David Galley

Laurent Pierrot

Nicolas Enjalbert

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 337 (2009) 639–644An entrainment model for the turbulent jet in a coflowNicolas Enjalbert ∗,1, David Galley, Laurent Pierrot 2Saint-Gobain Recherche, 39, quai Lucien-Lefranc, BP 135, 93303 Aubervilliers, FranceReceived 6 January 2009; accepted after revision 4 September 2009Available online 26 September 2009Presented by Jean-Baptiste LeblondAbstractThe entrainment hypothesis was introduced by G.I. Taylor to describe one-dimensionally the development of turbulent jetsissuing into a stagnant or coflowing environment. It relates the mass flow rate of surrounding fluid entrained into the jet to thecharacteristic velocity difference between the jet and the coflow. A model based on this hypothesis along with axial velocityassumed to follow a realistic Gaussian distribution is presented. It possesses an implicit analytical solution, and its results arecompared and shown to be fully equivalent to previously published models that are rather based on a spreading hypothesis. Allof them are found to be in agreement with experimental results, on a wide range of downstream positions and for various coflowintensities. To cite this article: N. Enjalbert et al., C. R. Mecanique 337 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméUn modèle d’entraînement pour le jet turbulent dans un écoulement co-courant. L’hypothèse d’entraînement a été introduitepar G.I. Taylor afin de décrire unidimensionnellement le comportement des jets turbulents se déchargeant dans un environnement aurepos ou co-courant. Elle consiste à relier le flux de masse du fluide environnant entraîné dans le jet à la différence de vitesse entrele jet et l’écoulement co-courant. On présente dans cet article un modèle fondé sur cette hypothèse et celle d’un profil de vitesseaxiale moyenne Gaussien. Il possède une solution analytique implicite ; ses résultats sont équivalents aux modèles précédemmentpubliés qui s’appuient plutôt sur une hypothèse d’élargissement. Tous ces modèles sont en accord avec les résultats expérimentaux,disponibles sur un grand domaine de positions le long du jet et pour différentes intensités de l’écoulement co-courant. Pour citercet article : N. Enjalbert et al., C. R. Mecanique 337 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Fluid mechanics; Turbulent jets; EntrainmentMots-clés : Mécanique des fluides ; Jets turbulents ; Entraînement* Corresponding author.E-mail addresses: enjalbert@coria.fr (N. Enjalbert), david.galley@saint-gobain.com (D. Galley), laurent.pierrot@saint-gobain.com(L. Pierrot).1 Present address: CORIA – CNRS and INSA de Rouen, Technopôle du Madrillet, BP 8, 76801 St-Étienne-du-Rouvray, France.2 Present address: Saint-Gobain Oberland, Oberlandstrasse, 88410 Bad Wurzach, Germany.1631-0721/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2009.09.005640 N. Enjalbert et al. / C. R. Mecanique 337 (2009) 639–6441. IntroductionTurbulent jets are encountered in countless situations, such as industrial furnaces or propulsion engines. They sharesimilarities with other turbulent structures such as thermal plumes and pool fires, where not inertia, but buoyancy isthe source of momentum. Jets involve a wide range of time and length scales: microscopic turbulent phenomena areat the core of their structure, but for an intuitive, global understanding of their overall behavior, the point of viewof larger scales must be taken. In this Note only the steady state, and the region where the flow is fully developed,far from the jet origin, are considered. The potential core, a zone right downstream of the nozzle where the averagevelocity is uniform, is not taken into account. The jet is only characterized by its initial mass and momentum flowrates.The jets considered here are assumed of constant density and non-reacting. They constitute a very interesting text-book case because, although the problem is quite simplified by these restrictive assumptions, theory and experimentsprovide tools to easily extend it to non-constant density [1] or reacting situation [2–4].Observation shows that as a jet develops, its width increases and its velocity decreases. The core phenomenon ofthis evolution is the entrainment, which is the incorporation of fluid from the surroundings into the jet by turbulenteddies generated by the shear existing between the two regions. As is thoroughly presented by Morton et al. in [5],the entrainment hypothesis applied to a jet released into a still atmosphere states that the total inflow across the edgeof the jet is proportional to a characteristic velocity of the jet at the level of inflow, and depends additionally on thegeometry of the considered problem.The purpose of this Note is to present a one-dimensional model of the behavior of turbulent jets based on thishypothesis. Its results