Turbulence in Buoyant Jets using an Integral Flux FormulationZone Of Flow Establishment (ZOFE)...

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Turbulence in Buoyant Jets using an Integral Flux Formulation Jonathan T. Schwalbe, David B. Gouldey, and Joseph A. Hopper The MITRE Corporation, McLean, VA 22102, USA (Dated: December 10, 2012) The turbulent kinetic energy (κ), and eddy dissipation rate () for an axisymmetric turbulent buoyant jet are numerically determined. An outline of the derivation of the governing equations for these quantities from the time-averaged Navier-Stokes Equations is given. The Boussinesq approx- imation is adopted. An integral model is used to reduce the system of partial differential equations to one of ordinary differential equations by assuming Gaussian cross sections and quantities corre- sponding to the flux of κ and are found. This turbulence model is coupled with a well known model for the mean-field behavior of buoyant jets. Results from the integration of the system of equations are given as well as comparisons to experimental data for air/air jets which show agreement with the predicated output of the model. PACS numbers: I. INTRODUCTION The dynamics of the effluent air from the exhaust stack of a powerplant, manufacturing facility, or other struc- ture can have an impact on the environment, municipal planning, as well as aviation [1]. In the latter half of the last century, up through today, much work has been done to model this behavior. An excellent summary can be found in [2]. While many techniques have been adopted to examine this phenomena, of primary interest to this report are the integral models that have been created to model the behavior of jets and plumes [3, 4]. Through the past couple decades, these integral models have become more complex and have evolved to deal with increasingly general environmental conditions [5, 6]. Integral models can be derived by assuming a cross- sectional profile for the average quantities. Typical pro- files leveraged are either the Gaussian distribution [2], which fits well with observation or a top-hat profile [7], which in many cases is less complicated. These models typically only calculate mean fluxes and report turbulent quantities by using constitutive equations. The experi- ments of Papanicolaou and List [8] show that turbulent kinetic energy accounts for 16% of the momentum and Jirka [2] points out that turbulent momentum and scalar fluxes are typically about 10% of the mean fluxes. Many studies do not directly evolve the turbulent quantities using their respective transport equations. The justification for this is that the turbulent quanti- ties are proportional to the mean quantities so there is no need for a separate representation [2]. However, Wang [9] argues that the turbulent quantities may vary between different regions of the buoyant jet and with different environmental conditions. Wang accounts for the turbu- lent quantities by treating them as an empirically varying percentage of the mean quantities, whereas other studies have used a fixed percentage [10]. The purpose of this paper is to extend the integral method of Reference [2] by adding evolution equations for the turbulent kinetic energy (κ) and eddy dissipation rate (). These equations are coupled to those of the mean flow to provide accurate predictions of the turbu- lent kinetic energy and the eddy dissipation rate in a va- riety of conditions. To the authors knowledge no integral model exists in the literature that evolves the turbulent kinetic energy separately or calculates the eddy dissipa- tion rate using an integral flux formulation. In Section II, the problem statement will be given and relevant equa- tions, coefficients, and boundary conditions will be devel- oped. For the sake of brevity, many of the details are left to the cited literature (in particular the equations for the mean flow behavior), leaving the novel turbulence equa- tions to this presentation. In Section III the results of the integration of the evolution equations will be compared to experimental data. II. PROBLEM FORMULATION Consider a cylindrical exhaust stack from a facility such as a power plant or manufacturing installation which has diameter D depicted in Figure 1. For the purposes of this study the height of the exhaust stack is of no relevance. For the remainder of this paper, this will be referred to as ”the stack”. s b x z y D ua FIG. 1: A schematic of the problem.

Transcript of Turbulence in Buoyant Jets using an Integral Flux FormulationZone Of Flow Establishment (ZOFE)...

