departure from linearity but also a more pronounced selfsimilarity among jet apertures at least for the upper layer. The overall aperture of the jet is equivalent or larger than in the case of free jets. Thus, the present Coanda jets with free-surface effects exhibit some self-similar behavior for lateral wall distances H~ [ 5 which resembles an inclined free jet, disregarding the initial part in which the jet turns toward the lateral wall.

6 Comments and conclusions Fig. 19 Coanda Jet: decaying term of the jet centerline horizontal velocity as a function of lateral wall distance

dependence from H~ is deduced by using a more complex fit, the hyperbolic tangent with a very good accuracy. It appears that there exist a threshold which separates two different domains for the H~ dependence, and this threshold value is again around H~ ¼ 5 as for the virtual origin. However, in comparison with the free jet (in which

The main results reported in this paper confirm that the presence of a lateral wall strongly deviates a jet from its rectilinear path toward the wall itself. This phenomenon takes place as a result of non-symmetric pressure fields at the jet boundaries even though the unsteady establishment of the effect is still questioned. In any case, the net consequence is that the jet fluid is displaced toward the side of the field where the lateral wall is placed as also does a potential passive scalar present in the jet stream. Thus, in

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the framework of coastal engineering, suitable marine structures could be positioned to avoid pollutant enhancements at given locations. The experiments are performed on a large tank in which geometrical similarity among effective river mouth and laboratory model relative dimensions is attained: mouth size (in situ in the order of 100 m, in laboratory 0.05 m), shoreline extension (about 5 km and 2 m) and marine structures (around 1 km and about 0.5 m). Simultaneously, dynamical similarity on buoyancy forces is maintained by considering Froude numbers from 0.05 and 0.5 (typical river mouth exit velocities from 1 to 10 m/s and sea depths from 10 to 50 m). This is obtained by changing both water depth in the tank and jet outlet velocity. Of course, the dynamical similarity of viscous forces is not attained due to the limiting laboratory model size so that Reynolds numbers are limited between 5000 and 10000. Therefore, this choice allows to consider in reasonable detail the effect of the water free surface on the behavior of a free jet and on modified offset jets. Velocity field measurements are performed by means of Feature Tracking which is part of the class of velocimetry techniques called Optical Flows. These methods allow to derive the velocity in time of suitable ‘‘features’’ with a much larger data density in comparison with classical Particle Tracking Velocimetry, i.e. similar to Particle Image Velocimetry (PIV). Thus, the derived velocity field has a spatial resolution sufficiently high to resolve the large velocity gradients observed in shear layer flows. The average measurements performed on the well-known free jet configuration without any marine structure (i.e. lateral wall) give results in agreement with available data from other authors at similar Reynolds numbers. In particular, the scaling behaviors of decaying jet centerline velocity and increasing jet aperture are examined in connection to the development of self-similar solutions. The presence of water free-surface at tested Froude numbers gives rise to some change in centerline velocity decay which is slower than in plane jets. This is due to the simultaneous effects of the bottom tank wall and the upper free-surface which limit jet spreading along the vertical allowing it only horizontally, thus limiting ambient fluid entrainment. The offset jet is obtained by placing a lateral wall on one side of the jet at different lateral distances (the other side is undisturbed as in the free jet case). From average velocity fields at all wall distances, an inclined offset jet configuration is obtained and lines of maximum velocity, half width velocity and zero velocity are determined. Even if the overall behaviors look different, a detailed analysis allows to characterize the offset jet and to evaluate the possible presence of scaling laws for jet centerline velocity and jet aperture. The results indicate that for lateral wall distances larger than about 5 jet widths such a scaling

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exists, either in jet centerline velocity decay along the maximum velocity line or in jet aperture. The scaling law in these conditions is linear for centerline velocity and slightly non-linear for jet aperture. In comparison with the free jet, Coanda jets with free surface have a higher deceleration and a larger overall aperture. Acknowledgments The authors would like to thank Dr. Luca De Antoniis and Dr. Massimo Falchi for the help in data acquisitions.

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