AIAA Scitech, Kissimmee, Florida, USA

Streamwise Vortices in Plane Mixing Layers

Originating from Laminar or Turbulent Initial

Conditions

W. A. McMullan∗ and S. J. Garrett†

Dept. Engineering, University of Leicester, LE1 7RH, United Kingdom

Large Eddy Simulations of the plane mixing layer have been performed, for the purposeof educing the streamwise vortex structure that may exist in these flows. Both an initially-laminar and initially-turbulent mixing layer are considered in this study. The initially-laminar flow originates from Blasius profiles with a white noise fluctuation environment,whilst the initially turbulent flow has an inflow condition obtained from an inflow turbulencegeneration method. Both simulations produce good mean flow statistics when comparedwith reference experimental data. The simulations capture the change in growth ratewhen the initial conditions are either laminar or turbulent. Flow visualisation imagesdemonstrate that both mixing layers contain organised turbulent coherent structures, andthat the structures contain rows of streamwise vortices distributed across the span of themixing layer. Ensemble averaging of the cross-plane data, however, shows no evidence forstatistically stationary streamwise vortices in either simulation.

I. Introduction

The plane turbulent mixing layer is commonly referred to as a statistically two-dimensional flow. Ex-perimental work has shown that the mixing layer rolls up into primary vortices owing to the action of theprimary Kelvin-Helmholtz (K-H) instability. These vortices are then responsible for governing the entrain-ment of fluid into the layer, and the growth of the layer as a whole. At modest Reynolds numbers, it hasbeen shown that the primary vortices pair together, with the growth of the mixing layer occurring as aresult of the interaction.1 At higher Reynolds number, the coherent structures embedded in the turbulentflow2 undergo continuous growth between interactions,3 with recent experimental evidence demonstratingthat this continuous growth is linear in nature, with interactions between coherent structures contributingnothing to the overall growth of the mixing layer.4 For mixing layer flows originating from initially turbulentboundary layers, it is not entirely clear whether coherent structures form in the flow.14 It is well-known,however, that the initially turbulent mixing layer displays a reduction in its rate of growth when compared toits initially laminar counterpart.15,17 The cause for this change in growth remains an open research question.

Whilst early flow visualisations demonstrated that the coherent structures in the flow were essentiallyquasi-two-dimensional, subsequent experimental studies presented evidence of a secondary, streamwise struc-ture in the mixing layer.5 Extensive research into the origin and evolution of these streaks demonstrated thatthe streamwise vortex strength and spacing can vary significantly between experimental facilities.6−11 Quan-titative measurements by Bell & Mehta11 showed that, for initially laminar conditions, stationary streamwisevortices form following the roll-up of the mixing layer, and evolve slowly with increasing downstream dis-tance. These stationary vortices persist far into what would normally be considered the fully-developedturbulent flow.12 Subsequent measurements of mixing layers originating from turbulent boundary layersshowed no particular evidence for stationary streamwise vorticity.13

∗Lecturer, AIAA Senior Member.†Reader. AIAA Member.Copyright c© 2015 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free

license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by thecopyright owner.

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In recent years, advances in supercomputing power have permitted high-fidelity simulations of mixinglayers to be performed. The temporal mixing layer has been studied extensively, with the effect of initialconditions on the flow documented in detail.18−21 Temporal mixing layers, however, do not predict theasymmetric entrainment ratio noted in experiments,22 and are beyond the scope of interest of this study.Spatially-developing mixing layer simulations are less common in the literature, with most of the publishedliterature on the topic being produced in the past decade. Recent Direct Numerical Simulation (DNS) studieshave shown that computing power is now sufficient to simulate the turbulent region of the mixing layer flow ingreat detail.23,24 Most of the published work focuses on mixing layers which originate from initially laminarconditions, with simple boundary conditions used to model the inflow to the computational domain.25−27 Thehyperbolic tangent function commonly used to describe the inflow of the spatially-developing flow producesinconsistencies in the initial development of the flow, and the overall entrainment into the mixing layer,28−31

when compared to simulations using prescribed boundary layer profiles for the inflow.Very few reported simulations of spatially-evolving mixing layers which originate from a turbulent high-

speed boundary layer are present in the research literature. This is likely due to the difficulty in producinga time dependent, turbulent inflow condition for numerical simulation methods such as Large Eddy Simu-lation and Direct Numerical Simulation. Recent research by Sandham & Sandberg32 suggested that Brown& Roshko structures2 may not be present in the initial region of a mixing layer which develops from aturbulent boundary layer. However, a previous study has shown the potential for Large Eddy Simulation,combined with an appropriate inflow generation technique, to capture the reduction in growth rate caused byinitially-turbulent boundary layers.33 In that study, coherent structures were found to emerge some distancedownstream of the splitter plate. The flow statistics presented in the study were in good agreement withexperimental data, and warranted further investigation.

