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On the sensitivity of planar jets Péter Tamás Nagy∗, György Paál Department of Hydrodynamic Systems, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary

a r t i c l e

i n f o

Article history: Available online 19 October 2016 Keywords: Compound matrix method Orr–Sommerfeld equation Spatial stability investigation of the planar jet Most sensitive part of the planar jet

a b s t r a c t Planar jets are widely researched ﬂows. Many aspects of their instability have been well-known for a long time. One of them is the observation that the jet is more sensitive to disturbances near the oriﬁce than elsewhere. This paper aims at giving an explanation to this hitherto unexplained phenomenon. To this end, linear stability investigations on various velocity proﬁles were carried out using the Orr–Sommerfeld (OS) equation. The velocity proﬁles were provided by analytical approximations and numerical computational ﬂuid dynamics (CFD) simulations. A special method called compound matrix method (CMM) was used for solving the OS equation to obtain suﬃciently accurate results. The adaptation of the method for symmetric jets was brieﬂy derived. The stability of various velocity proﬁles was compared based on the local spatial growth rate. The results of the comparison clearly show that velocity proﬁles near the oriﬁce are more unstable than the downstream ones. The outcomes also show that the velocity proﬁle close to the oriﬁce and the wall thickness of the nozzle have impact on the stability properties of the ﬂow there. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Propagation of instability waves along a planar jet plays a crucial role not only in wind instruments, e.g. organ pipes or ﬂutes, but also in active ﬂow control. For about one and a half century, authors have taken it for granted that the “most sensitive part of the jet” to external disturbances is at the immediate vicinity of the oriﬁce. In this paper, we are asking the question why that so, and attempt to give an answer. The research on excited jets began in the middle of the 19th century. One of the ﬁrst papers was published by Leconte (1858). He recognised that a gas-burner ﬂame could be excited by a musical tone, referring to Tyndall (1857) and Plateau (1857). 10 years later, Barrett (1867) observed the sensitivity of jets independently of the previous authors. Continuing his research, Tyndall (1867) showed that the sensitivity of ﬂames was independent of the combustion process. Tyndall (1875) was the ﬁrst researcher who summarised his observations on sensitive jets in his book on sound (Chapter VI.). This is the ﬁrst publication in which the vicinity of the oriﬁce was deﬁned as the most sensitive part. The next important milestone was laid by Ridout (1878) and Rayleigh (1888), who both pointed out the importance of the direction of the impinging sound. In addition, using standing waves, Rayleigh

∗

Corresponding author. Fax: +36 1 463 3091. E-mail addresses: [email protected] (P. Tamás Nagy), [email protected] (G. Paál). http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2016.09.017 0142-727X/© 2016 Elsevier Inc. All rights reserved.

discovered that the ﬂame/jet can be excited only by velocity ﬂuctuations (at antinodes) and not by pressure ﬂuctuations (at nodes). Rayleigh also described that the stopcock placed close to the oriﬁce had a strong effect on the sensitivity of the jet. The most likely, modern explanation is that the stopcock modiﬁes the velocity proﬁle inside the pipe. At the beginning of the 20th century, researchers started to investigate unignited jets independently of ﬂames. The next important paper belongs to Brown (1935), who carried out detailed measurements, made beautiful photographs, and summarised the contemporary knowledge about the sensitivity of jets (the effect of pressure and the direction of the sound). What Tyndall (1875) had also observed for circular jets, Brown validated for planar jets: the most sensitive part is the region close to the oriﬁce. “It is well known that the sensitive region in a gaseous stream is in the immediate neighbourhood of the oriﬁce. This can be demonstrated by lowering a shield near the jet and between it and the loudspeaker: no change in the vortex-formation can be observed until the shield cuts off the sound waves from the last millimetre or so at the oriﬁce.” Furthermore, he stated that “The fact that shielding the stream of gas from the sound has absolutely no effect, except when the shield covers the ﬁrst millimetre or so the nearest to the oriﬁce, shows that the vibrations of sound play no part in the subsequent development of the undulations produced here.” He recognised that the phenomenon of acoustical sensitivity in jets is not related to turbulence already existing in the stream when it issues from the oriﬁce. The velocity below which the jet remained stable was a decreasing function of the amplitude of the excitation.

