Research Article Analysis of Flow Evolution and …downloads.hindawi.com/journals/mpe/2015/891894.pdfa study in a backward-facing step con guration is shown in Barrios-Pi na et al

Research ArticleAnalysis of Flow Evolution and Thermal Instabilities in theNear-Nozzle Region of a Free Plane Laminar Jet

Hector Barrios-Piña,1 Hermilo Ramírez-León,2 and Carlos Couder-Castañeda3

1Tecnologico de Monterrey, Avenida General Ramon Corona 2514, 45201 Zapopan, JAL, Mexico2Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas Norte 152, Colonia San Bartolo, 07730 Ciudad de Mexico, DF, Mexico3Centro de Desarrollo Aeroespacial del Instituto Politecnico Nacional, Belisario Domınguez 22, 06010 Ciudad de Mexico, DF, Mexico

Correspondence should be addressed to Carlos Couder-Castaneda; [email protected]

Received 6 November 2014; Accepted 13 December 2014

Academic Editor: Oluwole Daniel Makinde

Copyright © 2015 Hector Barrios-Pina et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This work focuses on the evolution of a free plane laminar jet in the near-nozzle region. The jet is buoyant because it is driven bya continuous addition of both buoyancy and momentum at the source. Buoyancy is given by a temperature difference betweenthe jet and the environment. To study the jet evolution, numerical simulations were performed for two Richardson numbers:the one corresponding to a temperature difference slightly near the validity of the Boussinesq approximation and the other onecorresponding to a higher temperature difference. For this purpose, a time dependent numerical model is used to solve the fullydimensional Navier-Stokes equations. Density variations are given by the ideal gas law and flow properties as dynamic viscosity andthermal conductivity are considered nonconstant. Particular attention was paid to the implementation of the boundary conditionsto ensure jet stability andflow rates control.Thenumerical simulationswere also reproduced by using theBoussinesq approximationto find out more about its pertinence for this kind of flows. Finally, a stability diagram is also obtained to identify the onset of theunsteady state in the near-nozzle region by varying control parameters of momentum and buoyancy. It is found that, at the onsetof the unsteady state, momentum effects decrease almost linearly when buoyancy effects increase.

1. Introduction

Jet flow occurs in a variety of industrial applications such aspulverization, thermal isolation, film cooling, solid straight-ening, welding in aerodynamic, and hydrodynamic fields. Forthese applications, there exist two kinds of jet flow configura-tions: the axisymmetric jet, characterised by a circular nozzle,and the plane jet, characterised by a rectangular nozzle. Suchtypes of jet flows suddenly are three-dimensional and showsymmetric conditions that are object of different studies. Thestudy of a two-dimensional jet flow configuration generally isone of the stages to better understand some phenomena thatgovern three-dimensional jet flows. In terms of numericalmodelling, three-dimensional jet flows clearly are extremelymore complex and demand long computing time.

Some works on laminar jets are limited to analyticalor numerical solutions of the governing equations in thedeveloped region.The solutions use a change of variables withthe aim of ignoring the discharge conditions at the nozzle

orifice [1]. For nonisothermal free laminar jets, works havebeen devoted to study the plume away from its source [1–5]and in the initial near-nozzle region [1, 6].

In reality, jet flows are turbulent and have been analysedboth experimentally [7–9] and numerically [10–12]. Otherexperimental studies have analysed turbulent jet flow withvariable density by considering twomixed fluids [13]. An ana-lytical solution for the transition region, with buoyancy forcesand inertial forces of same order of magnitude, has beenaddressed by Yu et al. [14] in laminar regime. These authorshave introduced new parameters and used a change ofvariables. The equations they obtained were solved by takinginto account the boundary conditions and the two integrationconstants derived from the momentum conservation and theheat flux. These two constants replace the discharge condi-tions at the nozzle orifice andwere verified byMhiri et al. [15].

Despite the number of studies on plane jet flow, itscomplexity still makes the necessity of studying the influence

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 891894, 11 pageshttp://dx.doi.org/10.1155/2015/891894

2 Mathematical Problems in Engineering

of various physical parameters related to the jet evolution.When taking into account density variations, the problembecomes even more complex due to the interaction ofbuoyancy-momentum. Therefore, the jet flow with variabledensity is still a large open subject and this work focuseson the study of the evolution of such a flow in the near-nozzle region.The jet flow configuration studied here is a freevertical plane jet, discharged froma rectangular nozzle, wherethe aspect ratio width/thickness is considered significant. Fora nozzle of width 𝑙 and thickness 𝑒, the lateral expansion of thejet can be neglected when the aspect ratio is 𝑙/𝑒 ≥ 20. Thus,the flow can then be considered two-dimensional on average[16]: the average flow values in a given plane are identical inall other parallel planes. The present jet flow configurationis a kind of embedded or immersed jet, as it emerges in anenvironment constituted of the same fluid but colder. The jetflow is also free, since the boundary conditions theoreticallyare considered to be at the infinity.

