ROLE OF INITIAL CONDITIONS IN THE EVOLUTION OF ANAXISYMMETRIC TURBULENT JET DUE TO GEOMETRICAL

EFFECTS ON THE NEAR-FIELD COHERENCE

Massimiliano BredaDepartment of Aeronautics

Imperial CollegeSouth Kensington Campus, London, SW7 2AZ

mb2014@ic.ac.uk

Oliver R. H. BuxtonDepartment of Aeronautics

Imperial CollegeSouth Kensington Campus, London, SW7 2AZ

o.buxton@imperial.ac.uk

ABSTRACTIn this paper the role of initial conditions in affect-

ing the evolution of an axisymmetric turbulent jet is ex-amined. The jet’s near-field large-scale coherent structures(the so called Kelvin-Helmholtz vortex rings) are manipu-lated with the aid of non-circular exit geometries of identi-cal open area D2

e . Hot-wire anemometry and 2D-2C planarPIV were completed between the exit of the jet and 26 Dedownstream, to study the effect of round, square and fractalgeometries on the coherent structures and on the jet’s evo-lution both spatially and temporally. It is found the fractalgeometry breaks up the jet’s coherent structures leading toa delay of the overall jet development, meaning some ofthe jet properties can be rescaled in terms of eddy turnovertimes. Moreover, non-equilibrium dissipation is found forall jets at issue, indicating it takes place for this shear flowindependently of the initial conditions. This property iscommon to axisymmetric wakes and grid-generated flowstoo.

INTRODUCTIONRecent studies have been focused on understanding

how the initial conditions of a shear flow affect its evolutiontowards the self-similar state. Since the work of Townsend(1976), it has been generally accepted that across its evo-lution a shear flow would become asymptotically indepen-dent of the initial conditions. This was backed by the workof Antonia & Zhao (2001), who found that a “top-hat” anda fully developed axisymmetric round pipe jet would be-come self-similar after the same streamwise distance. How-ever, George (1989) argued that the self-similar state wouldbe linked to the coherent structures. The work of Antonia& Pearson (2000) hinted in this direction since the authorsfound that different axisymmetric wake generators had dif-ferent mean energy dissipation rates. In the present study,the role of the initial conditions on the evolution of an ax-isymmetric turbulent jet is studied further by manipulating

the coherent structures of an axisymmetric jet with the aidof noncircular exit geometries.

The latter have been shown to significantly affect thenear-field state of an axisymmetric jet. Gutmark et al.(1989), investigating triangular, square and elliptical jets,showed that a combination of corners and flat sides com-bined the small scale turbulence generated at the cornerswith the coherent structures at the flat sides. Shakouchi &Iriyama (2013) showed that a large wetted perimeter led toan increased mixing between the turbulent jet and the irro-tational free stream. Therefore in order to further explorethe effect of the exit geometry on the coherent structures, asquare and fractal orifice are compared to a round one. Thelatter fractal geometry has been previously studied on theperimeter of axisymmetric wake generators where a break-up of the coherent structures and a reduction of the sheddingenergy were observed (Nedic et al., 2015). Having shownthe reduction in energy content of the coherent structures,Nedic et al. (2013b) examined the self-similar region ofthe flow and found novel scaling exponents for the spatialevolution of the wake’s width and centreline velocity de-cay. This was attributed to the presence of non-equilibrium(NE) dissipation (refer to Vassilicos (2015) for a completedescription of the topic), where the inter-scale energy trans-fer is not instantaneously in equilibrium with the turbulentkinetic energy dissipation rate. When non-equilibrium dis-sipation is present, the dissipation coefficient Cε is not con-stant but a function of local Reynold number Reλ f

(eq. (1))and of the global one ReG (eq. (4)), as shown in eq. (2).

