THE EUROPEAN PHYSICAL JOURNAL E

Regular Article

Laminar and Intermittent ﬂow in a tilted heat pipe E. Rusaouen1 , X. Riedinger1,2 , J.-C. Tisserand1 , F. Seychelles1,3 , J. Salort1 , B. Castaing1 , and F. Chill` a1,a 1 2 3

Laboratoire de Physique de l’Ecole Normale Sup´erieure de Lyon UMR5672 - 46 all´ee d’Italie, 69364 Lyon Cedex 07, France College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter, EX4 4QF, UK CNRS UMR 6614 - CORIA, Universit´e de Rouen, Site Universitaire du Madrillet, 675, avenue de l’Universit´e, BP 12, 76801 Saint Etienne du Rouvray Cedex, France Received 29 April 2013 and Received in ﬁnal form 16 September 2013 c EDP Sciences / Societ` Published online: 29 January 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. Heat transfer measurements performed by Riedinger et al. (Phys. Fluids, 25, 015117 (2013)) showed that in an inclined channel, heated from below and cooled from above with adiabatic walls, the ﬂow is laminar or intermittent (local bursts can occur in the laminar ﬂow) when the inclination angle is suﬃciently high and the applied power suﬃciently low. In this case, gravity plays a crucial role in the characteristics of the ﬂow. In this paper, we present velocity measurements, and their derived tensors, obtained with Particle Image Velocimetry inside the channel. We, also, propose a model derived from a jet interpretation of the ﬂow. Comparison between experiment and model shows a fair agreement.

1 Introduction Free convection has a peculiar status within hydrodynamic ﬂows. The driving force, gravity, is constant, and is not subject to unavoidable ﬂuctuations occurring with artiﬁcial drivings. It allows to deﬁne model experimental ﬂows with particularly clean driving and boundary conditions. Moreover, most of natural ﬂows originate in free convection mechanisms, which raises the practical interest of these model ﬂows. Additionally, convective ﬂows are involved many practical problems, such as ventilation of closed environments or heat extraction in industrial processes. Ventilation of rooms has been schematized using single or multiple plumes in cavities [1], while for example cooling of solar cell has been studied using channel with heated walls as, for instance, in [2]. Within the last decade, a new kind of ﬂow has been introduced: a channel in which a buoyancy forcing is introduced directly in the bulk. A hot chamber is bound to the lower part of the channel and a cold chamber to the upper part. In this system hot and cold plumes enter and mix inside the channel, far from the injection boundary [3]. This ﬂow is useful to mimic aeration in gallery, quarry pits [4] or mixing in oil wells, see for example [5]. When buoyancy destabilization is triggered by a temperature diﬀerence, it results in a heat ﬂux Qz and a longitudinal temperature gradient −β in the tube. A systematic study of the dependence of the heat ﬂux Qz versus β in this ﬂow both in vertical or inclined channel revealed a very rich behavior, with many regimes depending on the tilt angle ψ between a

e-mail: [email protected]

the channel axis and the gravity g and the heat power carried by the ﬂow [6]. Particularly, for an inclination ψ greater than 40◦ and an applied power ranging from 0 W to 43 W, two regimes can be observed: one “Laminar”, and the other “Intermittent”. The velocity proﬁles in these regimes are inconsistent with the model of inﬁnitely long channel, valid in the other regimes. This yields [6] to suggest that they consist in two independent laminar jets, which destabilize through intermittent bursts in the “Intermittent” regime. But it remains to precisely understand why the ﬂuid between the two jets is not carried by the jets, through the viscosity. The purpose of this paper is to analyze the velocity proﬁles and the stress tensor in these regimes and to develop our jet model for the “Laminar” one. After a brief presentation of the experimental set-up, sect. 2, we ﬁrst present the velocity and stress tensor proﬁles in the channel obtained from Particle Image Velocimetry (sect. 3). In [6], we have interpreted these proﬁles as corresponding to two jets, a cold one and a hot one, and we gave a dimensional order of magnitude for the width of these jets to justify this interpretation. Here, in sect. 4, we derive the full jet velocity proﬁle, and show that it nicely ﬁts the experimental data, and further justiﬁes the independence of the two jets. This derivation gives us the temperature proﬁle, which was not measured. We examine in sect. 5 some natural questions raised by the solution. Finally, in sect. 6, we analyse the velocity ﬂuctuations in the Intermittent regime to obtain information on the intermittent bursts, which are the unstable modes of the laminar proﬁle, before concluding with sect. 7.

