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Propulsion and Power Research 2016;5(1):22–33

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ORIGINAL ARTICLE

Nozzle geometry variations on thedischarge coefficient

M.M.A. Alama,n, T. Setoguchia, S. Matsuob, H.D. Kimc

aInstitute of Ocean Energy, Saga University (IOES), 1, Honjo, Saga-shi, Saga 840-8502, JapanbDepartment of Advanced Technology Fusion, Saga University, JapancDepartment of Mechanical Engineering, Andong National University, Korea

Received 14 February 2014; accepted 16 November 2015Available online 12 February 2016

KEYWORDS

Boundary layer;Compressible flow;Reynolds-averagedNavier–Stokes(RANS);Shear layer;Sonic lines;Supersonic core

tional Laboratory foe (http://creativecom

16/j.jppr.2016.01.00

hor. Tel.: (880) 8043

alam@me.saga-u.ac

esponsibility of Natia.

Abstract Numerical works have been conducted to investigate the effect of nozzlegeometries on the discharge coefficient. Several contoured converging nozzles with finiteradius of curvatures, conically converging nozzles and conical divergent orifices have beenemployed in this investigation. Each nozzle and orifice has a nominal exit diameter of12.7� 10�3 m. A 3rd order MUSCL finite volume method of ANSYS Fluent 13.0 was used tosolve the Reynolds-averaged Navier–Stokes equations in simulating turbulent flows throughvarious nozzle inlet geometries. The numerical model was validated through comparisonbetween the numerical results and experimental data. The results obtained show that the nozzlegeometry has pronounced effect on the sonic lines and discharge coefficients. The coefficientof discharge was found differ from unity due to the non-uniformity of flow parameters at thenozzle exit and the presence of boundary layer as well.& 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

r Aeronautics and Astronautics. Produmons.org/licenses/by-nc-nd/4.0/).

2

156244.

.jp (M.M.A. Alam).

onal Laboratory for Aeronautics

1. Introduction

Nozzles are found encountering in a wide variety ofengineering applications, mainly to generate jets [1–4], flowmetering [5–7], and sprays [8,9]. The accurate prediction ofthe compressible nozzle flows is still challenging for theaerodynamicist, and achieves increasing importance since

ction and hosting by Elsevier B.V. This is an open access article under the

Nomenclature

a sound speed (unit: m/s)Ae area at nozzle exit (unit: m2)Cd discharge coefficientCp specific heat at constant pressure (unit: J/(kg �K))De diameter at nozzle exit (unit: m)Dm diameter of Mach disk (unit: m)E total energy per unit mass (unit: J/kg)F inviscid flux vectorsG viscous flux vectorsH total enthalpy per unit mass (unit: J/kg)H vector for source termsi unit vector in the x-directionj unit vector in the y-directionk unit vector in the z-directionk turbulent kinetic energy per unit mass (unit: J/kg)l0 location of minimum jet section (unit: m)Lm location of Mach disk (unit: m)Ls length of supersonic core (unit: m)m mass flow rate (unit: kg/s)M Mach numberp pressure (unit: Pa)q heat flux (unit: W/m2)Q dependent vector of primary variablesr radius (unit: m)R radius of curvature (unit: m)Re Reynolds numberT temperature (unit: K)

Ur reference velocity (unit: m/s)w electromagnetic energy density (unit: J/m3)ui, uj Cartesian mean velocity components (unit: m/s)vx, vy, vz Cartesian velocity components in x-, y- and z-direc-

tions (unit: m/s)

Greek symbols

β conic divergent angle (unit: degree)δ boundary layer thickness (unit: m)δsh shear layer thickness (unit: m)γ ratio of specific heatsμ dynamic viscosity (unit: Pa � s))υ kinematic viscosity (unit: m2/s)θ conic convergent angle (unit: degree)ρ density (unit: kg/m3)ℜ gas constant (unit: J/(kmol �K))τ shear stress (unit: Pa)ω specific dissipation rate (unit: s�1)

Subscripts

0 stagnation pointb ambientt turbulentx x-coordinatey y-coordinatex z-coordinate

Nozzle geometry variations on the discharge coefficient 23

the nozzle performance is significantly influenced by itsinlet geometry. The flow emanating from nozzle exit servesas the initial conditions for the downstream jet flows. Thus,the studies on nozzle geometric effect are becoming a majorinterest for compressible and incompressible nozzle flows.