Reλ f=

λ f u′

ν(1)

Cε ≡εL

u′3∼

Re1/2G

Reλ f

6= constant

∀Reλ f= Reλ f

(x) 6= constant

(2)

1

10C-2

Here, ε is the dissipation, u′ the root-mean-square (r.m.s.)of the streamwise velocity fluctuations and L the integrallength scale. In order to study how the exit geometry af-fects the near-field coherent structures and the subsequentjet’s evolution, two dimensional-two component (2D-2C)planar particle image velocimetry (PIV) and hot-wire con-stant temperature anemometry (CTA) are performed.

METHODOLOGYThe study is conducted in a newly designed jet facility

at Imperial College London. The flow is conditioned with ahoneycomb and two screens of decreasing mesh size, beforeit enters a 5th order polynomial contraction. Preliminarytesting showed the exit flow had a sharp “top-hat” mean ve-locity profile. Three different orifices of thickness 0.1 mmand of identical open area (D2

e), as shown in fig. 1, are at-tached to the nozzle. Apart from the jet exit shape, all theconditions of the jets are kept the same. Three Reynoldsnumbers of ReG = 3,000 (Ue = 2.98m/s), ReG = 10,000(Ue = 9.93m/s) and ReG = 31,000 (Ue = 30.80m/s) werechosen, where two different regimes of Kelvin-Helmholtzvortices were expected in order to test the ReG-dependence.

The flow is studied between 0 and 26De with two-dimensional two-component (2D-2C) planar PIV, with thelaser sheet placed across the centreline of the jets. A seededlow speed co-flow (speed of ≈ 0.1m/s) was added to avoidbiasing the experiments as background fluid is entrainedinto the jet. In total, 1,500 images were acquired per jetconfiguration at various streamwise locations. The fieldsof view were 9De × 6.7De. An initial interrogation areaof 64× 64 pixels was used to process the images recur-sively with a final window size of 16× 16 pixels. A 50% overlap between adjacent interrogation areas was usedleading to a final vector spacing of 0.03De. Following thepreliminary results, highly resolved planar 2D2C PIV to in-vestigate the velocity gradients was performed in the nearand far-field (23-26 De, where the Reynolds stresses be-come self-similar). This time, a final interrogation area of12x12 pixels with 50 % overlap was used. This led to avector spacing of 0.013 De and a spatial resolution between3 and 4 Kolmogorov length scales (η) and ranged between0.08-0.12 longitudinal Taylor length scales (λ f ). The test-ing campaign was concluded with single hot-wire constanttemperature anemometry (CTA) in the near- and far-field atReG = 10,000 and ReG = 31,000, with a sampling rate of100 kHz, which was subsequently low-pass filtered at 30kHz. Gold-plated 55P01 Dantec hot wires were used, theratio between wire length and diameter was 250 (wire diam-eter 5 µm, spatial resolution 1.25mm ≈ 10η), the overheatratio was set at 0.8 and the square wave response was above40 kHz.

De =√

Exit area Ue =Volumetric flow rate

Exit area(3)

ReG =UeDe

ν(4)

RESULTSIn order to understand how the exit geometry affects

the large scale coherent structures (the Kelvin-Helmholtz

vortex rings), the two-point spatial correlation Rvx of theradial velocity component v in the streamwise direction xwas calculated as a function of the stremawise incrementξ across the jet shear layer (defined as the peak of stream-wise fluctuations), where the coherent structures reside inthe near-field, as shown in eq. (6). The ensemble average〈∗〉 indicates an average in both time and space (within a3× 3 vector box surrounding location (x,r) to improve thestatistical convergence).