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Eur. Phys. J. E (2014) 37: 4 2

ld Co

"upper wall"

d=

m 5c z

x

U* and U*x (cm/s) z

1.5

te pla

1 0.5 0 −0.5 −1 −1.5

cm 20

ψ

−2

−20

−10

0

10

20

x (mm)

t Ho

te pla

Fig. 1. Sketch of the cell, showing the deﬁnition of the coordinates, and the two jets.

2 Experimental set-up The set-up has been described in [7], where supplementary details can be found. It is sketched in ﬁg. 1. For the sake of completeness, we recall below the main characteristics of the experiment. The measurement part is a channel, made of PMMA (PolyMethylMethAcrylate), connecting two chambers, a hot chamber at the bottom and a cold chamber at the top. The lateral walls of the channel are as adiabatic as possible. Particle Image Velocimetry (PIV) indeed requires the removal of the isolation on these walls, but PMMA itself is a good insulator. The channel has a square section, with d × d = 5 × 5 cm inner dimensions, 20 cm long, ﬁlled with deionized water. Each square-ending section of the channel is inscribed in the circular opening of the corresponding chamber. The gravity acceleration g remains inside the plane of the front walls. The two chambers at its ends are conical (height 10 cm, largest diameter 20 cm and smallest one 7 cm), based by copper plates (diameter 20 cm, thickness 2 cm). The hot (lower) one is Joule heated (power P ) by a spiral resistor, 20 Ω. The cold (upper) one is temperature regulated by a water bath. Basically, we impose a heat ﬂux through the hot chamber and the device response with a temperature gradient. All the measurements are done at an average temperature of 25 ◦ C (P r = 6.1), close to the room temperature, to minimize possible heat leaks.

3 PIV results The velocity ﬁeld is measured by Particule Image Velocimetry (PIV). A one watt green continuous laser beam (DPSS-Laser System, 532 nm, Melles Griot), is enlarged with a cylindrical lens to produce a laser sheet. The ﬂuid (deionized water) is spreading with hollow glass particles (Sphericel 110P8, LaVision, GmbH), 10 micrometers in

Fig. 2. The symetrized velocity proﬁle for an inclination angle of ψ = 50◦ , and an applied heat power of 19 W (blue, continuous line), 43 W (red, dashed line).

average diameter. The recording and the batched processing are performed with the software provided by Davis LaVision. The images are grouped by series of 3 (40 ms between the images and 2 velocity ﬁelds for each series). The series are separated by 10 s. A record consists in 1800 series (which corresponds to 2 hours and 50 minutes). 3.1 Laminar ﬂow The laser sheet is positioned in the central part of the channel at the position y = 0 and the velocity ﬁelds are calculated on a window of 90 mm of height and 50 mm wide. The velocity ﬁelds have a z-spatial resolution of 62 points from −50 mm to 40 mm around z = 0, and a xspatial one of 33 points from −25 mm to 25 mm around x = 0. From these ﬁelds, we obtain the proﬁles by averaging over time and z. The average on z is computed from z = −10 mm to z = 10 mm for the velocity. Indeed, as we shall see, this proﬁle is expected to slowly depend on z, and we should mix diﬀerent proﬁles by averaging on the whole range of z. The other kinds of proﬁles are averaged from z = −50 mm to z = 40 mm. They should thus be interpreted as a z-average of the various possible proﬁles. In ﬁg. 2 we show the longitudinal z- and x-components of the velocity proﬁle for an inclination angle ψ = 50◦ , and applied powers P = 19 W and 43 W ∗ Ux,z = Ux,z t,z .