Page 1: Turbulence in Buoyant Jets using an Integral Flux FormulationZone Of Flow Establishment (ZOFE) described in [2], where the uniform e ux ow turns to fully developed jet ow. The top

Turbulence in Buoyant Jets using an Integral Flux Formulation

Jonathan T. Schwalbe, David B. Gouldey, and Joseph A. HopperThe MITRE Corporation, McLean, VA 22102, USA

(Dated: December 10, 2012)

The turbulent kinetic energy (κ), and eddy dissipation rate (ε) for an axisymmetric turbulentbuoyant jet are numerically determined. An outline of the derivation of the governing equations forthese quantities from the time-averaged Navier-Stokes Equations is given. The Boussinesq approx-imation is adopted. An integral model is used to reduce the system of partial differential equationsto one of ordinary differential equations by assuming Gaussian cross sections and quantities corre-sponding to the flux of κ and ε are found. This turbulence model is coupled with a well known modelfor the mean-field behavior of buoyant jets. Results from the integration of the system of equationsare given as well as comparisons to experimental data for air/air jets which show agreement withthe predicated output of the model.

PACS numbers:

I. INTRODUCTION

The dynamics of the effluent air from the exhaust stackof a powerplant, manufacturing facility, or other struc-ture can have an impact on the environment, municipalplanning, as well as aviation [1]. In the latter half of thelast century, up through today, much work has been doneto model this behavior. An excellent summary can befound in [2]. While many techniques have been adoptedto examine this phenomena, of primary interest to thisreport are the integral models that have been created tomodel the behavior of jets and plumes [3, 4]. Through thepast couple decades, these integral models have becomemore complex and have evolved to deal with increasinglygeneral environmental conditions [5, 6].

Integral models can be derived by assuming a cross-sectional profile for the average quantities. Typical pro-files leveraged are either the Gaussian distribution [2],which fits well with observation or a top-hat profile [7],which in many cases is less complicated. These modelstypically only calculate mean fluxes and report turbulentquantities by using constitutive equations. The experi-ments of Papanicolaou and List [8] show that turbulentkinetic energy accounts for 16% of the momentum andJirka [2] points out that turbulent momentum and scalarfluxes are typically about 10% of the mean fluxes.

Many studies do not directly evolve the turbulentquantities using their respective transport equations.The justification for this is that the turbulent quanti-ties are proportional to the mean quantities so there isno need for a separate representation [2]. However, Wang[9] argues that the turbulent quantities may vary betweendifferent regions of the buoyant jet and with differentenvironmental conditions. Wang accounts for the turbu-lent quantities by treating them as an empirically varyingpercentage of the mean quantities, whereas other studieshave used a fixed percentage [10].

The purpose of this paper is to extend the integralmethod of Reference [2] by adding evolution equationsfor the turbulent kinetic energy (κ) and eddy dissipationrate (ε). These equations are coupled to those of the

mean flow to provide accurate predictions of the turbu-lent kinetic energy and the eddy dissipation rate in a va-riety of conditions. To the authors knowledge no integralmodel exists in the literature that evolves the turbulentkinetic energy separately or calculates the eddy dissipa-tion rate using an integral flux formulation. In Section II,the problem statement will be given and relevant equa-tions, coefficients, and boundary conditions will be devel-oped. For the sake of brevity, many of the details are leftto the cited literature (in particular the equations for themean flow behavior), leaving the novel turbulence equa-tions to this presentation. In Section III the results of theintegration of the evolution equations will be comparedto experimental data.

II. PROBLEM FORMULATION

Consider a cylindrical exhaust stack from a facilitysuch as a power plant or manufacturing installation whichhas diameter D depicted in Figure 1. For the purposesof this study the height of the exhaust stack is of norelevance. For the remainder of this paper, this will bereferred to as ”the stack”.

s

b

x

z

yD

ua

FIG. 1: A schematic of the problem.

mastro
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The discharge from the stack is moving at a velocity U0,and has a uniform density of ρ0. An ambient uniformvelocity field may be present (ua) which will orient theplume at an angle of θ(s) with respect to the x-axis wheres is a coordinate representing the length of the centerlineof the plume. The ambient conditions are given by den-sity ρa(z), pressure pa(z), and temperature Ta(z). Thereexists a regions directly above the stack, known as theZone Of Flow Establishment (ZOFE) described in [2],where the uniform efflux flow turns to fully developed jetflow. The top of the ZOFE is where boundary conditionsare to be specified.