P. Tamás Nagy, G. Paál / International Journal of Heat and Fluid Flow 62 (2016) 114–123

Fig. 1. A plane jet with a self-similar Bickley velocity proﬁle U = sech (y ). 2

According to his research, no clear maximum jet velocity exists for instability. Brown made further important observations. He varied the excitation frequency and he measured the minimum value below which the jet remained quiescent. At the same time, he was unable to determine the upper frequency limit for instability. The next essential step in the development of the theory of planar jets was the work of Bickley (1937). He derived an analytical formula (Bickley-proﬁle) for the velocity proﬁle generated by an inﬁnitely long line source, assuming self-similarity. This approximates the plane jet velocity proﬁle in the self-similar region very well (Fig. 1). Andrade (1941) made experiments with sensitive ﬂame and “water-into-water” jet with a circular oriﬁce and he concluded that the investigations of unignited, single phase planar jets are essential to understand the previously mentioned ﬂow conﬁgurations. He used the hydro-magnetic analogy for investigating the two-dimensional displacement of the jet and the distribution of vorticity. Furthermore, he thoroughly investigated the sensitivity of two different jets (a circular and a planar jet). His ﬁndings conﬁrmed the increased sensitivity of the jets near the oriﬁce. Savic (1941) was the ﬁrst who solved the Rayleigh equation for the plane jet and found a good agreement with the experimental results of Brown (1935) in terms of wavelength and wave velocity. Curle (1957) and Tatsumi and Kakutani (1958) later solved the Orr–Sommerfeld equation and published the neutral stability curve for the analytical Bickley proﬁle. They concluded that the critical Reynolds number (Re), below which the ﬂow is linearly stable, was 4.0 at a non-dimensional wavenumber 0.2; the lower branch of the neutral stability curve tended rapidly to zero (as can be assumed for every free shear ﬂow (Boiko et al., 2012)). Moreover, if the Reynolds number is larger than around 200, the neutral stability curve does not really change any more with Re. This means that here the linear stability of the Bickley jet becomes independent of the Reynolds number and the simpler Rayleigh equation can be used instead of the Orr–Sommerfeld equation. Sato (1960) also solved the Orr–Sommerfeld equation but he used the mean-velocity proﬁles from his measurements instead of the theoretical Bickley-proﬁle. Mattingly and Criminale (1971) compared the varicose and sinuous disturbances in the spatial stability of Bickley proﬁle using the inviscid Rayleigh equation. They concluded that the “symmetric disturbances are the more highly unstable as determined by their spatial ampliﬁcation factors”. They also investigated the Reynolds stress distribution and found that the energy is transferred from the main motion to the disturbances along mean ﬂow streamlines on either side of the jet centerline but not on the centerline because there the stress is zero. Later many researchers attempted to reduce the assumptions of the Orr–Sommerfeld equation. Some of these will be mentioned, without attempting to give an exhaustive review. Varapaev et al. (1973) derived a modiﬁed Orr–Sommerfeld equation, which considered the transversal velocity component with the help of the stream function. According to his results, the critical Reynolds

115

number (deﬁned as Curle, Tatsumi and Kakutani) is around 14–15. A similar attempt was made by Bajaj and Garg (1977), who also took into account the transversal mean velocity. Their modiﬁed equation is more usable because instead of the stream function they used the mean transversal velocity proﬁle in the equation, which can be easily exported from numerical simulations. Garg (1981) continued the research and developed a weakly nonparallel model that took into consideration the effects of the transversal velocity component and the streamwise variations of the basic ﬂow, the disturbance amplitude, the wavenumber, and the spatial growth rate. In this case, the growth rate depends not only on the streamwise and transverse co-ordinates but also on the particular ﬂow variable. Finally, he presented his results in terms of the growth rate of the kinetic energy at x = 0 and found the critical Reynolds number at the oriﬁce to be 21.6. His ﬁnding agreed very well with the experimental observation of Sato and Sakao (1964), according to which Bickley jet is stable to inﬁnitesimal disturbances below a Reynolds number of 20. However, the Bickley-proﬁle becomes a good approximation only 6–8 diameters from the oriﬁce. Tam (1995) omitted the parallel ﬂow assumption with the help of curvilinear co-ordinates. He refrained from introducing the usual non-dimensional parameter, the Reynolds number because there is neither characteristic length nor velocity in this problem. He contended that “the ﬂow is unstable, regardless of Reynolds numbers, however deﬁned.” Nolle (1998) solved the Rayleigh equation numerically for the Bickley jet and validated it by measurements. He also suggested analytic expressions to describe the transition of the mean ﬂow of the jet from the top-hat to the Bickley proﬁle. These proﬁles are used in this paper. Jo and Kim (2002) studied the stability of the parallel and non-parallel jet in a wide range of Reynolds numbers. They solved the vortex transport equation for the mean ﬂow and its linearised form for the stability investigations. They identiﬁed a new nondimensionalisation scheme including the streamwise distance and the Reynolds number. They also carried out a global stability analysis and obtained good agreement with the local one. Although in our paper only the parallel ﬂow approximation was used with a different calculation of the basic ﬂow ﬁeld and different boundary conditions, new results were reached, for example related to the varicose mode or the relation of analytical and numerical proﬁles. In the same year, Atassi and Lueptow (2002) developed a nonlinear “long wave” model. The model describes the exponential and linear growth of disturbances. The paper deals with asymmetric mean velocity proﬁles as well, approximated by a piecewise linear function. Since the model is inviscid, it describes the stability of the ﬂow only for a “top-hat” jet at high Reynolds numbers. Bechert (1988); Bechert and Stahl (1988), and Kerschen (1997) analytically investigated the receptivity of the ﬂow in the vicinity of the nozzle but in these papers the jet was modelled as a shear layer, which makes it diﬃcult to compare the results near the oriﬁce to those far away where the Bickley-proﬁle has already developed. In our paper, the stability investigations were carried out while considering the complete jet proﬁle to inﬁnity in transversal direction. This method enables us to compare the stability properties of the ﬂow near the oriﬁce and far from the oriﬁce. At the same time, Hsiao and Huang (1990) validated the theory of the evolution of coherent structures in planar jets by measurements. According to this idea, these structures develop from an initial instability wave. After the amplitude of this wave saturates, the ﬁrst subharmonic starts growing and becomes the most ampliﬁed instability wave. These experiments suggest that the properties of the initial instability wave have a large impact on the development of the unstable structures in the ﬂow. Nowadays, numerous studies focus on the modiﬁcation of the planar jet by actuators placed near the oriﬁce. Recently