(1)

We obtain Ux t,z 10−2 Uz t,z , which means that it could correspond to a misalignment between the channel and the camera of less than one degree. We thus consider that our measurement of the mean x-component velocity is not signiﬁcant, and we only discuss the mean zcomponent, and the ﬂuctuations of both. Qualitatively, at a low applied power, here 19 W, the slope of the proﬁle around x = 0 is much smaller than the average one, whereas close to the walls the proﬁle is divided into two symmetrical parts. This behavior looks very similar to what can be expected for two independent

Eur. Phys. J. E (2014) 37: 4

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−3

6

x 10

5

2 τ xx/Ut

4 3 2 1 0

−20

−10

0 x (mm)

10

20

Fig. 3. The transverse velocity variance τxx normalized by Ut2 = (−gz αP/cp d)2/3 for an inclination angle ψ = 50◦ and two diﬀerent powers 19 W (blue, continuous line), 43 W (red, dashed line).

Fig. 4. Instantaneous longitudinal velocity map corresponding to the occurrence of a perturbation in the hot jet. The inclination angle is ψ = 50◦ and the applied power is 43 W.

jets. Our vision is as follows: the water cooled by the cold plate accumulates in the lower part of the cold chamber, then pours along the lower wall of the channel. The symmetrical occurs in the hot chamber: the water heated by the hot plate accumulates in the upper part of the hot chamber, then pours along the upper wall of the channel. Increasing the applied power results in a center slope much closer to the mean one. This suggests that increasing the applied power decreases the independence of the jets. We deﬁne the transverse velocity variance as (2) τxx = (δux )2 = (Ux − Ux t,z )2 . Being the average of a quadratic quantity, τxx is sensitive to the noise on our Ux measurement. This noise is difﬁcult to distinguish from the physical contributions, and thus to evaluate, but it should be approximately constant with x. The variance τxx characterizes the interactions between the two jets. Figure 3 shows the proﬁle of τxx . A local minimum is located around x = 0 (center of the cross section of the channel) for both powers. This fact suggests that the two parts of the channel are independent. Moreover, the value of this local minimum increases when the applied power is increased. This conﬁrms the previous observations on the evolution of the velocity proﬁle. 3.2 Intermittent destabilization Thermal results [6] distinguish the Laminar regime, where the longitudinal temperature gradient β depends on the applied power P as β ∝ P 4/5 , from the Intermittent regime where β is constant, independent of P . Particle Image Velocimetry (PIV) shows that, in the Intermittent regime, turbulent periods altern with laminar ones. However, PIV also shows that perturbations of the laminar ﬂow can occur within the Laminar regime. An instantaneous map of the longitudinal velocity ﬁeld with a perturbation in the hot jet is shown in ﬁg. 4. For comparison, an example of the instantaneous map of the longitudinal

Fig. 5. Instantaneous longitudinal velocity map when the ﬂow is laminar. The inclination angle is ψ = 50◦ and the applied power is 43 W.

velocity ﬁeld when the ﬂow is laminar is shown in ﬁg. 5. Both correspond to an applied power P = 43 W. It is clear from ﬁg. 4 that in this case the perturbation is located in the hot jet and the cold one remains nearly unperturbed. We also observed perturbations in the cold jet. Indeed, for P = 43 W, the perturbations are rather frequent, and occasionally, a hot and a cold perturbation interact. However, all single perturbations have the same step function shape. For P = 19 W, the perturbations where too small to be detected on this Uz map. However, we know that some perturbations occur. Indeed, those perturbations create turbulent velocity ﬂuctuations which are related to the transverse Reynolds stress. Similarly to the transverse velocity variance, the transverse Reynolds stress is deﬁned as τxz = δux δuz = (Ux − Ux t,z )(Uz − Uz t,z ) . (3)

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Eur. Phys. J. E (2014) 37: 4 −3

5

hot jet. The coordinates are deﬁned in ﬁg. 1. The problem is similar to that of a rising laminar plume, and several such situations have been studied in the past (plumes in stagnant ﬂuids) [8,9]. As shown in ﬁg. 5 the x variations of longitudinal velocity are much more rapid than the z ones. This is true for all the other quantities as transversal velocity or momentum transfer and allows a Prandtl-like approximation, where the coordinates x and z can be considered as having diﬀerent dimensions. The equations veriﬁed by the velocity U and the temperature θ are

x 10

2 τxz/Ut

0

−5

−20

−10

0 x (mm)

10

20

Fig. 6. The transverse Reynolds stress τxz normalized by Ut2 = (−gz αP/cp d)2/3 for an inclination of 50◦ and applied powers 19 W (blue, continuous line) and 43 W (red, dashed line).