Several works reported information on aerodynamicfeatures of jets and flow with various inlet-boundaryconditions and nozzle geometries. Matsuo et al. [10]performed numerical study to investigate the effect ofnozzle geometry on the sonic line and characteristics ofthe supersonic air jets. Two contoured converging nozzles,two conically converging sharp-edged nozzles (451 and751) and a sharp-edged orifice were employed in theirstudy. Otobe et al. [11] investigated the near-field structureof highly underexpanded sonic jets using three nozzlegeometries (cylindrical straight nozzle, 751 convergenceconical nozzle and 451 divergent orifice), and they proposedan empirical relation of diameter of Mach disk in terms ofthe pressure ratio, regardless of the nozzle geometry.Menon and Skews [12] conducted a numerical study onunderexpanded sonic jets issuing from nozzles with con-toured inlet, 451 conical inlet and an orifice inlet under arange of pressure ratio between 2 and 10. Hatanaka andSaito [13] conducted experimental and numerical studies toinvestigate the effect of nozzle geometry on the structure ofsupersonic free jets for three simple nozzle geometries overa wide range of pressure ratios up to 90. However, most of

the above research works concentrated mainly on the shockand Mach characteristics of jets. In another study, Yu et al.[14] performed numerical simulations to investigate theeffects of geometry variations on flow through nozzles.Four nozzle configurations were considered in the study: abaseline nozzle and three modified (extended, grooved andringed) nozzles. The turbulence characteristics of incom-pressible flow through nozzles at Reynolds number ofapproximately 50,000 were investigated in their study. Onlyvery few studies have reported, till date, on the performanceof nozzle in terms of discharge coefficients. Hebber et al.[15] conducted an analytical study to obtain a simple,explicit and analytical expression for the discharge coeffi-cients of conical convergent nozzle operating under varyingpressure ratios. Cruz-Maya et al. [16] performed study tocharacterize the discharge coefficients in the venturi sonicnozzle considering the viscous and multidimensional effectsof the fluid flow as uncoupled phenomenon.

Since the main purpose of this research is to investigatethe effect of nozzle geometries on the performance in termsof discharge coefficients, five cylindrical, four conicalconvergent nozzles and eight conical divergent orifices withvarying radius of curvatures, convergent and divergentangles, respectively, have been used. Sonic lines and theirinflections were analyzed to examine the effect of flowparameter at nozzle exit on the discharge coefficient. Basedupon the computed results, the nozzle geometry has

M.M.A. Alam et al.24

significant influences on the discharge coefficient, profile ofsonic lines and on the nozzle downstream flow features. Jetboundaries, shear layers and lengths of supersonic core ofjets were investigated to clarify the performance of nozzleson the downstream flows.

2. Numerical model and computationalmethodology

2.1. Governing equations

Computational fluid dynamics (CFD) investigations wereperformed to simulate the turbulent flows through nozzlewith various geometries. The flow under study was treatedas compressible, viscous and supersonic at underexpandedconditions. The governing equations used in the presentcomputations are axisymmetric, compressible Reynoldsaveraged Navier–Stokes (RANS) equations [17], given by:

Γ∂∂t

ZVQdV þ

IF�G½ �U dA¼

ZVHdV ð1Þ

where, F and G are the inviscid and viscous flux vectors instandard conservation form, Q is the dependent vector ofprimary variables, and the vector H contains source terms.

