Navier-Stokes computation of some gas mixture problems in ...scientiairanica.sharif.edu/article_3645_7c763b2e... · E-mail address: [email protected] (I. Zahmatkesh) calculations

Scientia Iranica B (2015) 22(1), 187{195

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Navier-Stokes computation of some gas mixtureproblems in the slip ow regime

I. Zahmatkesha;�, H. Emdadb and M.M. Alishahib

a. Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran.b. School of Mechanical Engineering, Shiraz University, Shiraz, Iran.

Received 11 November 2012; received in revised form 11 April 2014; accepted 26 April 2014

KEYWORDSGas mixture;Rare�ed ow;Slip ow regime;Navier-Stokesequations;Nozzle;Microchannel.

Abstract. Several kinetic descriptions have already been utilized for the simulation ofrare�ed gas mixture ows. Although such developments are important, Navier-Stokescomputation can �nd extended use in engineering applications. Recently, the authorshave derived new velocity-slip and temperature-jump boundary conditions for slip owsof gas mixtures. Appealing to these new boundary conditions, Navier-Stokes computationof rare�ed gas mixture problems has become feasible. In the present contribution, theproposed conditions, in conjunction with the Navier-Stokes equations and an equationfor the conservation of species, are solved for some binary gas mixture problems in theslip ow regime. Applications include low pressure ow in a converging-diverging nozzle,wall-cooling of a nozzle under rare�ed condition, and parallel mixing in a microchannel.Simulation results are presented in terms of the distributions of overall Knudsen number,Mach number, pressure, temperature, and concentration.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

The physics of rare�ed single-component gas ows hasbeen extensively studied in the past, and a number ofinteresting phenomena, which vanish at the continuumlimit, have been observed. In practice, however, onedeals with gas mixtures more often than with singlegases and, thereby, attention must be paid to thesimulation of such ows. In gas mixture ows, severalnew phenomena can lead to a somehow surprisingbehavior, compared with single-component cases. Asan example, di�usion can be observed due to imposedpressure, temperature, and concentration gradients.Additionally, concentration gradients may lead to thetransfer of heat. Mixture problems also require dras-tically higher computational e�ort, as a result of thelarge number of additional parameters involved in their

*. Corresponding author. Tel.:+98 915 5109399E-mail address: [email protected] (I.Zahmatkesh)

calculations. Consequently, it is not surprising to seethat rare�ed gas mixture ows have not been analyzedextensively in the past. Some recent works are asfollows.

Yan and Farouk [1] simulated the mixing problemof two parallel gas streams entering a microchannelinitially separated by a splitter plate. They used theDirect Simulation Monte Carlo (DSMC) method andconcluded that the mixing decreases with increasingthe inlet-outlet pressure di�erence, and that the mixingprocess for nearly specularly-re ective walls is muchslower than for perfectly di�using walls. Such aproblem was further analyzed by Wang and Li [2]that clari�ed the importance of main ow velocity aswell as temperature on the degree of gas mixing. Arecent analysis of this ow problem was undertaken byDarbandi and Lakzian [3].

The developed ow of binary mixtures of noblegases between parallel plates driven by small pressure,temperature, and concentration gradients was analyzedby Naris et al. [4]. They utilized the McCormack model

188 I. Zahmatkesh et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 187{195

of the Boltzmann equation [5] and presented some ow features for the whole range of gas rarefaction.Concerning their results, they concluded that the hard-sphere molecular interaction model did not providereliable results when mixture ow problems were stud-ied. This problem was again analyzed by Kosuge andTakata [6]. They constructed a ux database for binarymixtures of noble gases that covers the entire rangeof Knudsen number and concentration, and a widerange of temperature. Later, Naris et al. [7] extendedtheir previous formulation (i.e. [4]) to developing owsthrough microchannels.

