Large Eddy Simulations of 2-D and 3-D Spatially

Developing Mixing Layers

C. S. Martha∗, G. A. Blaisdell,† and A. S. Lyrintzis‡

School of Aeronautics and Astronautics

Purdue University

West Lafayette, IN 47907

A complete understanding of the noise generation mechanisms is prerequisite to reducingaircraft jet noise. A computationally intensive large eddy simulation (LES) can directlypredict the noise. For this purpose, the Ansys-Fluent LES tool is evaluated for its modelingaccuracy by studying a canonical problem. A mixing layer at a Reynolds number of 720is simulated using LES in two- and three-dimensions without the splitter plate walls andis studied extensively to gain insights of the flow physics. The effects of inflow forcing andthe buffer zone at the domain exit incorporated in 2-D LES are investigated. The primaryand secondary instabilities of the 2-D mixing layer are captured well using random inflowforcing. The buffer zone is found to help the prediction of the Reynolds stresses near theexit boundary. The sensitivity of the 2-D results to time-step, sampling time and grid is alsostudied. The 2-D grid is optimized before constructing a mesh for 3-D LES. An improvedvortex method (VM) algorithm, implemented for inflow forcing in 3-D, is found to capturethe translative instability and the stream-wise vortices of the mixing layer. The energyspectrum of the LES computations indicate that the second-order-accurate bounded centraldifferencing scheme of Ansys-Fluent, used in the present study, is adequate for simulatingturbulent flows.

Nomenclature

BCD Bounded Central DifferencingCFL Courant - Friedrichs - LewyDNS Direct Numerical SimulationFTC Flow-Through CycleLES Large Eddy SimulationRANS Reynolds Averaged Navier StokesVM Vortex Method

I. Introduction

Computationally intensive techniques, such as large eddy simulation(LES) and hybrid RANS-LES, arepromising in developing the jet noise reduction methods as they offer better insight into the noise generationmechanisms unlike the experiments. These techniques have become feasible for complex flows with the recentimprovements in the computing power of the processor chips. The current work deals with developing anLES methodology using Ansys-Fluent. The methodology is evaluated by modeling a canonical problem,i.e. a planar mixing layer.

A mixing layer is formed at the interface of two fluid streams moving one over the other with differentvelocities. The term ‘planar’ signifies that the interface is a plane. It represents an important class of free

∗Graduate Research Assistant, Student Member AIAA.†Associate Professor, Associate Fellow AIAA.‡Professor, Associate Fellow AIAA.

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49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-830

Copyright © 2011 by C.S. Martha, G.A. Blaisdell, and A.S. Lyrintzis. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

shear flows. Mixing layers are a common occurrence in many engineering applications involving chemicallyreacting flows, scalar mixing and jets, etc. A clear understanding of the characteristic features of the mixinglayers is prerequisite to control the process of mixing in such applications.

The mixing layer is unstable for certain wavy disturbances.1 The instability modes of the mixing layer canbe classified broadly into two categories: a) two-dimensional modes, formed as a consequence of span-wiseinvariant disturbances, that cause vortex sheet roll-up(primary instability) and subsequent pairing(secondaryinstability) downstream, b) three-dimensional, formed when the perturbations have span-wise variation, thatresult in helical pairing,2 translative instability,3 and stream-wise vortices. The ‘translative instability’ ofa mixing layer bends the cores of the span-wise vortices and is found to be most unstable for span-wise

wavelengths approximately 23

rdof the spacing between the 2-D span-wise vortex rollers. This instability

is attributed to the development of counter-rotating stream-wise(rib) vortices, which eventually leads tothe three-dimensionality of the mixing layer. The rib vortices develop in the braids between the spanwisevortex rollers4 and extend from the bottom of one roller to the top of its neighbor. The interaction ofthese instabilities leads to the transition of the mixing layer to turbulence resulting in self-similarity furtherdownstream. The linear mixing layer growth rate and collapse of the profiles of appropriately scaled meanvelocity and turbulent quantities are indicators of the self-similarity.