The aim of the present study is to elucidate the nature of the streamwise vortex structure that exists in theplane mixing layer. High-fidelity Large Eddy Simulations are performed against the reference experimentaldata of Browand and Latigo.17 For the simulations where the high-speed side separating boundary layer isturbulent, the recycling and rescaling method of Xiao et al.34 is used to generate a turbulent inflow condition.Cross-stream measurements at several streamwise locations are recorded in order to directly compute thestreamwise vorticity and secondary shear stress in the flow. This mean statistical information can then beused to quantify the presence of the streamwise vortices in the simulations.

II. Numerical Methods

The spatially filtered equations for conservation of momentum and mass for a uniform density fluid are

∂ui∂t

= −1

ρ

∂p

∂xi+

∂

∂xj(−uiuj + 2νSij) (1)

Sij =1

2

( ∂ui∂xj

+∂uj∂xi

)(2)

∂ui∂xi

= 0 (3)

These equations are discretised on a staggered mesh. The viscosity ν can consist of both a molecular anda subgrid component, ν = νm + νsg, if a subgrid-scale model is used. In this study the Wale Adapting LocalEddy-viscosity (WALE) model35 is utilised, with the eddy viscosity, νsg, calculated by

νsg = (Cw∆)2(Sd

ijSdij)

3/2

(SijSij)5/2 + (SdijS

dij)

5/4(4)

where Sdij = 1

2 (g2ij+g2ji)− 13δij g

2kk, gij = ∂u/∂x, and Cw is a model constant specified a priori. The WALE

model is attractive for the simulation of free shear flows with initially laminar conditions, as it predicts zeroeddy viscosity in the presence of pure shear. It has been shown in other work by the author that this modelproduces improved plane mixing layer predictions when compared to the standard Smagorinsky model.36

Temporal advancement of the governing equations is performed by the Adams-Bashforth method. In thismethod, a provisional velocity field is obtained through

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u∗i = uni + ∆t(3

2Hn

i −1

2Hn−1

i

)(5)

with

Hi =∂

∂xj(−uiuj + 2νSij) (6)

The provisional velocity u∗i does not satisfy continuity, and is updated to the actual velocity at the nexttime step, un+1

i , by the pressure solver. The pressure field is solved implicitly by the use of the continuityequation. The provisional velocity field of u∗i is used to derive the actual velocity by including the gradient

of an unknown pressure field pn+12 , such that

un+1i = u∗i −∆t

∂pn+12

∂xi(7)

As the new velocity field must have zero divergence, a Poisson equation can be found for the pressurefield between the present and next time step

∇2pn+12 =

1

∆t

∂u∗i∂xi≡ R (8)

As the current code requires that one spatial dimension be periodic, a Fourier transform can be performedon equation 8 to give a sequence of Helmholtz problems for each wavenumber kz

∂2p

∂x2+∂2p

∂y2− k2z p = R (9)

A multi-grid method is used to enhance the speed of convergence of the solution. A standard convectiveoutflow condition is used at the downstream end of the computational domain.

For the purposes of flow visualisation a passive scalar is also introduced into the flow domain, which isgoverned by the equation

∂ξ

∂t=

∂

∂xi

(− uiξ + α

∂ξ

∂xi

)(10)

where α is the diffusivity, which contains both a molecular and a subgrid component, α = αm + αsg,if a subgrid-scale model is used. With a subgrid scale model employed, the subgrid diffusivity is set toαsg = νsg/0.7. The scalar is discretised on the staggered mesh at the cell centre, and a third-order upwindingscheme is used to calculate the scalar flux between cell faces. The Adams-Bashforth method is used tointegrate the scalar field forward in time, with a method of discretisation very similar to equations 5 - 6.