Sharipov et al. [8] calculated heat ux magnitudesthrough rare�ed binary mixtures of noble gases con-�ned between parallel plates. Their solution was basedon the McCormack model [5] and led to the conclu-sion that using the hard-sphere molecular interactionmodel, accurate heat uxes would not be resulted.Recently, DSMC simulation of this problem was un-dertaken by Strapasson and Sharipov [9]. Comparisonof the results of the two approaches demonstratedthat, in small temperature di�erences, the McCormackmodel provided reliable results with an uncertainty of2%.

Raines [10] solved the Boltzmann equation forthe ow of binary mixtures between parallel platesand presented some ow aspects for a wide range ofKnudsen number.

A lattice Boltzmann model was derived for mi-croscale gas ows of binary mixtures by Guo et al. [11].The proposed model utilizes a collision operator withmultiple relaxation times.

Szalmas and Valougeorgis [12] analyzed the owof gas mixtures through triangular and trapezoidalmicrochannels. They solved the McCormack kineticmodel [5] and presented some ow features for thewhole range of gas rarefaction.

Steady, one-dimensional ow of a binary mixtureof hard-sphere gases in contact with an absorbingplanar wall was studied by Frezzotti et al. [13] throughthe DSMC method. Recently, they extended theirinvestigation to two-dimensional circumstances [14].

Although such developments are important, theNavier-Stokes description can �nd extended use inengineering applications due to its (i) simplicity andfamiliarity, (ii) computational e�ciency, (iii) applica-bility to monatomic, as well as polyatomic, gases, (iv) exibility for possible extension to multicomponentproblems, (v) suitability for simple, as well as complex,geometries.

In spite of that, to the authors' knowledge,Navier-Stokes equations have not been employed forthe computation of microscale gas mixture ows, sincelittle have been known about the near wall behaviorof the gas mixture. Some models have been proposedfor velocity-slip, temperature-jump, and concentration-

jump in rare�ed mixture ows over catalytic surfaces(e.g. [15,16]). Nevertheless, the inclusion of surfacereaction has made them quite complex. Meanwhile, formost gas mixtures, no reaction occurs on the surface,and near wall behavior is not so sensitive to local owproperties.

Recently, Zahmatkesh et al. [17] derived newvelocity-slip and temperature-jump boundary condi-tions for gas mixture ows under rare�ed conditions.Appealing to these boundary conditions, the Navier-Stokes computation of gas mixture ows in the slip ow regime has become feasible.

To continue this e�ort, in the present contribu-tion, the proposed boundary conditions, in conjunctionwith Navier-Stokes equations and an equation for theconservation of species, are solved for some binary gasmixture problems in the slip ow regime. Applicationsinclude low pressure ow in a converging-diverging noz-zle, wall-cooling of a nozzle under rare�ed condition,and parallel mixing in a microchannel. Since previousinvestigations in the context of rare�ed gas mixtureshave been restricted to straight microchannels, the�rst originality of the current study goes back to theanalysis of more complex geometries. This contribu-tion can also be regarded as the �rst Navier-Stokescomputation of gas mixture problems under rare�edconditions.

2. Conservation equations

For simplicity of exposition, binary mixtures in twospace dimensions are analyzed in this contribution.Accordingly, uid dynamics of the binary systemsare described by Navier{Stokes equations, with anadditional equation, expressing the conservation ofmass for the �rst component. This additional equation,in conjunction with mass conservation for the wholemixture, implies the conservation of mass for thesecond component as well. In a generalized curvilineartwo-dimensional coordinate system, the conservationequations are:

@ ~Q@t

+@~F@�

+@ ~G@�

=@~Fv

@�+@ ~Gv

@�; (1)

where:

Q =

8>>>><>>>>:E��u�v�c

9>>>>=>>>>; ; F =

8>>>><>>>>:(E + p)u

�u�u2 + p�uv�uc

9>>>>=>>>>; ;

G =

8>>>><>>>>:(E + p)v

�v�uv

�v2 + p�vc

9>>>>=>>>>; ; Fv =

8>>>><>>>>:u�xx + v�xy � qx

0�xx�yxjx

9>>>>=>>>>; ;

I. Zahmatkesh et al./Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 187{195 189

Gv =

8>>>><>>>>:u�xy + v�yy � qy

0�xy�yyjy

9>>>>=>>>>; : (2)

