Control of planar jet using asymmetric nozzle lip

15th International Conference on Fluid Control, Measurements and Visualization 27-30 May 2019, Naples, Italy

Extended Abstract ID:XXX or Paper ID:XXX 1

Control of planar jet using asymmetric nozzle lip

Takashi Noguchi1,*, Katsuya Hirata1, Yuya Otomine1, Ryo Ogura1, Shuhei Yasuda1,

Tomotaka Motoki1

1Department of Mechanical Engineering, Doshisha University, Kyoto, Japan

*corresponding author: [email protected]

Abstract We investigate the planar jet from an two-dimensional nozzle with asymmetry in lip lengths. Experiments are conducted at Re = 1000, 3000, 6000. The aspect ratio of the nozzle exit is fixed to 300, which is sufficiently large to obtain two-dimensional well-developed turbulent jet. The lip length l varies in a wide range of 0 – 20h, where h denotes the gap of the nozzle exit. Using a hot-wire anemometer, we examine mean velocity and turbulence intensity profiles at both near and far downstream. For each lip length, the asymmetric planar jet shows characteristics similar to the symmetric jets. However, with increasing lip length, the planar jet exhibits oscillatory variation in deflection angle. Furthermore, we attempt to propose an empirical formula for the deflection angle of the jet. Keywords: Jet, Turbulent Mixing, Two-Dimensional Flow, Nozzle, Air conditioning, Air Curtain

1 Introduction Turbulent jets are simple and useful device for mixing/diffusion enhancements (Hirata et al. (2009) [1] and Funaki et al. (2009) [2]), and they are widely utilized in various engineering aspects such as chemical reactors, heat exchangers, burners/combustors, and air conditioners.

Most of the past studies concern a circular jet, namely, a jet from a nozzle with a circular cross section. On the other hand, we have been recently focusing our interest upon non-circular jets, to achieve more effective mixing/diffusion enhancements. Among such non-circular jets, a planar jet has fundamental and practical importance, for its applicability for various fields such as (1) drying or cooling of plastic films and fabrics, (2) cleaning, draining, or drying of manufacturing products, (3) flow controls inside burners and at furnace inlets and (4) efficient and smart air conditioning devices such as air curtains and air screens. Recently, another application of a planar jet is seen in stealth airplanes and drones for the purposes of (1) infrared-ray stealth, (2) radio stealth and (3) noise reduction, as well as (4) possible high-performance maneuvering by the control of nozzle geometry (Malla & Gutmark (2016) [3]).

Here we focus on turbulent free planar jets at high Reynolds numbers. So far there have been several researches concerning the turbulent and free planar jet (Förthmann (1936) [4], Görtler (1942) [5], Zijnen (1958) [6], Quinn (1992) [7], Mi et al. (2005) [8] and Deo (2005) [9], Krothapalli et al. (1981) [10], Zaman (1999) [11]). In most of practical applications, the nozzle for planar jet has a rectangular shape with a limited aspect ratio and hence the jet soon comes towards the round jet due to the end-effect. To investigate fundamental characteristics of a planar jet, we need to use a rectangular nozzle with as large aspect ratio as possible.

Moreover, in most of researches on the turbulent planar jet the nozzles were symmetrical and there have been only a few concerning a jet emitted from an asymmetrical nozzle (Horne et al. (1981) [12], Husain & Hassain (1983) [13], Kiwata et al. (2009) [14], Hirata et al. (2010) [15] and Malla & Gutmark (2016) [3]). So our knowledge about asymmetrical planar jets is not enough yet, in spite of their growing needs in various applications as (1) increase or decrease of the flow entrainment or the streamwise growth of flow rate, (2) promotion or suppression of the decay of flow velocity, (3) control of the jet direction, (4) realization of oscillatory or pulsatory jets, and (5) generation of the asymmetrical flow fields concerning temporal mean velocity, turbulence and so on.

In the present study, which is an extension of our previous study at Re = 6000 (Hirata et al. (2010) [15]), we examine the turbulent planar jet from an asymmetrical two-dimensional nozzle whose asymmetry is introduced by a lip (or a ramp) attached on one side of the nozzle. Among


Paper ID:153 2

various control methods for planar jet such as (1) asymmetrical nozzle geometry, (2) insertion of a downstream object or a nozzle-surface object and (3) addition of secondary jets, the lip-length control has advantages of both (1) simple geometry with less control parameters and (2) easier change of control parameter values. We examine especially the influence of the lip length upon characteristics of turbulence and temporal-mean flow, at Re from 1000 to 6000, which is a parameter range much wider than our previous study [15]. The aspect ratio AR of the nozzle exit is fixed to 300. The lip length l varies in a wide range of 0 – 20h, where h is the height (gap) of the nozzle exit. Using a hot-wire anemometer, we measure the mean velocity and the turbulence intensity at various downstream sections, in order to reveal fundamental characteristics of the jet in both near and far downstream. We also attempt to propose an empirical formula for several characteristics of the jet.