Experimental evidence shows that the growth rate of a mixing layer can be greatly manipulated by forcingthe mixing layer near a subharmonic of the most-amplified frequency (or fundamental frequency) at a verylow forcing level.5 The development of a mixing layer is highly sensitive to the boundary conditions. Thesplitter plate geometry,6 the free-stream turbulence intensity7 and the velocity ratio8,9 affect the evolutionof mixing layer. The planar mixing layer has been investigated experimentally by Wygnanski and Fiedler,10

Spencer and Jones11 as well as Bell and Mehta.12 It has been studied computationally by Rogers andMoser,4,13 Stanley and Sarkar,14 and Uzun,15 among others.

The mixing layer is computationally investigated by two different approaches: a) temporally evolvingmixing layer, and b) spatially evolving mixing layer. In the first approach, the evolution of flow structuresis investigated as they convect downstream with respect to an observer sitting on top of these structures.Therefore, the changes in flow structures are seen in time in this approach. The second approach deals withthe study of the flow structures as they convect downstream with respect to a fixed frame of reference. Inthis approach, the changes in flow structures are seen in space.

In the present study, the LES computational methodology is validated by modeling 2-D and 3-D spatiallydeveloping mixing layers at a Reynolds number of 720 based on the velocity difference and the visualthickness. The walls of the splitter plate are not modeled to keep the computational cost low.

II. Computational Procedure

The commercial CFD tool, Ansys-Fluent (version 6.3.26) is used in the present study. The tool is basedon cell-centered finite volume discretization. The detailed description of its LES solver and the numericalschemes can be found in reference.16

II.A. Parameters of the Mixing Layer

The planar mixing layer is simulated by specifying the following hyperbolic tangent velocity profile for thestream-wise mean velocity at the inlet

u(y) =U1 + U2

2+

U1 − U2

2tanh

(

2y

δω(0)

)

; (1)

and the mean cross-stream velocity, v(y) = 0. The terms U1, U2 represent the velocities of high-speed andlow-speed streams, respectively. And δω(0) is the initial vorticity thickness, which is defined as δω(0) =

U1−U2

| ∂u∂y |max

. This velocity profile is a a good approximation to the flow over a splitter plate after the wake

effects have vanished. The convective velocity of the large-scale eddies of the mixing layer is, Uc = U1+U2

2 =

0.375c∞; and the relative convective Mach number of the mixing layer is Mc = U1−U2

2c∞= 0.125, where c∞ is

the speed of sound at ambient temperature. The Reynolds number based on the initial vorticity thickness

and the velocity difference across the mixing layer is, Re = (U1−U2)δω(0)ν = 720. The velocity ratio of the

mixing layer is given by, η = U1−U2

U1+U2

= 13 .

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II.B. Computational Mesh

II.B.1. 2-D Grid

The computational domain extends from 0 to 350δω(0) in the stream-wise (x) direction and from −300δω(0)to 300δω(0) in the cross-stream (y) direction. The number of grid points in the x, y directions are 576and 575, respectively. The nodes are stretched in the y-direction from the origin and mirrored about thecenterline of the mixing layer. The grid stretching ratio in the y-direction is ∼ 1.01 (or ∼1%). The minimumgrid spacing in the y-direction is 0.16δω(0) around the centerline, resulting in 7 points across the mixing layer

at the inlet. This spacing is about 1.5 times the mixing length scale, computed from l = C3

4

µk0

3

2

ǫ0using a 2-D

RANS simulation, within the mixing layer.17 The computational domain has a physical domain extendingfrom 0 to 200δω(0) in the x-direction. A buffer zone (or sponge region) is attached downstream to avoid thepossible contamination of the flow field by the reflected vorticity waves caused by the exit boundary. Thephysical domain has 500 nodes that are uniformly spaced at ∆x = 0.4δω(0) in the x-direction. The bufferzone has 77 nodes that are stretched in the x-direction with a stretching ratio of approximately 1.04. Thismesh is similar to the one used by Uzun15 for DNS of a spatially developing 2-D mixing layer.