In this study, the fully dimensional Navier-Stokes equa-tions are solved and no simplifying assumption is consideredon the role of pressure. The numerical simulations werealso reproduced by using the Boussinesq approximation tofind out more about its pertinence for this kind of flows.Different authors have focused on the criteria to validatethe Boussinesq approximation. One of the most cited worksabout this subject is the work of Gray and Giorgini [17],who have shown that the Boussinesq approximation is welladapted to natural convection flows under small temperaturegradients and negligible compressibility effects. For the caseof air, regarding density variations with temperature, Grayand Giorgini have shown that the Boussinesq approximationis true for temperature differences approximately less than28.6 K. More recently, other authors have carried out somestudies with the aim of defining the limitations of theBoussinesq approximation [18, 19].

A time dependent two-dimensional numerical model isused in the present work, where density variations are givenby the ideal gas law and flow properties as dynamic viscosityand thermal conductivity are considered nonconstant. Thenumerical approach is based on a second-order finite differ-ence formulation and the coupled set of equations is solved byusing an iterative predictor-corrector procedure at each timestep. Some numerical tests were also carried out to define theappropriate boundary conditions since stability andflow ratescontrol must be ensured.

2. Numerical Model

2.1. Governing Equations. The governing equations for con-servation of mass, momentum, and energy, in Cartesiancoordinates, are used in the following form:

𝜕𝜌

𝜕𝑡+𝜕 (𝜌𝑢𝑖)

𝜕𝑥𝑖

= 0, (1)

𝜕 (𝜌𝑢𝑖)

𝜕𝑡+𝜕 (𝜌𝑢𝑖𝑢𝑗)

𝜕𝑥𝑗

= −𝜕𝑃

𝜕𝑥𝑖

+𝜕𝜏𝑖𝑗

𝜕𝑥𝑗

− 𝜌𝑔𝛿𝑖2, (2)

𝜕𝑇

𝜕𝑡+𝜕 (𝑢𝑖𝑇)

𝜕𝑥𝑖

= (1 −𝑟

𝑐V)𝑇

𝜕𝑢𝑖

𝜕𝑥𝑖

−𝛾

𝜌𝑐𝑝

𝜕𝑞𝑖

𝜕𝑥𝑖

, (3)

where 𝑢𝑖is the velocity vector in the 𝑥

𝑖direction (in m/s),

𝑃 is the pressure (in Pa), 𝜌 is the density (in kg/m3), 𝑇 is thetemperature (in K),𝑅 is the specific gas constant (in J/(kg K)),𝑐V is the specific heat at constant volume (in J/(kg K)), 𝑐

𝑝is the

specific heat at constant pressure (in J/(kg K)), 𝛾 is the ratio ofheat capacities, and the viscous stress tensor 𝜏

𝑖𝑗(inN/m2) and

the heat flux 𝑞𝑖(in W/m2) along the 𝑥

𝑖direction are defined,

respectively, as

𝜏𝑖𝑗= 𝜇[

𝜕𝑢𝑖

𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖

−2

3𝛿𝑖𝑗

𝜕𝑢𝑘

𝜕𝑥𝑘

] , (4)

𝑞𝑖= −𝜅

𝜕𝑇

𝜕𝑥𝑖

, (5)

where𝜇 is the dynamic viscosity (in kg/(m s)), 𝜅 is the thermalconductivity (in W/(m K)), and 𝑖 = 𝑗 = 1, 2.

Viscous dissipation in energy equation is ignored accord-ing to low values of velocity. Additionally, the ideal gas law isused to compute density variations as

𝑃 = 𝜌𝑟𝑇, (6)

where 𝑟 = 287 J/(kg K) for air. Because of the temperature dif-ferences, the dynamic viscosity (𝜇) and thermal conductivity(𝜅) are computed by the Sutherland law:

𝜇 (𝑇) = 𝜇0(𝑇

𝑇0

)

3/2 𝑇0+ 𝑆𝜇

𝑇 + 𝑆𝜇

,

𝜅 (𝑇) = 𝜅0(𝑇

𝑇0

)

3/2𝑇0+ 𝑆𝜅

𝑇 + 𝑆𝜅

,

(7)

where 𝜇0= 𝜇(𝑇

0) and 𝜅

0= 𝜅(𝑇

0) are the dynamic viscosity

and the thermal conductivity at 𝑇0. For air, the following

values are considered: 𝜇0= 1.68 × 10

−5 kg/(m s), 𝜅0= 2.43 ×

10−2W/(m K), 𝑇

0= 273K, and the Sutherland constants are

𝑆𝜇= 110.4K and 𝑆

𝜅= 194K. This choice is valid for a range

of temperature values from 167K to 1900K approximately.