A. Governing equations

The flow of fluid out of the exhaust stack is governedby conservation of mass, momentum, and thermal energy,along with an equation of state relating density and tem-perature. In order to make this problem tractable, theconvention is to separate the steady (Ui) and fluctuating(ui) components of the velocity field. The Latin sub-script i indicates spatial components of a vector; unlessotherwise noted, summation is assumed for all repeatedindices. For completeness, some of the relevant equationswill be given here. Much of the derivation of the govern-ing equations will be omitted from this presentation; thereader is referred to [11, 12] for complete details. The re-sulting time-averaged equations for conservation of mass,momentum, and thermal energy respectively are

∂ρUi∂xi

= 0, (1)

ρD

Dt(Ui) =

∂xj

(µ∂Ui∂xj− ρuiuj

)− ∂p

∂xi+ ρgi, (2)

and

cpρD

Dt(T ) =

∂xi

(λ∂T

∂xi− cpρuiT ′

)(3)

where D/Dt is the material derivative. In Eq. (2) µ is themolecular viscosity of the fluid. In Eq. (3) cp is the spe-cific heat at constant pressure and λ is the diffusivity ofthe medium. The term uiuj represents an unknown cor-relation between fluctuating velocities originating fromthe nonlinearity of the Navier-Stokes equations and theaveraging process. This term is commonly known as theturbulent Reynolds’ stress. The thermal energy equationalso contains an unknown term representing the trans-port of heat due to turbulent processes, ρuiT ′. Here T ′

is the turbulent fluctuation in the temperature.

In order to close this system, governing equations forthe quantities uiuj and uiT ′ must be formulated.

1. Equations for κ and ε

The turbulent kinetic energy (κ), and eddy dissipationrate (ε) are defined as,

κ =1

2uiui, (4)

and

ε = −ν ∂ui∂xl

∂ui∂xl

, (5)

where ν is the molecular kinematic viscosity. After ma-nipulation of the time-averaged and time-dependent gov-erning equations [11, 13], the following evolution equa-tions are found for κ and ε,

D

Dt(κ) =

∂xi

(csκ

εuiul

∂κ

∂xl

)− P +G− ε, (6)

and

D

Dt(ε) =

∂xi

(cεκ

εuiul

∂ε

∂xl

)+ cε1

ε

κ(P +G)− cε2

ε2

κ(7)

The terms P and G respectively represent the productionof Reynolds’ stress and the production or destruction ofbuoyancy and are given by

P = −uiuj ∂Ui∂xj, (8)

G = −βgiuiT ′. (9)

where β = (1/ρ)(∂ρ/∂T ) is the volumetric expansion co-efficient where the derivative is taken at constant pres-sure. The reader should note that in writing Eq. (7)without a flux Richardson number, a vertical shear layeris assumed. In Eqs. (6) and (7) several empirical param-eters are present which are determined from experiments.The values of these parameters are discussed in AppendixB. In addition to Eqs. (6) and (7), three algebraic ex-

pressions for uiuj , uiT ′, and T ′2 can be found in [11] andare given in Appendix A.