2 Experimental Configuration and Method

Figure 1 shows the model, namely, a planar jet ejected into an open-space of stationary air from a two-dimensional nozzle with the only asymmetric dimension l, which is the difference in the lengths of the upper and lower edges of the nozzle, which is hereinafter called lip length. In the present study, the upper edge is longer than the lower, with 0 < l < 20h, where h (= 1.5×10-3 m) is the gap of the nozzle exit and is taken as the characteristic length scale. The upper and lower edges of the nozzle are designated as lip-side (L) and no-lip side (NL), respectively.

Figure 1 also shows the coordinate system, together with important physical parameters. The origin of the coordinate O is at the nozzle exit (at NL) on the midgap plane and on the midspan plane. The coordinate is Cartesian with x axis being streamwise, y axis spanwise, and z axis transversal.

Supposing the two-dimensionality of statistical flow field, we consider only the midspan plane at y = 0. Hence the flow velocity u is effectively a function of x, z and t. The flow velocity on the midgap plane is written as U(x, t). Subscript “0” denotes the location of the nozzle exit. Both temporal-mean flow umean and root-mean-square flow urms are functions of x and z alone. The half width bu mean of the velocity profile is defined as the width at whose boundaries umean = (umean)max/2, where a subscript “max” denotes the maximum value of u at each x.

The geometric control parameter of the jet we consider is the (reduced) lip length l/h. As a kinetic control parameter, we consider the Reynolds number Re defined as𝑅𝑒 ≡ 𝑈&ℎ/𝜈, where the characteristic velocity scale U0 is the temporal-mean velocity at the origin, and v is kinematic viscosity.

When we discuss the streamwise variations of the flow, we use a flow rate Q the temporal-mean local flow rate per unit span through a transverse section at each x. We define an aspect ratio AR of the nozzle exit by w/h, where w is the spanwise width of the nozzle exit. The nozzle has an aspect ratio AR (≡ w/h) of 300.

Figure 2 shows the details of the nozzle geometry. Due to the machining imperfection, the nozzle has not achieved a perfect symmetry even for l = 0; that is to say the thickness eL at the end of the upper half of the nozzle is slightly different from the thickness eNL at the end of the lower half of the nozzle, except for l = 3.3h. This slight asymmetry in nozzle geometry could bring some asymmetry in the jet but it was adjustable as will be shown later.

Table 1 summarizes the values of main experimental parameters. All the experiments are conducted for 0 < l/h < 10 (0 – 20 in some cases) and for Re =1000, 3000 and 6000. We non-dimensionalized the dominant frequency f of the flow-velocity fluctuation as the Strouhal number St ≡ fh/𝑈&. In addition to Re and St, we might see 𝑅𝑒* (≡ 𝑈&𝜃/𝜈), 𝑆𝑡(𝑙)(≡ 𝑓𝑙/𝑈&) and 𝑆𝑡(𝜃)(≡𝑓𝜃/𝑈&), for reference, where 𝜃 denotes the boundary-layer momentum thickness at the nozzle exit.

Air is driven by a blower into a sufficiently long straight pipe with a constant cross-section area of 1.8 m in length and 56 mm in inner diameter. Air is issued out of the asymmetrical two-


Paper ID:153 3

dimensional nozzle into stationary open space. No sidewalls were attached to the span-end of the nozzle. Using a hot-wire anemometer with an I-type probe whose temperature effect was compensated by an I-type cold-wire probe, we measured u at various locations downstream of the nozzle exit. The insertion angle of the I-type probe was determined so as to minimize the disturbance by the probe. Temporal mean values were obtained by averaging over 20 s or longer.

Fig. 1 Model and coordinate system.

Fig. 2 Details of nozzle geometry, with w = 450 mm and AR = 300.


Paper ID:153 4

Table 1. Experimental parameters.

3. Results and Discussion

3.1 Mean-Velocity Profiles

In the preliminary experiments [15], we have confirmed two-dimensionality of the flow from the present nozzle for Re = 6000. More specifically, mean-velocity profiles in y (span-wise) direction, showed good two-dimensionality for whole span of tested nozzle. Hence we discuss only the results at midspan (at y/h = 0).

Figure 3 shows typical transverse profiles of umean. Figure 4 is the same as Fig. 3, but axes are re-scaled and shifted. Specifically, these figures are for l/h = 5.0 and Re = 1000 at several values of x/h. At the nozzle exit (at x/h = 0), a clear potential core is seen at -0.5 < z/h < 0.5. The absence of data in 0.5 < z/h < 2.7 in the profile at x/h = 0 is due to the existence of the lip. The flow was very slow in a wide range of |z/h| > 2. The slow flow is considered to be related to the entrainment of ambient fluid into the jet. As x/h increases from zero, the profile becomes gradual, that is, its peak becomes low and its tail becomes wide. As well as the profile at x/h = 0, the profiles at x/h ≠ 0 accompany the slower flow related with the entrainment. In addition, we can see that the profile center tends to shift to the positive z/h direction with increasing x/h. This will be discussed in Fig. 7. More specifically, the profile center shifts to positive or negative z/h direction depending upon l/h.