2.2. Numerical Approach. The solution method is an adapta-tion of a fully implicit formulation based on a time-accuratealgorithmproposed by Sewall andTafti [20].The local densityvariation is coupled with both temperature and pressurevariations, whereas other properties are only coupled withtemperature variations. No low Mach number assumption isused in this formulation. Sewall and Tafti [20] successfullyapplied this method to natural convection in a differentiallyheated square cavity flow and in a Poiseuille-Benard flowwithlarge temperature differences.

The numerical approach herein is based on a second-order finite difference formulation with a fully staggeredarrangement.The primitive variables as pressure, density, andtemperature are located at the middle of the cell and velocitycomponents are shifted at the middle of the cell sides. Thecoupled set of equations is solved using an iterative predictor-corrector procedure at each time step. During the predictionstep, the momentum conservation equation (2) and the

Mathematical Problems in Engineering 3

Table 1: Algorithm steps.

Step Computed or updated variable By applying1 𝑢

𝑚

𝑖, 𝑇𝑚, and 𝑃

𝑚𝑢𝑚

𝑖= 𝑢𝑛

𝑖, 𝑇𝑚 = 𝑇

𝑛, and 𝑃𝑚= 𝑃𝑛

2 Temperature 𝑇𝑚+1 Equation (3)3 Intermediate density 𝜌∗ Equation (6)4 Intermediate velocity 𝑢∗

𝑖Equation (2)

5 Pressure correction 𝜑 Equation (8)6 Pressure 𝑃𝑚+1 𝑃

𝑚+1= 𝑃𝑚+ 𝜑

7 Corrected density 𝜌𝑚+1 Equation (6)

8 Corrected velocity 𝑢𝑚+1𝑖

𝑢𝑚+1

𝑖=

𝜌∗𝑢∗

𝑖

𝜌𝑚+1−

𝛿𝑡

𝜌𝑚+1

𝜕𝜑

𝜕𝑥𝑖

The convergence is verified by the averaged L2 norm as follows: if ‖𝜓𝑚+1 −𝜓𝑚‖ ≤ 𝜔 is false, then𝜓𝑚 = 𝜓𝑚+1 and return to Step 2; else,𝜓𝑛 = 𝜓𝑚+1 and proceedto next time step (Step 1). Here 𝜓 = {𝑢𝑖, 𝜌, 𝑇} and 𝜔 = 1 × 10

−8.

energy equation (3) are advanced implicitly using the Crank-Nicolson method for the advection and diffusion terms. Thedivergence term in (3) is treated implicitly. The numericalsolution of these equations is considered satisfactory whenthe residual, summed over the whole calculation domain, issmaller than 1 × 10

−8.The coupling between the mass and the momentum

conservation equations is completed through a pressure cor-rection (𝜑), determined from the following ellipticHelmholtzequation:

𝜕2𝜑

𝜕𝑥𝑖𝜕𝑥𝑖

−𝜑

𝛿𝑡2𝑟𝑇𝑚+1=

𝜌∗− 𝜌𝑛

𝛿𝑡2+

1

𝛿𝑡

𝜕 (𝜌∗𝑢∗

𝑖)

𝜕𝑥𝑖

, (8)

where 𝛿𝑡 is the discrete time step (in s) and the superscripts𝑛 and 𝑚 denote, respectively, time (or outer) and inneriteration.The symbol (∗) means an intermediate or predictedvalue. The solution of the elliptic Helmholtz equation (8) isobtained using a V-cycle multigrid method with the strongimplicit procedure ILU as smoother [21]. The numericalsolution of this equation is assumed to be converged whenthe normalized residual, summed over the whole calculationdomain, is smaller than 1 × 10

−5. The different steps ofthe algorithm are summarized in Table 1. The numericalapproach is further documented in Barrios-Pina [22] anda study in a backward-facing step configuration is shownin Barrios-Pina et al. [23]. The numerical simulations werecarried out on a cluster IBM-xeon, 14xbi-quadX3550 3GHZ,16 and 32GbRAM.

3. Geometry of the Flow and Initial Conditions

Figure 1 illustrates the configuration of the jet flow understudy. The flow is a laminar coflowing plane jet from anozzle of thickness 𝑒. The length 𝐿 and the height 𝐻 limitthe study domain. The fluid of the jet and the ambient isconsidered air. The jet temperature 𝑇jet is greater than theambient temperature 𝑇

𝑎(positive buoyancy). The Reynolds

A

B

Coflow U∞ Coflow U∞

C

D

x1

x2

L

e

Ta

Ujet

Tjet

g

H

Figure 1: Geometry of the plane jet flow of study.

and Richardson numbers which characterise the flow aredefined as

Re =Δ𝑈jet𝑒

]jet,

Ri =𝑔𝑒 (1 − 𝑆)

