On the sensitivity of planar jets

On the sensitivity of planar jets

International Journal of Heat and Fluid Flow 62 (2016) 114–123 Contents lists available at ScienceDirect International Journal of Heat and Fluid Flo...

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International Journal of Heat and Fluid Flow 62 (2016) 114–123

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff

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

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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.

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(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

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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.

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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-

121

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



(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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to the literature value Recrit ≈ 4.0 for symmetric modes, independently of the profile shape. In the case of antisymmetric modes the values were larger and depended strongly on the shape of the profile. The non-dimensional growth rate is virtually independent of Re if Re ࣡ 10 0 0 in all the cases. The reason for the higher growth rate close to the orifice is twofold. The first reason is that the velocity profiles which are closer to the top hat profile have higher non-dimensional spatial growth rates. These profiles are closer to the orifice. The other one is that the local length scale grows continuously from the orifice and the growth rate is inversely proportional to the local length scale for the same non-dimensional velocity profile. Furthermore, an explanation was given why the antisymmetric modes can appear in experiments in the vicinity of the nozzle. Based on our calculations, it seems that they can appear only if the shear layer is very thin and it can also be influenced by the thickness of the nozzle wall. References Andrade, E.N.D.C., 1941. The sensitive flame. Proc. Phys. Soc. 53 (4), 329–355. doi:10. 1088/0959-5309/53/4/301. Atassi, O.V., Lueptow, R.M., 2002. A model of flapping motion in a plane jet. Eur. J. Mech. B Fluids 21, 171–183. doi:10.1016/S0997-7546(01)01176-1. Bajaj, A.K., Garg, V.K., 1977. Linear stability of jet flows. J. Appl. Mech. 44, 378. doi:10.1115/1.3424087. (September 1977). Barrett, W., 1867. Note on “sensitive flames”. Philos. Mag. Ser. 4 33 (222), 216–222. doi:10.1080/14786446708639773. Bechert, D., 1988. Excitation of instability waves in free shear layers. part 1. theory. J. Fluid Mech. 186, 47–62. doi:10.1017/S0 0221120880 0 0 035. Bechert, D.W., Stahl, B., 1988. Excitation of instability waves in free shear layers part 2. experiments. J. Fluid Mech. 186, 63–84. doi:10.1017/S0 0221120880 0 0 047. Ben Chiekh, M., Ferchichi, M., Béra, J.C., 2011. Modified flapping jet for increased jet spreading using synthetic jets. Int. J. Heat Fluid Flow 32 (5), 865–875. doi:10. 1016/j.ijheatfluidflow.2011.06.004. Bickley, W.G., 1937. The plane jet. Philos. Mag. Ser. 7 23 (156), 727–731. doi:10.1080/ 14786443708561847. Boiko, A.V., Dovgal, A.V., Grek, G.R., Kozlov, V.V., 2012. Physics of transitional shear flows: instability and laminar-Turbulent transition in incompressible near-wall shear layers. Fluid Mech. Appl. 98, 1–264. doi:10.1007/978- 94- 007- 2498- 3. Brown, G.B., 1935. On vortex motion in gaseous jets and the origin of their sensitivity to sound. Proc. Phys. Soc. 47 (4), 703–732. doi:10.1088/0959-5309/47/4/314. Criminale, W.O., Jackson, T.L., Joslin, R.D., 2003. Theory and Computation of Hydrodynamic Stability. Cambridge University Press doi:10.1017/CBO9780511550317. Curle, N., 1957. On hydrodynamic stability in unlimited fields of viscous flow. Proc. R. Soc. A Math. Phys. Eng. Sci. 238 (1215), 489–501. doi:10.1098/rspa.1957.0013. Garg, V.K., 1981. Spatial stability of the non-parallel bickley jet. J. Fluid Mech. 102, 127. doi:10.1017/S0022112081002577.

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