B. Round Buoyant Jets

In this section, the equations governing the mean flowand turbulent dynamics will be specialized for the casewhen the buoyant jet is round and the initial flux is op-posite the direction of gravity. The model developedhere is valid beyond the ZOFE, where it is assumedthat both the Reynolds (Re = U0D/ν) and Grashof(Gr = g(ρa − ρ0)D3/ρ0ν

2) numbers are sufficiently highso that the flow is entirely turbulent and all viscous ef-fects can be neglected. Characteristic velocity and lengthscales have been chosen as U0, and D, respectively. Acylindrical coordinate system is adopted where s is thevertical coordinate and r is the radial coordinate. U willdenote the component of the velocity in the direction of

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the main flow, and V in the radial direction. Buoyant jetscontain regions depending on whether the flow is inertiaor buoyancy dominated. The jet region is dominated byinertia while the plume region is dominated by buoyancy.With uniform velocity and density profiles at the effluxpoint, the Froude number is Fr = U2

0 ρ0/gD(ρa − ρ0)and gives the relative importance of buoyancy and iner-tia. The characteristics of the environment into whichthe effluent air is released plays a role in determining thebalance of inertia and buoyant forces.

Due to the nature of this shear layer flow the followingboundary layer approximations is made,

V � U,∂

∂s� ∂

∂r(10)

Both static pressure and the pressure gradient in themean flow direction are independent of the radial coordi-nate and hence dp/ds = −ρag. Additionally, only steadyprocesses are considered here and turbulent transportdominates over molecular transport. These assumptionsgreatly reduce the complexity of the governing equationsfor the mean flow and turbulent quantities.

1. Flux Formulation

At this point in the development, the system is com-posed of five partial differential equations and five auxil-iary algebraic expression needed to close the system (thethree in Appendix A broken into u and v components).These partial differential equations are converted to aset of ordinary differential equations by first assuming aGaussian profile across each cross section of the flow,

U = uce−r2/b2 + ua cos(θ), T = Tce

−r2/(σT b2)

κ = κce−r2/(σκb2), ε = εce

−r2/(σεb2),(11)

where the subscript c denotes the value on the centerlineof the flow, and integrating each term across the flow,∫

A

(·)dA (12)

where A is the cross section area element and dA =rdrdθ. The quantity b(s) denotes the radial distancewhere the velocity has decreased by a factor of e−1uc,where uc is the centerline velocity of the plume. Apply-ing the assumptions from the previous section and inte-grating each term using Eq. 12 the following equationsare found which govern the dynamics of the flux of κ,and ε

d

ds(Σ) =CµK1

κ2cεcu2c +K2βg

κ3cε2cucTc−

K3β2g2

κ4cε3ct2c −

1

2b2εc,

(13)

and

d

ds(Φ) =Cµcε1M1κcu

2c −M2cε1

κ2cεcuc−

M3cε1κ3cε2c− cε2M4b

2 ε2c

κc.

(14)

The quantities Σ and Φ represent the flux of κ and εthrough a cross section of the flow. The point-wise quan-tities are computed using

Σ = b2(uc

σκ1 + σκ

+ uaσκ cos(θ)

)κc, (15)

Φ = b2(uc

σε1 + σε

+ uaσε cos(θ)

)εc. (16)

The expressions for the quantities Cµ, {Ki}, and {Mi}can be found in Appendix B. The algebraic expressionsfrom Appendix A have been incorporated in Eqs. (13)and (14). The equations governing the dynamics of themass, momentum, and temperature flux from which themean centerline velocity (uc) and temperature (Tc) arederived can be found in [2] and are not reproduced here.

2. Initial Conditions

In general, initial conditions for a buoyant jet at apower plant or manufacturing facility are given at the topof the stack. This is the beginning of the ZOFE howeverthe domain of the system in question here begins at theend of the ZOFE. Conditions for the mean flow quantitiesat the top of the ZOFE are given in [2]. What is neededtherefore are conditions for the turbulent quantities oncethe fully turbulent flow is established. Following the dis-cussion in [8], the quantities u/U = b1 and v/U = b2 areconstant throughout the flow. This yields the followingconstant quantity

κ =1

2(b21 + 2b22)U2 = CκU

2 (17)

where Cκ = 0.06. The initial conditions for κ at thebeginning of the turbulent flow is therefore κ0 = 0.06U2