𝑆Δ𝑈2jet,

(9)

where 𝑔 is the gravity, ]jet is the jet kinematic viscosity (inm2/s), Δ𝑈jet = 𝑈jet − 𝑈co, 𝑈jet is the jet velocity (in m/s), and𝑈co is the coflow velocity (in m/s).The relative jet density 𝑆 isdefined by the relationship jet density 𝜌jet to reference density𝜌𝑟(in kg/m3).The coflow velocitymust be smaller than the jet

velocity to not affect its natural expanding into the ambient.Therefore, we considered a relationship 𝑈co/Δ𝑈jet = 0.1 [10].A coflow velocity is considered here to ensure numericalstability and to control jet propagation out of the domain.

For the initial thermal state, the aerostatic equilibrium ofthe atmosphere is considered, where pressure, temperature,and density change with the elevation. From the equation offluid statics,

𝜕𝑃𝑎

𝜕𝑥2

+ 𝜌𝑎𝑔 = 0, (10)


and from the following expression of density,

𝜌𝑎= 𝜌𝑟(𝑃

𝑃𝑟

)

1/𝛾

, (11)

the vertical variations of pressure, density, and temperatureas function of elevation are written as

𝑃𝑎= 𝑃𝑟(1 −

𝑥2

ℎ0

)

𝛾/(𝛾−1)

,

𝜌𝑎= 𝜌𝑟(1 −

𝑥2

ℎ0

)

1/(𝛾−1)

,

𝑇𝑎= 𝑇𝑟(1 −

𝑥2

ℎ0

) ,

(12)

where the index 𝑟 implies properties at ground level definedat 𝑥2

= 0 (properties of reference), the index 𝑎 denotesproperties of the adiabatic atmosphere in steady state, and ℎ

0

is defined as ℎ0= 𝛾𝑃𝑟/[(𝛾 − 1)𝜌

𝑟𝑔].

This state of reference, which is adiabatic and aerostatic,considers an environment where pressure, density, and tem-perature are not constants and varywith elevation, as happensin the real lower atmosphere.

4. Boundary Conditions

From Figure 1, three different kinds of boundary conditionsare identified: the inflow boundary condition (AD), theoutflow boundary condition (BC), and the lateral boundaryconditions (AB and CD). The treatment of boundary con-ditions is not trivial in this kind of flow configuration. Thecontrol of flow rates through the open boundaries and thepressure condition at these boundaries are the cause of thedifficulty. Numerous works in the literature have focused onthis topic; however, there is not a general methodology toensure good prior boundary conditions.

The characteristic boundary conditions applied byThompson [24, 25] and later discussed and modified byPoinsot and Lele [26] are widely used for the treatment ofopen boundaries, especially when acoustic waves appear inthe flow like in supersonic flow regimes [27].The basic idea ofcharacteristic boundary conditions is to split the convectiveterms in the boundary-normal direction into several waveswith different characteristic velocities and then expressunknown incoming waves as a function of known outgoingwaves. Thus, to apply this kind of boundary conditions, thestrength of the waves entering the computational domainmust be determined, which is not trivial [28]. Moreover, thecharacteristic boundary conditions particularly work well innumerical algorithms based on density, which generally useexplicit schemes. For algorithms using implicit formulationsbased on pressure, like the present numerical model, theimplementation of the characteristic boundary conditions ismore complex.

Another kind of boundary conditions widely used for thetreatment of open boundaries is the absorption boundarycondition known as PML (Perfectly Matched Layer) [29].

The PML technique considers an absorbing zone which istheoretically reflectionless for multidimensional linear wavesat any angle of incidence and any frequency. Althoughthe PML technique itself is relatively simple when it isviewed as a complex change of variables in the frequencydomain, it is important to note that, to yield stable absorbingboundary conditions, the phase and group velocities of thephysical waves supported by the governing equations mustbe consistent and be in the same direction. Thus, specialtreatment needs to be performed to improve absorptionof waves or the different flows that impinge the base ofthe PML at grazing incidence and travel inside the PMLover long time periods. Moreover, to implement the PMLboundary condition, additional mesh points are necessary toconsider the absorbing region. The fact of considering moremesh points will result in a CPU-time increase, which isnot desirable, especially for fine mesh numerical simulations.Some other techniques have been developed to improvethe implementation of the PML boundary conditions, forexample, convolutional formulation of the Perfectly MatchedLayer [30].

Given the complexities mentioned above, the subsonicregime of the present jet flow, and expected negligible acous-tic wave effects, some numerical tests using typical boundaryconditions like Neumann, Dirichlet, and convective wereperformed to define the one to be imposed on each side ofthe domain (see Figure 1). Thus, the boundary conditionsretained here are the following.