In ﬁg. 6, we show τxz normalized as in [6] by the quantity Ut2 = (−gz αP/cp d)2/3 , where cp is the volumetric heat capacity. As Ut is the expected amplitude of the turbulent velocity [6], this quantity gives an estimate of the probability of occurrence of the perturbation. Several remarks can be made. For P = 19 W, we saw in ﬁg. 3 that τxx is small. Being the average of a positive quantity, it could be due to the measurement noise, which is diﬃcult to precisely estimate. However, the contribution of this measurement noise should average to zero in τxz . We see in ﬁg. 6 that τxz has the same order of magnitude as τxx , at least within the jets. This conﬁrms our opinion that these values are physical, due to the perturbations above mentioned, not to the measurement noise. In these regions within the jets, the normalized values of τxz for P = 19 W and 43 W are similar, within a factor 2. On the opposite, the absolute value of τxz in the center of the channel (x = 0) is much larger for P = 43 W than for 19 W. This corresponds to a much larger momentum ﬂux from one jet to the other. We infer from this that this momentum transfer is due to the interactions between hot and cold perturbations, and that this mechanism is responsible, at larger applied power, to the transition to turbulence in the Intermittent regime. The independence of the jets is thus much better realized for P = 19 W than for 43 W.

4 Non-dimensional equations In this section, our goal is to propose a model that accounts for the counter-current convective ﬂow, evidenced by [6] from both PIV and calorimetric measurements, i.e. the presence of two independent jets. We consider here the 2D problem of two opposite jets in a channel with adiabatic walls. This situation, as it will be shown, represents quite well our ﬂow in the central part of the channel where it can be considered invariant for y translation. As the jets are independent, we can consider a solution for each of them, the global solution will be given from the superposition of the two. To be speciﬁc, let us consider the

Ux

∂p ∂ 2 Uz ∂Uz ∂Uz + Uz =− − gz αθ + ν , ∂x ∂z ρ∂z ∂x2

(4)

Ux

∂p ∂ 2 Ux ∂Ux ∂Ux + Uz =− − gx αθ + ν , ∂x ∂z ρ∂x ∂x2

(5)

Ux

∂2θ ∂θ ∂θ + Uz =κ 2, ∂x ∂z ∂x

(6)

∂Uz ∂Ux + = 0, ∂x ∂z

(7)

θ = T − Tm with Tm mean temperature at which the ﬂuid is maintained. Here Tm = 25 ◦ C, p is the pressure and ρ the ﬂuid density. g is the gravitational acceleration, ν, the kinematic viscosity, κ the thermal diﬀusivity and α the isobaric thermal expansion coeﬃcient of the ﬂuid. The upper wall (see ﬁg. 1) being at the abscissa xw , the boundary conditions are Uz (x = xw ) = 0, Uz (x → −∞) = 0, ∂θ/∂x(x = xw ) = 0, ∂θ/∂x(x → −∞) = 0. Neglecting the heat conductivity, the longitudinal heat ﬂux is Qz = cp Uz θ, where cp is the volumetric heat capacity. The integral xw P (8) Uz θ dx = D= 2c pd −∞ is a constant, independent of z. We consider here that the width of the channel is much larger than the width of the jet, and that the center of the channel can be considered as x → −∞ for the hot jet. Note that we have two jets, a cold one and a hot one and that the section of the channel is S = d2 so that the mean heat ﬂux through the channel is Q = dP2 = 2cp D d. Considering the above remarks, if ∂p/ρ∂x balances with gx αθ (eq. (5)), then ∂p/ρ∂z is negligible against gz αθ (eq. (4)). The hot jet has its virtual origin at z = −L, whose value will be precised when comparing to experiments. With respect to this virtual origin, the longitudinal coordinate is Z = L + z. With the remaining pertinent parameters, x, z, go = −gz (the z component of gravity), D, α (the isobaric thermal expansion coeﬃcient), ν (the kinematic viscosity), κ (the heat diﬀusivity), we can construct – a velocity along z Uo =