0 .According to the work of [14], the dimensionless quantity

Cε = 3b0.5ε/(2κ)3/2 (18)

is constant throughout the flow where b0.5 is the half-velocity radius. The quantities b and b0.5 are related bythe constant b3 = b/b0.5 ≈ 1.2. Leveraging Eq. 17 andrearranging the expression for Cε yields

ε =23/2

3Cεb3C

3/2κ

(U3

b

)= 0.018

(U3

b

). (19)

The initial condition for ε at the beginning of the fullyestablished turbulent flow is ε0 = 0.018U3

0 /b0. It shouldbe noted that the authors of [14] referenced multiplevalues of Cε; this may change the initial value of ε toε0 = 0.019U3

0 /b0.

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III. RESULTS

In this section, results from integration of the equa-tions for the mean flow [2] coupled to Eqs. 13 and 14using MATLAB’s ODE45 routine and the initial condi-tions discussed in the previous section are obtained. Theevolution of κ, and ε are shown with varying initial condi-tions. Experiments have been performed which measurethe turbulent fluctuations of the mean velocity field (u′i)which can be used as a point of comparison for numericalsolutions to the governing equations.

A. Dynamics of κ and ε

Parameters of interest are the diameter of the exhauststack, the velocity of the effluent at the exit point of thestack, and the difference in temperature of the effluentand the ambient conditions. Calm ambient conditionswill be assumed. Figure 2 shows the effect of varyingthe stack diameter on the evolution of κ and ε. Varyingthe stack diameter has a greater effect on ε than it doeson κ. The variance scaled by the square of the meanof the distributions of κ and ε at a height of 100 diam-eters above the stack are 0.0049 and 1.11, respectively,indicating that the eddy dissipation rate of the systemis more sensitive to changes in exit stack diameter thanthe turbulent kinetic energy. To further investigate the

κ

ε

s/D

FIG. 2: Evolution of κ and ε (inset) with a varying stackdiameter. The stack diameter was varied from 2m to 47m insteps of 5m. The arrows indicate the direction of increasingstack diameter. Relevant model parameters were cε1 = 1.44,

cε2 = 2.02, c1T = 3.0, c2T = 0.5, c3T = 0.8, R = 0.8, σ1/2κ =

1.343, σ1/2ε = 1.505, and σ

1/2T = 1.20.

response of the system to changes in stack diameter, Fig-ure 3 shows changes in κ and ε, scaled by their initialvalues at the top of the ZOFE, at a downstream distanceof 100D as a function of stack diameter. With respect tothe scaled variables, the anticipated tradeoff of decreas-ing κ with increasing ε is seen here.

κ/κ0 ε/ε0

D

FIG. 3: Values of κ (solid) and ε (dashed) scaled by their ini-tial values at an downstream distance of 100 stack diametersabove the exit point for varying stack diameters. Relevantmodel parameters and initial values are the same as those inFigure 2.

Figure 4 illustrates the system’s response to a varyingeffluent velocity at the stack exit point. It is seen that in-creasing the exit velocity increases the turbulent kineticenergy and eddy dissipation rate. Increasing excess efflu-ent temperature causes an monotonic increase in κ at allelevations above the stack. This is not the case with ε.Figure 5 shows the change in κ and ε at two elevationsabove the exit point of the stack. At a particular (andsufficient) height above the stack, ε exhibits a local max-imum as a function of excess temperature. Figure 6 givesthe excess initial temperature where the local maximumoccurs as a function of vertical location. For the valuesof the model parameters chosen for this study (see cap-tion from Figure 2), the minimum location where a localmaximum exists is approximately s/D ≈ 16.