Inflow Conditions (AD). For the vertical velocity component𝑢2and for temperature, hyperbolic tangent profiles were

imposed:

𝑢2(0 ≤ 𝑥

1≤ 𝐿, 0) =

𝑈jet + 𝑈co

2+Δ𝑈jet

2tanh(𝑥1

2𝜃) ,

𝑇 (0 ≤ 𝑥1≤ 𝐿, 0) =

𝑇jet + 𝑇𝑟

2+Δ𝑇jet

2tanh(𝑥1

2𝜃) ,

(13)

where 𝑇jet is the jet temperature (in K), 𝑇𝑟is the reference

temperature at ground level (in K), Δ𝑇jet = 𝑇jet − 𝑇𝑟, 𝜃 is the

momentum thickness (in m), and is imposed to be 𝑒/20. Inthis way, the initial profiles adopt a flattened shape and aresymmetric with respect to the jet axis.The horizontal velocitycomponent 𝑢

1is considered to be zero.

For pressure, the equation of fluid statics is imposed:

𝜕𝑃

𝜕𝑥2

= −𝜌𝑔, (14)

and density is obtained from the ideal gas law.

Outflow Boundary (BC). At the upper boundary, convectiveconditions, also known as Sommerfeld radiation conditions,are considered. Convective conditions relate the temporalderivative and the normal derivative of the unknown incase of boundaries far away from sources and normal tothe propagating waves. Its application in cases of nonper-pendicular wave incidence and boundaries close to sourcesis doubtful, and therefore, this boundary condition has to


be understood as an approximate boundary condition toavoid wave reflection in such cases. However, the numericalimplementation of this boundary condition is not difficultcompared with the boundaries mentioned before. The upperboundary condition is considered as follows:

𝜕𝜙

𝜕𝑡+ 𝐶𝜙

𝜕𝜙

𝜕𝑥2

= 0, (15)

where 𝜙 = {𝑢1, 𝑢2, 𝑇} is the variable computed at the

boundary and 𝐶𝜙is the phase velocity. To estimate the phase

velocity, the Orlanski method [31] is used. First the phasevelocity is computed locally with

𝐶𝜙= −

[𝜙𝑛

𝑁2−1− 𝜙𝑛−2

𝑁2−1]

[𝜙𝑛𝑁2−1

+ 𝜙𝑛−2

𝑁2−1− 𝜙𝑛−1

𝑁2−2]

Δ𝑥𝑁2

2Δ𝑡, (16)

where𝑁2 denotes the last mesh point along the 𝑥2direction,

normal to the boundary, and 𝑛 is the time iteration. Once thephase velocity is computed, the following expression is usedto calculate 𝜙 at 𝑛 + 1:

𝜙𝑛+1

𝑁2=

[1 − (Δ𝑡/Δ𝑥𝑁2

) 𝐶𝜙]

[1 + (Δ𝑡/Δ𝑥𝑁2

) 𝐶𝜙]𝜙𝑛−1

𝑁2

+2 (Δ𝑡/Δ𝑥

𝑁2) 𝐶𝜙

[1 + (Δ𝑡/Δ𝑥𝑁2

) 𝐶𝜙]𝜙𝑛

𝑁2−1,

(17)

where 0 ≤ 𝐶𝜙< Δ𝑥𝑁2

/Δ𝑡must be satisfied. At the limit 𝐶𝜙=

Δ𝑥𝑁2

/Δ𝑡, expression (17) yields

𝜙𝑛+1

𝑁2= 𝜙𝑛

𝑁2−1, (18)

and when 𝐶𝜙= 0,

𝜙𝑛+1

𝑁2= 𝜙𝑛−1

𝑁2, (19)

or

𝜙𝑛+1

𝑁2= imposed value. (20)

In this way, no information comes from the internal solution;therefore, this condition must be considered as informationimposed from outside (by imposing an appropriate value ora value from the previous time step). It is noticeable thatfor each variable 𝜙 corresponds to an expression of form(15); thus, there exist different phase velocities that must becomputed separately for each variable.

To keep consistency with the inflow boundary conditionfor pressure, the equation of fluid statics is still imposed:

𝜕𝑃

𝜕𝑥2

= −𝜌𝑔, (21)

and density is obtained from the ideal gas law.