Here, � is the mass density of the mixture and c is themass fraction of the constituent, �. Moreover, u, v, qx,and qy are the components of velocity and heat uxvector. �ij is the viscous stress tensor and E is thetotal energy per unit volume. Meanwhile;

~F =1J

(F�x + G�y) ; ~G =1J

(F�x + G�y) ;

~Q =1J

Q; ~Fv =1J

(Fv�x + Gv�y);

~Gv =1J

(Fv�x + Gv�y): (3)

with:

J =@(�; �)@(x; y)

= �x�y � �y�x: (4)

In this formulation, j is the molecular mass ux vector,which is de�ned as [18]:

j = �Dbrc+ kTr lnT + kpr ln pc: (5)

Here, the �rst term represents di�usion due to con-centration gradients, while the contributions associatedwith temperature gradients and pressure gradientsare incorporated in the second and the third terms,respectively. The coe�cient, D, is the binary di�usioncoe�cient. Meanwhile, kT and kp can be calculatedfrom the mixture composition as [19]:

kT = 0:0663

"m�m�

(cm� + (1� c)m�)2

#; (6)

kp = (m� �m�)c(1� c)�cm�

+1� cm�

�: (7)

These coe�cients become in uential in problems withlarge temperature/pressure gradients. During com-putations, it must be borne in mind that kT is notvanishing in the absence of one of the constituents.

The viscous stress tensor and the heat ux vectorare obtained from:

�xx = ��

43@u@x� 2

3@v@y

�; (8)

�xy = �yx = ��@u@y

+@v@x

�; (9)

�yy = ��

43@v@y� 2

3@u@x

�; (10)

qx = ��@T@x

+2Xi=1

hijx;i; (11)

qy = ��@T@y

+2Xi=1

hijy;i; (12)

with � and � being the thermal conductivity andviscosity of the gas mixture, respectively, and hi beingthe partial speci�c enthalpy of the ith species. It isnoteworthy that the second terms in the right-handside of Eqs. (11) and (12) represent the contributionof inter-di�usion on the transfer of heat. Here, jx;iand jy;i are the components of the molecular mass uxvector for the i's species. For instance:

jx;� = �D�@D@x

+kTT@T@x

+kpT@p@x

�: (13)

The evolution equations are supplemented by the idealgas equation of state, which expresses pressure, p, interms of mass density, �, speci�c internal energy, e,mass fraction, c, and the ratio of the speci�c heats, :

p(�; e; c) = ( (c)� 1)�e; (14)

where:

(c) =cpcv

=cp;�c+ cp;�(1� c)cv;�c+ cv;�(1� c) : (15)

Here, cv;�, cv;� , cp;� and cp;� are the speci�c heatsat constant volume and constant pressure for thecomponent species, respectively.

The conservation equations, in conjunction withthe proposed velocity-slip and temperature-jumpboundary conditions (that will be described in theforthcoming section), are solved invoking the explicit ux di�erencing of Roe's scheme [20]. In this method,the inviscid terms are calculated based on the eval-uation of eigenvalues and eigenvectors of the Eulerequations. The viscous terms are also discretized usinga central di�erencing. Moreover, an upwinding methodis applied to the equation of the continuity of thespecies. During the current computations, transportcoe�cients of the constituents are taken from Bergmanet al. [21], while the Wassiljewa's correlation [22] isadopted for the thermal conductivity of the mixture.