II.B.2. 3-D Grid

An optimized 2-D grid is obtained by decreasing the cross-stream domain size to −50δω(0) to 50δω(0) andincreasing the cross-stream grid stretching ratio to 1.10 using the 2-D baseline mesh. The quality of the LES

results is not compromised by the optimized mesh.17 The optimized mesh contains roughly 1/8th of the gridpoints in the 2-D baseline mesh. This mesh is extruded in the span-wise direction to construct a 3-D baselinemesh. The number of grid points in the span-wise direction is 51. The grid points in the stream-wise andcross-stream directions are 576 and 73, respectively. The total number of grid points is approximately 2.15million. The size of the computational domain is 350δω(0) × 100δω(0) × 20δω(0). The minimum spacingsaround the mixing layer centerline are (∆x, ∆y, ∆z) = (0.4δω(0), 0.16δω(0), 0.4δω(0)). The length of thephysical zone is same as that of the 2-D baseline grid.

II.C. Boundary Conditions and Inflow Forcing

The mixing layer is simulated artificially by specifying a hyperbolic tangent velocity profile and accordinglya velocity inlet boundary condition is used at the inlet. The pressure outflow boundary condition is used atthe exit of the buffer zone with a back pressure of 101325 Pa. The top and bottom boundaries are modeledas inviscid walls. The span-wise boundaries in the 3-D simulation are modeled as periodic.

Inflow forcing is critical for LES computations as the evolution of the flow-field is sensitive to the per-turbations at the inlet. The forcing methods used to emulate the turbulence generated off the walls of thesplitter plate in the 2-D and 3-D simulations are discussed briefly. The velocity profile and the inflow forcingare implemented using user defined functions(UDF’s) within Fluent.

II.C.1. 2-D Simulations

The forcing based on random perturbations15,18 is used in the 2-D simulations to supply realistic inflowboundary conditions to the LES solver. The perturbations are forced only on the cross-stream velocity atthe inflow boundary using the following Gaussian distribution

v(y) = ǫαUcexp

(

− y2

∆y20

)

, (2)

where ǫ is a random number between −1 and 1, α = 0.0045, and ∆y0 is the grid spacing around the centerline of the mixing layer. A random number is assigned to ǫ after each time-step resulting in the variation ofamplitude of the Gaussian profile. The implemented inflow forcing algorithm ensures that the amplitude ofthe Gaussian profile is the same for all the 2-D runs at a given time-step. In the present study, ∆y0 is set to0.16δω(0) in all the 2-D simulations to better compare the flow-field.

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II.C.2. 3-D Simulations

The random vortex method(VM),19 in-built within Fluent to generate span-wise varying perturbations atthe inlet, is found to be unsuitable for the current study.17 Therefore, an improved version of the VMalgorithm is implemented using a UDF to force perturbations in 3-D. Two pairs of counter-rotating stream-wise vortices with a length of 25δω(0), emulating the hair-pin vortices developed within the splitter platewalls, are used to generate velocity perturbations at the inlet. As these vortices pass through the inlet withtime, two new pairs of vortices with the same length are generated with opposite circulation. These vorticesare shown in figure 1. The vortices move randomly only along the span-wise direction with 75% of the meaninlet velocity as the 3-D instabilities of the mixing layer are sensitive mainly to the span-wise wavelengthof the perturbations. The size of each vortex is 1.5δω(0) and the average spacing between the vortices ineach pair is 7.5δω(0). The most unstable span-wise wavelength corresponding to the development of thetranslative instability3 is about 6.6δω(0) for the current mixing layer. The amplitude of the perturbationsat the inlet is about 0.08% of the center line velocity.