Lateral Conditions (AB and CD). The lateral conditionsimplemented here allow the contribution of mass acrossboundaries and also the pressure control. These bound-ary conditions are known in the literature as traction-free

boundary conditions. This kind of boundary conditionshas been used for incompressible jet flows [32, 33]. In thepresent work, the traction-free boundary conditions havebeen reformulated to consider compressibility. The principleis

𝜎𝑖𝑗⋅ 𝑛𝑗= 0, (22)

where the stress tensor 𝜎𝑖𝑗is

𝜎𝑖𝑗= −𝑃𝛿

𝑖𝑗−2

3𝜇𝜕𝑢𝑘

𝜕𝑥𝑘

𝛿𝑖𝑗+ 𝜇(

𝜕𝑢𝑖

𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖

) , (23)

and 𝑛𝑗is the unit normal to the boundary.Thus, the boundary

conditions for the velocity components are𝜕𝑢1

𝜕𝑥1

=𝑃

2𝜇+1

3

𝜕𝑢𝑘

𝜕𝑥𝑘

, (24)

𝜕𝑢2

𝜕𝑥1

= −𝜕𝑢1

𝜕𝑥2

. (25)

The boundary condition for pressure is obtained from thefollowing mass conservation equation

𝜕𝜌

𝜕𝑡+𝜕 (𝜌𝑢𝑖)

𝜕𝑥𝑖

= 0, (26)

combined with (24) as follows:

𝑃 = −2𝜇

𝜌(𝜕𝜌

𝜕𝑡+𝜕𝜌𝑢2

𝜕𝑥2

+ 𝑢1

𝜕𝜌

𝜕𝑥1

) −2

3𝜇𝜕𝑢𝑘

𝜕𝑥𝑘

. (27)

The traction-free boundary conditions are stable when thecell Reynolds number is less than 2 at the boundary [32].This restriction can be satisfied by mesh refinement at thenear boundary region. However, it may not be necessary forthe present configuration because of the low velocity valuesobtained away from the jet influence.

The lateral boundary condition imposed for temperatureis the homogeneous Neumann boundary condition of theform

𝜕𝑇

𝜕𝑥1

= 0. (28)

Finally, density is obtained from the ideal gas law.

5. Initial Conditions andSimulation Parameters

For the numerical simulations, 𝐿 and 𝐻 are 𝐿 = 15𝑒 and𝐻 = 20𝑒, respectively. The mesh resolution is 195 × 259 cellpoints and is refined in shear zones and near boundaries (seeFigure 2).

The initial conditions given by (12) are imposed for initialpressure, density, and temperature, respectively. The velocitycomponents are zero all over the computational domain:𝑢1(𝑥1, 𝑥2) = 𝑢

2(𝑥1, 𝑥2) = 0. Convergence is supposed to

be reached when 𝜔 is less than 1 × 10−5 and the time step

is 1 × 10−4 s, in all cases.

The state of reference at ground level defined at 𝑥2= 0

is 𝑃𝑟= 1.013 × 10

5 Pa, 𝑇𝑟= 293K, 𝜌

𝑟= 𝑃𝑟/(𝑟𝑇𝑟), and 𝑔 =

9.81m/s2.


Table 2: Physical parameters of the numerical simulations.

Case Δ𝑇jet (K) 𝑆 Re RiS1 30 0.9071 300 0.0364S2 100 0.7455 200 0.1383

0.0 5.0 10.0 15.0 20.0

x1/e

0.0

5.0

10.0

15.0

x2/e

Figure 2: Numerical grid used for the flow jet configuration.

6. Results and Discussions

The numerical simulations (cases S1 and S2) focus on theanalysis of the jet evolution by taking into account twodifferent Richardson numbers. The first one corresponds toa temperature difference Δ𝑇jet where the pertinence of theBoussinesq approximation may be questionable. The secondcase corresponds to a temperature difference significantlyhigher, where the use of the Boussinesq approximation isno more applicable. The interest of this analysis is to testthe behaviour of the boundary conditions and to evaluateby comparison of results the relevance of using the presentnumerical model against the Boussinesq approximation.

Results for the analysis of instabilities at the near-nozzleregion are finally shown, where a stability diagram wasobtained to specify the onset of the unsteady state. Forthis reason, some numerical simulations were carried outby considering the Reynolds and the temperature differenceΔ𝑇jet, as control parameters.

6.1. Analysis of the Jet Evolution. The physical parametersconsidered in the numerical simulations are shown in Table 2.

We observe that for the physical parameters considered inthe numerical simulations the flow is unsteady and presentstwo-dimensional sinusoidal oscillations, known as sinuousmode. Thus, the following discussion is based on averagedvalues calculated with an integration time of 10 s.

Figure 3 shows the time-averaged vertical velocity at thejet centreline for cases S1 and S2, Figures 3(a) and 3(b),respectively. In both cases, the centreline vertical velocityremains constant in a region close to the nozzle. In thisregion, called potential cone, the inertial forces are dominantover the buoyancy forces. The length of the potential cone

in case S2 is smaller than in case S1. To a higher height,the centreline velocity increases to a maximum magnitude,which implies that buoyancy forces become dominant overthe inertial forces: this zone is known as plume zone. Themaximum velocity obtained in case S2 is greater than in caseS1. Beyond the plume zone, the centreline velocity decreaseswith height, which implies that inertial forces then becomepreponderant. By comparing these results against thoseobtainedwith the Boussinesq approximation, we observe thatthe velocity is overestimated when using this approximation.The difference between the maximum velocity magnitudesin case S1 (Figure 3(a)) is less important than in case S2(Figure 3(b)).