3. The proposed boundary conditions

The proposed velocity-slip and temperature-jumpboundary conditions for a binary system take the form


of [17]:

us � uw =2� ��

�overall@u@n

��s

+34��T

@T@s

��s; (16)

Tg�Tw = 22� �T�T

bcs�� + (1� cs)��c�s�vs

hcs �+1

2 cv;� + (1� cs) �+12 cv;�

i @T@n

��s;(17)

with s and n being the streamwise and normal coor-dinates, respectively. �overall is the overall Knudsennumber of the gas mixture:

�overall = cs�� + (1� cs)�� : (18)

Moreover, us and uw are the slip and wall velocities,Tg and Tw are the gas and wall temperatures, and is the ratio of speci�c heats. Subscript s standsfor surface conditions. Coe�cients � and �T are thetangential-momentum-accommodation coe�cient andthe thermal-accommodation coe�cient, respectively.�vs is the mean molecular speed of the gas mixture:

�vs =�

8�kTsm

�0:5

: (19)

Meanwhile, � is the viscosity coe�cient of the gasmixture that can be determined in terms of the mixturecomposition, as:

�=��

1+G��

M�

M�

��1

+��

1+G��

M�

M�

��1

;(20)

with:

G�� =

�1 +

q��=��

pM�=M�

�2

2p

2(1 +M�=M�); (21)

G�� =

�1 +

q��=��

pM�=M�

�2

2p

2(1 +M�=M�): (22)

Here, M� and M� denote molar masses.

4. Simulation results

In this section, simulation results are presented forthree distinct gas mixture problems in the slip owregime. Nitrogen and oxygen are regarded as the mix-tures constituents. The current applications includelow pressure ow in a converging-diverging nozzle,wall-cooling of a nozzle under rare�ed condition, andparallel mixing in a microchannel. The accuracy of

Figure 1. Simulated velocity pro�les at two di�erentsections along the microchannel compared with DSMCresults.

the employed Navier-Stokes solver for the descriptionof these ow problems in the continuum ow regime hasbeen established during our previous investigations [23-26]. Now, the validity of this solver in the computationof the slip ow regime is examined. For this purpose,and in the absence of any analytical or experimentaldata in the current ow problems, a He-Ne ow in a1 � 4 �m microchannel is computed, and the corre-sponding results are compared with those of the DSMCsimulation of Wang et al. [27]. Here, the inlet andoutlet pressures are taken to be 300 kPa and 100 kPa,respectively. Moreover, a temperature of 300 K and aconcentration of 0.2 are assigned to the in owing gasstream. Under this circumstance, the overall Knudsennumber of the mixture is about 0.1 in the outgoinggases, and a slip ow establishes in the microchannel.Figure 1 compares the simulated velocity pro�les at twodi�erent sections along the microchannel with thoseof the DSMC method. The centerline distributionsof pressure are also indicated in Figure 2. Inspectionof the �gures demonstrates that the two approacheslead to almost the same simulation results. Thisprovides con�dence to the developed Navier-Stokessolver for further studies in rare�ed circumstances.Consequently, in the forthcoming sections, it is usedto simulate the aforesaid gas mixture problems in theslip ow regime.

4.1. Low pressure ow in aconverging-diverging nozzle

As the �rst example, a low pressure gas mixture owin a converging-diverging nozzle is regarded. Theschematic of the nozzle is depicted in Figure 3. The


Figure 2. Simulated centerline distribution of pressurecompared with DSMC results.

Figure 3. Schematic of the �rst mixture problem.

Table 1. Speci�cations of the �rst ow problem in Case I.

Inlet stagnation pressure pt;i = 7000 PaInlet stagnation temperature Tt;i = 750 KInlet concentration ci = 0:5Exit pressure pe = 5000 PaLength of the nozzle L = 2 mmSize of the inlet and exit 2H = 2 mm

Inlet areaThroat area 1.18Thermal condition of the wall Tw = 750 KWall accommodation coe�cients � = �T = 0:85

computations are undertaken for three distinct casesin the slip ow regime. Speci�cations of the �rst case(Case I, henceforth) are illustrated in Table 1. Underthis circumstance, the overall Knudsen number of themixture remains below 0.001 over the entire ow �eld.Consequently, this case represents the lower limit of thegas rarefaction in the slip ow regime.

In regard to the second case (Case II), the degreeof gas rarefaction is increased by assigning pt;i =

700 Pa and pe = 500 Pa at the boundaries. Otherspeci�cations are maintained identical to Case I. Thismakes the maximum value of the overall Knudsennumber about 0.01. The last case (Case III), however,corresponds to a ow in the upper limit of the gasrarefaction in the slip ow regime (i.e., with Kn� 0:1).This is attained by prescribing 70 Pa and 50 Pa at theinlet and outlet, respectively.