For the simulation requiring a turbulent inflow condition, the recycling and rescaling method of Xiaoet al.34 is employed. This method requires virtual domains to be placed upstream of the main simulationdomain, in which the turbulent flow-field is generated. The flow in the virtual domain then provides theinflow condition for the main simulation domain. This method has been used extensively in recent research,being used to specify inflow conditions for plane mixing layers,34,33 axisymmetric jet flows,37 and multiphaseflow.38 In all of these flow configurations, the turbulent inlet condition produced main simulation flow-fieldswith accurate flow statistics.

Case U1 θ1 (mm) U2 θ2 (mm)

LIS 25.6 0.457 5.19 0.86

TIS 25.6 0.81 5.19 0.86

Table 1. Flow Simulation Parameters.

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III. Simulation Set-up

The experiments of Browand and Latigo17 form the reference dataset for the present study. In those ex-periments, both initially-laminar and initially-turbulent mixing layers were studied, with constant freestreamvelocities maintained between experimental runs. The initial conditions of the mixing layer were reasonablywell-documented, making these experiments a good candidate for replication by numerical simulation. Inaddition, several useful statistical quantities were reported, which will be used to asses the quality of thesimulation data.

The flow conditions of the experiments are reported in Table II. The velocity ratio, R, is defined as

R =U1 − U2

U1 + U2(11)

where U1 and U2 are the freestream velocities of the high- and low-speed streams at the splitter platetrailing edge respectively. The initial velocity ratio parameter of the flow is R = 0.66 for both experiments.As the lower wall of the experimental facility was fixed horizontally, a small pressure gradient was establishedin the flow. This pressure gradient had the effect of decreasing the low-speed side freestream velocity, thusincreasing R with increasing streamwise distance. Statistical quantities obtained in the experiments weretherefore presented as a function of local freestream velocities.

The Laminar Inflow Simulation (LIS) and Turbulent Inflow Simulation (TIS) share common parameters;the WALE model constant takes a value of Cw = 0.56, the time step of the simulation is 6.0 ×10−7s, and thesimulations are run for 1,200,000 time steps. The initial 260,000 time steps are used to propagate the initialcondition through the computational domain and achieve a statistically stationary flow field. Statisticalsamples and flow visualisation outputs are recorded during the remaining time steps.

A. Laminar Inflow Simulation (LIS)

The computational domain for the untripped simulation begins at the downstream end of the splitter plate,with no solid geometry included in it. The domain extends 0.75 × 0.61 × 0.18 (m) in the streamwise,cross-stream, and spanwise directions respectively. It has been shown elsewhere that this spanwise domainextent is sufficient to prevent confinement of the flow.40 The plane of the splitter plate is located at themid-point of the lateral domain, as was the case in the experimental facility.

The domain is discretised into 768 × 256 × 256 cells, resulting in an overall cell count of 50.3 million cells.The mesh is refined near the splitter plate, and grid-stretching is employed in both the streamwise and cross-stream directions in order to reduce the overall computational cost of the simulations. The minimum gridspacing in the streamwise and lateral directions are ∆xmin = 0.0002m and ∆ymin = 0.00004m respectively.The cell spacing in the spanwise direction is uniform.

The experimental data of Browand & Latigo demonstrated that the flow at the trailing edge of the splitterplate was laminar in nature in their untripped experiment. The measured mean streamwise velocity profileshowed that the laminar boundary layers were very near to the Blasius form. In Case LIS, Blasius profileswhich match the momentum thickness of each stream are applied at the inflow plane of the simulation.Pseudo-random white noise of a magnitude which matches the reference data are superposed onto the inletprofile at each time step. The method applied here is very similar to that employed in previous studies ofthe idealised mixing layer by the author.28,29 In those studies, it was found that the transition to turbulence,and the coherent structures in the turbulent flow were readily captured.

B. Turbulent Inflow Simulation (TIS)

The reference experimental data recorded the mean streamwise velocity profile, and its associated velocityfluctuation distribution, in the region of the trailing edge of the splitter plate. No cross-stream or spanwisevelocity fluctuation profiles were recorded in the experiment. As the inflow generation method also requiresprofiles of the cross-stream and spanwise r.m.s. velocity fluctuations, these profiles are obtained from theDNS data of Spalart.41 Case TIS, therefore, is not necessarily an exact numerical replication of the referenceexperiment as a complete description of the experimental initial conditions has not been provided.

