Inlet flow disturbance effects on direct numerical simulation ofincompressible round jet at Reynolds number 2500Zejing Hua,b, Yongliang Zhanga, Zuojin Zhua,* a Faculty of Engineering Science, University of Science and Technology of China, Hefei 230026, Chinab Anhui Xiang Shui Jian Pumped Storage Co., LTD, Wuhu 241083, China

H  I  G  H  L  I  G  H  T  S

• The inlet flow disturbance effects on direct numerical simulation (DNS) of incompressible round jet at Reynolds number 2500 are reported.• In the DNS, a six order biased upwind difference scheme (BUDS) for spacial discretization of nonlinear convective terms in the Navier-Stokesequations is used.• Centerline velocity and its root square value are compared with some existing results for the BUDS validation.• The contours of time-circumferential mean factor of swirling strength intermittency are shown.

A  R  T  I  C  L  E      I  N  F  O A  B  S  T  R  A  C  T

 

Article history:Received 20 July 2018Received in revised form 16 August 2018Available online 10 September 2018  

Keywords:Incompressible round jetJet core coneBiased upwind finite differenceSwirling strength intermittency factorAccurate projection method

 

This letter reports inlet flow disturbance effects on direct numerical simulation of incompressibleround jet at Reynolds number 2500. The simulation employs an accurate projection method inwhich a sixth order biased upwind difference scheme is used for spatial discretization of nonlinearconvective terms, with a fourth order central difference scheme used in the discretization of thedivergence of intermediate velocity. Carefully identifying reveals that the inlet flow disturbance hassome influences on the distribution pattern of mean factor of swirling strength intermittency. Withthe increase of inlet disturbance magnitude jet core cone slightly shortens, observable differencesoccur in the centerline velocity and its fluctuations, despite the negligible impacts on the leastsquare fitted centerline velocity decay constant (Bu) and distribution parameter (Ku) for velocityprofile in self-similar region.

 

©2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical andApplied Mechanics. This is an open access article under the CC BY-NC-ND license

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

 

In many practical applications turbulent jets play crucialroles, for instance in the thermodynamical mechanisms of com-bustion, heat and mass transfer. As a prototype of free flow, tur-bulent jet has been studied frequently, there are a large numberof publications and reviews. Some early reviews were reported[1-3], with a recent review given in Ref. [4].

To confirm the behavior of jet similarity at least asymptotic-ally, experiments were conducted by Ref. [5-7]. But the univer-sality of jet similarity can still depend on other factors, such asthe initial conditions [8]. Using three distinct acceleration

schemes of linear, quadratic, and exponential to increase thenozzle exit velocity by an order of magnitude, Zhang et al. [9]studied acceleration effects on turbulent jets in a series of flowvisualization experiments. They found that as the flow acceler-ated, a discernible front was established. Based on the scaling ofcenterline velocity in steady jets, a model appears to correctlypredict the time dependence of the front position. Some othersexperimental results were reported in Refs. [10-14]. For instance,Abdel-Rahman et al. [10] measured the velocity field of turbu-lent round air jet flows to study the relative influence of using awall at the jet exit plane on the jet behaviors and characteristics.The dispersion and mixing of passive scalar (temperature) fluc-tuations in a turbulent jet was studied by Tong and Warhaft [14],

 

  

* Corresponding author.E-mail address: zuojin@ustc.edu.cn (Z.L. Zhu).

Theoretical & Applied Mechanics Letters 8 (2018) 345-350

 

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Theoretical & Applied Mechanics Letters

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http://dx.doi.org/10.1016/j.taml.2018.05.0092095-0349/© 2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under theCC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

for which the centerline velocity fluctuation at Reynolds num-ber 18000 was reported.