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Ben Chiekh et al. (2011) determined the desired frequency to generate antisymmetric modes in a planar jet using synthetic jets. On the other hand, how the geometry conditions inﬂuence the development of the preferred modes in unexcited cases is still unrevealed. Another interesting recent paper is that of Oberleithner et al. (2014) which considers cylindrical jets, yet it contains several relevant ﬁndings. They perform PIV experiments using a variable amplitude excitation. Their weakly nonparallel stability model works very well except very near to the nozzle and far downstream. They attribute the former discrepancy to the interaction of the stability wave with the nozzle, while the latter one is attributed to the reversal of energy ﬂow direction between the stability wave and the mean ﬂow. They found that there is practically no turbulence in the wave growth region; turbulence appears only in the decay region. The main question tackled in this paper is why the jet is more sensitive to disturbances near the nozzle exit than elsewhere. The basic equation to describe the stability of parallel, incompressible ﬂows is the Orr–Sommerfeld (OS) equation which was used in the current work. A special method called compound matrix method (CMM) (Ng and Reid, 1979) was used for solving the equation in order to provide accurate results. Another advantage of this method is that the boundary conditions at inﬁnity can be simply prescribed. The adaptation of the method for unlimited symmetric and antisymmetric ﬂows is described brieﬂy in Section 2. Two sets of velocity proﬁles were investigated: proﬁles provided by analytical expressions and by CFD-simulations. The latter ones describe the ﬂow at the oriﬁce more precisely. They are discussed in more detail in Section 3. In Section 4, the solution procedure is explained and the results of linear stability investigations are presented in Section 5. Finally, concluding remarks are made in Section 6.

prescribed at two locations. We have to ﬁnd the α − ω parameter pairs which fulﬁl the boundary conditions. Another problem is that this fourth order differential equation usually becomes stiff. Let us investigate the solution in the far ﬁeld, at y → ∞, where U (y ) = U (y ) = 0 holds for jets. Here, the differential equation (Eq. (1)) can be simpliﬁed to

2. The OS equation and the compound matrix method

y→∞

2.1. The OS equation

These conditions can be fulﬁlled only if a2 (y ) = a4 (y ) ≡ 0 because (λ2, 4 ) > 0 (where (.) means the real part of the expression). Let us introduce six new functions

φ (iv) − 2α 2 φ + α 4 φ = iRe (αU − ω )(φ − α 2 φ ) − αU φ , (1)

where φ (y) can be the amplitude of the dimensionless transversal perturbation velocity (v(x, y, t ) = φ (y ) exp[i(α x − ωt )]) or the amplitude of the stream function. In this paper, the former one was used. U(y) is the non-dimensional velocity proﬁle in the equation, α is the non-dimensional wavenumber, and ω is the nondimensional angular frequency. (The speciﬁc quantities for nondimensionalisation are given in Section 3.2.) Here means differentiation with respect to y. If we take the limit of Eq. (1) at Re → ∞, then the inviscid case can be obtained. This equation is known as the Rayleigh-equation,

(2)

and it was solved for Bickley proﬁle by Nolle (1998). This equation is not used here but a comparison was made between Nolle’s results and our large Reynolds number case. 2.2. The compound matrix method Usually the solution of the OS equation leads to a boundary value and eigenvalue problem because boundary conditions are

(3)

This equation has four different characteristic roots. λ1,2 = ∓α

and λ3,4 = ∓Q where Q = α 2 − iReω. (Sometimes U (y → ∞ ) = Uin f = 0 because of the entrainment of the ﬂow. In these cases,

Q = α 2 + iRe(αUin f − ω ).) For large Reynolds numbers |λ3, 4 | |λ1, 2 |, which makes the problem stiff. One possibility to solve the problem is the compound matrix method (CMM), which is the best one according to Sengupta (2012) for hydrodynamic problems. It provides accurate results and at the same time it can be easily implemented. This method was developed for the OS equation by Ng and Reid (1979). Later, Sengupta (2012) adapted this method to the Blasius proﬁle where some of the boundary conditions are prescribed at inﬁnity. In this paper, his idea is followed with the adaptation of the method to symmetric plane jets. The general solution of Eq. (1) can be written in the following form:

φ = a 1 φ1 + a 2 φ2 + a 3 φ3 + a 4 φ4 ,

(4)

where φ j (y) are the fundamental solutions and

φ j (y ) ∝ eλ j y as y → ∞.

(5)

In the next step, the boundary conditions have to be prescribed at y → ∞. The ﬂuctuating velocities are assumed to be zero far from the jet,

lim

The OS equation is the fundamental equation to investigate the linear stability of parallel, incompressible ﬂows. It is derived from the continuity equation and the Navier–Stokes equations. The parallel ﬂow assumption means that the velocity distribution does not change in the ﬂow direction. This assumption is globally valid if the ﬂow has solid, parallel boundaries (developed channel or pipe ﬂow with parallel walls). In some other cases, this assumption can be accepted as approximately valid if the ﬂow is investigated locally, as is done in this paper. The OS equation is given by

(αU − ω )(φ − α 2 φ ) − αU φ = 0

φ (iv) − 2α 2 φ + α 4 φ = iRe ω (φ − α 2 φ ).