F¼ ρv; ρvvx þ pi; ρvvy þ pj; ρvvz þ pk; ρvHh i T

ð2Þ

G¼ 0; τxi; τyi; τzi; τijvj þ q� � T ð3Þ

Q¼ p; vx; vy; vz; T� � T ð4Þ

Here ρ, v, and p are the density, velocity, and pressure ofthe fluid, respectively. τ is the viscous stress tensor, and q isthe heat flux. In the above equations, H is the total enthalpyper unit mass and is related to the total energy E byH ¼ E þ p=ρ, where E includes both internal and kineticenergies. The preconditioning matrix Γ is included in Eq.(1) to provide an efficient solution of the present axisym-metric compressible flow. This matrix is given by:

Γ ¼

θ 0 0 0 ρTθvx ρ 0 0 ρTvxθvy 0 ρ 0 ρTvyθvz 0 0 ρ ρTvx

θH�δ ρvx ρvy ρvz ρTH þ ρCp

26666664

37777775

ð5Þ

where ρT is the derivative of density with respect totemperature at constant pressure. δ¼1 for an ideal gasand δ¼0 for an incompressible fluid. The parameter θ isdefined as:

θ¼ 1

U2r

� ρTH þ ρCp

� � ð6Þ

In Eq. (6), the reference velocity Ur is chosen such thatthe eigenvalues of the system remain well conditioned withrespect to the convective and diffusive timescales and Cp isthe specific heat at constant pressure.

To close governing equations, the k–ω SST (shear stresstransport) which is a two equation eddy-viscosity turbu-lence model [18–20] were employed in the computation.This turbulence model is an effective blend of the robustand accurate formulation of the Wilcox's k–ω model in thenear-wall region with the free-stream independence of thek–ε model in the far field. The turbulent kinetic energy kand the specific dissipation rate ω are determined by thefollowing transport equations.

∂∂t

ρkð Þ þ ∂∂xi

ρkuið Þ ¼ ∂∂xj

Γk∂k∂xj

� �þ ~Gk�Yk ð7Þ

∂∂t

ρωð Þ þ ∂∂xi

ρωuið Þ ¼ ∂∂xj

Γk∂ω∂xj

� �þ Gω�Yω þ Dω ð8Þ

In these equations, ~Gk represents the generation ofturbulence kinetic energy due to mean velocity gradients,Gω represents the generation of ω, Dω is the cross-diffusionterm. σk and σω are the turbulent Prandtl numbers for k andω, respectively, and given by;

σk ¼1

F1=σk;1 þ 1�F1ð Þ=σk;2ð9Þ

and

σω ¼ 1F1=σω;1 þ 1�F1ð Þ=σω;2

ð10Þ

The turbulent viscosity μt is computed as;

μt ¼ρk

ω

1

max 1=α�; SF2=α1=ω� � ð11Þ

where S is strain rate magnitude, defined as S� ffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

pand Sij ¼ 1

2∂uj∂xi þ

∂ui∂xj

, α¼ α1

α�α0þRet=Rω

1þRet=Rω

, and Ret ¼ ρk

μω.

Blending functions F1 and F2 are given by;

F1 ¼ tanh Φ41

� � ð12Þ

Φ1 ¼ min max

ffiffiffik

p

0:09ωy;500μρy2ω

� �;

4ρkσω;2Dþ

ωy2

� �ð13Þ

Dþω ¼ max 2ρ

1σω;2

1ω

∂k∂xj

∂ω∂xj

; 10�10

� �ð14Þ

and

F2 ¼ tanh Φ22

� � ð15Þ

Φ2 ¼ max 2

ffiffiffik

p

0:09ωy;500μρy2ω

� �ð16Þ

where y is the distance to the next surface and Dþω is the

positive portion of the cross-diffusion term. The cross-diffusion term Dω is defines as;

Dω ¼ 2 1�F1ð Þρσω;2 1ω

∂k∂xj

∂ω∂xj

ð17Þ

Model constants are given as;