2:43£ 103

To study the inflow condition effect on far field self-similarregion of a round jet, the direct numerical simulation (DNS) of aspatially developing free round jet at a low Reynolds number of

was performed by Boersma et al. [15]. They found theevidence in support of the suggestion by Ref. [8] that the detailsof self-similarity depend on the initial conditions, implying thatthere may exist no universally valid similarity scaling for the freejet.

A brief literature review has been reported elsewhere [16]. Inthe present study, to present a more specific insight of turbulentintermittency from views relating to variations of vortical struc-tures, prediction of swirling strength intermittency factor withDNS of incompressible jet at Reynolds number 2500 is carriedout. For this purpose, a sixth order biased-upwind differencescheme (BUDS) and a fourth order central difference scheme aredeveloped and used in an accurate projection numerical meth-od for the DNS. The BUDS is also used in seeking the 1+1 Bur-gers equation under definite conditions to inspect its applica-tion feasibility at first and then used in the DNS of the jet flow.The numerical results have been checked by a step ratio ap-proach [17, 18 ], and validated by the existing experimental andDNS results, hence the checking of grid independence is omit-ted. We will start our description from governing equations andterminate by several concluding remarks.

U0

In Fig. 1(a), a sketch of the jet geometry and the cylindricalcoordinate system are illustrated. Let us assume that the fluid isemitting the computational domain through an orifice in a wallwith a jet speed . The fluid flow enters into free space withoutconfining walls. While moving downstream, the jet cross sectionincreases due to entrainment, ambient fluid has to enter the jetregion [19]. This entrainment is generated by small pressure dif-ferences between the surrounding and the jet flow. The flow isassumed to be isothermal and incompressible and is thus gov-erned by the following equations based on the conservation ofmass and momentum:

r ¢ = 0; (1)

@

@t+r ¢ ( ) = ¡ 1

½rp+ ºr2 ; (2)

p ½º

= (v;w;u)

where is the velocity vector, is the pressure, is the massdensity, and is the kinematic viscosity of the fluid, both takento be constant. In a cylindrical coordinate system, ,

the nonlinear convective terms can be defined by

H 1 = v@v@r+ w

µ@v

r@µ¡ w

r

¶+ u

@v@z

;

H 2 = v@w@r+ w

µ@wr@µ

+vr

¶+ u

@w@z

;

H 3 = v@u@r+ w

@ur@µ

+ u@u@z

: (3)

P1 =

µ@2

@r 2+

1r

@

@r

¶P2 =

1r 2

@2

@µ2P3 =

@2

@z2

D = P1+ P2+ P3

Further label , , and ,

use , the diffusion terms can be defined by

D 1 = º

µDv¡ 2

r 2

@w@µ

¡ vr 2

¶; D 2 = º

µDw+

2r 2

@v@µ¡ w

r 2

¶;

D 3 = ºDu: (4)

Then, the continuity Eq. (1) and momentum equation (2) can bere-written as

@v@r+

1r@w@µ+

@u@z+

vr= 0;

@v@r+ H 1 ¡ D 1 = ¡

@p@r

;

@w@t+ H 2 ¡ D 2 = ¡

@pr@µ

;

@u@t+ H 3 ¡ D 3 = ¡

@p@z

: (5)

The computational domain and its boundaries are shown inFig. 1(b). The three types of boundaries: the inflow, outflow, andlateral boundary, require distinguished boundary condition.

At the inflow boundary of the jet all three velocity compon-ents are specified, and the pressure is left free. For the specifica-tion of the velocity, we distinguish between the area inside andoutside the orifice, i.e.,

v =

8><>:+A 0r 0:5 ¢ cos(1:5µ); in the ori¯ce 0; on nozzle wall;

@2v@z2

= 0; otherwise;

w =

8><>:¡A 0r 0:5 ¢ sin(1:5µ); in the ori¯ce 0; on nozzle wall;

@2w@z2

= 0; otherwise;

u =

8<:U0; in the ori¯ce 0; on nozzle wall;@u@z= 0; otherwise; (6)

 

z

r

U0

a

z

r

o

h

Lateralboundry

Inflowboundary

r m

Orif

ice

U0ro = D/2

zm

Outflowboundary

b

 

r zµ rm = 5D zm = 45D

Fig. 1.   a A sketch of the jet geometry plus cylindrical coordinate system, denotes the radial direction, denotes the axial direction, with thecircumferential direction omitted. b A sketch of the computational domain. , and .