φ = ylim φ = 0. →∞

(6)

η1 =φ1 φ3 − φ1 φ3 ,

(7)

η2 =φ1 φ3 − φ1 φ3 ,

(8)

η3 =φ1 φ3 − φ1 φ3 ,

(9)

η4 =φ1 φ3 − φ1 φ3 ,

(10)

η5 =φ1 φ3 − φ1 φ3 ,

(11)

η6 =φ1 φ3 − φ1 φ3 ,

(12)

and take the derivatives with respect to y and substitute them into the OS equation (Eq. (1)), which leads to the following differential equation system (column vectors are denoted by and matrices by ): =

⎡

0 ⎢0 ⎢0 η = A=η = ⎢ ⎢0 ⎣ b2 0

1 0 b1 0 0 b2

0 1 0 0 0 0

0 1 0 0 b1 0

0 0 1 1 0 0

⎤

0 0⎥ 0⎥ ⎥η, 0⎥ ⎦ 1 0

(13)

where b1 := 2α 2 + iRe(αU − ω ) and b2 := α 4 + α 2 iRe(αU − ω ) + iα ReU . In this case, all the new functions have the same exponential growth rate for y → ∞,

η1,∞ ∝ (−Q + α )e−(α+Q )y ,

(14)

P. Tamás Nagy, G. Paál / International Journal of Heat and Fluid Flow 62 (2016) 114–123

η2 , ∞ ∝ ( Q 2 − α 2 ) e − ( α + Q ) y , η3,∞ ∝ (−Q + α )e 3

3

− ( α +Q )y

(15) ,

(16)

η4,∞ ∝ (−α Q 2 + α 2 Q )e−(α+Q )y ,

(17)

η5 , ∞ ∝ ( α Q 3 − α 3 Q ) e − ( α + Q ) y ,

(18)

η6,∞ ∝ (−α 2 Q 3 + α 3 Q 2 )e−(α+Q )y .

(19)

η −Q + α

e ( α +Q )y .

(20)

Here, we get the limit at y → ∞ for the normalised function,

⎡

η

∞

⎤

1 ⎢ − (α + Q ) ⎥ ⎢α 2 + α Q + Q 2 ⎥ ⎥, := ⎢ ⎢ ⎥ αQ ⎣ ⎦ −α Q (α + Q ) 2 2 α Q

(21)

=

=

lim

φ = y→−∞ lim φ = 0.

(22)

(23)

The other choice is when the modes are split into symmetric (sinuous, denoted by s) and antisymmetric (varicose, denoted by a) modes. Here, symmetry means that the perturbation velocity is symmetric to the centreline of the jet,

φs (0 ) =

φs (0 ) = 0.

(24)

(In the literature, this condition is sometimes called antisymmetric because the displacement of the jet is antisymmetric so that the jet visually appears antisymmetric.) The antisymmetricity is interpreted similarly,

φa (0 ) = φa (0 ) = 0,

(29)

This expression is known as characteristic or dispersion relation for the symmetrical disturbances. The same calculation can be made for the antisymmetric modes. In this case, the dispersion relation can be obtained as

(30)

The original problem is simpliﬁed to an initial value problem, starting at inﬁnity. Eq. (22) is to be solved with the initial condition Eq. (21) and the parameter pairs which fulﬁl Eq. (29) or (30). These parameters are unknown; they can be determined by a “shooting” method introduced brieﬂy in Section 4. When this procedure is completed, the eigenvalue problem is solved and the eigenfunctions can be calculated. The original φ function can be calculated in four different ways (Ng and Reid, 1979) from the compound variables; we chose

η1 φ − η2 φ + η4 φ = 0.

(31)

3.1. Analytical velocity proﬁles

If the differential equation system (22) with the initial conditions Eq. (21) is solved, then the boundary conditions are automatically fulﬁlled in the far ﬁeld. Since a fourth order differential equation needs four boundary conditions, two further boundary conditions are still necessary. For jets, two options are available. The ﬁrst one is that the perturbation velocities are assumed to be zero in the far ﬁeld on the other side of the jet, y→−∞

5 (0 ) = 0. Ds ( α , ω ) = η

3. Analytical and numerical velocity proﬁles

which are the initial conditions there. The normalisation of the functions modiﬁes the matrix in the differential Eq. (13) into

η = [A + (α + Q ) I ] η.

(28)

This is identical to the ﬁfth component of η being 0 at y = 0,

2 (0 ) = 0. Da ( α , ω ) = η

Let us normalise the η functions with the limit of the ﬁrst variable as y → ∞ :

η :=

(φ1 φ3 − φ1 φ3 )|y=0 = 0.

117

(25)

meaning that the jet appearance is symmetric. Both choices can be implemented but the latter one is computationally cheaper and easier to interpret, therefore this was used in the current work. Now, we have to transform these boundary conditions for the compound variables. First, let us consider the symmetric case. Applying the boundary conditions (24) on Eq. (4) leads to

a1 φ1 (0 ) + a3 φ3 (0 ) = 0,

(26)

a1 φ1 (0 ) + a3 φ3 (0 ) = 0.