(go αD)2 Z ν

1/5 ,

(9)

Eur. Phys. J. E (2014) 37: 4

Page 5 of 8 1.5

Vo =

go αDν 2 Z2

1/5 ,

(10)

– a temperature θo =

D4 go αν 2 Z 3

1/5 ,

(11)

Velocity profile (cm/s)

– a velocity along x

1 0.5 0 −0.5 −1

– and a length along x δ=

ν3Z 2 go αD

−1.5

1/5 ,

Ux = −Vo v(ξ, P r),

θ = θo t(ξ, P r). (13) Diﬀerentiating versus x is diﬀerentiating versus ξ (denoted by ) with the factor −1/δ. For the z diﬀerentiation, we have to consider both the prefactor, and the dependence of ξ at ﬁxed x. So, eq. (7) becomes ∂ξ Vo ∂Uo v + u + Uo u = 0. δ ∂z ∂z

(14)

As

0 x (mm)

10

20

Fig. 7. Comparison between the experimental velocity proﬁle and theoretical one for an inclination of 50◦ and applied power of 19 W; blue continuous line: experimental one, red dashed line: theoretical one.

the last relation between u, v, and t can be written as 2 3 1 t . vt − ut − ξut = 5 5 Pr The sum condition (8) becomes ∞ ut dξ = 1,

(22)

(23)

0

For an easy numerical resolution, it is convenient to deﬁne u1 = u and t1 = t . We then have the following ﬁrst-order equations: u = u1 ,

Uo ∂Uo = , ∂z 5Z

∂ξ 2ξ =− , ∂z 5Z

Vo Uo = , δ Z

(15)

we get 2 1 ξu − u. 5 5 Similarly, eq. (4) becomes v =

(16)

go αθo =

Uo2 , Z

1 2 u1 = −t + vu1 + u2 − ξu1 u, 5 5 2 1 v = ξu1 − u, 5 5 t = t 1 ,

Vo Uo ∂Uo 2 ∂ξ ν vu + Uo u + Uo2 u u = go αθo t + Uo 2 u . δ ∂z ∂z δ (17) Considering that ν Uo , = 2 δ Z

(18)

we can write

2 3 t1 = P r vt1 − ut − ξut1 , 5 5

(19)

Finally, eq. (6) becomes Vo θo ∂θo ∂ξ κ vt + Uo ut + Uo θo ut = θo 2 t . δ ∂z ∂z δ 3 θo ∂θo =− , ∂z 5Z

κ=

ν , Pr

(20)

(21)

(24)

with the initial conditions t(0) = to , u1 (0) = ato , u(0) = 0, v(0) = 0 and t1 (0) = 0. to and a are chosen such to verify the condition (23) and to give u approaching asymptotically zero at inﬁnity. For P r = 6.1, we ﬁnd to 2.00,

1 2 vu + u2 − ξu u = t + u . 5 5

As

−10

(12)

ξ = (xw − x)/δ, and P r = ν/κ are the only nondimensional parameters. As we shall see, the normalizing length δ is much smaller than the width of the channel d. So each jet can be treated as if the other one were at inﬁnity. We can then deﬁne u, v, t as Uz = Uo u(ξ, P r),

−20

a 0.795.