346 Zejing Hu et al. / Theoretical & Applied Mechanics Letters 8 (2018) 345-350

U0 A 0

U0=p

Du

ur

w ¡A 0r 0:5 ¢ sin(1:5µ)

A ¤ = A 0

pD =U0

where is the axial velocity in the orifice, it is a constant. isthe magnitude of inlet flow disturbance, has the unit of .By assuming the main flow streamwise gradient of is vanished,the velocity component in the orifice can be simply derivedfrom the continuity equation if is set as . Toseek the effects of the artificially assumed magnitude of inletflow disturbance, in this letter we will use the normalizedparameter to distinguish the DNS scenarios.

v

r = rm

At the lateral boundary, by assuming the radial derivative forradial velocity component is zero, using the so-called traction-free boundary condition [15, 20], we can derive that the secondorder derivative should be vanished, that means for ,

@v@r= 0;

@2w@r 2

= 0;@2u@r 2

= 0: (7)

r = 0At the flow centerline, the jet symmetry requires that for

,

v = 0;@w@r= 0;

@u@r= 0: (8)

z = zm

At the outflow boundary we use a so-called convectiveboundary condition [15], for as

@v@t= ¡U

@v@z

;@w@t= ¡U

@w@z

;@u@t= ¡U

@u@z

; (9)

Uwhere is mean velocity over the outflow boundary. Equation(9) is discretized, with first-order discretization in space andtime.

When circumferential period is assumed to be , the period-ic boundary condition in the circumferential direction is givenby

(r; µ+ ; z) = (r; µ; z): (10)

Such assumption permits to use less grids in the circumferentialdirection as compared that for the case without the period halfshortening.

¹ Á n´ (v;w;u) = (H 1;H 2;H 3)

T

= (D 1;D 2;D 3)T

The governing Eq. (5) of jet flow were discretized by a finitedifference method in a staggered grid system. An accurate pro-jection method [21] is used for the DNS. Now just a brief descrip-tion is described below. Let the intermediate velocity vector, thepressure potential, and the time level be , , and , respect-ively. Define , , and

then letn+1 = ¹ ¡¢trÁ; (11)

¹we can calculate by

¹ ¡ n

¢t+ n+1=2 ¡ n = ; (12)

pand calculate pressure by

pn+1=2 = Á; (13)

Áwhere the pressure potential satisfies the Poisson's equation

r2Á = r ¢ ¹=¢t; (14)

n+1=2

¹

the terms are calculated explicitly using the second orderAdams-Bashforth formula [22]. As mentioned above, theconvective terms in the governing equations are spatiallydiscretized by the sixth order BUDS [see in Fig. 2(a) and (b)],with the viscous diffusion terms by 2nd-order centraldifference scheme. A fourth order central difference scheme isused in the discretization of the divergence of intermediatevelocity . Details of the BUDS and the fourth order centraldifference scheme are reported in [16]. The Poisson's equation ofpressure potential is solved by the approximate factorization onemethod [23].

DU0 º

h Drm = 5D zm = 45D

Nr = 86 Nz = 450Nµ = 37

Computational parameters are shown in Table 1. The jetReynolds number Re is defined by orifice diameter , orifice ve-locity , and kinematic viscosity of fluid . The orifice wallthickness is assumed to be 0.025 . Respectively, the lateraland outflow boundaries are given by and ,with the mesh numbers in radial and axial directions set at

and . While the mesh number in circumferen-tial direction is set at , as we have made the assumptionof period half shortening.