(27)

This new linear algebraic equation system has a non-trivial solution if and only if the determinant of the associated matrix of the system of equations is zero,

Schlichting (1933) and Bickley (1937) derived a velocity proﬁle for a plane jet with the assumption of a constant, line momentum source and self-similar ﬂow. This velocity proﬁle is a good approximation for jets far from the oriﬁce (Fig. 1). According to Nolle’s experiments (Nolle, 1998), this proﬁle can be observed from roughly 8 times the oriﬁce size downstream. The Bickley proﬁle in nondimensional form is given as

U = sech (y ). 2

(32)

Two special velocity proﬁles can be distinguished at the oriﬁce. A parabolic proﬁle develops if the nozzle is a long, parallel channel. The other one is the “top-hat” proﬁle, in which the boundary layer in the nozzle and the shear layer in the jet are very thin. It develops in the case of short, convergent nozzles. An analytical approximation for the transitional velocity proﬁles is available only in the case of top-hat outﬂow in (Nolle, 1998),

U = sech (yn ). 2

(33)

The velocity proﬁles, which are closer to the oriﬁce and are thus steeper, are approximated by larger “n” parameters. In this paper, the n = 1, 2, 3 cases were investigated (Fig. 2), where n = 1 is identical with the original Bickley proﬁle. The non-dimensional ﬂow rates for various proﬁles were calculated for comparison. The result can be seen in Table 1. The maximum difference compared to the Bickley proﬁle was less than 5%. 3.2. Numerical velocity proﬁles It was necessary to carry out CFD simulations because in the case of the parabolic velocity proﬁle there are no analytical formulae for the transition proﬁles between the parabolic and the Bickley proﬁles. Furthermore, in the case of the top-hat proﬁle, the parameter n can take only whole numbers and the transitional proﬁles between them cannot be described analytically by this expression. The CFD simulations were carried out on a ﬁne, structured mesh by ANSYS CFX. The size of the oriﬁce was δ = 1 mm, the cell-size was 0.05 mm in the core domain which was 24 mm long (Fig. 3). Another domain was deﬁned around the core one with continuously increasing cell sizes in order to reduce the effect of boundaries. On the upstream side of the simulation domain, two

118

P. Tamás Nagy, G. Paál / International Journal of Heat and Fluid Flow 62 (2016) 114–123 Table 1 The comparison of the dimensionless ﬂow rates between analytical proﬁles. Proﬁle

Dimensionless ﬂowrate [-]

The difference from the Bickley-prof. [%]

Bickley sech2 (y2 ) sech2 (y3 ) “Top-hat”

2 1.9056 1.9138 2

0 −4.7 −4.3 0

entrainment process has already begun in the vicinity of the oriﬁce. At other boundaries “opening” was prescribed, except at the nozzle where the pre-deﬁned top-hat or parabolic velocity proﬁles were implemented. The simulation was carried out at Reglob = 100. Reglob is the global Reynolds number in the ﬂow deﬁned as

Reglob = Fig. 2. The analytical approximations of the velocity proﬁle close to the oriﬁce 2 (U = sech (yn )).

different boundary conditions were deﬁned. One is a no-slip wall (prescribed zero velocity) which represents a thick-walled nozzle, blocking streamwise entrainment around the nozzle. The other one is an “opening” (prescribed zero pressure allowing in- and outﬂow) which describes an inﬁnitely thin-walled nozzle in which the

δ UMean , νair

(34)

where νair = 1.545 · 10−5 m2 /s is the kinematic viscosity of air at 25 °C and UMean = 1.545 m/s is the mean velocity at the oriﬁce which can be calculated based on the previous parameters. The global Reynolds number describes the whole ﬂow. In contrast, the local Reynolds number belongs to a given cross-section and describes the local velocity proﬁle (similarly to Jo and Kim, 2002). To be consistent with the analytical proﬁles, the speciﬁc quantities for the local Reynolds number are deﬁned as

Fig. 3. The simulation domain and boundary conditions in the CFD calculations.

P. Tamás Nagy, G. Paál / International Journal of Heat and Fluid Flow 62 (2016) 114–123

119

Fig. 4. The (a) the difference to Bickley proﬁle (ε ), (b) the length (Lˆ), (c) the maximum velocity Uˆmax , (d) the local Reynolds number along the mean ﬂow direction (xˆ) at various outﬂow conditions.

Re(xˆ) =

Lˆ (xˆ) UˆMax (xˆ)

νair

,

(35)

where as UˆMax (xˆ) is maximum velocity in a certain cross-section, Lˆ (xˆ) is the width (y) where the velocity is equal to 0.42 UˆMax (xˆ), coming from the evaluation of Bickley proﬁle at y = 1, since 2 sech (1 ) ≈ 0.42. With these deﬁnitions, the previously deﬁned analytical and numerical proﬁles are non-dimensionalised in the same way. The further non-dimensional quantities were calculated as

α = αˆ Lˆ, ω = ωˆ

(36) Lˆ

UˆMax

,

(37)

ˆ denotes dimensional quantities. where The velocity ﬁeld was exported to Matlab for further investigations. First, the non-dimensionalisation procedure was carried out. The speciﬁc length and velocity values were determined for each

cross-section and the local Reynolds numbers were calculated. It can be seen in Fig. 4 that ﬂows with different outﬂows at the exit (parabolic, “top-hat”) have different velocity and length scales even far from the oriﬁce. The maximum velocity is always a bit higher in the parabolic case (b), while the width of the jet is always a bit larger (c) in the top-hat case. These two effects almost cancel each other during the calculation process of the local Reynolds number: the curves run close to each other in every case. The continuous increase of the local Reynolds number (d) is also observable, which is a known phenomenon also known (White, 2006). The effect of the wall on the development of Bickley proﬁle can also be investigated. Let us deﬁne the parameter ε which describes the non-dimensional difference of the velocity proﬁle to the selfsimilar Bickley proﬁle as

1 ε (xˆ) = 2Lˆ (xˆ)Uˆmax (xˆ)

∞

−∞

2 Uˆ (xˆ, y ) − Uˆmax (xˆ) sech (y/Lˆ (xˆ) )

2

dy.