(25)

The function u(ξ) reaches its maximum for ξ 0.92, so the maximum of Uz is approximately at a distance δ from the wall. The symmetrized solution is shown and compared to the experiment in ﬁg. 7. We used the value of Z as an adjustable parameter. Indeed, at the bottom of the channel, the rising jet is not inﬁnitely thin, which is equivalent to a virtual origin upstream, at an adjustable distance L from the middle of the channel, where our measurements are taken. We had to choose Z = L = 60 cm,

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Eur. Phys. J. E (2014) 37: 4

Temperature Profile (oC)

1

where Φ(Z) = Uo (Z)δ(Z) = (ν 2 go αDZ 3 )1/5

0.5

and z has its zero at the middle of the channel. In the measurement zone, |z| never exceeds 5 cm L = 60 cm. From eq. (24), and above, we have 3 ∞ u dξ (30) v∞ = − 5 0

0

−0.5

and −1

(29)

−20

−10

0 x (mm)

10

20

Fig. 8. The theoretical temperature proﬁle (blue), θ = θo t (see deﬁnitions (11) and (13)), with the same parameters as in ﬁg. 7. θo u is shown for comparison (red dashed line). The Prandtl number is 6.1 (T = 25 ◦ C).

which is indeed larger than the total length of the channel (20 cm). Note that the quantity Uo2

3 dΦ (Z = L) = dZ 5

go αD (26) δ ν is independent of Z. Thus we cannot adjust the maximum velocity and the width of the jet independently. The good agreement we have is thus highly signiﬁcant. With this value of Z, we have Uo = 2.1 · 10−2 ms−1 , and δ = 5.05 · 10−3 m. The corresponding temperature proﬁle is shown in ﬁg. 8. We see that, due to the relatively large value of the Prandtl number (P r = ν/κ = 6.1), the temperature proﬁle is thinner than the velocity one. The hot and cold part are well separated. Note that, logically enough, the inﬂexion point of the hot velocity proﬁle occurs where the hot temperature proﬁle ends. The transverse temperature diﬀerence, from one side to the other, is rather high, comparable to the total temperature diﬀerence between the ends of the channel. It stresses that the channel length cannot be considered as large in this regime: the hot jet reaches the cold chamber nearly at the same temperature it enters the channel. =

5 Discussion The solution we have found has some apparent drawbacks which deserve a discussion. First, we can note that ˙ (27) M = Uz dx is not zero, as it should, nor independent of z in our model. Indeed, taking into account the two jets ∞ M˙ = u dξ (Φ(Z = L + z) − Φ(Z = L − z)) 0 ∞ dΦ (Z = L), (28) u dξ z 2 dZ 0

Thus

ν 2 go αD L2

1/5

M˙ = −2Vo v∞ z.

=

3 Vo . 5

(31) (32)

To correct this spurious mass ﬂow, we should introduce a secondary (e.g. cosine) ﬂow, whose z component is πx πx π π = Vo v∞ z cos . (33) U2 (x, z) = − M˙ cos 2d d d d Let us recall that Vo = (δ/L)Uo Uo . In the case we examined, |U2 |/Uo does not exceed 2·10−2 , which explains why we can neglect this secondary ﬂow z component. Such is not the case for the corresponding x component V2 (x). The incompressibility eq. (7) implies πx ∂U2 ∂V2 =− =⇒ V2 (x) = −Vo v∞ sin , (34) ∂x ∂z d whose amplitude is equivalent to our solution −Vo v. Indeed, if δ d, V2 (x) can be considered as constant across the jets. This constant exactly compensates the other jet contribution in the symmetrized solution, and really makes the two jets independent. The second point is connected to our fundamental question about the behavior of the ﬂuid between the jets. Looking at ﬁg. 8, we see that in the center of the channel, the buoyancy term is essentially zero, and the ﬂuid should behave as in a Couette ﬂow, i.e. it should be carried by the jets by viscosity. However, this is not so, just due to the x component of the velocity, Ux , which reaches a ﬁnite value out of the jet. Indeed, while laminar, the ﬂow is not in the Stokes limit and a quadratic term can balance the viscous one. For instance, out of the hot jet, eq. (4) becomes ∂ 2 Uz ∂Uz =ν . (35) Ux ∂x ∂x2 Approximating Ux with a constant gives Ux (xw − x) (36) Uz ∝ exp − ν decreasing when going away from the wall. As Ux is of order Vo , the length ν/Ux is close to ν/Vo = δ. This totally explains the surprising three inﬂexion points in the longitudinal velocity proﬁle. The longitudinal velocity is zero out of the jets, while it would linearly depend on x in a Couette ﬂow, to make the viscous stress ν∂Uz /∂x constant.