are studied and linked with those of different models, recently published and relying on differenthypotheses, as well as with experimental results. The case of a free round jet is first discussed, as an illustration forintroducing the entrainment hypothesis. Then the hypothesis is extended to the problem of a non-zero coflow velocityand a new model formulated. In a concluding section, its results are compared with those of previous models and withexperimental results.2. The entrainment hypothesis applied to the free jet caseWe consider an incompressible jet issuing through a circular nozzle into a stagnant atmosphere of the same fluidwhose spatial extent is infinite. The jet is characterized by its initial mass flow rate Q0 and momentum flow rate M0.Characteristic initial velocity and radius can be assessed, respectively as M0/Q0 and Q0/√πρM0. The Reynoldsnumber Re = (M0/πρν2)1/2 is assumed large enough for the flow to be fully turbulent at the considered distancesfrom the nozzle. In its expression ρ denotes the fluid density and ν the kinematic viscosity. The average velocityprofile at the injection is assumed to be axisymmetric. A cylindrical coordinate system may then be used, with x theaxial position, originating from the nozzle, and r the radial distance from the nozzle axis. The quantities consideredin this paper are average quantities. The entrainment model involves the intrinsic characteristics of the jet, mass andmomentum fluxes through successive constant-x sections of the jet. Given the constant-density assumption, one mayremove ρ from their expression and manipulate the specific fluxes, denoted by q(x) and m(x):q(x) = 2π∞∫0u(x, r)r dr, m(x) = 2π∞∫0u(x, r)2r dr (1)The corresponding quantities q(x,R) and m(x,R) integrated over the finite interval [0;R] instead of [0;∞) are alsointroduced; they verify q(x,R) → q(x) and m(x,R) → m(x) when R → ∞.As mentioned above, we study the region far from the nozzle, x  q0/√πm0. A key element of the modelling ofthe jet is to assume that, there, the average velocity field is self-similar. This is widely observed in experiments [6,7]and therefore a usual assumption. The axial average velocity profile may then be written u(x, r) = um(x)f (r/b(x)),where um(x) and b(x) are respectively the average centerline velocity and a characteristic transverse dimension ofthe jet at abscissa x. Here, f is the profile function which only depends on the self-similar variable η = r/b(x). Itis possible to determine f theoretically, through assumptions for a closure of the RANS transport equations [7]. Buthere the experimentally observed Gaussian profile is a priori assumed: f (η) = exp(−η2).N. Enjalbert et al. / C. R. Mecanique 337 (2009) 639–644 641The model system equations are now derived. The integration of the Gaussian profiles leads to the following limitexpressions of the flow rates when r → ∞, in terms of um and b:q(x, r) → q(x) = πb(x)2um(x), m(x, r) → m(x) = 12πb(x)2um(x)2 (2)The equations verified by q(x) and m(x) remain to be established. Momentum and mass balances are carried outon a cylindrical set [x;x + dx] × [0;R]. As far as the mass balance is concerned, one describes the entrainmentof fluid into the jet by introducing ε(x,R) the entrained flow rate per unit length at a given abscissa x into an x-oriented cylinder of radius R. It is defined in such a way that the mass balance reads dq(x,R)/dx = ε(x,R) and tendsto dq(x)/dx = ε(x) when R → ∞. The entrainment increases with the jet velocity, and the entrainment hypothesis isformulated at this stage by assuming that ε(x) is proportional to um (as historically introduced in [5]). Dimensionally,a length has to be involved in the expression of ε, and a natural choice for it is a characteristic perimeter of the jet,2πb. Finally, the entrainment assumption may be writtendq(x)dx= d(πb(x)2um(x))dx= 2απb(x)um(x) (3)where α is a coefficient assumed to be constant, called the ‘entrainment coefficient’.In turn, the momentum conservation written on the same set reads dm(x,R)/dx = u(x,R)dq(x,R)/dx, where therhs is the lateral input of momentum due to the entrainment. As the terms of the rhs product tend respectively to zeroand the finite value ε(x), making R tend to infinity leads todm(x)dx= d(πb(x)2um(x)2)dx= 0 (4)This expresses the overall conservation of momentum of the jet along its course.The solution for system (3)–(4) readsb(x) = 2α(x − x0), um(x) =√m0/π2α(x − x0) (5)where x0 is the abscissa of a ‘virtual origin’ of the jet [8]. The well-known behavior of the turbulent jet far from itsorigin, with a linear spreading b ∼ x and a velocity decreasing as 1/x, is found. The constant α was reliably estimatedby Ricou and Spalding [1] through a study of the mass flow rate evolution; they give a relation which in terms of theinitial momentum flow rate reads dq/dx ∼ 0.16√πm0: applied to (3) and (5) in the limit