V = v+ v′ (5)

Rvx(x,r,ξ ) =〈v′(x,r)v′(x+ξ ,r)〉

〈v′2(x,r)〉1/2〈v′2(x+ξ ,r)〉1/2(6)

As shown in fig. 2a, the fractal geometry suppresses theanti-correlation of Rvx, which is usually associated with thevorticity of the Kelvin-Helmholtz vortex rings, suggestingthey have been broken-up. This may be seen in the PIV im-ages in fig. 3, where the fractal jet shows no signs of largecoherent structures. Moreover, the turbulent kinetic energyproduction region of the fractal jet was found to be extendedcompared to the round and square jets, as shown in fig. 2b.This indicated that the break up of the coherent structuresaltered the initial development of the jet, suggesting thatthe fractal jet development may be delayed further down-stream compared to the other two cases. As stated by Hunt(1994), the turbulent structure of a shear flow retains “mem-ory” of the initial conditions, since the eddy convection timeis linked to its growing time scale. Hence, despite irrota-tional fluid being entrained into the jet stream, the initialconditions still affect the development of the eddies throughthe large-scale structures far downstream. Therefore, the jetphysics is revisited in terms of the advection time scale (re-lated to the mean velocity) and the number of eddy turnovertimes undergone whilst the flow advects a distance x.

Nex(x) =Advection time

Eddy turnover time=∫ x

1De

u′2(ξ )1/2

Lux(ξ )u(ξ )dξ . (7)

Here, the integration is initiated at x/De = 1 since nei-ther the velocity fluctuations nor the velocity correlationsare well defined at the “top-hat” jet exit. In the near-field,the calculations are performed at the shear layer (found asthe peak of r.m.s. velocity fluctuations). From the hot-wiredata, the integral length scale Lux is calculated integratingthe autocorrelation of the velocity fluctuations in time delayτ from τ = 0 to τ = T , where T → ∞. From a sensitiv-ity analysis, T = 20τ ′ (where τ ′ is the time delay for thefirst zero-crossing of the correlation) was found to be suffi-cient. The autocorrelation in time was converted into spaceby multiplying by the mean velocity and assuming the Tay-lor’s hypothesis.

R(x,τ) =u′(x, t)u′(x, t + τ)

u′2(x)(8)

2

10C-2

Lux(x) = u(x)∫ T

0R(x,τ) dτ (9)

Now the jet properties are revisited in terms of Nex , to findout which ones are unique and which ones are simply de-layed by the break-up of the coherent structures.

First the entrainment rate of background fluid into theturbulent jet is considered. Manipulating the near-fieldcoherent structures has been shown to significantly affectthe entrainment rate and mixing of irrotational flow intothe jet’s turbulent stream (Brown & Roshko, 1974) and asa consequence its evolution towards the self-similar state.Here, the entrainment rate is evaluated in terms of the lo-cal spatial gradient of the volume flux Q in the streamwisedirection x, defined in Craske & van Reeuwijk (2014) as afunction of the entrainment parameter α , the characteristicjet velocity um and the characteristic jet width rm. The en-trainment parameter α contains contributions from the tur-bulent kinetic energy production and from the energy, mo-mentum and volume fluxes. It is also consistent with theentrainment model of Kaminski et al. (2005).

∂Q∂x

= 2αrmum. (10)

Due to the finite size of the field of view, the integrationof the terms in table 1 is done up to the radial location R0where the inwards radial velocity v is 1% of the streamwisecentreline velocity ucl .

v(x,R0) = 0.01ucl(x) (11)

Table 1: Terms of the entrainment parameter equation

Term Equation

Volume Flux Q = 2∫ R0

0 urdr

Momentum Flux M = 2∫ R0

0 u2rdr

Characteristic jet width rm = QM1/2

Characteristic jet velocity um = MQ

Turbulence Production δm = 4u3

mrm

∫ R00 u′v′ ∂u

∂ r rdr

Energy Flux γm = 4u3

mr2m

∫ R00 u3rdr

Entrainment parameter α =− δm2γm

+ Q2M1/2

∂

∂x (lnγm)

It is shown that the fractal geometry leads to a reduc-tion of the entrainment parameter α , however when ∂Q/∂xis evaluated in terms of eddy turnover times, all jets tendasymptotically to the same trend after approximately 3 Nex ,as shown in fig. 4. Hence, it is found that despite the differ-ent exit geometries, all jets have a comparable entrainmentrate when rescaled accordingly after an initial “transient”geometry-specific period of 3-4 large-scale eddy turnovers.In terms of mixing, at ReG = 3,000 the round jet has thehighest spreading rate, while the fractal jet the lowest. AtReG = 10,000, all jets have a comparable spreading rate,

however due to the different location of the virtual origin(assuming the jet evolves from a single point source of mo-mentum), the round jet is always the widest in the far-fieldwhile the fractal jet the narrowest.