(38)

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ε is plotted in Fig. 4(a). It can be seen that when the wall boundary condition was used, the self similar Bickley proﬁle did not develop because of the insuﬃcient entrainment. In the case of “opening” boundary condition (representing a thin-walled nozzle) after x 6 − 8 the Bickley proﬁle will evolve, as noted by Nolle (1998) based on his experiments. This process is a bit faster in the case of parabolic outﬂow. Here, the advantage of the usage of nondimensional proﬁles is highlighted. If x > 8, the stability results for the non-dimensional Bickley proﬁle can be used after redimensionalisation. If the dimensional form of the stability equations had been used, the whole solution procedure of the OS equation should have been repeated instead of a simple redimensionalisation. 4. The solution method The following parameters should be determined for the OrrSommerfeld equation. The ﬁrst one is the Reynolds number that could be a speciﬁed value (in a real ﬂow, numerical velocity proﬁles) or an arbitrary value (in general, analytical velocity proﬁles). The task is to determine the remaining two parameters: the wave number (α ) and circular frequency (ω), which fulﬁl the condition in Eq. (29) or (30), depending on the symmetry of the mode. In other words, the roots of D(Re, U, ω, α ) = 0 have to be determined. In the general, case both α and ω can be complex numbers but this means innumerable solutions. Here, the spatial stability analysis was carried out, the amplitude of the wave grows only in space, α = αr + αi is a complex number, while ω is real. The spatial growth rate is μs = −αi . It is possible that the amplitude of the disturbance wave grows both in space and time but comparisons with experiments show a very good agreement with the spatial stability analysis in the case of planar jets (Criminale et al., 2003; Nolle, 1998). The initial condition Eq. (21) for the differential equation system Eq. (22) is prescribed at y → ∞. A suﬃciently large y value has to be chosen such that the boundary conditions are approximately fulﬁlled and Eq. (5) is true. In the case of analytical pro2 ﬁles, y = 12 was chosen where sech (12 ) ≈ 10−10 . In the case of numerical proﬁles, the velocity in the far ﬁeld is never zero because of numerical errors and the entrainment. In these cases, Uinf was taken into account during the calculation of Q as described in Section 2.1. The differential Eq. (22) was solved by Runge–Kutta–Cash–Karp method with adaptive step size technique where the tolerated absolute and relative errors were set to 10−10 . The critical point during the solution procedure is to determine the ﬁrst eigenvalue pair. Here, we followed the idea of Sengupta (1992). In the ﬁrst step the Reynolds number was ﬁxed at 100 in the case of analytical proﬁles and to the calculated Reynolds number in the case of numerical proﬁles. ω was also ﬁxed at 0.1 in both cases. A ﬁne grid was made in the α -plane α r ∈ [0, 2] and αi ∈ [−2, 1] and the differential equation was solved for each α parameter on the grid. The investigation on a larger area is not necessary because a lower α i means very rapid ampliﬁcation which is not observed experimentally. At the same time, large positive values mean rapidly decaying waves that are not interesting from our point of view. After that the (D ) = 0 and (D ) = 0 (meaning the real and imaginary part of the expression), contour lines were plotted and the intersection point of these lines was detected. These points are the eigenvalues corresponding to different modes and they are sorted according to ascending α i values because lower values mean more rapid growth. It can be assumed that if the Re and ω parameters are changed slightly, the corresponding α values change only slightly. After one point was determined, the next point was calculated by slightly modifying the parameters and using the Newton–Raphson method for which the initial guess is the previous solution. With this technique, all the eigenvalue pairs can

Fig. 5. The growth rate of the amplitude of the perturbation velocity in the case of analytic Bickley proﬁle for various Reynolds numbers and in the inviscid case (Rayleigh). Table 2 The critical Reynolds numbers of analytic velocity proﬁles. Mode