The virtual domain is placed on the high-speed side stream and has an extent of 14δ. It is separatedfrom the trailing edge of the splitter plate by a small region of length δ. This is done to permit the flowto develop naturally along the splitter plate prior to its separation and formation of the mixing layer. The

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recycling plane is located 1.2δ upstream of the trailing edge of the splitter plate. A small region upstream ofthe trailing edge of the splitter plate on the low-speed side is added to the computational domain, to allowthe flow to develop naturally at the trailing edge. The recycling method is not used in the low-speed laminarstream.

The specification of the mesh for the Turbulent Inflow Simulation presents a significant computationalchallenge. There are conflicting requirements between the turbulent boundary layer and mixing layer regionsof the flow. For the turbulent boundary layer, a highly-resolved near-wall region is necessary to capturethe viscous sublayer, whilst in the mixing layer region it is important that the spanwise domain extent issufficient to prevent any confinement of the turbulence structure that develops in this direction.40 To allowfor direct comparisons with the Laminar Inflow Simulation, the grid distribution in the mixing layer regionis unchanged. The grid distribution in the boundary layer region matches that of the first streamwise planein the mixing layer domain - yielding a posteriori estimations of the non-dimensional near-wall grid spacingof ∆x+ ≈ 15, ∆y+ ≈ 1.5, and ∆z+ ≈ 50.

The mesh distribution of the mixing layer domain remains unchanged from that of Case LIS. The inclusionof the extra region computational domain upstream of the trailing edge adds 33.35 million cells to thecalculation, resulting in a total of 83.65 million cells. The inflow condition for the high-speed stream isobtained using the recycling method described above. The low-speed side inflow condition is provided byimposing a Blasius velocity profile at the inflow plane, with white noise disturbances of magnitude matchingthose found in the experimental facility superposed onto the mean profile at each time step. The low-speedinflow velocity profile is chosen such that the momentum thickness of the boundary layer matches that ofthe experiment at the trailing edge.

IV. Results

The boundary layer mean flow statistics obtained from the inflow generator have been described else-where,33 and are not repeated here. The generation technique produces satisfactory predictions of theboundary layer, given the uncertainty in the initial conditions of the experimental conditions, and the re-sulting mixing layer flow agrees well with the reference data.

In the reference experiment, the fixed lateral walls of the test section produced an adverse pressuregradient in the flow. This resulted in a decrease in the low-speed side freestream velocity with increasingstreamwise distance from the splitter plate. Where flow statistics are normalised by the velocity differenceacross the mixing layer, ∆U = U1 − U2, the local values of the freestream velocities are used.

A. Mean Mixing Layer Flow Statistics

The integral thickness of the mixing layer is commonly determined through the momentum thickness, θ.This quantity is defined as

θ =1

∆U2

∫ ∞−∞

(U1 − ut)(ut − U2)dy (12)

where ut is the mean streamwise velocity. The normalised momentum thickness distribution in bothsimulations is shown in Figure 1. The quantity is normalised by the initial momentum thickness, θi, whichis assumed to be equal to the high-speed stream boundary layer momentum thickness. There is extremelygood agreement between simulation and experimental data in both simulations. The tailing off of the profilestowards the downstream end of the computational domain is frequently reported in simulations of spatially-developing shear layers23,24,36 and is caused by the alteration of the entrainment characteristics of coherentstructures as they pass through the outflow boundary.40 The reduction in the growth of a mixing layeroriginating from an initially-turbulent high-speed side boundary layer is reproduced well in the Case TIS. Inthe experiment, the rate of growth of the initially-turbulent mixing layer relaxed towards that of its initiallylaminar counterpart downstream of x/θi ≈ 400, and there is evidence for this relaxation in Case TIS. Thecomputational domain, however, is too short to verify that this switch is captured correctly in the simulation.

The mean streamwise velocity, recorded in the self-similar region of both simulations, agrees well with theexperimental data, as shown in Figure 2a. The streamwise velocity fluctuation profile is also well-predictedby both simulations (Figure 2b). The mean flow statistics presented here indicate that the mixing layer

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originating from both initially-laminar and initially-turbulent boundary layers can be accurately reproducedby LES.