From Fig. 4 we can confirm the similarity in transverse profiles of umean. Specifically, the figure is normalized using the local maximum velocity (umean)max for the ordinate, and using both a shifted coordinate 𝑧3 instead of z and a local half width 2bu mean instead of h for the abscissa. In Fig. 4 is plotted the empirical formula proposed by van der Hegge Zijnen (1958) [6] for a symmetrical nozzle. We can find that the present profiles agree well with Ref [6] for x/h ≥ 10.0, although the profile at x/h = 0 is obviously different from the others. We can find that, at 𝑧3/ bu mean ≥ 1.5, all the profiles at x/h ≥ 10.0 tend to differ from Ref [6]. Considering the difference in nozzle geometry between the present and Ref [6], we regard that this discrepancy is due to the entrainment of ambient fluid into the jet. The above features are shown in the other nozzles with 0 < l/h < 10.0. Comparing with [15] for Re = 6000, we can confirm that effect of Re is negligible for mean velocity in the range.

h (m) 1.5×10-3

l (m) 0.0, 3.0×10-3, 5.0×10-3, 6.0×10-3, 7.5×10-3, 9.0×10-3, 11×10-3, 12×10-3, 15×10-3, 30×10-3

w (m) 450×10-3

U0 (m/s) 10, 30, 60

𝜃 (m) 0.03×10-3 – 0.06×10-3

l/h 0, 2.0, 3.3, 4.0, 5.0, 6.0, 7.3, 8.0, 10, 20

AR (= w/h) 300

Re 1000, 3000, 6000

𝑅𝑒* 31 – 222


Paper ID:153 5

Fig. 3 Mean-velocity profiles in the z direction at midspan (y/h = 0) for l/h = 5.0 and Re = 1000.

Fig. 4 Re-normalized mean-velocity profiles in the z direction at midspan (y/h = 0) for l/h = 5.0 and Re = 1000. A

coordinate z′ is so shifted that 𝑢6789 = (𝑢6789)68; at z′=0.

3.2 Streamwise Decay of Mean Velocity

Figure 5 shows the streamwise distributions of the maximum-mean-velocity (umean)max, for nine values of l/h, in order to observe the streamwise variation of the jet decay. Panels (a) and (b) represent the results for Re = 1000 and 3000, respectively. Figure 5 also shows the theory by Tollmien (1945) (in Rajaratnam (1976) [16]) for two-dimensional free jet, the experiments by Mi et al. (2005) [8] for symmetrical nozzles, and the experiments by Kiwata et al. [12] for asymmetrical nozzles, for reference. Table 2 summarizes these researchers’ experimental parameters. As will be discussed later, we should note that most of the experiments by Mi et al. (2005) [8] were conducted at not very large AR.

In far-downstream at x/h ≳ 6, the present streamwise distributions are almost similar to the Tollmien’s theory, and then with those for Re = 6000 [15]. That is, all the results collapse on the Tollmien’s theory, being independent of the values of l/h and Re. Of course, this is consistent with Mi et al. (2005) [8], whose result also approaches to the Tollmien’s theory with increasing AR.

Next, we examine near-downstream at x/h ≲ 6. According to Rajaratnam (1976) [16], the potential core of a two-dimensional jet exists at x/h ≤ 6. All the present results are consistent with this, being independent of the values of l/h, since (umean)max/U0 is approximately equal to unity at x/h ≲ 6.


Paper ID:153 6

Finally, in figure 6, we show the streamwise distributions of (umean)max for Re = 1000 with the abscissa of x/De instead of x/h in order to compare with other researchers’ results, whose parameters are again summarized in Table 2. Note that both axes are logarithmic.

According to Mi et al. (2005) [8], we see (1) the potential core zone, (2) quasi-plane-jet zone, (3) the transition zone and (4) quasi-axisymmetric-jet zone, in sequence, as x/De increases from zero. The larger AR is, the smaller the critical value of x/De where the quasi-plane-jet zone appears and the wider the range of x/De for the quasi-plane-jet zone is. In the quasi-plane-jet zone and the quasi-axisymmetric-jet zone, (umean)max/U0 is proportional to x-1/2 and x-1, respectively.

All the present data at x/De ≳ 1 collapse on a common straight line which is in proportion to x-1/2, because of high AR as 300. This suggests good two-dimensionality of the jet for all the l/h. In addition, we can confirm a consistency with Mi et al. (2005) [8]; namely, the present value of (umean)max/U0 is always smaller than Mi et al. at each x/De. This is reasonable, if we remind the high AR as 300, which is larger than Mi et al., being independent of the values of l/h. It seems difficult to compare the above results with Kiwata et al. (2009) [14], due to their different velocity profile caused by much different nozzle geometry with a far-upstream contraction.

If we compare Fig. 7 for Re = 1000, 3000 and 6000, we can confirm not a qualitative but a quantitative influence of Re. That is, the amplitude of the jet bias tends to increase with decreasing Re (see subsection – 3.9 for quantitative discussion). All the present data at 0.4 < x/De < 1 collapse on another common straight line which is in proportions to x-0.8. This is also considered to be another influence of Re, if we remind that all the data for Re = 6000 is in proportions to x-1/2 even at 0.3 < x/De < 1 [15].

(a) Re = 1000

(b) Re = 3000

Fig. 5 Streamwise distributions of the maximum mean-velocity at midspan (y/h = 0) for 0 < l/h < 10.