σk;1 ¼ 1:176; σω;1 ¼ 2:0; σk;2 ¼ 1:0; σω;2 ¼ 1:168

Nozzle geometry variations on the discharge coefficient 25

α1 ¼ 0:31; βi;1 ¼ 0:075; βi;2 ¼ 0:0828

α�1 ¼ 1; α1 ¼ 0:52; α0 ¼19; β�1 ¼ 0:09

Rk ¼ 6; Rω ¼ 2:95; ζ� ¼ 1:5

2.2. Numerical methods

The density-based algorithm in ANSYS Fluent 13.0 [17]was used to solve the preconditioned governing equations.A fully implicit method was implemented on the presentspatial domain. The convective fluxes were formulatedusing the Roe's flux difference splitting scheme [21], andthe 3rd order accuracy of this scheme was conceived fromthe original MUSCL (monotonic upstream-centered schemefor conservation laws) [22] finite volume scheme that is ablend of central differencing and second-order upwindschemes, in which the physical domain is subdivided intonumerical cells, and the integral equations were applied toeach cell. Second-order central difference scheme was usedfor viscous terms.

2.3. Computational conditions

Schematics of the computational domain used in thepresent study are illustrated in Figure 1. Nozzles with thefollowing geometry variations were studied:

(1) Addy [23] nozzle which is a cylindrical straight nozzleis composed of a convergent curved entrance wallhaving radius of curvature R¼De followed by straightwall with a length of 0.4De (Figure 2(a)).

Figure 1 Schematics of computational domain and boundaryconditions.

(2) Cylindrical nozzles with variable radius of curvaturesR¼6, 12, 18, 24 and 30 mm (Figure 2(b)).

(3) Conical nozzles with convergence angles θ¼301, 451,601 and 751 (Figure 2(c)).

Figure 2 Schematics of nozzle configurations. (a) Addy nozzle,(b) cylindrical convergent nozzles, (c) conical convergent nozzles, and(d) conical divergent orifices.

Figure 3 Mesh dependency test results.

Table 1 Detailed about various nozzle configurations.

Nozzle types DescriptionAddy nozzle Circular arc, R¼De; straight¼0.4De

R/10�3 mCylindrical convergentnozzles

6R 612R 1218R 1824R 2430R 30

θ/degreeConical convergentnozzles

30T 3045T 4560T 6075T 75

β/degreeConical divergentorifices

30D 3035D 3540D 4045D 4550D 5060D 6075D 7590D 90

M.M.A. Alam et al.26

(4) Conical divergent orifices with divergence anglesβ¼301, 351, 401, 451, 501, 601, 751 and 901(Figure 2(d)).

The detail information about these nozzle inlet config-urations are presented in Table 1. The nominal diameter ofthe nozzle exit and orifice in all cases was kept constant atDe¼12.7� 10�3 m (characteristics length). The symbolicrepresentations for various nozzles as presented in Table 1were used in the discussion for convenience.Dry air as working fluid was driven at Reynolds number

of Re¼5.92� 105 through the nozzles. The nozzle pressureratio p0/pb (ratio of total pressure p0 to the back pressure pb)was kept constant at 6.2. The upstream total temperature T0and total pressure p0 were maintained constant at 298.15 Kand 101.3 kPa, respectively, through the whole computa-tions. The boundary conditions used were the inlet totalpressure and the outlet static pressure at the upstream anddownstream of the computational domain, respectively. Thesymmetric conditions at the axis of the nozzle were used toreduce the computational effort for the full domain, and theadiabatic no-slip conditions were applied to the solid walls.To ensure the computational domain independent solu-

tions, the upstream domain was extended to the distance of100De upstream from the nozzle exit, and the downstreamwas extended to the distance of 100De and 50De in the x- andr-directions, respectively. A structured grids system wasemployed in computations. The typical grids system empl-oyed in the present computation is shown in Figure 1.Several preliminary computations have been performed withdifferent sizes of mesh to investigate the grid dependence ofthe solution. The location Lm and diameter Dm of Machdisk were calculated for those grids systems. The calculated