The jet physics are evaluated further by looking at theterms of the time averaged momentum equation (TAME),split into radial (eq. (12)) and streamwise (eq. (13)) direc-tions as in Chao & Sandborn (1966).

v∂v∂ r

+u∂v∂x

+∂v′2

∂ r+

1r(v′2−w′2)+

∂u′v′

∂x=

− 1ρ

∂ p∂ r

+ν

(∂ 2v∂ r2 +

1r

∂v∂ r− v2

r2 +∂ 2v∂x2

) (12)

v∂u∂ r

+u∂u∂x

+∂u′v′

∂ r+

u′v′

r+

∂u′2

∂x=

− 1ρ

∂ p∂x

+ν

(∂ 2u∂ r2 +

1r

∂u∂ r

+∂ 2u∂x2

) (13)

The data are taken in the same region 2− 5De, howeverthey are plotted in terms of eddy turnover time. Note thatthe terms containing the pressure gradients and the out-of-plane fluctuations w′ could not be computed directly fromthe data, and were thus derived from the residual of the bal-ance of the various terms. Overall the terms that remainsignificant throughout the jet’s evolution are the pressureterms, the streamwise mean velocity gradient, the Reynoldsshear stress and its gradient. It is found that the radiallyevolving terms of the mean momentum follow the sametrend after 3-4 Nex , as shown in fig. 5b, since the noncir-cular jets are evolving towards an axisymmetric state. Sim-ilar findings had been seen by Nedic et al. (2013a), whichshowed axisymmetric wakes generated by both square andfractal plates would quickly return to axisymmetry after aninitial development region. However, the jets show differenttrends for the streamwise mean momentum terms, as shownin fig. 5a, indicating the forcing through a noncircular ge-ometry has a larger effect on the evolution of the stream-wise mean flow. The influence of the fluctuating velocity isfound to be minor compared to the mean flow. Hence, boththe TAME and the entrainment rate indicates that the jetshave an initial development region, which lasts between 3-4Nex , where the jet evolves towards an axisymmetric state.After this distance, all jets evolve in a comparable fashionradially, irrespectively of the jet exit.

Overall, the noncircular forcing is found to signifi-cantly affect the properties of the jets only for a small “de-veloping” region near the jet exit. Now, their subsequentevolution towards the self-similar state is examined to eval-uate if the argument of Townsend (1976) (the jets becomeasymptotically independent of the initial conditions) or ofGeorge (1989) (the self-similar state is linked to the coher-ent structures) holds. If it is assumed that classical dissi-pation holds (Cε ∼ constant), the self-similar scalings arewidely reported in Tennekes & Lumley (1972). A prelim-inary analysis indicated that even if it was assumed thatCε =Cε (x), the same scalings for centreline velocity ucl andthe jet half-width r1/2 would be found due to the constantmomentum flux of the jet. This was found to be the caseexperimentally. Therefore, traces of non-equilibrium dissi-pation were looked for in the microscales, dissipation and

3

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dissipation coefficient, where the following scalings wereexpected:

η ∼ x ε ∼ x−4 Cε ∼ constant (14)

The region probed was between 24 and 26 De downstream.As observed in fig. 6a, ε ∼ xα , where α ' −3, which islower than the value predicted by the classical theories. Thisreflects on the Kolmogorov length scale which scales asη ∼ x0.8. Additionally, calculating Cε from both the PIVand the CTA data, it is found Cε is a function of both Re1/2