B. Comparison to experimental data

Burattini et al [14] created a jet by passing air througha round orifice into a large laboratory room with no cross-flow (ua = 0). The experiment was set up in such a waythat there was no buoyancy difference between the jetand the ambient environment. This ensures that the vol-

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κ

ε

s/D

FIG. 4: Evolution of κ and ε (inset) with a varying effluentexit velocity, which was varied from 5m/s to 50m/s in stepsof 5m/s. The arrows indicate the direction of increasing exitvelocity. Other relevant model parameters and initial valuesare the same as those in Figure 2.

umetric expansion coefficient (β) is constant. Velocitymeasurements were taken using a hot wire anemometerand a single set of initial conditions was used for all ofthe data. The diameter of the orifice was 55mm, theefflux velocity was 35.1m/s and the viscosity of the airwas 1.5 × 10−5m2/s. The effect of varying initial condi-tions on experimental outcomes can be found in Refer-ence [15]. Figure 7 shows the measured values of κ, andε from [14] as well as the result of the model solutionfor these quantities. The difference between the modelresults and experimental measurements for κ and ε wason average 0.69% and 1.56% with standard deviations of3.08% and 6.37%, respectively. For both parameters theerror never went over 9.8%.

The absence of a transport equation for ε in Reference[12] left the system incomplete without an empirical re-lationship for ε. An example of such a relationship is

ε = cκκ3/2

Lκ(20)

where Lk is the integral length scale for κ [12]. As ε isknown from the solution of its transport equation, Eq.20 it can be leveraged to yield cκ or Lκ. Integral lengthscales for u, and v are determined from similar equationsto that in Eq. 20, Li = ciκ

3/2/ε. The empirical constantsci are chosen to best fit the data. The data taken fromRef. [14] and the result from the model solutions areshown in Figure 8.

κ

κ

ε

ε

T/Ta

s/D = 10

s/D = 20

FIG. 5: Values of κ (solid) and ε (dashed) with a varyingexcess effluent temperature (T/Ta), at 10 (bottom) and 20(top) stack diameters above the top of the stack. Other rel-evant model parameters and initial values are the same asthose in Figure 2.

T/Ta

s/D

FIG. 6: Excess initial temperature (scaled by the ambient)of local maximum value of ε as a function of altitude abovethe exit point of the stack. Relevant model parameters andinitial values are the same as those in Figure 2.

IV. CONCLUSIONS

This study investigated the evolution of turbulent ki-netic energy and eddy dissipation rate of a round turbu-

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κ, ε

s/D

FIG. 7: A comparison of κ− ε model and the measured datafrom Reference [14]. The solid line and square symbols arethe modeled and measured κ, respectively. The dashed lineand triangle symbols are the modeled and measured ε, re-

spectively. Relevant model parameters were σ1/2κ = 1.343,

σ1/2ε = 1.505, and σ

1/2T = 1.20

lent buoyant jet, commonly formed above exhaust stacksat power plants or manufacturing facilities. Large enoughReynolds and Grashof numbers were assumed such thatthe flow can be treated as being entirely turbulent. Aboundary layer approximation was made, and it wasassumed that turbulent transport dominates moleculartransport. Lastly, the system was reduced from partialto ordinary differential equations by assuming the formof the relevant fields is Gaussian and integrating acrossa cross section to yield an integral flux model. The re-sult was a set of two coupled differential equations whichcouple as well with a model for the mean characteristicsof the flow adopted with modification from Reference [2].

Integration of the system of equations yields resultsfor the mean flow consistent with previous works [2].The response of the system to changes in initial condi-tions (effluent temperature and velocity), as well as phys-ical conditions (stack diameter) were investigated for il-lustrative purposes. Scenarios were devised to comparewith experimental measurements of turbulent quantities.The results of Burattini et al [14] experiments for turbu-lent fluctuations in the velocity field and derived integrallength scales compared favorably against the predictedoutput of the derived model.