Figure 4 illustrates the decay curve at the centreline ofthe time-averaged temperature. We observe in both cases aregion where the temperature remains constant. This is theregion of the temperature cone. According to Figures 4(a)and 4(b), the height of the temperature cone is almost thesame for both cases. The Boussinesq approximation slightlyoverestimates this height in case S1 and it is even moresignificant for case S2.

Concerning the effects of boundary conditions, in case S1both centreline velocity and temperature normally developto the outflow (the upper boundary). However, in caseS2 where the temperature difference is large, there is anevident problem with the boundary conditions due to theunexplained increase of velocity and temperature.This effect,inherent to such boundary conditions, could be dissipatedeither bymesh refinement near boundary or by implementinga buffer zone.

Figure 5 shows the time-averaged vertical velocity profilesat three different heights: at the nozzle level, 5𝑒, and 10𝑒. Anearly flat shape characterises the lower velocity profile. Awayfrom the nozzle, the velocity profiles show a Gauss-bell form,which gradually expand laterally as height increases. For aheight of 5𝑒, the velocity increases at the jet centre, and thenit decreases, as can be observed in the 10𝑒 velocity profile.

For both cases, in the near-nozzle region up to 5𝑒, thevelocity profiles obtained with the Boussinesq approximationare similar to those computed without approximation. How-ever, as height increases beyond 5𝑒, the maximum velocitymagnitude at the jet centre is overestimated by the Boussinesqapproximation for case S1. Moreover, for case S2 the profileobtained by the Boussinesq approximation is more flattenedthan that computed without approximation.

Longitudinal temperature profiles are shown in Figure 6.Temperature profiles imposed at the inflow adopt similarflat shape compared to the velocity profile. In both casesS1 and S2, the temperature profiles gradually spread asheight increases; see Figures 6(a) and 6(b), respectively. Thetemperature at the jet centre also decreases with height. Thetemperature profiles obtained with the Boussinesq approxi-mation are similar to those obtained without approximationin the near-nozzle region (5𝑒); however, an overestimation oftemperature at the jet centre is evident, especially when usingBoussinesq approximation for case S2. Far away from thenozzle, the temperature profiles obtained with and withoutthe Boussinesq approximation are closed for case S1. On theother hand, a significant difference between both profiles


Δu2/Δ

Uje

t

10−1

100

101

0.8

1.0

1.2

1.4

1.6

x2/e

BoussinesqNon-Boussinesq

(a) Case S1, Ri = 0.0364Δu2/Δ

Uje

t

10−1

100

101

0.8

1.0

1.2

1.4

1.6


x2/e

(b) Case S2, Ri = 0.1383

Figure 3: Centerline time-averaged velocity.

ΔT/Δ

Tje

t

0.2

0.4

0.6

0.8

1.0

10−1

100

101


x2/e

(a) Case S1, Ri = 0.0364

ΔT/Δ

Tje

t

0.2

0.4

0.6

0.8

1.0

10−1

100

101


x2/e

(b) Case S2, Ri = 0.1383

Figure 4: Centerline time-averaged temperature.

obtained for case S2 can be observed in Figure 6(b). Theprofile obtained with the Boussinesq approximation showstwo peaks or maximum temperature values, whereas theprofile without approximation has a Gaussian shape with asingle maximum temperature value.

Finally, Figure 7 shows the instantaneous temperaturefields obtained without approximation for both cases. Thepresence of the temperature cone which rises in the near-nozzle region is observed. In the plume zone, the instabilities

develop in a sinuousmode,where the plume ismore extendedfor case S2.

6.2. Analysis of the Onset of Instabilities at the Near-NozzleRegion. This section focuses on the unsteady-state onsetat the near-nozzle zone. For this purpose, a number ofsimulations were carried out by considering two controlparameters: the Reynolds number, which characterises theintensity of inertial forces, and the temperature difference


Δu2/Δ

Uje

t

5 6 7 8 9 10 11 12 13 14 15

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x1/e

0e Boussinesq5e Boussinesq10e Boussinesq

0e

5e

10e

(a) Case S1, Ri = 0.0364Δu2/Δ

Uje

t

5 6 7 8 9 10 11 12 13 14 15

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


0e

5e

10e

x1/e

(b) Case S2, Ri = 0.1383

Figure 5: Time-averaged vertical velocity profiles at different heights.