Paper ID:153 7

Fig. 6 Streamwise distributions with logarithmic scales of the maximum mean-velocity at midspan (y/h = 0), for AR = w/h = 300, 0 < l/h < 10 and Re = 1000. Note that the distance is normalized by an equivalent diameter De ≡ (4wh/π)1/2

for comparison.

3.3 Streamwise Variation of Jet Bias on Mean-Velocity

Figure 7 shows the streamwise distributions of a jet bias (zu mean)max on mean-velocity, namely, a local mean-velocity-profile center, for several values of l/h. We define (zu mean)max as the value of z where the transverse mean-velocity profile attains the maximum (umean)max at each x, as shown in Figure 1. Error bars in Figure 7 show the boundaries of the region where umean ≥ 0.95 U0. This region could be approximately regarded as the potential core.

At first, when we consider the jet bias for l/h = 0, we can clearly confirm the streamwise growth of the jet bias suggested in Figure 3. Specifically, the jet bias is slightly negative and is almost in proportion to x/h. Namely, the jet direction is not horizontal, but somewhat downward. This is considered to be related with the incompleteness of the symmetry in nozzle geometry, even for l/h = 0. In addition, the jet axis is almost linear.

The jet biases for l/h = 2.0 and 3.3 show the above two features such as (1) the downward deflection and (2) the spatial linearity of the jet axis. However, from a quantitative viewpoint, we can find an effect of l/h upon the jet bias, if we compare the results for l/h = 0. Namely, by the l/h effect, the large l/h becomes, the more downward the jet deflection is.

In contrast with the jet biases for l/h = 0, 2.0 and 3.3, the jet biases for l/h = 4.0 and 5.0 indicate such a different feature as the jet deflection is upward.

Again, the jet biases for 6 < l/h < 20 shown the same two features for l/h = 0 as (1) the downward deflection and (2) the spatial linearity of the jet axis. We can find an effect of l/h upon the jet bias. Namely, the jet biases for 6 < l/h < 20 almost coincide with that for l/h = 0.

In summary, when we compare the results with different l/h values, we can classify all the results into three categories; that is, (1) ones for l/h = 2.0 and 3.3, (2) ones for l/h = 4.0 and 5.0, and (3) the others for l/h = 0 and 6.0 < l/h < 20. The first, second and third ones are characterized by strongly-downward, upward and weakly-downward jet biases, respectively. However, the jet axis is almost straight at any time for all the categories (the jet bias linearly increases/decreases with increasing x/h), being independent of l/h.

If we compare Fig. 7 for Re = 1000, 3000 and 6000, we can confirm not a qualitative but a quantitative influence of Re. That is, the amplitude of the jet bias tends to increase with decreasing Re (see Subsection 3.9 for quantitative discussion).

Finally, we consider the near downstream at x/h < 10, where figure 7 is not appropriate owing to condensed results at x/h < 10. To conclude, we can again confirm that the present results coincide


Paper ID:153 8

with Rajaratnam (1976) [16], as the potential core of a two-dimensional jet exists at x/h ≤ 6 for all the tested l/h and Re.

Concerning the shape of the potential core, it seems difficult to find out any clear l/h effects. Especially for l/h = 0, the upper and lower outer boundaries of the potential core are fairly symmetrical about the horizontal axis (zu mean)max/h = 0. This suggests that the nozzle axis is accurately installed parallel to the horizontal axis, while the jet bias exists in the downstream even for l/h = 0.

(a) Re = 1000

(b) Re = 3000

(c) Re = 6000

Fig. 7 Streamwise distributions of jet bias z(u mean)max at midspan (y/h = 0) for 0 < l/h < 20.


Paper ID:153 9

3.4 Streamwise Growths of Half Width on Mean-Velocity and of Flow Rate

Figure 8 shows the streamwise distributions of the half width 2bu mean on mean-velocity profile at midspan (y/h = 0) for several values of l/h, and for Re = 1000. In each figure, both the axes are normalised by h. For reference, the figure also shows the theory by Rajaratnam (1976) [16] as 2bu mean = 0.20x. (1)

At x/h ≲ 10, 2bu mean/h is almost unity, being independent of l/h. This is consistent with the existence of the potential core. At x/h ≳ 10, 2bu mean/h linearly increases with increasing x/h. Moreover, we can see that all the results almost collapse on the theory by Rajaratnam for a two-dimensional free jet, being independent of l/h. To be strict, all the results are slightly larger than the Rajaratnam’s theory. The results for Re = 1000 are the same as those for Re = 3000, and 6000 (not shown). Then, we can conclude that the influence of Re upon 2bu mean/h is almost negligible.

Figure 9 shows the streamwise distribution of the local temporal mean (volumetric) flow rate Q per unit span at midspan (y/h = 0) for 0 < l/h < 10. Panels (a) and (b) are for Re = 1000 and 3000, respectively. In each figure, the abscissa is normalised by h, and the ordinate is normalized by Q0, which denotes the flow rate from the nozzle exit. For reference, the figure also shows the theory by Albertson et al. (1950) [17] for a two-dimensional free jet as q/Q = 0.44(2x/h)1/2 (2) and the experiments by Kiwata et al. (2009) [14] for asymmetrical nozzles.