n=1

n=2

n=3

Sym Antisym

4.0 88.2

4.2 26.6

4.2 22.1

be determined for one mode, starting from one known intersection point in the α plane. In this paper, only the ﬁrst mode was investigated. 5. Results and discussion 5.1. Results for the analytical proﬁles Our results for the Bickley proﬁle are compared to those of Nolle (1998) to verify our calculations for a high Reynolds number. The comparison at various Reynolds numbers can be seen in Fig. 5. At Re = 10 0 0, our results were almost identical to those of Nolle, which means that the stability properties of the Bickley proﬁle above this number are independent of the Reynolds number. In Curle (1957) and Tatsumi and Kakutani (1958), the threshold Reynolds number above which viscosity does not inﬂuence stability properties was suggested to be 200 (Fig. 5). If we reduce the Reynolds number, the non-dimensional growth rate also decreases, except at low frequencies. The critical Reynolds numbers below which the growth rate is always negative (the perturbation decays) and the ﬂow is linearly stable were calculated and are presented in Table 2. The critical Reynolds number of the Bickley proﬁle is Recrit ≈ 4.0. This number was also predicted by Curle (1957) and Tatsumi and Kakutani (1958) and was also validated also by Bajaj and Garg (1977). This number is almost the same for all symmetrical modes at all n parameters, as shown in Fig. 6. There, the neutral stability curves belonging to the zero growth rate of perturbations were plotted in various cases. Of course, the results should be interpreted carefully at such low Reynolds numbers because the validity of the parallel mean ﬂow assumption is uncertain. The next observation is that the lower branch of the neutral curves tends rapidly to the ω = 0 axis in the case of all the symmetrical modes as also predicted in Curle (1957) and Tatsumi and Kakutani (1958) for the Bickley proﬁle. (In these cases, the lower branch of the neutral curve is so close to the axis that it cannot be seen in the ﬁgure.) This is actually valid for every free shear ﬂow (unlimited ﬂows) (Boiko et al., 2012). At the same time, the antisymmetric disturbances grow only at higher Reynolds numbers and Recrit ≈ 88.2 is quite large in the case of Bickley proﬁle compared to the other proﬁles. The neutral curves tend much slower to the ω = 0 axis in the case of antisymmetric disturbances.

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Fig. 6. The neutral curves for symmetric and antisymmetric modes in the case of 2 analytic proﬁles (U (y ) = sech (yn )).

Fig. 7. The growth rate of amplitude of perturbation velocity at Re = 100 in the 2 case of analytic proﬁles (U (y ) = sech (yn )).

At a given Reynolds number, the growth rates are much higher for larger n parameters, except at low frequencies, as plotted in Fig. 7. These results show us that the growth of the perturbation velocity wave is more rapid close to the oriﬁce (for large n parameters). Another very interesting result is revealed when the symmetric and antisymmetric disturbances are compared to each other. It has been known for a while that at n = 1 the growth rate of the symmetric disturbances are much larger than that of the antisymmetric modes (Mattingly and Criminale, 1971). In the ﬁgure, the growth rate of the antisymmetric disturbances at Re = 100 is almost zero or negative, while the growth rate of symmetric disturbances is positive below a given angular frequency. (Of course, at higher Reynolds numbers the growth rate of antisymmetric disturbances becomes positive in a certain frequency range but the growth rate of the symmetric modes always remains larger. This is the reason why the symmetric disturbances are usually observable in experiments, except if antisymmetric excitation is applied, as stated by Criminale et al., 2003.) A surprising and new result can be observed at larger n parameters. For n = 2, the growth rates of disturbances is quite close to each other at higher angular frequencies (ω > 0.5), furthermore at higher Reynolds number Re > 200, there is a narrow range ω ≈ 0.6 − 0.7 where the antisymmetric modes have larger growth rates (not plotted here). When the parameter “n” is further increased to 3, the angular frequency domain where the growth rate of the antisymmetric modes is larger than that of the symmetrical ones becomes wider. Here, the growth rate of the antisymmetric mode is deﬁnitely larger than that of the symmet-

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Fig. 8. The non-dimensional growth rates of the amplitude of perturbation velocity (symmetric) for numerical velocity proﬁles at various distances from the oriﬁce. The boundary conditions of the base ﬂow at the upstream side were: wall and a prescribed velocity proﬁle (a) parabolic, (b) “top-hat”.

ric one. Moreover, the maximum growth rate of the antisymmetric disturbances is higher than that of the symmetric mode. The largest growth rate will overwhelm the others and will dominate in natural modes when the jet is not excited. This means that antisymmetric modes can and will dominate in the vicinity of the nozzle in natural modes. The same result was already experimentally observed by Thomas and Goldschmidt (1985), who measured the frequency of symmetric and antisymmetric modes with the help of the autocorrelation technique at a larger Reynolds number (Re = 60 0 0). (There, the symmetry was deﬁned in the opposite way, based on the displacement of the jet.) Finally, the non-dimensional angular frequencies corresponding to the maximum growth rate are higher at larger “n” parameters, suggesting that the dominating frequencies of disturbances in experiments without excitation are higher in the case of steeper basic velocity proﬁles (e.g. closer to the oriﬁce). 5.2. Results for the numerical proﬁles The results were similar in the case of numerical proﬁles. In Fig. 8, the non-dimensional growth rates are plotted at various distances, with wall boundary conditions (except noted otherwise). (The justiﬁcation for the usage of the wall boundary condition is that this reproduces the experimental observation of Thomas and Goldschmidt, 1985 about the existence of antisymmetric modes in the vicinity of the nozzle exit better.) The growth rates were higher for velocity proﬁles which are closer to the oriﬁce in both cases (parabolic, “top-hat”), except at low frequencies. The same observation was made when the “opening” boundary condition was applied. It can be concluded that the growth of the disturbances is larger close to the oriﬁce independently of the type of the nozzle or the thickness of the nozzle wall (“opening” or wall boundary condition). At the same time, it is clearly visible that the growth rate is much larger in the case of the top-hat proﬁle. The maximum growth rate at xˆ = 0.1 mm is μmax = 0.622 (ω = 0.738) in this case, while it is only μmax = 0.395 (ω = 0.330) in the case of the parabolic proﬁle. This difference vanishes far from the oriﬁce as the Bickley proﬁle evolves; it is less than 5% if xˆ > 4 mm. It would also be interesting to investigate which outﬂow conditions cause a larger growth rate, which also translates to a more “unstable” ﬂow ﬁeld and how the antisymmetric disturbances behave in the vicinity of the oriﬁce. The growth rate of various outﬂow conditions and different modes (symmetric, antisymmetric) is plotted in Fig. 9(a) close to the oriﬁce (xˆ = 0.1 mm). The growth

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Fig. 9. The non-dimensional growth rates of the perturbation velocity for numerical velocity proﬁles at xˆ = 0.1 mm. The boundary conditions of the base ﬂow at the upstream side were: wall (a) or “opening” (b) and two inlet velocity proﬁles (parabolic or “top-hat”).