II.D. Numerical Schemes and Simulation Procedure

The highest Mach number in the flow field is 0.5, which makes the flow mildly compressible, correspondingto the high-speed flow stream. Therefore, the flow is modeled as incompressible in the present work. Thefiltered, unsteady Navier-Stokes equations of the incompressible flow are solved using the pressure-basedsolver. The sub-grid scale stresses of the momentum equation are modeled using dynamic Smagorinskymodel. The second-order-accurate bounded central differencing(BCD) scheme, which is a blend of purecentral differencing and the second-, first-order upwind schemes, is used for spatial discretization. Thepressure-implicit with splitting of operators (PISO) scheme is used to couple pressure and velocity. ThePRESTO! (PREssure STaggering Option) scheme is employed for pressure interpolation in the momentumequation. The gradients are computed using the ‘Green-Gauss Cell-Based’ method. The flow is dampedwithin the buffer zone implicitly through the artificial dissipation introduced by the coarser mesh in thebuffer. In order to maintain high temporal accuracy, a second-order implicit time-stepping scheme is usedfor time advancement with 20 inner iterations.

The best practices for an LES as described by Georgiadis20 have shaped the simulation procedure em-ployed in the present study. The time-step size is 10−7 s for all the simulations corresponding to thenon-dimensional time-scale, Uc∆t

δω(0) = 0.1. The corresponding CFL number is 0.25 in the 2-D and 3-D baseline

simulations. All the simulations are single precision and advanced in time for 12 flow-through cycles(FTC’s),where a FTC determines the time a flow particle resides within the computational domain. The first 4 cyclesaccount for the transients to exit the domain and the next 8 are used for gathering the flow statistics atevery time-step. A steady RANS solution is obtained prior to the start of simulation using the standard k−ǫturbulence model.16 A spectral synthesizer16 is used to superimpose turbulent velocity fluctuations over theRANS simulation velocity field. The flow-field is then used as an initial condition to start the present LESruns.

III. Results and Discussion

The 2-D baseline simulation takes about 3.5 days on 16 processors and the 3-D simulations requires about11 days on 64 processors for 12 FTC’s or 42,000 time-steps.

III.A. 2-D Simulations

The mixing layer growth rate and Reynolds stresses, obtained after sampling the flow for 8 FTC’s, arecompared with the data available in the literature in table 1. The Reynolds stresses of the 2-D baseline casematch well with the 2-D DNS results of Stanley14 and Uzun.15 The overprediction of σyy, due to the absenceof energy transfer to the third dimension in the 2-D simulation, compared with the experiments is also seenin the table. The 2-D mixing layer growth rate is slightly overpredicted compared to the DNS results. Thegrowth rate and the stresses are found to have reached the statistically stationary values as confirmed bythe invariance of the flow statistics with sampling periods of 8 and 12 FTC’s.17

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III.A.1. Sensitivity to Time-step

The sensitivity of the mixing layer growth rate to the time-step is studied by reducing the time-step to5 × 10−8s. The reduced time-step matches that of the 2-D DNS of Uzun’s mixing layer.15 Two additionalsimulations, denoted as LowT SF 1 and LowT SF 2, are performed with the lowered time-step with samplingfrequencies(SF) of 1 and 2 time-steps. The time tags of the flow samples considered in these simulations areshown in table 2. The growths of the mixing layer obtained with these runs are listed in table 3. As shownin table 3, the LowT SF 1 matches Uzun’s mixing layer growth rate better. The LowT SF 2 simulation doesnot improve the growth rate, suggesting that the sampling frequency affects the mean flow statistics with alower time-step.