Eur. Phys. J. E (2014) 37: 4

Page 7 of 8 −3

6 Destabilization of the laminar jet

4

In this section, we examine the way the laminar proﬁle destabilizes. It occurs through localized perturbations of the velocity ﬁeld, we call bursts. We shall examine the position and the approximative shape of these bursts.

x 10

τxx/U2t

3

2

6.1 Position In ﬁg. 4 we show the axial velocity ﬁeld corresponding to one of these bursts in the hot jet. The applied power is 43 W and the inclination angle is ψ = 50◦ . As previously said, the perturbation is clearly localized in the hot jet, the cold jet being nearly unperturbed, which suggests that the instabilities of each jet are decoupled. The value of τxx in the center of the channel is minimum. τxx presents two maxima, approximately located at the inﬂexion points of the velocity proﬁle. We know that these inﬂexion points are critical for the destabilization of laminar proﬁles of inviscid ﬂuid ﬂows [10]. The position of the perturbation, we shall precise below, thus suggests an inertial mechanism for the destabilization of the laminar ﬂow.

1

−20

−10

0 x (mm)

10

20

Fig. 9. The primitive of the 19 W, 50◦ transverse Reynolds stress τxz (red dashed-dotted line) compared to the corresponding transverse velocity variance τxx (blue continuous line), normalized by Ut2 = (−gz αP/d)2/3 .

6.2 Shape To get an idea of the shape of the perturbation, let us assume that the corresponding perturbation of the current function can be written as φ(x)χ(z), that is as the product of a x function and a z function (separated coordinates). As far as we could see, the perturbations look as thin elongated structures almost parallel to the z axis, which justiﬁes this approximation. Then, the x (δux ) and z (δuz ) components of the velocity perturbation are δux = −φ(x)χ (z)

and

δuz = φ (x)χ(z).

(37)

To obtain its contribution to the proﬁle of τxx , we average (δux )2 between z− and z+ , which is our observation range z+ φ(x)2 δτxx = χ2 (z) dz (38) z+ − z− z− while for the τxz proﬁle, we average δux δuz φ(x)φ (x) z+ δτxz = χ(z)χ (z) dz. z+ − z− z−

(39)

Assuming that both integrals are not zero, it means that τxz should be proportional to the derivative of τxx . The transverse Reynolds stress τxz , ﬁg. 6, is indeed rather consistent with the derivative of τxx . Integrating τxz gives a reasonable approximation of τxx , as shown in ﬁg. 9 (dashed line). τxz crosses zero approximately at the point where τxx is maximum. Comparing ﬁgs. 9 and 6, we can even infer that z+ χ(z)χ (z) dz = χ2 (z+ ) − χ2 (z− ) (40) 2 z−

Fig. 10. Tentative longitudinal velocity map when a perturbation occurs. The parameters are chosen so to compare with ﬁg. 4.

is positive for the cold jet, and negative for the hot one, that is of the sign opposite to the velocity. This is surprizing if we think of these perturbations as convected at the unperturbed velocity at their center. In such a case, χ2 (z+ ), on average, should be equal to χ2 (z− ). Here, we obtain that χ2 downstream is smaller than χ2 upstream. We could imagine that these perturbations are reminiscences of turbulences at the channel entrance, which damp during their convection. The perturbation of the current function looks as a step function along z, and a localized one along x. To set the ideas, and to evaluate the position and width of the experimental perturbation shown in ﬁg. 4, we choose φ as a Gaussian, and χ as a hyperbolic tangent. In ﬁg. 10, we

Page 8 of 8

show such a perturbation, with (x − xp )2 , φ(x) = exp − 2b2 z − zp . χ(z) = A 1 − tanh c