Firstly, we see panel (a). For all l/h, Q/Q0 monotonically tends to increase with increasing x/h. Even from a quantitative point of view, all the results for every l/h are almost the same and are always smaller than Ref [17], being independent of x/h.

Secondly, we see panel (b). Q/Q0 monotonically tends to increase with increasing x/h for all l/h. Even from a quantitative point of view, all the results for every l/h are almost the same and are always slightly larger than Ref [17]., being independent of x/h. These results for Re = 3000 are the same as those for Re = 6000 [15].

Finally, the comparison between panels (a) and (b) suggests that the flow entrainment can be suppressed not by the lip but by viscosity, when we remind the result at a low Re as 1000 where Q/Q0 always smaller than Ref [17], and the results at high Re as 3000 and 6000 where Q/Q0 always slightly larger than Ref [17]. To summarize this and previous Subsections 3.1 – 3.4, we can see a distinctive influence of l/h only concerning (1) the jet bias, but cannot see it concerning (2) the maximum mean-velocity, (3) the half width or (4) the temporal mean local flow rate. In addition, we can see three influences of Re concerning (1), (2) and (4) for Re≲1000, except for (3). It seems difficult to directly explain the influence of Re together with the influence of l/h concerning (1), although the two influences of Re concerning (2) and (4) seems consistent in the context of fluid viscosity enhancement with decreasing Re. From a practical point of view, as the distinctive influence of l/h concerning (1) is important and useful, we will discuss again in Subsection 3.9, together with the effect of Re concerning (1).


Paper ID:153 10

Fig. 8 Streamwise distributions of the half width 2bu mean of mean-velocity profile at midspan (y/h = 0) for 0 < l/h <10

and Re = 1000.

(a) Re = 1000

(b) Re = 3000

Fig. 9 Streamwise distributions of temporal mean local flow rate Q at midspan (y/h = 0) for 0 < l/h < 10.

3.5 Turbulence Intensity Profiles


Paper ID:153 11

In this and following Subsections 3.5 – 3.8, we consider streamwise distributions of quantity as turbulence intensity urms instead of umean in Subsections 3.1 – 3.4, based on the measurements by a hot-wire anemometer at various values of x/h and z/h on the midspan plane (at y/h = 0).

Figure 10 shows typical examples of transverse profiles of urms in the z direction at y/h = 0. Specifically, the figures are for l/h = 5.0 and Re = 1000 at several values of x/h. Figure 11 is the same as Fig. 10, but axes are scaled and shifted. A coordinate z′ is shifted so thtat umean = (umean)max at z′= 0.

First, we see Fig. 10. At the nozzle exit (at x/h = 0), a sharp peak exists at z/h = -0.5, which corresponds to a shear layer on the potential-core boundary. At x/h = 5.0, another sharp peak can be observed at z/h = 0.5, where the peak value of urms/(umean)max is smaller than that at z/h = -0.5. As x/h further increases from 5.0 to about 10, the two peak values of urms/(umean)max increase. However, as x/h increases from about 10, both the two peak values decreases. On the other hand, the width in the z/h direction between the two peak values decreases. Furthermore, the width in the z/h direction between the two peaks of urms/(umean)max monotonically increase as x/h increases. In addition, we can confirm both (1) the profiles asymmetry and (2) the profile-center bias. The former (1) can be represented by the difference between the two peak values of urms/(umean)max, and vanishes at a large x/h as 100. Till the vanishing, the superiority in the value of urms/(umean)max usually switches between the two peaks. (This will be discussed in Figs. 12 and 13.) The latter (2) intensifies with increasing x/h, as well as the profile of umean/U0. Both the above features (1) and (2) are commonly seen for other values of l/h and Re.

Second, we see Fig. 11. We can confirm the similarity in transverse profiles of urms. Specifically, the figure is normalized using the local maximum turbulence intensity (urms)max for the ordinate, and using both a modified coordinate 𝑧3 instead of z and a local half width 2bu mean instead of h for the abscissa as well as Fig. 4. We can find that the present profiles agree well with one another at x/h ≥ 10.0, although the profile at x/h = 0 is obviously different from the others. The above results resemble the results [15] for Re = 6000. So, we expect that Re effect is negligible (see later for precise discussion).

Fig. 10 Turbulence-intensity profiles in the z direction at midspan (y/h = 0) for l/h = 5.0 and Re = 1000.


Paper ID:153 12

Fig. 11 Re-normalized turbulence-intensity profiles in the z direction at midspan (y/h = 0) for l/h = 5.0 and Re =

1000. A coordinate z′ is so shifted that 𝑢6789 = (𝑢6789)68; at z′=0.

3.6 Streamwise Variation of Jet Bias on Turbulence Intensity

Now, we consider the streamwise distributions of a jet bias (zu rms)max on turbulence-intensity on each side, namely, a local turbulence-intensity-profile center for several values of l/h. We define (zu

rms)max as the value of z where the transverse turbulence intensity profile attains the maximum (urms)max on the lip or no-lip side at each x. Figures 12 and 13 show the streamwise distributions of this jet-turbulence-peak bias (zu rms)max. Figure 12 represents for Re = 1000, and Fig. 13 presents for Re = 6000. In each figure, panels (a) and (b) are on the lip side and on the no-lip side for 0 < l/h < 20, respectively.