R s

R s

S max = 1%R s

R s

R s

R s

S max

The seventh column of Table 1 shows the step ratio calcu-lated by the approach of Smirnov et al. [17, 18]. The step ratio refers to the ratio of maximal allowable number of time steps forthe problem and the actual number of time steps used to obtainthe result. For , using the time step shown in the sixthcolumn of Table 1, equals 3.158 which is larger than unity. Asreported [17, 18], characterizes reliability of results to determ-ine the limit of the simulations. The higher the value of , thelower the accumulated error is. As approaches unity, the er-ror tends to a maximum allowable value . As the reliabilityof numerical results is evaluated by a step ratio approach [18],the relevant verification is done in comparison with existingmeasured and calculated data, the DNS is conducted withoutchecking grid independence.

Using the accurate projection method given above, the DNSof the incompressible jet at Reynolds number 2500 was carriedout in a personal computer with a memory of 3.2 GB and CPUfrequency 3.30 GHz. The DNS is started from a postulated lamin-

 

a For ui ≥ 0,uz = [eui − 4 + dui − 3 + cui − 2 + bui − 1 + aui + fui + 1 + gui + 2 + o (ε6)]

b For ui < 0,uz = [eui + 4 + dui + 3 + cui + 2 + bui + 1 + aui + fui − 1 + gui − 2 + o (ε6)]

e d c b a f g

ui − 4 ui − 3

zi − 4 zi − 3 zi − 2 zi − 1 zi + 1zi

ui − 2 ui − 1 ui ui + 1 ui + 2

g f a b c d e

zi − 2 zi − 1 zi + 1 zi + 2 zi + 3zi

ui − 2 ui − 1 ui ui + 1 ui + 2 ui + 3 ui + 4

 

(@u=@z) zi ui ¸ 0 ui < 0Fig. 2.   A sketch for discretizing at grid with a sixth BUDS, a for , b for .

Zejing Hu et al. / Theoretical & Applied Mechanics Letters 8 (2018) 345-350 347

D=U0 t0 = 10

D=U0

100=¢t = 62500

ar velocity field, as soon as the flow patterns have appearedsome turbulent properties, the instantaneous flow field is savedand the time in the unit of is changed and reset at .For statistical analysis of jet flow field, further simulations arecarried out for 100 time scales to achieve DNS data. Such afurther simulation needs CPU time about 64.5 h. As the total stepnumber is , the per time step CPU time is about3.715 seconds.

t µ

Uc=U0

z=D

U0=Uc

The DNS predicts mean values of velocities on the basis oftime-circumferential ( - ) average. The inverse of mean center-line velocity in the unit of orifice velocity ( ) plotted as afunction of distance to the orifice ( ) is shown in Fig. 3(a),where the yellow filled black circles show the DNS results ob-tained by solving the Navier-Stokes equations in spherical co-ordinates with a method briefly described by Ref. [15], thesquares and plus-symbols are the existing measured data [5, 7].For self-similar jet, as shown by blue deltas in Fig. 3(a), can be described empirically [7]

U0

Uc=

1B u

¢µ

z ¡ z0

D

¶; (15)

z0 B uwhere represents virtual origin, and is the decay constantwhich in general has a value between 5.7 and 6.1 depending onthe experimental setup, as reported by Ref. [15], also can be seenin Table 2.

z ¸ 12 z0 = 5:0B u = 6:051; 6:038 A ¤ = 0:0618; 0:1

Using least squares method to analyze the present DNS datain the far region , when , it is estimated that

and 5.846 respectively for ,and 0.1382 labeled by DNS1, DNS2, and DNS3. The decay con-stant is excellently well predicted in comparison with that value5.9 predicted by Boersma et al. [15] and the experimental data5.9 of Hussein et al. [7].