results for five representative meshes are shown in Figure 3.Here, the data of location Lm and diameter Dm of Mach diskare normalized by the nozzle exit diameter De and Δymin

represents the grids closest to the wall. Based on theseresults, the type IV mesh which was used for the simulationsis the suitable and economic one to simulate the problem.The grids in type IV mesh are 70 in� 80 in the nozzle regionand 100 in� 170 in the jet plume region and Δymin arelocated at 0.0004De away from the wall. The grids weredensely clustered in the boundary layers in order to providemore reasonable predictions.

A solution convergence was obtained when the residualsfor each of the conserved variables were reduced below theorder of magnitude 4. Another convergence criterion was tocheck the conserved quantities directly through the compu-tational boundaries. The net mass flux was investigatedwhen there was an applicable imbalance through the com-putational boundaries.

3. Results and discussion

3.1. Comparison with the experimental results

The validation of the present computational work wascarried out through a comparison of the computed under-expanded supersonic free jets with our experimental results[24]. In both experiment and numerical works jets wereissued from the cylindrical straight nozzle [23]. The rangeof pressure ratio p0/pb was varied between 3.8 and 6.2. Theupstream total temperature T0 and pressure p0 in thenumerical study were the same as that with the experiment,and values were 298.15 K and 101.3 kPa, respectively.

Figure 4 shows a comparison of the computed iso-densitycontours with the shadowgraph pictures of underexpandedsupersonic jets. The jet at p0/pb¼3.8 is slightly underexpanded(see Figure 4(a)), and the resulting weak shock waves thathave the regular reflection at point ‘P’. While the jet flow atp0/pb¼6.2 is strongly underexpanded at the nozzle exitcompared with the case shown in Figure 4(a), and barrelshocks are formed due to the difference in pressures betweenthe underexpanded and ambient gases. These shocks reflect on

Figure 5 Discharge coefficients for nozzle with varying geometries.

Figure 4 Comparison of computed iso-density contours with theexperimental results. (a) p0/pb¼3.8 and (b) p0/pb¼6.2.

Nozzle geometry variations on the discharge coefficient 27

the jet axis. Consequently, Mach disk is formed near the jetaxis. At the interaction of Mach disk with barrel shock, atriple-point is formed. In the downstream of the triple point theslip line is observed. As the pressure ratio increases, the shockcells extend stream-wise that result in an increase of the lengthof supersonic core of jets. The predicted iso-density contoursare nearly the same as the results obtained by the experiment.In addition, the present computations predicted the diameterand location of Mach disk to be 0.515 and 1.526, respectively,while the experimental values are 0.521 and 1.545, respec-tively. Thus, it can easily be mentioned that the presentcomputations predicted the jet and shock structures of under-expanded supersonic jets with a good accuracy.

3.2. Discharge coefficients and sonic lines

The discharge coefficient Cd is an important character-istics of a nozzle. It is defined as the dimensionless ratio ofthe actual mass flow rate to the ideal mass flow ratecorresponding to one-dimensional isentropic flow for thesame upstream stagnation conditions, and is given by,

Cd ¼m actmideal

ð18Þ

In the equation, m ideal for an ideal compressible gasesflow through cylindrical and conical convergent nozzles canbe calculated as follows [25]:

mideal ¼Aep0ffiffiffiffiffiT0

pffiffiffiffiffiγ

ℜ

rM 1þ γ�1

2M2

� �� γþ12 γ � 1ð Þ

ð19Þ

Whilst mideal for an ideal gases through conical divergentorifices can be estimated by the following equation [26].

m ideal ¼ Ae

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ρ0p0 γ= γ�1ð Þ� �

p=p0� �2=γ� p=p0

� � γþ1ð Þ=γh ir

ð20Þwhere, Ae is the area at nozzle exit, ℜ is the gas constant, γis the ratio of specific heats, M is the flow Mach number, pis the pressure and T is the temperature. Subscript 0 refers tothe stagnation condition. The mass flow rate is a maximumfor nozzles at chocked condition (M¼1). The dischargecoefficient differs from unity due to the nonuniformity offlow parameters (such as Mach number) at the nozzle exitand presence of boundary layer (viscous effect).