The ow cases are simulated using 10500 (70 �150) nodes based on a grid re�nement study. Inwhat concerns the centerline distribution of Machnumber, simulation results for the three cases of thegas rarefaction are illustrated in Figure 4. Here, results

Figure 4. Centerline distribution of Mach number in the�rst mixture problem.


obtained without the rare�ed boundary conditions (i.e.,with no-slip/no-temperature jump conditions) are alsoprovided for comparison. Obviously, in all of thecurrent cases, the centerline Mach number increasesalong the nozzle and a maximum value is attainedat the downstream side of the throat. This proceedswith a slight decrease in the diverging part of thenozzle. The behavior is attributed to strong viscouse�ects in these low Reynolds ow �elds. Notice thatwith an increase in the overall Knudsen number, thecenterline Mach numbers diminish. This occurs despitethe pressure ratios that are maintained identical here.The physical reasoning for this behavior is the fact that,as the degree of gas rarefaction increases, velocity-slipbecomes more pronounced and, thereby, more uniformvelocity pro�les establish in each section through thenozzle. The aforesaid variations in the Mach numbermake the centerline distributions of the temperatureand pressure as depicted in Figures 5 and 6.

Comparing the results obtained with the proposedboundary conditions and those obtained without suchconditions, in the three cases of gas rarefaction, is in-structive. Evidently, as the gas becomes more rare�ed,the discrepancies that appear between the results of thetwo approaches enhance, and, thereby, the in uence ofthe implemented rare�ed boundary conditions becomesmore pronounced.

4.2. Wall-cooling of a nozzle under rare�edcondition

As the second example, wall-cooling of a converging-diverging nozzle under rare�ed condition is analyzedhere. For this purpose, the same nozzle geometry asin the preceding example is chosen. Figure 7 depictsthe schematic of this ow problem. It is assumed thatnitrogen enters from the inlet of the nozzle and oxygen,as a coolant, is blown from the walls downstream of thethroat. Flow conditions for this case are illustrated inTable 2.

The problem is solved employing a 155�335 grid.Figure 8(a) presents the distribution of overall Knudsennumber in the nozzle. Obviously, the lowest values

Table 2. Speci�cations of the second ow problem.

Inlet stagnation pressure pt;i = 460 PaInlet stagnation temperature Tt;i = 500 KInlet concentration ci = 1Coolant x-velocity uc = 100 m/sCoolant y-velocity vc = �5 m/sCoolant static temperature Tc = 300 KCoolant concentration cc = 0Exit pressure pe = 300 PaThermal condition of the wall AdiabaticWall accommodation coe�cients � = �T = 0:85

Figure 5. Centerline distribution of temperature in the�rst mixture problem.

of the Knudsen number appear in the vicinity of theoxygen blowing. This occurs since molecular collisionsare more frequent between the oxygen molecules thatare larger than the nitrogen ones. Notice that the over-all Knudsen number lies between 0.0095 and 0.0145,which, again, falls into the slip ow regime.

Simulation results, in terms of the distributionsof concentration, Mach number, and temperature,are plotted in Figure 8(b) to (d). Concerning thecontour plots of concentration in Figure 8(b), theview of the observer is directed towards the coolantthat is blown and convected downstream. Inspec-tion of Figure 8(c) demonstrates that due to oxygen


Figure 6. Centerline distribution of pressure in the �rstmixture problem.

blowing at the diverging part, the e�ective shape ofthe nozzle is converging instead of the converging-diverging nozzle. Moreover, temperature distributionprovided in Figure 8(d) indicates that high temper-ature gradients appear in the vicinity of the diverg-ing part of the nozzle walls, wherein the coolant isblown.

4.3. Parallel mixing in a microchannelAs the last example problem, simulation results arepresented for the mixing ow system depicted inFigure 9. Here, the parallel streams of oxygen and

Figure 7. Schematic of the second mixture problem.