When the velocity profile is assumed to be Gaussian,UUc= exp(¡K u´

2): (16)

[z=D 2 (24; 40)]By averaging the axial velocity over the range ,

 

++

++

++ +

+ ++ ++

+ ++

+

0 10 20 30 40

1

2

3

4

5

6

7

U0/U

c

a

z/D

HCGWFBBNBu = 6.038DNS1DNS2DNS3

+ + + ++

++

++

++

+ ++ +

00

1.0

0.8

0.6

0.4

0.2

U/U

c

b

0.200.150.100.05h = r/ (z − z0)

+

DNS3DNS2DNS1Ku = 75.59BBNPLHCG

 

z=D t µ

´ = r=(z ¡ z0)

Fig. 3.   a Inverse of centerline velocity in the unit of orifice velocity versus the distance to orifice . b The - averaged mean velocity profilesscaled by the local centerline velocity as function of . Note that BBN= Boersma, Brethouwer and Nieuwstadt [15], HCG= Hussein,Capp and George [7], WF= Wygnanski and Fiedler [5].

Table 1   Parameters of the round jet flow used in the DNS.

R e = U0D=º h=D zm=D rm=D Nr £ Nµ£ Nz ¢t R s*

2:5£ 103

0.025 45 5 86£ 37£ 450 1:6£ 10¡3 3.518

¤R s S max = 1% n = 100=¢t is calculated using and , details were reported by Smirnov et al. [18]

Table 2   Some parameters obtained from different experiments and DNS.

Ref. WF[5] Rodi [2] HCG[7] PL[6] BBN[15] DNS1 DNS2 DNS3

R e ¼ 105 8:7£ 104 ¼ 105 1:1£ 104 2:4£ 103 2:5£ 103 2:5£ 103 2:5£ 103

B u 5.7 5.9 5.8 6.1 5.9 6.051 6.038 5.846

K u - - - 75.2 76.1 75.09 75.59 74.17

z0 3 - 4.0 - 4.9 5.0 5.0 5.0

WF= Wygnanski and Fiedler; HCG= Hussein, Capp and George; PL= Panchapakesan and Lumley; BBN= Boersma, Brethouwer andNieuwstadt.

348 Zejing Hu et al. / Theoretical & Applied Mechanics Letters 8 (2018) 345-350

K u

K u = 75:2K u = 76:1

by means of a least-squares method, the numerical data by DNSshow that the distribution parameter =75.09, 75.59, and 74.17for DNS1, DNS2 and DNS3 respectively, see in Fig. 3(b), anexcellent agreement as compared with the experimental data( ) of Panchapakesan and Lumley [6], and DNS data( ) in Ref. [15].

Uc0=Uc

z=D = 18:97

z0 ¼ 0

Uc0=Uc

3:0£ 104 8:6£ 104

z=D · 12z=D > 12 » 0:24

Uc0=Uc

For the root mean squares (rms) of centerline velocity de-noted by , some existing experimental and DNS results to-gether with the rms distributions in the axial direction from thepresent DNS are shown in Fig. 4(a), with the temporal evolutionof centerline velocity at shown in Fig. 4(b). It isnoted that the velocity in the orifice was turbulent in the experi-ment of Tong and Warhaft [14], and Abdel-Rahman et al. [10]. Asreported [15], this turbulent inflow will lead to a much smallervirtual origin (probably ). On the other hand, the meas-ured data of Fellouah, Ball and Pollard [13] show an almost thesame axial distribution of as that reported previously by[12], even though the Reynolds number of the measured jet flowis , less than the jet flow Reynolds number ( )in the experiment of Xu and Antonia [12]. The rms based on theDNS of Boersma et al. [15] are shown by cyan-filled blue circles,which illustrates a satisfactory distribution in the region close toorifice ( ), but in the region far from the orifice( ), the rms curve oscillates around a value ( ).For the rms values of calculated based on the presentDNS1, DNS2 and DNS2, there is a good agreement with the ex-isting measured and DNS data. The present DNS reveals that in-let flow disturbance can impose an observable influence the axi-