III.A.2. Effect of Inflow Forcing

The growth rate and center of the 2-D mixing layer obtained after 12 FTC’s with and without inflow forcingare shown in figures 2(a) and 2(b). It is evident from figure 2(a) that the growth of the mixing layer isgreatly influenced by the inflow forcing as seen in the literature. The perturbations at the inlet cause themixing layer to roll up quickly at about x = 60δω(0) resulting in linear growth region with a higher slope.The mixing layer is mostly laminar when it is not forced and eventually begins to grow faster due to thevortex roll-up caused by the numerical errors involved in the simulation. Figure2(b) shows the tendency ofthe high-speed flow stream to move into the low-speed stream as suggested by the curving down of the centerof the mixing layer. The tendency is higher when the inflow forcing is present. The Reynolds stresses beginto grow sooner in the computational domain with inflow forcing. The profiles of Reynolds shear stress(σxy)at various downstream stations, as shown in figure 3, indicate the effect of forcing on the flow statistics.The instantaneous vorticity contours at the end of baseline simulation with random inflow forcing, shown infigure 4(a), clearly shows the primary and secondary instabilities of the mixing layer leading to the vortexroll-up and pairing. The vortex pairing is found to occur at random locations within the computationaldomain due to the random inflow forcing.

III.A.3. Effect of Buffer Zone

As the flow is damped implicitly within the buffer zone, the effectiveness of such a method is examined here.An LES case is run without any buffer zone and retaining the same physical zone of the baseline mesh. Theinstantaneous vorticity contours after t = 4.2× 10−3s are compared with that of the baseline simulation andare shown in figure 4. It can be seen that there is virtually no difference between the two cases. However,it is found that the absence of the buffer zone causes the Reynolds shear stress to be overpredicted locallynear the exit boundary, as depicted in figure 5. The other components of the Reynolds stresses do not showany exit boundary effect and match closely with those of the baseline simulation. The growth rate of themixing layer is also unchanged as the time-history of the evolution of the mixing layer is the same as seen infigure 4. The buffer zone is retained in the subsequent 2-D and 3-D simulations to improve the predictionof the Reynolds shear stress near the exit boundary.

III.A.4. Grid Resolution Study

The baseline mesh is refined in x- and/or y- directions by factors of√

2 and 2 resulting in four additionalsimulations. A fifth simulation is run with decreasing the x-resolution by a factor of 2. All the cases utilizesame amplitude for the Gaussian perturbation profile at a given time-step to facilitate the comparison of theeffect of grid resolution alone. The instantaneous vorticity contours corresponding to t = 4.2 × 10−3s areshown in figure 6. It can be deduced from figures 6(a), 6(b) that the contours are nearly identical when they-resolution is doubled. Figures 6(a) and 6(c) indicate that the time history of the vortex evolution changeswhen the grid is better resolved in the stream-wise direction. The resolution of smaller scales with the refinedmesh may have caused the flow to evolve differently in these cases. The effect of grid refinement in boththe directions is shown in figures 6(d) and 6(e). The evolution of the vorticity is more or less unchangedfor x < 170δω(0). The second vortex from x = 200δω(0) undergoes pairing with the third when the gridis refined by a factor of

√2, whereas it undergoes pairing with the first vortex from x = 200δω(0) when

the grid is refined by a factor of 2 in both directions. It is believed that the increased smoothness of theGaussian profiles, which depends on the y-resolution at the inlet, may have contributed to such a pairingbehavior with the refined meshes. The Reynolds stresses are found to increase slightly as the grid resolution

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is increased, whereas the mixing layer growth rate approximately stays the same. Figures 6(a) and 6(f)indicate that the coarsened mesh dissipates the vortices for x > 70δω(0). This suggests that the baselinemesh is adequate for resolution of the large-scale structures of the flow.

III.B. 3-D Simulations

The Reynolds stresses and growth rate obtained after 12 FTC’s are compared with the experiments in table 1.The growth rate obtained with the Fluent’s in-built vortex method(VM) for forcing is higher compared tothe experiments. The growth rate obtained with the improved VM matches the experiments better. TheReynolds stresses, except σzz, are over predicted with the improved VM. It is believed that the presence ofthe hair-pin vortices during the entire course of the simulation may have caused the overprediction of theturbulent stresses. The span-wise averaged Reynolds stresses are plotted at different stream-wise locationsin figure 7. It can be seen from figure 7 that the Reynolds stresses collapse for x > 140δω(0) suggesting self-similarity. The profiles of scaled velocity are found to collapse well for x > 100δω(0). The one-dimensionalenergy spectrum of the stream-wise velocity perturbations, obtained from Taylor’s hypothesis of frozenturbulence, is plotted in figure 8(a) corresponding to the velocity history obtained at x = 200δω(0). The gridcut-off, computed by assuming 9 stream-wise points per wavelength, is also shown in the figure. The figureindicates that the energy decay rate of − 5