ΔT/Δ

Tje

t

0.0

0.2

0.4

0.6

0.8

1.0

5 6 7 8 9 10 11 12 13 14 15

x1/e


0e

5e

10e

(a) Case S1, Ri = 0.0364

ΔT/Δ

Tje

t

0.0

0.2

0.4

0.6

0.8

1.0

5 6 7 8 9 10 11 12 13 14 15

x1/e


0e

5e

10e

(b) Case S2, Ri = 0.1383

Figure 6: Time-averaged temperature profiles at different heights.

Δ𝑇jet, which characterises the intensity of buoyancy forces.To observe the occurrence of instabilities on the flow, a timesignal analysis of the velocity components was carried out atdifferent points distributed along the symmetry jet axis.

The Reynolds number for which the flow becomesunsteady, to a given temperature difference, is assumed whenthe maximum amplitudes of the velocity time series were

greater than 0.01%of the jet velocity at the nozzle,𝑈jet.Withinthe range of Reynolds numbers considered, only the sinuousmode was observed on the flow.

Figure 8 shows the stability diagram obtained from thenumerical simulations. The diagram graphs the Reynoldsnumber versus the temperature differences. The criticalReynolds number is about 273 for Δ𝑇jet = 0. Then, as


0 5 10 15 20

0

5

10

15

x1/e

x2/e

(a) Case S1, Ri = 0.0364

0 5 10 15 20

0

5

10

15

x1/e

x2/e

(b) Case S2, Ri = 0.1383

Figure 7: Instantaneous isotherms of the hot jet flow under study. 15 isotherms uniformly distributed between (a) 294K and 322K and (b)300K and 390K.

0 5 10 15 20 25 30

ΔTjet (K)

0

50

100

150

200

250

300

Re

Steady stateUnsteady state

Steadiness

Unsteadiness

Figure 8: Stability diagram, represented by the critical Reynoldsnumber as a function of the temperature difference Δ𝑇jet, whereRe = Δ𝑈jet𝑒/]jet.

the temperature difference increases, the buoyancy forcesbecome progressively dominant against the inertial forces.This fact causes the onset of the unsteady state for Reynoldsnumbers considerably lower than the isothermal criticalReynolds number. For a temperature difference about 25K,the flow is unsteady for a very low value of the Reynoldsnumber (Re ≈ 5). This allows us to conclude that beyondΔ𝑇jet = 25K, which corresponds to a high Richardsonnumber (Ri = 28.87), the flow is governed by buoyancyforces, whereas the inertial forces are nearly negligible. In

this case, the Reynolds number does not seem to be arelevant parameter to characterise the flow. Consequently,since the driving force is mainly due to buoyancy (purenatural convection), the use of the Grashof number seems tobe more appropriate to study such a flow.

The onset of instabilities shown in Figure 8 is supposedto be reached by considering some tolerance. This toleranceis illustrated by black dots and unfilled triangles. The blackdots denote the Reynolds number for which the flow is steady,whereas the unfilled triangles show the Reynolds number forwhich the flow is unsteady.The region under the black dots ischaracteristic of the steady flow regime and the region abovethe unfilled triangles corresponds to an unsteady flow regime.It can be observed that the Reynolds number based on the jetvelocity (𝑈jet) and the nozzle thickness (𝑒) loses its relevancein a system of pure natural convection.

7. Conclusions

A free vertical plane jet under large temperature gradients ispresented.The first tests allowed us to define the best adaptedboundary conditions for the flow under study. At the outflowboundary, the convective conditions were implemented andthe Orlanski method was used to compute the phase velocity.The lateral boundaries were treated by using traction-freeboundary conditions and were reformulated to considerdensity variations.

Two numerical simulations were conducted for studyingthe jet evolution and boundary conditions by using differentRichardson numbers. The convective conditions imposed atthe upper boundary (outflow) show an undesirable effect ina zone close to the boundary, especially when the Richardsonnumber is important. However, this undesirable effect canbe attenuated either by a mesh refinement near boundary orby implementing a buffer zone. In any case, this boundaryshould be pushed as far as possible from the nozzle.


A comparison with results from the Boussinesq approx-imation was also conducted to discuss the relevance of thepresent numerical model which solves the complete govern-ing equations. It appears that the Boussinesq approximationoverestimates the velocity and temperature. Disagreementsbetween the results obtained with and without Boussinesqapproximation were observed, especially when the temper-ature difference becomes significant (on the order of 100K).

Finally, an analysis of the onset of unsteadiness in thenear-nozzle region shows that the two control parame-ters need to be considered: the Reynolds number and thetemperature difference Δ𝑇jet. For the unsteady state, thecritical Reynolds number decreases almost linearly when thetemperature difference increases. Beyond Δ𝑇jet = 25K, theflow is essentially governed by the buoyancy forces and theinertial forces are nearly negligible.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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Research Article Analysis of Flow Evolution and …downloads.hindawi.com/journals/mpe/2015/891894.pdfa study in a backward-facing step con guration is shown in Barrios-Pi na et al