The computed discharge coefficients for cylindrical,conical convergent nozzles and conical divergent orificesare plotted against the radius of curvatures R, convergentangles θ and divergent angles β, respectively, in Figure 5.Here, it is mentioned that the discharge coefficient for Addynozzle is presented in the figure as reference. For cylindricalconvergent nozzles, the discharge coefficient increasesalong with R in the range of small R (r18 mm) butthe variation decreases in the range of large R (Z18 mm).For conical convergent nozzles with sharp corner at exit,the discharge coefficient significantly decreases with theincrease of convergent angles θ, and it can be given by alinear decay function of θ. It indicates that the performance

of conical convergent nozzles with small conical angles θ ispreferably higher than those with large angles θ. WhilstCd for conical divergent orifices increase almost linearlywith the increase of divergent angles β in the range of601oβr901, and the effect of divergent angles β isinsignificant on discharge coefficient in the range of451oβr601. The results Cd for β¼301, 351 and 401 are

Figure 7 Sonic lines for nozzle with varying geometries.(a) Cylindrical convergent nozzles, (b) conical convergent nozzles,and (c) conical divergent orifices.

Figure 6 Density contours and streamlines for conical divergentorifices. (a) β¼351 and (b) β¼451.

M.M.A. Alam et al.28

not presented here in the present discussion, though wehave conducted studies on them.The entrainment flows downstream of the divergent

orifices are restricted at small divergent angles β¼301,351 and 401, and this phenomenon can clearly be visualizedthrough the computed density contours and streamlines forconical divergent orifice with β¼351 presented in Figure 6(a). Besides, for β¼451 and on ward, the entrainment flowcomes contact with the flow issuing from the orifice, andthat results in the correct formation of jet and shockstructures, as shown in Figure 6(b). Thus, the resultspresented in the paper only for cases of fully developedflows (β¼451, 501, 601, 751 and 901) regardless ofconsidering the effect of divergent angles on the down-stream flows.In order to examine the effect of flow parameter such

as Mach number at nozzle exit on the discharge coefficient,the computed profiles of sonic lines for cylindrical, conicalconvergent nozzles and divergent orifices are illustrated inFigure 7(a), (b) and (c), respectively. For cylindrical con-vergent nozzles, sonic lines are curvilinear, and they arenotably influenced by the radius R. The sonic line forR¼6 mm is placed significantly downstream of the nozzleexit, whereas the sonic line moves upstream as R increases.For Addy nozzle, the profile of sonic line in viscous flowreaches downstream of the nozzle exit, whereas the inviscid

Nozzle geometry variations on the discharge coefficient 29

profile approaches to the nozzle exit. The viscous effect isdominant in the straight part of the Addy nozzle [23] andthat effect can clearly be noticed from the distribution ofboundary layer thickness near nozzle exit in Figure 8. Theboundary layer occupy a sizeable portion of the nozzle cross-section and acts like a converging–diverging section whichleads to shift virtual throat of this converging–divergingsection to some upstream location from the nozzle exit.

The profile of sonic lines for conical convergent nozzlesas shown in Figure 7(b) exhibit point of inflection and theflow nonuniformity may be substantial and it grows withincrease of the convergent angle θ. The sonic line trendstowards the downstream direction with the angle θ. Besides,the influence of divergent angles β on the sonic line profilesare insignificant for conical divergent orifices. The soniclines are intersect with the axis about an axial location of x/De¼0.2 downstream of the nozzle exit, and the profileshown in Figure 7(c) displays point of inflection in the samefashion like conical convergent nozzles.