3 is maintained for more than half-a-decade of wave numbers untilthe grid cut-off. This suggests that the present LES methodology of using second-order-accurate BCD issatisfactory for the current application. The stream-wise velocity correlation (Q11) is shown in figure 8(b).The correlation becomes zero as the separation distance is increased suggesting that the mixing layer canbe approximated as fully turbulent. The structure of the vortex cores corresponding to t = 4.2 × 10−3sfor the 3-D mixing layer is shown in figure 9. The translative instability of the span-wise vortex rollersaround x = 130δω(0) is observed. The counter-rotating stream-wise(rib) vortex pairs are also seen betweenthese vortex rollers in the computational domain. The mixing layer is predominantly three-dimensional forx > 150δω(0). The number of vortex pairs correlates with the number of modeled hair-pin vortices at theinlet. This suggests that the rib vortices develop due to the instabilities introduced by the hair-pin vorticesof the splitter plate walls.

IV. Conclusions

A mixing layer at a Reynolds number of 720 is studied in two- and three-dimensions without the splitterplate walls using Ansys-Fluent’s LES module. The 2-D simulations provided a platform to investigate someof the aspects of the LES runs, such as the inflow forcing, buffer zone, sensitivity of the results to time-stepand grid resolution. The inflow forcing significantly affected the evolution of the mixing layer and the growthrate. The random inflow forcing, incorporated in the current 2-D simulations, captured the vortex roll-upand the pairing. The buffer zone is found to improve the prediction of Reynolds shear stress near the exitof computational domain. The sampling frequency played a critical role in determining the growth rateof the mixing layer with lower time-steps. The grid resolution study indicated that the 2-D baseline meshis adequate for the resolution of the large-scales of the mixing layer. An improved vortex method (VM)algorithm, implemented for inflow forcing in 3-D, is found to capture the translative instability and the ribvortices of the mixing layer. The rib vortices are believed to have formed due to the instability introduced bythe hair-pin vortices of the splitter plate walls. The energy spectrum of the LES computations indicate thatthe second-order-accurate bounded central differencing scheme of Ansys-Fluent is adequate for simulatingturbulent flows.

Acknowledgments

This work was supported by the Rolls-Royce and Gulfstream Aerospace Corporations.

References

1Rayleigh, L., Sci. Papers, Vol. 1:474-87, Cambridge University Press, 1880.2Chandrsuda, C., Mehta, R. D., Weir, A., and Bradshaw, P., “Effect of free-stream turbulence on large structures in

turbulent mxing layers,” J. Fluid Mech, Vol. 85, 1978, pp. 693.

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3Pierrehumbert, R. T. and Widnall, S. E., “The two- and three-dimensional instabilities of a spatially periodic shearlayer,” J. Fluid Mech, Vol. 114, 1981, pp. 59–82.

4Rogers, M. M. and Moser, R. D., “The three-dimensional evolution of a plane mixing layer: the Kelvin-Helmholtz rollup,”J. Fluid Mech., Vol. 243, 1992, pp. 183–226.

5Ho, C.-M. and Huang, L.-S., “Subhormonics and vortex merrging in mixing layers,” J. Fluid Mech, Vol. 119, 1982,pp. 443–473.