First, we see Fig. 12. The jet-turbulence-peak bias for l/h = 0 shows the streamwise growth of the bias suggested in Fig. 10. The jet-turbulence-peak bias is positive (namely, the jet-turbulence-peak bias is deflected upward), and is almost in proportion to x/h (namely, the jet-turbulence-peak axis is almost linear). If the jet direction is horizontal, this seems consistent being related with the streamwise growth of a turbulence-peak width (will be shown in Subsection 3.8). The jet-turbulence-peak biases for l/h = 2.0 and 3.3 show the above two features such as (1) the upward deflection and (2) the spatial linearity of the jet-turbulence-peak axis. When we compare the results with that for l/h = 0, we see a l/h effect. Specifically, the large l/h becomes, the more downward the deflection of the jet-turbulence-peak bias is. The jet-turbulence-peak biases for l/h = 4.0 and 5.0 show the above two features (1) and (2), as well. From a quantitative viewpoint, in contrast with the jet-turbulence-peak biases for l/h = 0, 2.0 and 3.3, the jet-turbulence-peak biases for l/h = 4.0 and 5.0 indicate such a different feature as the jet-turbulence-peak bias is deflected largely-upward. The jet-turbulence-peak biases for 6 < l/h < 20 show the above two features (1) and (2), as well. From a quantitative viewpoint, the jet-turbulence-peak biases for 6 < l/h < 20 are again deflected moderately-upward. Then, the jet-turbulence-peak biases for 6 < l/h < 20 almost coincide with that for l/h = 0. In summary, when we consider the above effect of l/h upon the jet-turbulence-peak bias, we can classify all the results into three categories; that is, (1) one for l/h = 2.0 and 3.3, (2) one for l/h = 4.0 and 5.0, and (3) the other for l/h = 0 and 6 < l/h < 20. The first, second and third ones are characterized by the deflections of slightly-upward, largely-upward, and moderately-upward jet-turbulence-peak biases, respectively. However, the jet-turbulence-peak axis is almost straight at any time in all the categories (the jet bias linearly increases with increasing x/h), being independent of l/h.

When we see Fig. 13 for Re = 6000, we can see the same features as those in Fig. 12 for Re = 1000 mentioned above. From a quantitative viewpoint, comparing Fig. 13 for Re = 6000 with Fig. 12 for Re = 1000, we can confirm an influence of Re. That is to say, the amplitude of the jet-


Paper ID:153 13

turbulence-peak bias tends to decrease with increasing Re (see Subsection 3.9 for quantitative discussion).

At this stage, we can understand that all the above results in Figs. 12 and 13 are thoroughly consistent with those in Fig. 7, we will see conclusion as both the streamwise growth of turbulence-peak width and half width on turbulence-intensity linearly increase with increasing x/h being independent of l/h and Re.

Finally, we consider the near downstream at x/h ≲ 6. Results are at x/h < 10 as well as Fig. 7. To conclude, we can again confirm that the present results coincide with Rajaratnam (1976) [16], as the potential core of a two-dimensional jet exists at x/h £ 6 for all the tested l/h and Re.

(a) lip side

(b) no-lip side

Fig. 12 Streamwise distributions of jet-turbulence-peak bias (zu rms)max at midspan (y/h = 0) for 0 < l/h < 20 and Re = 1000.


Paper ID:153 14

(a) lip side

(b) no-lip side

Fig. 13 Streamwise distributions of jet-turbulence-peak bias (zu rms)max at midspan (y/h = 0) for 0 < l/h < 10 and Re = 6000.

3.7 Streamwise Growths of Turbulence Peak Width and Half Width on Turbulence-Intensity

Now, we consider the streamwise distributions of the width between jet-turbulence peaks and of the half width 2bu rms of turbulence-intensity profile, at midspan (y/h = 0) for several values of l/h. Figure 14 shows the streamwise distributions of a jet-turbulence bias difference between (zurms)maxL and (zurms)maxNL at y/h = 0 (for 0 < l/h < 10 and Re = 1000). Figure 15 shows the streamwise distributions of a half width 2bu rms for 0 < l/h < 10 and Re = 1000. In each figure, both the axes are normalised by h. For reference, each figure also shows the theory Eq. (1) by Rajaratnam (1976) [16].

To conclude, we can see a linearly increasing manner of both the half widths. More specifically, at x/h ≲ 10, {(zu rms)max L - (zu rms)max NL}/h and 2bu rms/h are almost unity, being independent of l/h and Re. This is consistent with the existence of the potential core. At x/h ≳ 10, {(zu rms)max L - (zu

rms)max NL}/h and 2bu rms/h increase linearly with increasing x/h. Moreover, we can see that all the results almost collapse on empirical formulae, being independent of l/h and Re (the result for Re = 3000 and 6000, which are not shown, are almost the same as those for Re = 1000). The formula for the turbulence-peak-intensity width is given by 2bu rms = 3/4(0.20x). (3) And the formula for the turbulence-intensity width is given by


Paper ID:153 15

2bu rms = 7/4(0.20x). (4) The former and the latter are smaller and larger than the Rajaratnam’s theory, respectively. We can confirm that all the results are well agree with these formulae.