Gand of Reλ f

, as shown in fig. 6b. This behaviour clearly out-lines the presence of non-equilibrium dissipation togetherwith classical scalings for centreline velocity and jet half-with. This was also concurrent to self-similar mean velocityand locally self-similar fluctuating Reynolds stresses pro-files (i.e. normalised by a local value of the Reynolds stressrather than of the centreline velocity). Comparable find-ings were found by Dairay et al. (2015) for axisymmetricwakes, indicating it is not a unique behaviour of axisymmet-ric jets, even though for that shear flow the presence of non-equilibrium dissipation was marked by novel scaling expo-nents for the wake width and velocity deficit (Nedic et al.,2013b). This is the first time traces of non-equilibrium dis-sipation have been found for an axisymmetric jet. More-over, this type of dissipation appeared for all jets at issue, in-dependently of the state of the coherent structures and of theinitial conditions. This indicates that the coherent structurescan affect the rate at which the jet develops, however themechanics far away from the jet exits remains unaffectedby them. It is noted, Hearst & Lavoie (2014) testing frac-tal grid generated flows showed that initially the flow wouldhave non-equilibrium dissipation but then it transitioned tothe classical Cε ∼ constant. Since the classical dissipationlaw has been documented for axisymmetric jets too, theymay eventually transition to the classical state, however thisdistance was not reached in the present experiment.

CONCLUSIONSThis study has analysed the role of the initial condi-

tions in affecting the evolution of an axisymmetric turbulentjet towards its self-similar state. Noncircular geometrieswere used to manipulate the near-field coherent structuresand the spatial and temporal properties of the jets were mea-sured with planar PIV and hot-wire anemometry. The frac-tal geometry is found to break-up the jet coherent structures,causing a delay of the overall jet development further awayfrom the jet’s exit. In fact, once the properties are re-scaledin terms of eddy turnover times, the terms involving the ra-dial evolution and entrainment are found to be comparablebetween the jets irrespective of the initial conditions. Thenoncircular forcing is found, however, to affect the stream-wise evolution of the mean momentum since the streamwiseterms of the TAME do not collapse in the region studied.Looking at the jet evolution of the self-similar state, the jetsare found to have the same scaling exponents for centre-line velocity and jet half-width as predicted by the classicaltheories. However, at the microscales, non-equilibrium dis-sipation is found, as shown by the new scaling exponents ofdissipation and of the Kolmogorov length scale η . More-over, this is found concurrent to locally self-similar profilesof the Reynolds stresses. Such properties are common toall jets studied, suggesting, that despite the very different

initial conditions, the evolution downstream of the jets isunaffected by the state of the coherent structures.

REFERENCESAntonia, R. A. & Pearson, B. 2000 Effect of initial condi-

tions on the mean energy dissipation rate and the scalingexponent. Physical Review E 62 (6), 8086–8090.

Antonia, R. A. & Zhao, Q. 2001 Effect of initial conditionson a circular jet. Experiments in Fluids 31 (3), 319–323.

Brown, Garry L. & Roshko, Anatol 1974 On density effectsand large structure in turbulent mixing layers. Journal ofFluid Mechanics 64, 775–816.

Chao, J. L. & Sandborn, V. A. 1966 Evaluation of themomentum equation for a turbulent wall jet. Journal ofFluid Mechanics 26, 819–828.

Craske, John & van Reeuwijk, Maarten 2014 Energy disper-sion in turbulent jets. Part 1: Direct simulation of steadyand unsteady jets. Journal of Fluid Mechanics 763, 500–537.

Dairay, Thibault, Obligado, Martin & Vassilicos,John Christos 2015 Non-equilibrium scaling lawsin axisymmetric turbulent wakes. Journal of FluidMechanics 781, 166–195.