6Dziomba, B. and Fiedler, H. E., “Effect of Initial Conditions on Two-Dimensional Free Shear Flows,” Journal of Fluid

Mechanics, Vol. 152, March 1985, pp. 419–442.7Patel, R. P., “Effects of Stream Turbulence on Free Shear Flows,” Aeronautical Quarterly, Vol. 29, Feb. 1978, pp. 33–43.8Mehta, R. D. and Westphal, R. V., “Near-Field Turbulence Properties of Single- and Two-Stream Plane Mixing Layers,”

Experiments in Fluids, Vol. 4, Sept. 1986, pp. 257–266.9Mehta, R. D. and Westphal, R. V., “Effect of Velocity Ratio on Plane Mixing Layer Development,” Proceedings of the

Seventh Symposium on Turbulent Shear Flows, Stanford Univ., Stanford, CA, Aug. 21-23 1989, pp. 3.2.1–3.2.6.10Wygnanski, I. and Fiedler, H., “The two-dimensional mixing region,” Journal of Fluid Mechanics, Vol. 41, No. 2, 1970,

pp. 327–361.11Spencer, B. W. and Jones, B. G., “Statistical investigation of pressure and velocity fields in the turbulent two-stream

mixing layer,” 1971, AIAA 71-613.12Bell, J. H. and Mehta, R. D., “Development of a two-stream mixing layer from tripped and untripped boundary layers,”

AIAA Journal , Vol. 28, No. 12, December 1990, pp. 2034–2042.13Rogers, M. M. and Moser, R. D., “The three-dimensional evolution of a plane mixing layer: pairing and transition to

turbulence,” J. Fluid Mech., Vol. 247, 1993, pp. 275–320.14Stanley, S. and Sarkar, S., “Simulations of Spatially Developing Two-Dimensional Shear Layers and Jets,” Theoret.

Comput. Fluid Dynamics, Vol. 9, 1997, pp. 121–147.15Uzun, A., 3-D Large Eddy Simulation for Jet Aeroacoustics, Ph.D. thesis, Purdue University, 2003.16Ansys-Fluent Inc., “User’s guide, FLUENT 6.3 Documentation,” 2006.17Martha, C. S., Large Eddy Simulations of 2-D and 3-D spatially developing mixing layers, Master’s thesis, Purdue

University, 2010.18Bogey, C., Calcul Direct du Bruit Aerodynamique et Validation de Modeles Acoustiques Hybrides, Ph.D. thesis, Labora-

toire de Mecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, France, 2000.19Mathey, F., Cokljat, D., Bertoglio, J. P., and Sergent, E., “Assessment of the vortex method for Large Eddy Simulation

inlet conditions,” Progress in Computational Fluid Dynamics, Vol. 6, No. 1/2/3, 2006, pp. 58–67.20Georgiadis, N. J., Rizzetta, D. P., and Fureby, C., “Large-Eddy Simulation: Current Capabilities, Recommended Prac-

tices, and Future Research,” 2009, AIAA 2009-948.

Figure 1. Schematic of the stream-wise vortex pairs (1 and 2) generated by the implemented VM forcing algorithm.The vortex pairs emulate hair-pin vortices developed within the boundary layer of a splitter plate wall.

Table 1. Comparison of the normalized peak Reynolds stresses and growth rates of 2-D and 3-D mixing layers withthe data in the literature.

Reωσxx

∆U2

σyy

∆U2

σzz

∆U2

σxy

∆U2

1η

dδω(x)dx Reference

- 0.031 0.019 0.0225 0.009 0.19 Experiment10

- 0.036 0.014 0.0225 0.013 0.16 Experiment11

1,800 0.032 0.020 0.022 0.010 0.163 Experiment12

720 0.040 0.084 - 0.023 0.15 2-D DNS14

720 0.048 0.078 - 0.012 0.15 2-D DNS15

720 0.044 0.080 - 0.015 0.17 Present 2-D LES

720 0.048 0.052 0.021 0.025 0.27 Present 3-D LES, Fluent VM

720 0.042 0.046 0.018 0.021 0.225 Present 3-D LES, Improved VM

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Table 2. Physical time of each sample in the 2-D baseline, LowT SF 1 and LowT SF 2 simulations. The value of n is28000 in all the simulations.