3.6 Streamwise Growth of Turbulence-Energy Integral

Now, we consider the streamwise distributions of a cross-streamwise integral Isq(u rms) of turbulence energy at midspan (y/h=0) for 0 < l/h < 10. Figure 16 shows the streamwise distributions of an integral Isq(u rms) of turbulence energy for 0 < l/h < 10. Panels (a) and (b) represent for Re = 1000 and 6000, respectively.

First, we look at Fig. 16(c). Turbulence-energy integral Isq(u rms) rapidly increases from zero, and approaches to a constant value of about 100, being independent of l/h. In far downstream as x/h ≳ 10, Isq(u rms) rapidly increases from zero, and approaches to a constant value of about 100.

Next, we see Fig. 16(a), which is more complicated than Fig. 16(b). More specifically, as well as Fig. 16(b), Isq(u rms) rapidly increase from zero in near downstream as x/h ≲ 10, and approaches to a constant value of about 100 in far downstream as x/h ≲ 10, and approaches to a constant value of about 100 in far downstream as x/h ≳ 10. To be exact, the approaching constant value for Re = 6000 in Fig. 16(b) is slightly larger than that of Re = 1000 in Fig. 16(a). This Re effect seems consistent, once we remind a result as the increasing temporal mean flow rate Q with increasing Re as shown in Fig. 9. We can confirm another Re effect in the downstream at x/h ≅ 10 – 100; namely, for a small Re as 1000, Isq(u rms) complicatedly fluctuates in the streamwise direction being independent of l/h. On the other hand, for a larger Re than 6000, Isq(u rms) rather monotonically increase and approaches to about 100, being independent of l/h and Re (see Fig. 16(b)).

(a) Re = 1000


Paper ID:153 16

(b) Re = 6000

Fig. 14 Streamwise distributions of a cross-streamwise integral Isq(u rms) of turbulence energy at midspan (y/h=0) for 0< l/h < 10.

3.7 Empirical Formula for Streamwise Variation of Jet Bias

Finally, we propose an empirical formula to predict the streamwise variation of jet bias, because the jet bias is most important, controllable and useful from a practical point of view. Figure 14 shows the gradient 𝐺AB of jet bias at midspan (y/h = 0) for 0 < l/h < 10. (zu mean)max/h = 𝐺AB x/h for 20 < x/h < 100, which is obtained from Fig. 7.

From Fig. 15, we see that the influence of l/h upon 𝐺AB is so complicated as the minimum, 𝐺AB appears for 2< l/h < 3 and as the maximum 𝐺AB appears for 4 < l/h < 5. For l/h > 6, 𝐺AB again approaches to zero, being independent of l/h and Re.

A remarkable influence of Re upon 𝐺AB only appears the values of the minimum and maximum 𝐺AB. More specifically, the minimum 𝐺AB tends to monotonically increase with increasing Re. On the other hand, the maximum 𝐺AB tends to monotonically decrease with increasing Re.

Now, considering the influences of l/h and Re, we propose an empirical formula to predict 𝐺AB. The formula is given by 𝐺AB = sech(l/h - φ) sin(l/h) (3) with a Reynolds-number factor φ = 4(1 - e-0.0015Re) (see Fig. 17). In Fig. 16, we can confirm good agreement with experiment, in spite of the complexity of the influences of l/h and Re. In order to further confirm the agreement of the proposed formula for 𝐺AB with experiments, Fig. 18 shows the comparison of empirical formula with experiment for 0 < l/h < 10 and Re = 3000. We can confirm good agreement again.


Paper ID:153 17

Fig. 15 Gradient 𝐺AB of jet bias at midspan (y/h = 0) for l/h = 0 – 10. (zu mean)max/h = 𝐺AB x/h for x/h = 20 – 100.

Fig. 16 Empirical formula to predict the gradient 𝐺AB of jet bias:

𝐺AB = sech(l/h - φ) sin(l/h) with a Reynolds number factor φ = 4(1 - e-0.0015Re).

Fig. 17 Reynolds number factor φ in the empirical formula for 𝐺AB.


Paper ID:153 18

Fig. 18 Comparison of empirical formula with experiment for 0 < l/h < 10

and Re = 3000.

4 Conclusion

We have achieved a sufficiently two-dimensional jet as the quasi-planar jet by Mi et al. (2005), at x/h < 100 for the reduced lip length 0< l/h < 20 and Re =1000 – 6000.

According to Hirata et al. (2010)[15], where experiments are conducted at l/h < 5.0 and Re = 6000, concerning the streamwise variations of jet bias and the streamwise growth of the half width, the case for l/h = 5.0 is considered to be exceptional. However, once we examine over wide ranges of l/h and Re in the present study, and have obtained the following conclusions.