Fig. 11. The maximum amplitude of the perturbation (a) and the corresponding frequency (b) along the mean ﬂow direction for various outﬂow conditions. The boundary conditions of the base ﬂow at the upstream side were: wall and a prescribed velocity proﬁle (parabolic or “top-hat”).

differences between various cross-sections. Therefore, the higher sensitivity at the exit is caused by two effects. The ﬁrst one is that the velocity proﬁles in the vicinity of oriﬁce induce larger growth rates of disturbances and the other one is the continuously increasing local length scale. Finally, the amplitude of the disturbances along the streamwise coordinate was calculated for various circular frequencies in the range of interest. The amplitude of the disturbances can easily be calculated as

A(xˆ, ω ˆ ) = A0 exp

Fig. 10. The dimensional growth rates of the amplitude of perturbation velocity (symmetric mode) in the case of numerical velocity proﬁles at various distances from the oriﬁce. The boundary conditions of the base ﬂow at the upstream side were: wall and parabolic velocity proﬁle.

−αi (ξˆ, ω ˆ )dξˆ ,

(39)

where A0 is the initial (xˆ = 0) amplitude of the disturbances that is chosen to be 1. The maximum amplitude along the angular frequencies was also determined as

Amax (xˆ) = max A(xˆ, ω ˆ ), ωˆ

rate of antisymmetric disturbances is larger than that of symmetric modes in the case of the top-hat proﬁle, as previously noted for analytical proﬁles. In Fig. 9 and to a lesser extent also in Fig. 7, it can be noticed that at low frequencies the growth rate of the antisymmetric modes is practically independent of the outﬂow conditions. When the “opening” boundary condition is applied, the growth rate of the antisymmetric mode is always very close to the growth rate of the symmetrical disturbances in the vicinity of the oriﬁce but always remains a bit smaller (Fig. 9(b)). This indicates that the thickness of the nozzle wall inﬂuences whether the antisymmetric mode is observable or not in a natural state of the jet. At the same time, in the case of parabolic outﬂow condition the growth rate of antisymmetric disturbances remains much below that of the symmetric modes. It can be concluded that antisymmetric disturbances will not appear in a natural mode of a jet with a parabolic outﬂow proﬁle. It seems that the conditions for the observability of the antisymmetric modes are the thick nozzle walls and the thin initial shear layers in the jet. The non-dimensional growth rates show us that the velocity proﬁles close to the oriﬁce are more “unstable” than far from there. During the non-dimensionalisation process, the length scale varied along the streamwise coordinate and the growth rate is inversely proportional to the speciﬁc length (Eq. (36)). It would also be interesting to investigate the redimensionalised growth rates because it would allow us the calculation of the total growth of the disturbances along the streamwise coordinate. The redimensionalised growth-rates are plotted as a function of the angular frequency at xˆ = 0.1 mm in Fig. 10. Here, the growth rate of disturbances are even larger close to the nozzle because the local width increases along the streamwise coordinate, further magnifying the

0

xˆ

(40)

at each cross-section with the corresponding angular frequency (ω ˆ max ) plotted in Fig. 11. In Fig. 11(a), we can see that the antisymmetric mode is a bit stronger close to the oriﬁce (xˆ < 0.6 mm) and the symmetric disturbances dominate further from the oriﬁce (xˆ > 1 mm) in the case of the top-hat outﬂow condition. In the other case (parabolic), the symmetric disturbances dominate everywhere. Furthermore, the amplitude of the disturbance grows only slightly, indicating that the antisymmetric mode cannot be effectively excited at such a low Reynolds number. If we compare the various outﬂow conditions, it can be clearly seen that the ﬂow with a top-hat velocity proﬁle at the oriﬁce is more sensitive, “unstable” compared to the parabolic proﬁle. The other observation is that the frequency of antisymmetric modes, at which the amplitude is the largest in each cross-section, is always larger than that of the symmetric modes. Moreover, generally the frequency corresponding to the maximum amplitude decreases monotonically along the streamwise coordinate, as was also found experimentally by Thomas and Goldschmidt (1985). 6. Conclusions In this paper, linear stability investigations of various velocity proﬁles were carried out using the OS equation solved by the CMM. The main goal of the investigation was to give an explanation why the ﬂow at the oriﬁce is more sensitive than elsewhere. The stability results showed that the disturbances grow more rapidly closer to the oriﬁce. These observations were valid for analytical velocity proﬁles as well as for numerical proﬁles obtained from CFD. In the case of analytical velocity proﬁles, the critical Reynolds numbers were determined which were very close

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