Baseline: t1 t2 t3 . . . . tn

LowT SF 2: t1 t2 t3 . . . . tn

LowT SF 1: t1 t1.5 t2 t2.5 t3 . . . . tn

Table 3. Mixing layer growth rates for the baseline, LowT SF 1 and LowT SF 2 simulations.

Case Growth rate, Deviation from ∆t, s Samplingdδω(x)

dx Uzun’s DNS15 time

Baseline 0.056 12% 10−7 8 FTC’s

LowT SF 2 0.055 10% 5 × 10−8 8 FTC’s

LowT SF 1 0.052 4% 5 × 10−8 8 FTC’s

Uzun’s 2-D DNS 0.050 - 6 × 10−8 6 FTC’s

x/δω(0)

δ ω(x

)/δ ω

(0)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

no forcingforcinglinear fit, slope = 0.0563

(a) 2-D mixing layer growth.

x/δω(0)

y c/δ

ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

no forcingforcing

(b) Center of the 2-D mixing layer.

Figure 2. The growth and center of the 2-D mixing layer with and without inflow forcing.

ξ

σ xy/∆

U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 190δω(0)

(a) No forcing.

ξ

σ xy/∆

U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100δω(0)x = 130δω(0)x = 160δω(0)x = 190δω(0)

(b) Random forcing.

Figure 3. Comparison of the profiles of σxy/∆U2 with and without inflow forcing for the 2-D mixing layer.

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x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(a) Baseline.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(b) Baseline without buffer zone.

Figure 4. Comparison of the instantaneous vorticity contours after t = 4.2 × 10−3 s with and without the buffer zone.

(a) Baseline (b) Baseline without buffer zone

Figure 5. Comparison of scaled σxy contours with and without the buffer zone.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(a) Revised baseline.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(b) Cross-stream resolution doubled.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(c) Stream-wise resolution doubled.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(d) x and y resolutions increased by√

2.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(e) x and y resolutions doubled.

x/δω(0)

y/δ ω

(0)

50 100 150 200-25

0

25

(f) Coarsened in stream-wise direction by a factor of 2

Figure 6. Effect of grid resolution on the evolution of the 2-D mixing layer. The contours correspond to the sameinstant of time.

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ξ

σ xx/∆

U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100δω(0)x = 120δω(0)x = 140δω(0)x = 160δω(0)x = 180δω(0)x = 200δω(0)

(a) σxx.

ξ

σ yy/∆

U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100δω(0)x = 120δω(0)x = 140δω(0)x = 160δω(0)x = 180δω(0)x = 200δω(0)

(b) σyy .

ξ

σ zz/∆

U2

-3 -2 -1 0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

x = 100δω(0)x = 120δω(0)x = 140δω(0)x = 160δω(0)x = 180δω(0)x = 200δω(0)

(c) σzz .

ξ

σ xy/∆

U2

-3 -2 -1 0 1 2 3-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

x = 100δω(0)x = 120δω(0)x = 140δω(0)x = 160δω(0)x = 180δω(0)x = 200δω(0)

(d) σxy .

Figure 7. Span-wise averaged scaled Reynolds stresses as predicted by 3-D LES using the improved VM.

κxδω(0)

E1(

κ xδω(0

))/∆

U2

δ ω(0

)

10-1 100 10110-8

10-6

10-4

10-2

100

Grid cut-off

(κxδω(0))−5/3

(a) 1-D energy spectrum.

x/δω(0)

Q11

/∆U

2

0 2 4 6 8 10 12-0.01

0

0.01

0.02

0.03

0.04

(b) The stream-wise velocity correlation (Q11).

Figure 8. 1-D energy spectrum and stream-wise velocity correlation for the 3-D LES using the improved VM algorithm.

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(a) High-speed flow side.

(b) Low-speed flow side.

Figure 9. Iso-surface of Q = 2.2× 10−2× (∆U/δω(0))2 colored by stream-wise vorticity obtained using the improved VM

corresponding to t = 4.2 × 10−3s. The computational domain is extended to 2 periods in the span-wise direction.

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