Power spectral density curves at various streamwise locations along the centreline of the mixing layer areshown in Figure 3 for each case. In Case LIS, distinct peaks are evident in the spectrum at x = 0.02m, whichcorrespond to the primary instability and its subharmonics. Further downstream the spectra approach the−5/3 roll-off indicative of fully developed turbulence. As was described elsewhere,40,33 the transition inCase LIS is precipitated by a pairing interaction of primary vortices, following the formation of secondarystreamwise structure. In Case TIS all spectra display a roll-off in the exponent of −5/3, indicating that fullyturbulent flow is present through the entire streamwise extent of the computational domain. The spectrumat x = 0.005m also demonstrates that the recycling and rescaling method produces a boundary layer flowthat is fully turbulent in nature, as the fluctuations along the centreline at this streamwise location originatein the turbulent boundary layer.

B. Flow Visualisation

It has been shown in previous studies that coherent structures in simulations of plane mixing layers canbe visualised through the spanwise averaging of a passive scalar.36 Representative instantaneous, spanwiseaveraged flow visualisations from the two simulations are shown in Figures 4 & 5 for Cases LIS and TISrespectively. Whilst caution must be exercised when analysing single time instants of flow visualisation, theimages presented in Figures 4 & 5 contain flow features that are readily observed throughout both simulations.In Case LIS, the flow rolls up into K-H vortices, which undergo a transition to turbulence following aninteraction between primary vortices. Beyond the mixing transition (x ≈ 0.12m in the image), quasi-two-dimensional coherent structures are present in the flow. In Case TIS, in contrast, the flow immediatelydownstream of the splitter plate does not display any evidence of K-H rollers. Previous work by the authorhas shown that the mixing layer immediately downstream of the trailing edge is highly irregular, and isinfluenced by the turbulence structure embedded within the separating turbulent high-speed side boundarylayer.33 Further downstream, it turbulent coherent structures of the Brown & Roshko form emerge in themixing layer. Qualitative inspection of sequences of flow visualisation output reveals that the evolution ofthe coherent structures in Case TIS is the same as those found in the post-transition region of Case LIS.

The coherent structures found in Cases LIS and TIS share topographical features. When viewed in aLagrangian frame through the subtraction of the convection velocity of the flow, Uc = 0.5(U1 + U2), thecoherent structures have a centre of rotation which is surrounded by two saddle points. These two saddlepoints, one upstream and one downstream of the centre of rotation, define the extent of the structure. Whilstthese coherent structures can be visualised in a spanwise-integrated sense, there is also an underlying three-dimensional structure contained in them. This three-dimensional structure can be investigated qualitativelyby interrogating the y− z passive scalar flow field. In Figure 6 the passage of a coherent structure through asampling plane in Case LIS shown. This structure proceeds through a sampling plane located at x = 0.45m.This structure is representative of all coherent structures passing, and was chosen for inspection as it doesnot undergo an interaction during its passage through the sampling plane. The y − z cuts of the flow inFigure 7 correspond to the spanwise-averaged images in Figure 6. Figures 6a & 7a record the flow pattern atthe downstream saddle point of the structure. The y−z image reveals a row of ‘mushroom-shaped eruptions’across the span of the mixing layer. These eruptions infer the presence of streamwise vortices in the mixinglayer. As the core of the structure passes through the sampling plane (Figures 6b,6c & 7a,7b), the row ofstreamwise structures is displaced downwards, and a new set of structures appears on the upper side of themixing layer. When the upstream saddle point passes through the sampling plane (Figures 6d & 7d), thelower row of structures disappears, and the upper row of structures displaces downwards to occupy the braidregion. This pattern of scalar distribution in the coherent structure is remarkably similar to that observedin both low- and high-Reynolds number mixing layers.9,8

A representative coherent structure from Case TIS is also analysed in this manner, as shown in Figures 8& 9. The sampling plane for this particular structure is located at x = 0.6m, but all structures share similarfeatures where they are present at any streamwise location in the flow. As the structure passes through thesampling plane, the y − z scalar distribution is very similar to that of the counterpart structure in CaseLIS; the braid region consists of a single row of streamwise structure, whilst the core region of the coherentstructure contains two rows of streamwise structure, located at the upper and lower extremes of the core.

The y − z images presented here provide qualitative evidence of the presence of streamwise vorticityin these simulations. However, these visualisation images do not produce any statistical information on

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the streamwise vorticity in the mixing layer. Statistical information pertaining to the streamwise vortexstructure is presented below.