Eur. Phys. J. E (2014) 37: 4

(41)

The best comparison between this tentative longitudinal velocity map and ﬁg. 4 (applied power P = 43 W) is obtained with A = 42 mm2 s−1 , b = 7 mm (the xwidth of the perturbation), and c = 20 mm (the z-width of the perturbation). The amplitude it contributes for τxz is thus of order A2 /bc = 13 mm2 s−2 . As a comparison, for P = 43 W, Ut2 = 18 mm2 s−2 . The position of the perturbation is zp = −20 mm, and xp = 16 mm, approximately at the inﬂexion point of the hot jet, as expected.

7 Conclusion In this paper, we have compared our PIV measurements of a laminar ﬂow in a convective pipe with the twodimensional solution of a plume along an inclined wall. We ﬁnd good agreement with the observed velocity proﬁle within two conditions. We have ﬁrst to take into account the ﬁnite width of our plume at the entrance of our channel, equivalent to a virtual origin upstream. Second, we have to limit our comparison to the lowest input heat power (< 20 W). For the highest heat power, even if the transverse Reynolds stress τxz is strongly reduced compared to the turbulent one, the longitudinal velocity proﬁle is perturbed by occasional bursts, occurring independently in the cold and in the hot jet. For the lowest heat power, the success of our model conﬁrms that the upward and downward-ﬂowing jets are independent of each other. This laminar ﬂow analysis works even though the ﬂow is perturbed intermittently by bursts, as revealed by the transverse Reynolds stress τxz . We analyzed these bursts, both statistically through their contribution to the Reynolds stresses τxx and τxz , and the examination of an instantaneous spot. It allowed us to propose a shape for these perturbations, in reasonable agreement with our observations. It is consistent with

their localization around the inﬂexion point of the unperturbed longitudinal velocity proﬁle, propagating at the velocity at this point, as for inviscid ﬂows, suggesting an inertial mechanism of destabilization. We observe these bursts within the Laminar regime, following the classiﬁcation of [6], based on the dependence between longitudinal heat ﬂux and temperature gradient, which shows that what makes the Intermittent regime is not simply the existence of these perturbations, but either their interaction, or the existence of another instability of the ﬂow. We acknowledge stimulating discussions with Laurent Chevillard. Thanks are also due to Y. Hallez, J.P. Hulin, J. Magnaudet, F. Moisy, D. Salin and J. Znaien for many useful discussions within the ANR project GIMIC. We thank region Rhone-Alpes for ﬁnancial support (Cible 2011, n◦ 2770). The post-doctoral position of X. Riedinger was supported by the ANR-07-BLANC-0181 project. Finally, the stimulating remarks of the referees are gratefully acknowledged.

References 1. P.F. Linden, Annu. Rev. Fluid Mech. 31, 201 (1999). 2. N. Bianco, B. Morrone, S. Nardini, V. Naso, Int. J. Heat Technol. 18, 23 (2000). 3. M. Gibert, H. Pabiou, J.C. Tisserand, B. Gertjerenken, B. Castaing, F. Chill` a, Phys. Fluids 21, 035109 (2009). 4. F. Perrier, P. Morat, J.L. Le Mouel, Phys. Rev. Lett. 89, 134501 (2002). 5. J. Znaien, F. Moisy, J.P. Hulin, Phys. Fluids 23, 035105 (2011). 6. X. Riedinger, J.-C. Tisserand, F. Seychelles, B. Castaing, F. Chill` a, Phys. Fluids 25, 015117 (2013). 7. J.C. Tisserand, M. Creyssels, M. Gibert, B. Castaing, F. Chill` a, New J. Phys. 12, 075024 (2010). 8. A. Linan, V.N. Kurdyumov, J. Fluid Mech. 362, 199 (1998). 9. B.R. Morton, G. Taylor, J.S. Turner, Proc. R. Soc. London 234, 1 (1956). 10. P. Huerre, M. Rossi, in Hydrodynamics and Nonlinear Instabilities, edited by C. Godr`eche, P. Manneville (Cambridge University Press, 1998).