x → ∞, it yields α  0.056.The entrainment hypothesis has been presented and applied to a simple jet case. It enables the writing of a modelwhich stays close to the core characteristics of the jet, namely the integral fluxes of mass and momentum. Only oneconstant is needed to parametrize the system and make it fit the experiments.3. Coflowing environmentThis section applies the entrainment hypothesis to the more complex case of a jet in a coflow. The former problemhypotheses are retained, except that the environment flow is now taken of uniform non-zero average velocity, denotedby u∞, and assumed small compared to the initial jet velocity: u∞ 	 m0/q0.Dilution by entrainment causes the jet velocity to tend towards u∞. As long as um  u∞, the jet does not heed theenvironment and behaves as a free round jet. The velocity decreases as 1/x; according to (5), um  u∞ at a transitiondistance of the order lm = √m0/π/u∞. Beyond, a so-called ‘wake regime’ is observed, where um −u∞ → 0 as x−2/3,and the jet width increases as x1/3. Such experimental results have been gathered by Wang [6] on a wide rangeof inlet and coflow conditions, at far-reaching downstream positions. They are plotted in Fig. 1 in the normalizedspace u∗ = um/u∞, b∗ = bu∞√π/me,0  b/lm and x∗ = xu∞√π/me,0  x/lm, where me,0 is the initial excess ofmomentum flux, defined later in this section. This non-dimensionalization allows a description of the jet behaviorindependent of the coflow velocity: indeed, all measurements by Wang collapse in this way with a 15% precision ontounique velocity and half-width curves. The asymptotes of these curves, describing the jet regime in the close regionand the wake regime in the distal region, are considered in the following as the reference for the experimental results.In this case it is relevant and supported by experimental results [6] to lay the self-similarity assumption upon theexcess velocity u − u∞ instead of the velocity itself. The same Gaussian similarity function f as earlier is taken.642 N. Enjalbert et al. / C. R. Mecanique 337 (2009) 639–644Fig. 1. Experimental measurements of the normalized average centreline velocity and normalized half-width of a jet in a coflowing environment.After [6].Fig. 1. Mesures expérimentales de la vitesse moyenne au centre normalisée et de la demi-largeur normalisée d’un jet dans un écoulement co-courant.D’après [6].Here, u(x, r) → u∞ when r → ∞, so that q(x, r) and m(x, r) do not tend to finite values anymore and the mass andmomentum balances must be carried out with more care.The mass flow rate through the surface r ∈ [0;R] may be decomposed as q(x,R) = π[R2 −b(x)2]u∞+δq(x,R)+qj (x,R), such that δq(x,R) → 0 and qj (x,R) → qj (x) = πb(x)2um(x) when R → ∞. The first term of this summay be understood as the contribution of the coflow to the total mass flow rate q , therefore qj may be considered asthe characteristic intrinsic mass flow rate of the jet. That way, the balance equation can be written dqj (x)/dx = ε(x).One frames an entrainment hypothesis similar to the former case, except that it now involves the velocity difference,in order to account for the fact that it is the shear that generates the entrainment phenomenon: ε = 2παb(um − u∞).Assuming that α is a constant, continuously relevant from the coflow jet to the free jet (where u∞ = 0) leads to thesame α  0.056. Finally, it holds thatdqj (x)dx= d[πb(x)2um(x)]dx= 2απb(x)[um(x) − u∞] (6)As far as the momentum balance is concerned, the relation dm(x,R)/dx = ε(x,R)u(x,R) may be obtained. Thisleads, as R → ∞, to the result that not m(x), but me(x) = m(x) − qj (x)u∞ is conserved. This quantity, of integralformulation∫ ∞0 2πu(u − u∞)r dr , may be referred to as the excess of momentum flow rate. With the consistentdefinition me,0 = m0 − u∞q0  m0, its conservation equation reads:πb(x)2[um(x) + u∞][um(x) − u∞] = 2me,0 (7)Eqs. (6) and (7), in terms of the non-dimensional variables, read:d(u∗b∗2)dx∗= 2αb∗(u∗ − 1), b∗2(u∗ + 1)(u∗ − 1) = 2 (8)It may be noted that the solution of this system has an implicit analytic expression, which may be obtained by definingan initial condition u∗(0), and then reads:[ −(v∗2 − 4v∗ + 1)3(v∗ + 1)3/2(v∗ − 1)5/2]u∗u∗(0)= 2αx∗ (9)The initial condition is established by setting the initial mass and excess of momentum flows to the inlet values, fromwhich comes, for the velocity, u∗(0) = 2(m0/q0)/u∞.The entrainment hypothesis, in the coflow case, leads as previously to a model whose formulation is tightly linkedto the physical behavior of the jet: the entrained mass flow rate is proportional to the shear between the jet and itsenvironment. It was seen that the jet regime behavior, similar to the free jet case, is well reproduced by it; the nextsection shows that the wake behavior can also be described successfully.N. Enjalbert et al. / C. R. Mecanique 337 (2009) 639–644 643Fig. 2. Comparison