C. Streamwise Vorticity

Experimental studies have shown that the quantitative data on streamwise vortices in the mixing layer canbe obtained from ensemble averaging of the flow across the span at various downstream locations.11 Moresophisticated experimental facilities have directly sampled y − z planes at various downstream locations todirectly compute the streamwise vorticity in the flow. Further, it has been found that for a flow contain-ing statistically stationary streamwise vortices, the secondary shear stress u′w′ can be used to educe thestreamwise vortex structure.11

In the current simulations, measurement stations are placed at x = 0.02, 0.05, 0.1, 0.15, 0.3, 0.45, and0.6m downstream of the splitter plate trailing edge. At each measurement station, the entire y − z flowfield is sampled at a rate of 1.67kHz, and 740 samples are obtained at each station. From these samples, anensemble average is obtained, and statistical properties of the flow are computed.

The locus of the centreline velocity, Uc, across the mixing layer span is plotted at each measurementstation in Figure 10 for both simulations. Both plots show that the centreline location does vary somewhatacross the span at each location, but also that these fluctuations are rather small in magnitude. There islittle evidence for a regular pattern in the plots for Case LIS at all streamwise locations. Case TIS showssome evidence of a regular pattern at x = 0.1, 0.15m, but beyond this point the distributions again becomesomewhat irregular.

The mixing layer thickness is defined as the vertical distance between the locations where U0.01 = U2 +0.01(U1 − U2), and U0.99 = U2 + 0.99(U1 − U2). The variation in mixing layer thickness across the span isplotted as a function of streamwise distance in Figure 11. In Case LIS, the thickness variation is extremelylow, and reaches a maximum at x ≈ 0.1m (the second measuring station). The maximum variation is 3.8%,significantly lower than the value reported in comparable experiments by Bell & Mehta.11 In the post-transition region the variability in mixing layer hovers at approximately 3%. Case TIS produces a lowermaximum variation of 2.8% at the second measurement station, and the variability relaxes to ∼2.5% forthe remainder of the computational domain. It is interesting to note that a three-dimensional, turbulenthigh-speed boundary layer produces a mixing layer that is more statistically two-dimensional than a mixinglayer originating from initially laminar conditions.

Contours of secondary shear stress, normalised by the square of the velocity difference between thefreestreams, are shown in Figures 12 and 13 at four measurement stations for Cases LIS and TIS respectively.For Case LIS, there appears to be evidence for some organisation in the peaks and troughs of the secondaryshear stress contours - alternating bands of positive and negative u′w′ are evident, but their spanwisescale is significantly larger than that of comparable experimental data.11 This may be due to the differinginitial conditions between the current simulation and the experimental data. Regardless, at subsequentmeasurement stations there is no particular evidence for regular peaks and troughs in the secondary shearstress contours across the span of the mixing layer. Instead, clumps of positive and negative secondary shearstress are present at all measurement stations, with no obvious pattern present in their distribution. Thescale of the clumps of secondary shear stress increases with streamwise distance, implying that the localscale of the streamwise structure increases with increasing streamwise distance, but it is not possible to drawany definitive conclusions regarding the spanwise wavelength of the streamwise structure from these images.Similarly, Case TIS shows no particular evidence of organised streamwise vortices at any of the measurementstation.

V. Conclusions

Large Eddy Simulations of plane turbulent mixing layers originating from both laminar and turbulenthigh-speed boundary layers have been performed. The simulations produce mean statistical data whichagrees well with reference experimental data. Flow visualisations from both simulations reveal that turbu-lent coherent structures of the Brown & Roshko2 form are present in the flow. Cross-stream visualisationsof the passive scalar provide quantitative evidence for secondary streamwise structures, which, when viewedin a spanwise integrated sense, comprise the primary coherent structure. Ensemble-averaged statistical datashows no particular evidence for statistically stationary streamwise vortices in the mixing layer simulation

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with laminar initial conditions. This is surprising, given that a large body of experimental research has pro-duced extensive data to demonstrate their existence in real flows.9,8,11 The reasons for the lack of organisedstreamwise vorticity in the initially laminar mixing layer simulation is currently under investigation. Theinitially-turbulent mixing layer also shows no evidence for statistically stationary streamwise vorticity, inagreement with prior experimental data.13,46

Acknowledgements

The simulations in this research were performed using ALICE, the University of Leicester High Perfor-mance Computing facility.