Uc0=Ucal distribution of .

r ¸ci

r ¸ = (¸cr + i¸ci)

As reported [24, 25], when the swirling-strength of filtered ve-locity gradient is denoted by , the complex eigenvalue of

is given by , the factor of swirling strength in-termittency (FSI) is defined by

FSI( ; t) =¸cip

¸2cr + ¸2

ci

: (17)

t µ

To measure jet flow intermittency, we use the mean factor ofswirling strength intermittency (MFSI) on the basis of time-cir-cumferential ( - ) average, since at any point of turbulent flow,as soon as the FSI is zero, the local flow should be laminar, but ifFSI is unity the local flow is fully turbulent. Hence a sub-gridmodel based on FSI for large eddy simulation has been de-veloped [24, 25].

r z

z=D · 12 z=D 2 (12; 24)r=D ¸ 3:5

z=D ¸ 24

z = 8:4 A ¤ =

To show the MFSI distribution patterns, contours in the -plane are given by Fig. 5(a-c). Carefully identifying indicates theinlet disturbance has some influences on the MFSI distributionpattern. It is seen that there is a more complex structure in theregion near the orifice ( ); in the region ,in the area close to lateral boundary , the MFSI islower. While in most areas of the region further far from the ori-fice , the MFSI is distributed relative uniformly and ingeneral takes a value in the range from 0.4 to 0.6, except for thenear centerline region. Particularly, the jet core displayed by theblue colored cone near the orifice has a relative short length, thecone ends at , 7.8, and 7.2 as 0.0618, 0.1 and 0.1382respectively.

In summary, the DNS results reveal that an intensity in-crease of inlet flow disturbances can shorten the jet core cone,

 

0 5 10 15 20 25 30 35 40 45

0.3

0.2

0.1

0

−0.1

u’c/U

c

z/D

++++++

+++++++++++++++++++++++++++++++

FBP XAACW TWBBN DNS1DNS2 DNS3

a

b

z/D = 18.97DNS1, A* = 0.0618DNS2, A* = 0.1DNS3, A* = 0.1382

t20 40 60 80

0

0.6

0.4

0.2

u c/U

0

100

 

z=Dz=D = 18:97

z=D = 18:97

Fig. 4.   a The root mean square values (rms) of axial velocity com-ponent measured by the local centerline velocity versus the dis-tance to the orifice . b Temporal evolution of centerline velocityat . Note that the vertical dash dot dot line in sunfigurea shows the axial position , and FBP= Fellouah, Ball andPollard [13], XA= Xu and Antonia [12], ACW= Abdel-Rahman, Chak-roun and Fahed [10], TW= Tong and Warhaft [14], BBN= Boersma,Brethouwer and Nieuwstadt [15].

 

4321

z/D5 10 15 20 25 30 35 40

r/D

a MFSI0.90.70.60.50.40.30.1

MFSI0.90.70.60.50.40.30.1

MFSI0.90.70.60.50.40.30.1

4321

z/D5 10 15 20 25 30 35 40

r/D

b

c

4321

z/D5 10 15 20 25 30 35 40

r/D

 

r z A ¤ = 0:0618 A ¤ = 0:1A ¤ = 0:1382

Fig. 5.   Contours of the mean factor of swirling strength (MFSI) inthe - plane, a DNS1, ; b DNS2, ; c DNS3,

.

Zejing Hu et al. / Theoretical & Applied Mechanics Letters 8 (2018) 345-350 349

larger space variations of mean factor of swirling strength inter-mittency occur in the region near the orifice, but in most areas ofthe jet flow region that are far from the jet orifice, the meanfactor has a value close to 0.5, with an uncertainty of about 0.1.

Acknowledgements

Here we are grateful to the financial support of the NationalNatural Science Foundation of China (11372303), to the invalu-able comments of Professor J.L. Niu at The Hong Kong Polytech-nic University.

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