The inflection of sonic lines for conical convergentnozzles and divergent orifices is measured through an angleα. The α is calculated as described in the typical schematicdiagram that illustrated in Figure 9. The computed angles ofinflection α for conical convergent nozzles and divergentorifices then graphed against convergent and divergent

Figure 9 Schematics of sonic line inflection for conical convergentnozzles.

Figure 8 Distributions of boundary layer thickness near the exit ofAddy nozzle.

angles θ and β in Figure 10(a) and (b), respectively. Theinflection angles α in both cases of conical convergentnozzles and divergent orifices can be given by linear growthfunctions of θ and β, respectively, as follows.

α ðdegÞ ¼ 0:1696� θ ðdegÞ þ 3:7763 ð21Þand

α ðdegÞ ¼ 0:041� β ðdegÞ þ 10:97 ð22ÞIn addition, the increment of the inflection angle α for

conical convergent nozzles is higher compared with thecase of conical divergent orifices.

3.3. Effects on the aerodynamic features of jets

The configuration of jet boundaries of supersonic jetsissuing from cylindrical, conical convergent nozzles anddivergent orifices are computed and presented graphicallyin Figure 11(a), (b) and (c), respectively. The location of jet

Figure 10 Inflection of sonic lines for conical convergent nozzlesand divergent orifices. (a) Conical convergent nozzles and (b) conicaldivergent orifices.

Figure 11 Jet boundaries for cylindrical, conical convergent nozzlesand divergent orifices. (a) Cylindrical convergent nozzles, (b) conicalconvergent nozzles, and (c) conical divergent orifices.

M.M.A. Alam et al.30

boundary is defined with the largest value of densitygradient (dρ/dr) on an arbitrary cross section normal tox-axis. The radius of curvature R and divergent β incylindrical convergent nozzles and conical divergent ori-fices exhibit no salient influence on the configuration of jetboundaries shown in Figure 11(a) and (b), respectively.Whereas, the present computed data for conical convergentnozzles shows that the jet boundary trends to shrink inwardwith the increase in the convergent angles θ, as shown inFigure 11(b). In addition, comparing the configuration of jetboundaries for conical convergent nozzles and divergentorifices the inward inflection of jet boundary is large fordivergent orifices. It indicates that the inward inflection isdependent on the nozzle geometry.

For conical convergent nozzles and divergent orifices, sincethe presence of curvilinear sonic lines result in the dependenceof the critical pressure drop on the value of convergent anddivergent angles θ and β, respectively. To make a clear insightabout this flow physics, we investigated efflux of a gas fromconical convergent nozzles and divergent orifices. A schematicview of efflux from a conical convergent nozzle is shown inFigure 12. From the schematic diagram, it can be seen that themaximum contraction takes place at finite distance of l0 slightlydownstream of the nozzle exit, where the jet is more or lesshorizontal. At this section, the cross-section area of fluid streamis minimum, streamlines are almost parallel and the pressuredrop across the jet is constant. This section is called as venacontracta. The reason for this phenomenon is that fluid stre-amlines cannot abruptly change direction. In the free jet, thestreamlines are unable to closely follow the sharp change ofangle in the nozzle wall. The converging streamlines follow asmooth path, which results in the narrowing of the jet observed.

The position of vena contracta l0 from the exit was computedand the data for the conical convergent nozzles and divergentorifices are plotted against the convergent and divergent anglesθ and β in Figure 13(a) and (b). In these cases, the distance l0increases with the increase of θ and β and can be given by thelinear functions those are presented in figures. For conicalconvergent nozzle, the slope of the linear function is largecompared with that of conical divergent orifices.

The shear layer in supersonic jets is one of the funda-mental characteristics of supersonic mixing layers. Sincethere exists expansion and compression waves in the jet

Figure 12 Schematics of the efflux from conical convergent nozzles.