between the different model results and the experimental asymptotes, in the normalized space, of velocity excess u∗ − 1 andhalf-width b∗.Fig. 2. Comparaison entre les résultats de différents modèles et les asymptotes expérimentales, dans l’espace normalisé, de l’excès de vitesse u∗ −1et de la demi-largeur b∗ .4. Comparison with experimental results and other modelsA comparison of the present model with the experiments and other models is conducted, through the study of theasymptotic behaviors in the jet and wake regimes. The solution of (8) displays the following asymptotic behaviors:Jet regime (u∗  1, x∗  1): u∗ ∼ (2αx∗)−1, b∗ ∼ 2αx∗Wake regime (u∗  1): u∗ − 1 ∼ (3αx∗)−2/3, b∗ ∼ (3αx∗)1/3 (10)First of all, the power laws agree with the experimental results, in both the close and far regions. Moreover, the valueα = 0.056 mentioned in Section 2 leads to a very close fitting of the model with the measurements: the discrepancy isalmost zero in the jet regime for velocity and of 5% for the half-width, and respectively of 14% and 6.5% for u∗ − 1and b∗ in the wake regime. This is illustrated in Fig. 2.These results can be compared with those of a different model, established by Lee and Chu [8]. It is based on adifferent approach, called a spreading hypothesis, in which an assumption is laid upon the half-width b∗ itself, leadingto:u∗ db∗dx= α′(u∗ − 1) (11)Along with a momentum conservation equation, this model leads to the correct power laws in the limit regimes, butthe prefactors can only be accurately predicted in both the jet and the wake regimes if one assumes a top-hat profile644 N. Enjalbert et al. / C. R. Mecanique 337 (2009) 639–644contradictory to experimental observations. Although the same precision as our model is reached, the grounds of thespreading model may be considered as stronger hypotheses: in a jet regime, the equation on the half-width comesdown to the assumption b∗ ∼ x, which is directly the solution one seeks. Another model, also based on a spreadinghypothesis, is provided by Wang [6]: a perfect collapse of its results upon the experimental asymptotes is achieved,through the use of three parametrizing constants. Fig. 2 presents the results of these two alternative models, too.5. ConclusionIn this Note, we have presented an application of the entrainment hypothesis to the case of a jet in a coflow.It presents two advantages: it is based on only one constant, α, which is numerically well determined. It makes it amodel as simple in terms of parametrization as Lee and Chu’s spreading model, and still leads to a good precision, that,to date, only the addition of more constants to tune the solution can improve (as Wang’s model does). The entrainmenthypothesis also leads to a solution that is directly correct, once a Gaussian profile is assumed, in both the jet and wakeregimes. Also, the assumptions formulated by the presented model involve only the intrinsic characteristics of the jet,namely the mass and momentum fluxes. Parameters such as the width or axial velocity are merely used as the outputof the model, to make comparisons with experiments and other models possible. This formulation is quite robust andmay be inserted into more complex jet modelling, for e.g. the jet in a crossflow, interacting jets, or reacting jets.AcknowledgementsThe authors gratefully acknowledge the members of the ‘Samovar’ working group at SGR for many helpful dis-cussions, and in particular Jean-Marc Flesselles who initiated this work.References[1] F. Ricou, D. Spalding, Measurements of entrainment by axisymmetric turbulent jets, J. Fluid Mech. 11 (1961) 21–32.[2] W. Hawthorne, D. Weddell, H. Hottel, Mixing and combustion in turbulent gas jets, in: Third Symposium (International) on Combustion, 1949,p. 266.[3] Theory of combustion modelling, in: J. Rhine, R. Tucker, Modelling of Gas-Fired Furnaces and Boilers, British Gas, London, 1991.[4] K.M. Tacina, W.J. Dahm, Effects of heat release on turbulent shear flows. Part 1. A general equivalence principle for non-buoyant flows and itsapplication to turbulent jet flames, J. Fluid Mech. 415 (2000) 23–44.[5] B.R. Morton, G.I. Taylor, J.S. Turner, Turbulent gravitational convection from maintained and instantaneous sources, Proc. R. Soc. Lond. Ser. AMath. Phys. Eng. Sci. 234 (1956) 1.[6] H.J. Wang, Jet interaction in a still or co-flowing environment, PhD thesis, Department of Civil Engineering, Hong Kong Univ. of Science andTechnology, Hong Kong, 2000.[7] C. Bailly, G. Comte-Bellot, Turbulence, CNRS Éditions, Paris, 2003.[8] J. Lee, V. Chu, Turbulent Jets and Plumes – A Lagrangian Approach, Kluwer Academic Publishers, Boston, 2003.

An entrainment model for the turbulent jet in a coflow

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An entrainment model for the turbulent jet in a coflow

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