George, William K. 1989 The Self-preservation of turbu-lent flows and its relation to initial conditions and coher-ent structures. Tech. Rep.. University at Buffalo, Buffalo,New York, USA.

Gutmark, Ephraim, Schadow, K. C. & Parr, T 1989 Noncir-cular jets in combustion systems. Experiments in Fluids7, 248–258.

Hearst, R. Jason & Lavoie, Philippe 2014 Decay of turbu-lence generated by a square-fractal-element grid. Journalof Fluid Mechanics 741, 567–584.

Hunt, Julian Charles Roland 1994 Atmospheric jets andplumes. In Recent Research Advances in the Fluid Me-chanics of Turbulent Jets and Plumes (ed. P. A. Davies &M. J. Valente Neves), pp. 309–334. London: NATO ASISeries.

Kaminski, Edouard, Tait, Stephen & Carazzo, Guillaume2005 Turbulent entrainment in jets with arbitrary buoy-ancy. Journal of Fluid Mechanics 526, 361–376.

Nedic, Jovan, Ganapathisubramani, Bharathram & Vassili-cos, John Christos 2013a Drag and near wake character-istics of flat plates normal to the flow with fractal edgegeometries. Fluid Dynamics Research 45 (061406).

Nedic, Jovan, Supponen, O., Ganapathisubramani,Bharathram & Vassilicos, John Christos 2015 Ge-ometrical influence on vortex shedding in turbulentaxisymmetric wakes. Physics of Fluids 27 (035103).

Nedic, Jovan, Vassilicos, John Christos & Ganapathisubra-mani, Bharathram 2013b Axisymmetric turbulent wakeswith new nonequilibrium similarity scalings. PhysicalReview Letters 111 (144503).

Shakouchi, Toshihiko & Iriyama, Shota 2013 Flow char-acteristics of submerged free jet flow from petal-shapednozzle. In 4th International Conference on Jets, Wakesand Separated Flows.

Tennekes, Henk & Lumley, John L. 1972 A First Coursein Turbulence, 1st edn. Cambridge, Massachusetts; Lon-don: M.I.T. Press.

Townsend, A. A. 1976 The Structure of Turbulent ShearFlow, 2nd edn. New York: Cambridge University Press.

Vassilicos, John Christos 2015 Dissipation in TurbulentFlows. Annual Review of Fluid Mechanics 47, 95–114.

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(a) Round (b) Square (c) Fractal

Figure 1: Jet exits

0 0.5 1 1.5 2 2.5

ξ/De

-0.2

0

0.2

0.4

0.6

0.8

1

Rvx

Round jet

Square jet, major

Square jet, minor

Fractal jet, major

Fractal jet, minor

(a)

4 5 6 7 8 9 10 11 120.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

(b)

Figure 2: (a) Two point correlation Rvx at 2 De at shear layer and (b) Turbulent production region at jet centreline atReG = 10,000.

(a) Round (b) Square (c) Fractal

Figure 3: PIV images with co-flow switched off, for visualisation purposes only.

5

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1 2 3 4 5-0.04

-0.02

0

0.02

0.04

0.06

0.08

(a)

0 1 2 3 4 5 6 7-2

0

2

4

6

8

10

1210-3

(b)

Figure 4: (a) Entrainment parameter and (b) local entrainment rate as a function of Nex at ReG = 10,000.

1 2 3 4 5 6 7-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

(a)

1 2 3 4 5 6 7-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

(b)

Figure 5: (a) Streamwise and (b) radial terms of the TAME at ReG = 10,000.

24 24.5 25 25.5 262.4

2.6

2.8

3

3.2

3.4

3.6

10-4

(a)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

0.2

0.4

0.6

0.8

1

1.2

0.24 0.26 0.28 0.3 0.32

0.2

0.25

0.3

(b)

Figure 6: (a) Dissipation rate ε at ReG = 10,000 and (b) dissipation coefficient Cε at ReG = 10,000 (black) and atReG = 31,000 (blue).

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