While employing empirical calculations to compute theturbulent quantities from the mean flow can give insightinto the dynamics of the turbulent motion, use of theevolution equations by employing an integral flux model

Li

s/D

FIG. 8: A comparison of integral length scales Lu, Lv, andLκ from the κ− ε model and the measured data from Refer-ence [14]. The solid line and square symbols are the modeledand measured Lu, respectively. The dashed line and trian-gle symbols are the modeled and measured Lv, respectively.The dotted line and diamond symbols are the modeled andmeasured Lκ, respectively. Relevant model parameters arethe same as those in Figure 7. The constants are cu = 0.40,cv = 0.24, and cκ = 0.024.

reduces their complexity to the point where the problemis very tractable to most numerical ODE routines.

Acknowledgments

The authors acknowledge thoughtful feedback from S.Adam Blake, and Michael J. Mills.

Notice

This work was produced for the U.S. Government un-der Contract DTFAWA-10-C-00080 and is subject to Fed-eral Aviation Administration Acquisition ManagementSystem Clause 3.5-13, Rights In Data-General, Alt. IIIand Alt. IV (Oct. 1996).

The contents of this document reflect the views of theauthor and The MITRE Corporation and do not neces-sarily reflect the views of the FAA or the DOT. Neitherthe Federal Aviation Administration nor the Departmentof Transportation makes any warranty or guarantee, ex-pressed or implied, concerning the content or accuracy ofthese views. c©2012 The MITRE Corporation. As such,the Government retains a nonexclusive, royalty-free right

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to publish or reproduce this document, or to allow othersto do so, for Government Purposes Only.

Approved for public release; distribution unlimited.12-4989.

APPENDIX A: ALGEBRAIC EXPRESSIONS

The following three algebraic expressions for uiuj ,

uiT ′, and T ′2 are given here for completeness. Derivationof these expressions begins with general transport equa-tions for these quantities. This study focuses on verticalbuoyant jets which represent a vertical shear layer. Asan approximation of these flows, convective and diffusivetransport of turbulent flux and stress can be neglecteddue to slowly evolving nature of the flow. Complete de-tails of this derivation can be found in [11]. The expres-sions are

uiuj = κ

[2

3δijc1 + c2 − 1

c1− 1− c2

εc1

(uiul

∂Ui∂xl

+

ujul∂Ui∂xl

+ βgiujT ′ + +βgjuiT ′)],

(A1)

ujT ′ =1

c1T

κ

ε

[uiul

∂T

∂xl− (1− c2T )ulT ′

∂Ui∂xl−

(1− c3T )βgT ′2],

(A2)

and

T ′2 = −2Rκ

εulT ′

∂T

∂xl. (A3)

APPENDIX B: AUXILIARY QUANTITIES

There are several quantities in Eqs. 13 and 14 whicharise from integration and manipulation of the governing

equations and are given in this section. The expressionsfor Ki are

K1 = 2σ2κ

(2− (σκ/σε) + 2σκ)2(B1)

K2 = 2h2(σ2κ/σT )

(3− 2(σκ/σε) + (σκ/σT ) + σκ)2(B2)

K3 = 2h3(σκ/σT )2

(4− 3(σκ/σε) + 2(σκ/σT ))2(B3)

K4 =1

2σε (B4)

and Mi are

M1 = 2σ2k

(1 + 2σk)2(B5)

M2 = 2h2(σ2κ/σT )

(2 + (σκ/σT )− (σκ/σε) + σκ)2(B6)

M3 = 2h3(σκ/σT )2

3 + 2(σκ/σT )− 2(σκ/σε)2(B7)

M4 =σκ

2 [2(σκ/σε)− 1](B8)

The values of hi are

h2 = Cµ1

c1T

[1 +

1

c1T(1−W )

1

W

], (B9)

h3 =4

3

1

c21TR(1− c3T )(1−W ) (B10)

where R is the gas constant, W = (1 − c2)/c1, Cµ =2W (1−W )/3, and the empirical constants ciT , Ciε andci are given in Reference [11]. However, as a result ofapplying Eq. 12 to Eqs 6 and 7, several of the experi-mentally determined parameters no longer remain in thegoverning equations (13) and (14).

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