Figure 8. Simulation results for the second mixtureproblem.

nitrogen, which are initially separated by a splitterplate, enter the mixing chamber. Speci�cations of this ow problem are illustrated in Table 3.

The problem is solved using 7171 (71 � 101)nodes based on a grid re�nement study. Figure 10(a)presents the distribution of overall Knudsen numberin the gas mixture. Evidently, as one proceeds alongthe microchannel, the gas becomes more rare�ed. Itis clear that the overall Knudsen number of thismixture problem remains below 0.02 over the entire


Table 3. Speci�cations of the third ow problem.

Inlet stagnation pressures pN2t;i = pO2

t;i = 15 kPa

Inlet stagnation temperatures TN2t;i = TO2

t;i = 300 K

Exit pressure pe = 7:5 kPa

Thermal condition of the wall Adiabatic

Wall accommodation coe�cients � = �T = 0:85

Height of the microchannel H = 100 �m

Aspect ratio of the microchannel L=H = 5

Figure 9. Schematic of the third mixture problem.

Figure 10. Simulation results for the third mixtureproblem.

ow �eld. Consequently, a slip ow establishes insidethe microchannel.

In what concerns the distributions of Mach num-ber, temperature, and concentration, simulation resultsare depicted in Figure 10(b) to (d). Inspection of Fig-ure 10(b) demonstrates that the ow is accelerated inaccordance with the density drop brought about by thedecrease in pressure, in a way that is consistent with themass conservation principle. As a consequence of this ow acceleration, a low temperature region graduallyforms in the downstream side of the microchannel,

which is obvious in Figure 10(c). The mixing pat-tern of the two parallel gas streams is illustrated inFigure 10(d). Previous observations of Zahmatkeshet al. [23] provide evidence that during the parallelmixing, the heavier species di�uses towards the othergas stream. This is attributed to the higher momentumof the heavier particles during molecular collisions inthe mixing layer. In spite of that, molecular massesof nitrogen and oxygen are almost identical, and thecurrent parallel mixing is not so intense.

5. Concluding remarks

In this contribution, recently proposed velocity-slip andtemperature-jump boundary conditions were adopted,and several Navier-Stokes computations were under-taken for some gas mixture problems in the slip owregime. Inspection of the presented results for the lowpressure nozzle ow demonstrated that the inclusionof the proposed boundary conditions has prominentconsequences on the establishment of the ow �elds.The method was also successful in the description ofwall-cooling of a nozzle under rare�ed condition andparallel mixing in a microchannel

The employed Navier-Stokes description can �ndextended use in the simulation of gas mixture ows inmicroelectromechanical systems due to its (i) simplicityand familiarity, (ii) computational e�ciency, (iii) ap-plicability to monatomic, as well as polyatomic, gases,(iv) exibility for possible extension to multicomponentproblems, (v) suitability for simple, as well as complex,geometries.

References

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Biographies

Iman Zahmatkesh obtained BS and MS degrees inMechanical Engineering from Ferdowsi University ofMashhad, Iran, and a PhD degree in the same subjectfrom Shiraz University, Iran. He is currently AssistantProfessor and Head of the Mechanical EngineeringDepartment at the Islamic Azad University, Mashhad,Iran. He has published about twenty ISI papers in areasof computational uid dynamics.

Homayoun Emdad earned his BS degree inAerospace Engineering from Washington University,USA, and MS and PhD degrees in the same subjectfrom Kansas University, USA. He is currently AssociateProfessor of Mechanical Engineering at Shiraz Univer-sity, Iran. His research interests are computational uid dynamics, aerodynamics, uid mechanics andtwo-phase ow analysis.

Mohammad Mehdi Alishahi earned his BS degreein Aerospace Engineering from Sharif University ofTechnology, Tehran, Iran, and MS and PhD degrees inthe same subject from the Massachusetts Institute ofTechnology, USA. He is currently Professor of Mechan-ical Engineering at Shiraz University, Iran. His mainresearch interests are computational uid dynamics,aerodynamics, and uid mechanics.

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