Figure 14 Distributions of shear layer thickness for conical conver-gent nozzles and divergent orifices. (a) Cylindrical convergent nozzles,(b) conical convergent nozzles, and (c) conical divergent orifices.

Figure 13 Distributions of l0 for conical convergent nozzles anddivergent orifices. (a) Conical convergent nozzles and (b) conicaldivergent orifices.

Nozzle geometry variations on the discharge coefficient 31

flowfield the exact estimation of the shear layer thickness isvery important. Hence, the definition of shear layer thick-ness using the 95%–5% criteria is applied to the cross-stream pressure data, and the measured of shear layerthickness δsh as function of axial distance x/De from thenozzle exit is presented in Figure 14(a), (b) and (c) forcylindrical, conical convergent nozzles and divergent ori-fices, respectively.

The thickness δsh is normalized by the nozzle exitdiameter De. The plot for cylindrical convergent nozzles,in Figure 14(a), suggests that the shear layer near nozzleexit (x/Der0.15) is significantly influenced by the radiusof curvatures R, and the thickness decreases with theincrease of R in this near exit region. Away from theexit (x/DeZ0.15) the thickness δsh follows fairly lineargrowth. Whilst, from Figure 14(b) and (c), the influence ofconvergent and divergent angles θ and β, respectively, onthe growth of shear layer is insignificant, and the shear layergrow linearly along the axial direction. The growth rate of

Figure 15 Supersonic core lengths of jets issuing from nozzles withvarying geometries.

M.M.A. Alam et al.32

shear layer is slower for conical divergent orifices comparedwith that of conical convergent nozzles.The length of supersonic core is the measure of physical

size of the jet. This length is determined by the axial pointfarthest downstream at which there exists a flow Machnumber of unity. The computed supersonic core lengths Lsfor cylindrical, conical convergent nozzles and divergentorifices are graphically plotted against the radius of curv-atures R, convergent angles θ and divergent angles β inFigure 15. For cylindrical convergent nozzles the length Lsis greatly affected by the radius R, and Ls decreases with theincrease of R. The length Ls for conical convergent nozzles,on the other hand, can be given by a linear growth functionof convergent angles θ. Whilst the supersonic core lengthsLs for conical divergent orifices increases linearly withsmaller slope in the range of divergent angles βr751 andbeyond this range (β4751) a rapid increment of the lengthLs is observed.

4. Conclusions

The present work dealt with the numerical simulations ofviscous turbulence flows through nozzles with varyinggeometry. Compressible RANS equations were solvedalong with SST k–ω turbulence model. The computationalcode was validated through comparison of the numericalresults with the experimental data. Several cylindrical,conical convergent nozzles and divergent orifices withvarying radius of curvatures, convergent and divergentangles, respectively, were used to investigate the geometryeffect on the nozzle performance in terms of the dischargecoefficient. Investigating the effect of flow parameters atnozzle exit the sonic lines and their inflection wereanalyzed. The discharge coefficient significantly influencedby the radius of curvatures, convergent and divergentangles. The sonic line moved towards nozzle exit as theradius of curvature increased, while it trend towards the

downstream direction with the increase of convergent angle.The influence of divergent angles on the profile of soniclines was insignificant. The inflection of sonic line foundincreasing with the increase of convergent and divergentangles. Significant influence of nozzle geometry was noti-ced on the configuration of jet boundaries, shear layers andlengths of supersonic core of jets. The geometry effect onjet boundaries was dominant for conical convergent noz-zles, and the jet boundary showed a trend to shrink inwardwith the increase of convergent angles. However, the influ-ence of convergent and divergent angles for conical con-vergent nozzles and divergent orifices, respectively, on theshear layer was insignificant. The length of supersonic coredecreased with the increase of radius of curvatures, andincreased with the increase of convergent and divergentangles. Finally, these findings surely could play importantrole in designing nozzles that are widely using in a varietyof engineering applications.

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