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27Yang W.B., Zhang H.Q., Chan C.K., Lau K.S., and Lin W.Y., “Investigation of plane mixing layer using large eddysimulation,” Computational Mechanics, Vol. 34, 2004, pp. 423-429.

28McMullan, W.A., Gao S., and Coats C.M., “A comparative study of inflow conditions for two- and three-dimensionalspatially developing mixing layers using Large Eddy Simulation,” International Journal of Numerical Methods in Fluids, Vol.55, 2006, pp. 589-610.

29McMullan, W.A., Gao S., and Coats C.M., “The effect of inflow conditions on the transition to turbulence in Large EddySimulations of spatially developing mixing layers,” International Journal of Heat and Fluid Flow, Vol. 30, 2009, pp. 1054-1066.

30Soteriou, M.C., Ghoniem, A.F., “On the Effects of the Inlet Boundary Condition on the Mixing and Burning in ReactingShear Flows,” Combustion and Flame, Vol. 112, 1998, pp. 404–417.

31Soteriou, M.C., Yang, X., “Inlet Condition Effects on Particle Dispersion in a Shear Layer,” Combustion Science andTechnology, Vol. 148, 199, pp. 59–92.

32Sandham, N.D., and Sandberg, R.D., “Direct numerical simulation of the early development of a turbulent mixing layerdownstream of a splitter plate,” Journal of Turbulence, Vol. 10, 2009, pp. 1–17.

33McMullan, W.A., “On the growth of a plane mixing layer from laminar or turbulent initial conditions,” AIAA Aviation,Atlanta, GA, USA, 2014, AIAA 2014-3096.

34Xiao, F., Dianat, M., and McGuirk, J.J., “A Recycling/Rescaling Method for LES Inlet Condition Generation,” Proceed-ings of the 8th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements, Marseille,France pp. 510–515.

35Nicoud, F., and Ducros, F., “Subgrid-scale stress modelling based on the square of the velocity gradient tensor,” Flow,Turbulence and Combustion, Vol. 62, 1999, pp. 183–200.

36McMullan, W.A., Gao, S., and Coats, C.M., “Organised large structure in the post-transition mixing layer. Part 2. LargeEddy Simulation, ” Journal of Fluid Mechanics, In Press.

37Wang, P.C., and McGuirk, J.J., “Large Eddy Simulation of high speed nozzle flows - assessment & validation of syntheticturbulence inlet conditions,” 20th AIAA Computational Fluid Dynamics Conference, Hawaii, Hi, USA, 2011, AIAA 2011-3555.

38Xiao, F., Dianat, M., and McGuirk, J.J., “LES of turbulent liquid jet primary breakup in turbulent coaxial air flow,”International Journal of Multiphase Flow, Vol. 60, 2014, pp. 103–118.

39Tenaud C., Pellerin S., Dulieu A., and Ta Phuoc L., “Large Eddy Simulations of a spatially developing incompressible3D mixing layer using the v − ω formulation,” Computers & Fluids, Vol. 34, 2005, pp. 67–96.

40McMullan, W.A., “Influence of spanwise domain on the Large Eddy Simulation of an idealised mixing layer,” AIAAAviation, Atlanta, GA, USA, 2014, AIAA 2014-3095.

41Spalart, P.R., “Direct simulation of a turbulent boundary layer up to Rθ = 1410, ” Journal of Fluid Mechanics, Vol.187, 1988, pp. 61–98.

42Biancofiore, L., “Crossover between two- and three-dimensional turbulence in spatial mixing layers,” Journal of FluidMechanics, Vol. 745, 2014, pp. 164–179.

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45McMullan, W.A., Gao S., and Coats C.M., “Analysis of the variable density mixing layer using large eddy simulation,”41st AIAA Fluid Dynamics Conference and Exhibit, Honolulu, HI, 2011, AIAA 2011-3424.

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x / θi

θ / θ

i

0 500 1000 1500 20000

5

10

15

20

25

30

35

40

45

50

LIS

TIS

Expt. lam

Expt. turb

Figure 1. Momentum thickness variation of the simulations.