All the re-normalized mean-velocity profiles at x/h ≥ 10.0 show good similarity with the theory by van der Hegge Zijnen (1958) [6] for a two-dimensional symmetrical nozzle: This corresponds to Hirata et al. (2010) [15] where the similarity is confirmed at x/h ≥ 6.7. As well, all the re-normalised turbulence-intensity profiles at x/h ≥ 10.0 show good similarity with one another.

At x/h ≥ 10, the jet is regarded to fully develop being independent of both l/h and Re, on the basis of (1) the similar profiles of mean-velocity and turbulence-intensity, (2) the unique streamwise-decay of the maximum mean-velocity (umean)max and the maximum turbulence-intensity (urms)max, (3) linearly-increasing manners of jet bases (zu mean)max and (zu rms)max in the streamwise direction, (4) linearly-increasing manners of half widths 2bu mean and 2bu rms. Flow rate as Q and turbulence-energy integral Isq(u rms). At x/h ≳ 10 influence of l/h is remarkable only upon jet biases like (zu mean)max and (zu rms)max, and not upon the maximum values, like (umean)max and (urms)max, jet widths like 2bu mean and 2bu rms. Flow rate Q is independent of l/h in both near and far downstream (at x/h = 10.0 – 100), but depends upon Re; namely, Q for Re = 1000 is smaller at each x/h than those for Re = 3000 and 6000 which almost coincide with Albertson et al. (1950). Turbulence-energy integral Isq(u rms) tends to be independent of l/h in a downstream as x/h ≳ 100. In the far downstream, Isq(u rms) becomes small with decreasing Re.

To be specific, all the streamwise decays of the maximum value (umean)max in the transverse mean-velocity profiles are close to the theory by Tollmien (see Rajaratnam (1976)[16]) at 10 < x/h < 100, being independent of l/h and Re. On the other hand, the maximum value (urms)max in the transverse turbulence turbulence-intensity profile grows in the streamwise direction at x/h ≲ 10, and decays in the streamwise direction at x/h ≳ 10, for all the cases. Thus, (urms)max always attains a peak value at x/h ≃ 10. As well as (umean)max, all the streamwise decays of (urms)max are identical with one another, being independent of l/h and Re. The influence of l/h upon the peak value of (urms)max is not negligible but complicated. And, the influence of Re upon the peak value is rather monotonical; namely, the peak value tends to decrease with increasing Re.


Paper ID:153 19

References

[1] Hirata, K., Funaki, J., Kubo, T., Hatanaka, Y., Matsushita, M. and Shobu, K. (2009) J. Fluid Science & Technology, JSME, vol. 4, No. 3, pp. 578 – 588.

[2] Funaki, J., Kobayashi, D., Shobu, K. and Hirata, K. (2009) J. Fluid Science & Technology, JSME, vol. 4, No. 3, pp. 673 – 686.

[3] Mall, B., and Gutmark, E., (2016) Proc. of AIAA 22th Aeroacoustics Conference, Lyon, France , pp. 3614 – 3633.

[4] Lasance C J M, Aarts R (2008) Synthetic jet cooling Part I: Overview of heat transfer and acoustic. In: Proceeding of the 24th, Annual IEEE Semiconductor Thermal Measurement and Management Symposium, pp 20–25.

[5] Förthmann, E. (1936) NACA Technical Memorandum, No. 789, pp. 1 – 18. [6] Van der Hegge Zijnen, B. G. (1958) Applied Scientific Research, Section A, vol. 7, pp. 256 –

276.

[7] Quinn, W. R. (1992) Experimental Thermal and Fluid Science, vol. 5, pp. 203 – 215. [8] Mi, J., Deo, R. C. and Nathan, G. J. (2005) Physics of Fluids, vol. 17, No. 6, pp. 068102.1 –

068102.4. [9] Deo, R.C. (2005) PhD Thesis, Adelaide Univ. [10] Krothapalli, A., Baganoff, D., and Karamcheti, K., 1981, Journal of Fluid Mechanics, Vol. 107,

No. 1, pp. 201 – 220.

[11] Zaman, K. B. M. Q., 1999, Journal of Fluid Mechanics, Vol. 383, pp. 197 – 228. [12] Horne, C., Jonkouski, G. and Karamcheti, K. (1981) Proc. of AIAA 7th Aeroacoustics

Conference, Maastricht, Netherlands, pp. 1 – 11. [13] Horne, C., Jonkouski, G. and Karamcheti, K. (1981) Proc. of AIAA 7th Aeroacoustics

Conference, Maastricht, Netherlands, pp. 1 – 11. [14] Kiwata, T., Kimura, S., Komatsu, N., Murata H. and Kim, Y. H. (2009) Journal of Fluid

Science and Technology, JSME, vol. 4, No. 2, pp. 268 – 278. [15] Hirata, K., Shobu, K., Murayama, T. and Funaki, T. (2010) Journal of Environment and

Engineering, JSME, vol. 5, No. 1, pp. 183 – 199. [16] Rajaratnam, N. (1976) Turbulent Jets, Elsevier, Amsterdam. [17] Albertson, M. L. et al. (1950) Trans. ASME vol. 115, pp. 639–664.

Documents

Control of planar jet using asymmetric nozzle lip