2(Uc ­ u

t) / ∆U

y /

θ

­1 ­0.5 0 0.5 1­15

­10

­5

0

5

10

15

LIS, x / θi = 1000

TIS, x / θi = 700

Expt.

(a) Mean streamwise velocity.

y / θ

u’ /

∆U

­15 ­10 ­5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

LIS, x / θi = 1000

TIS, x / θi = 700

Expt.

Experimental error

(b) Streamwise velocity r.m.s. fluctuation.

Figure 2. Mixing layer flow statistics. Experimental data recorded at x/θi= 1780 in the initially-laminar case,and x/θi= 1000 in the initially-turbulent case.

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f (Hz)

Powe

r Spe

ctra

l Den

sity

101 102 103 104 105

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

x = 0.02 mx = 0.05 mx = 0.10 mx = 0.15 m

-5/3

(a) Case LIS.

f (Hz)

Po

we

r S

pe

ctr

al D

en

sity

101

102

103

104

105

10­16

10­14

10­12

10­10

10­8

10­6

10­4

10­2

100

x = 0.005 m

x = 0.05 m

x = 0.15 m

­5/3

(b) Case TIS.

Figure 3. Power Spectral Density of streamwise velocity fluctuations.

x (m)

y (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Level<ξ>z:

10.01

30.08

50.15

70.22

90.29

110.36

130.43

150.5

170.57

190.64

210.71

230.78

250.85

270.92

290.99

_

Figure 4. Instantaneous spanwise averaged passive scalar distribution in Case LIS.

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x (m)

y (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Level<ξ>z:

10.01

30.08

50.15

70.22

90.29

110.36

130.43

150.5

170.57

190.64

210.71

230.78

250.85

270.92

290.99

_

Figure 5. Instantaneous spanwise averaged passive scalar distribution in Case TIS.

x (m)

y (m

)

0.3 0.4 0.5 0.6-0.1

-0.05

0

0.05

0.1

Level<ξ>z:

10.01

30.08

50.15

70.22

90.29

110.36

130.43

150.5

170.57

190.64

210.71

230.78

250.85

270.92

290.99

_

(a)x (m)

y (m

)

0.3 0.4 0.5 0.6-0.1

-0.05

0

0.05

0.1

(b)

x (m)

y (m

)

0.3 0.4 0.5 0.6-0.1

-0.05

0

0.05

0.1

(c)x (m)

y (m

)

0.3 0.4 0.5 0.6-0.1

-0.05

0

0.05

0.1

(d)

Figure 6. Passage of coherent structure through sampling plane at x = 0.45m in Case LIS.

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(a) (b)

(c) (d)

Figure 7. y − z passive scalar distribution in coherent structure shown in Figure 6.

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x (m)

y (m

)

0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1

Level<ξ>z:

10.01

30.08

50.15

70.22

90.29

110.36

130.43

150.5

170.57

190.64

210.71

230.78

250.85

270.92

290.99

_

(a)x (m)

y (m

)

0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1

(b)

x (m)

y (m

)

0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1

(c)x (m)

y (m

)

0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1

(d)

Figure 8. Passage of coherent structure through sampling plane at x = 0.60m in Case TIS.

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(a) (b)

(c) (d)

Figure 9. y − z passive scalar distribution in coherent structure shown in Figure 8.

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z (m)

y0 (

m)

0 0.05 0.1 0.15

­0.01

­0.005

0

x = 0.02m

x = 0.05m

x = 0.1m

x = 0.15m

x = 0.3m

x = 0.45m

x = 0.6m

(a) Case LIS

z (m)

y0 (

m)

0 0.05 0.1 0.15

­0.01

­0.005

0

x = 0.02m

x = 0.05m

x = 0.1m

x = 0.15m

x = 0.3m

x = 0.45m

x = 0.6m

(b) Case TIS

Figure 10. Centreline variation across the span of the mixing layer.

x (m)

va

ria

tio

n (

%)

0 0.2 0.4 0.60

2

4

6

8

LIS

TIS

Figure 11. Spanwise variation in mixing layer thickness in the simulations.

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(a) x = 0.05m (b) x = 0.1m

(c) x = 0.3m (d) x = 0.6m

Figure 12. Mean secondary shear stress contours in Case LIS.

(a) x = 0.05m (b) x = 0.1m

(c) x = 0.3m (d) x = 0.6m

Figure 13. Mean secondary shear stress contours in Case TIS.

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