# The effects of resolution and noise on kinematic features of fine-scale turbulence

- 420 Downloads
- 23 Citations

## Abstract

The effect of spatial resolution and experimental noise on the kinematic fine-scale features in shear flow turbulence is investigated by means of comparing numerical and experimental data. A direct numerical simulation (DNS) of a nominally two-dimensional planar mixing layer is mean filtered onto a uniform Cartesian grid at four different, progressively coarser, spatial resolutions. Spatial gradients are then calculated using a simple second-order scheme that is commonly used in experimental studies in order to make direct comparisons between the numerical and previously obtained experimental data. As expected, consistent with other studies, it is found that reduction of spatial resolution greatly reduces the frequency of high magnitude velocity gradients and thereby reduces the intermittency of the scalar analogues to strain (dissipation) and rotation (enstrophy). There is also an increase in the distances over which dissipation and enstrophy are spatially coherent in physical space as the resolution is coarsened, although these distances remain a constant number of grid points, suggesting that the data follow the applied filter. This reduction of intermittency is also observed in the eigenvalues of the strain-rate tensor as spatial resolution is reduced. The quantity with which these eigenvalues is normalised is shown to be extremely important as fine-scale quantities, such as the Kolmogorov length scale, are showed to change with different spatial resolution. This leads to a slight change in the modal values for these eigenvalues when normalised by the local Kolmogorov scale, which is not observed when they are normalised by large-scale, resolution-independent quantities. The interaction between strain and rotation is examined by means of the joint probability density function (*pdf*) between the second and third invariants of the characteristic equation of the velocity gradient tensor, *Q* and *R* respectively and by the alignments between the eigenvectors of the strain-rate tensor and the vorticity vector. Gaussian noise is shown to increase the divergence error of a dataset and subsequently affect both the *Q*–*R* joint *pdf* and the magnitude of the alignment cosines. The experimental datasets are showed to behave qualitatively similarly to the numerical datasets to which Gaussian noise has been added, confirming the importance of understanding the limitations of coarsely resolved, noisy experimental data.

## Keywords

Particle Image Velocimetry Direct Numerical Simulation Joint Probability Density Function Direct Numerical Simulation Data Velocity Gradient Tensor## Notes

### Acknowledgments

The authors would like to thank EPSRC for providing the computing resources on HECToR through the Resource Allocation Panel and for funding the experimental research through Grant No. EP/F056206. Funding from the Royal Aeronautical Society for ORHB is also greatly appreciated.

## References

- Antonia RA, Zhu Y, Kim J (1994) Corrections for spatial velocity derivatives in a turbulent shear flow. Exp Fluids 16:411–416CrossRefGoogle Scholar
- Ashurst WT, Kerstein AR, Kerr RM, Gibson CH (1987) Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence. Phys Fluids 30:2343–2353CrossRefGoogle Scholar
- Bermejo-Moreno I, Pullin DI, Horiuti K (2009) Geometry of enstrophy and dissipation, resolution effects and proximity issues in turbulence. J Fluid Mech 620:121–166zbMATHCrossRefGoogle Scholar
- Buxton ORH, Ganapathisubramani B (2010) Amplification of enstrophy in the far field of an axisymmetric turbulent jet. J Fluid Mech 651:483–502zbMATHCrossRefGoogle Scholar
- Chong MS, Perry AE, Cantwell BJ (1990) A general classification of three-dimensional flow fields. Phys Fluids A 2(5):765–777MathSciNetCrossRefGoogle Scholar
- Christensen KT (2004) The influence of peak-locking errors on turbulence statistics computed from PIV ensembles. Exp Fluids 36:484–497CrossRefGoogle Scholar
- Christensen KT, Adrian RJ (2002) Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy. Exp Fluids 33:759–769Google Scholar
- Donzis DA, Yeung PK, Sreenivasan KR (2008) Dissipation and enstrophy in isotropic turbulence: resolution effects and scaling in direct numerical simulations. Phys Fluids 20(045108):1–16Google Scholar
- Ganapathisubramani B, Lakshminarasimhan K, Clemens NT (2007) Determination of complete velocity gradient tensor by using cinematographic stereoscopic PIV in a turbulent jet. Exp Fluids 42:923–939CrossRefGoogle Scholar
- Ganapathisubramani B, Lakshminarasimhan K, Clemens NT (2008) Investigation of three-dimensional structure of fine-scales in a turbulent jet by using cinematographic stereoscopic PIV. J Fluid Mech 598:141–175zbMATHCrossRefGoogle Scholar
- Ganapathisubramani B, Longmire EK, Marusic I, Pothos S (2005) Dual-plane PIV technique to determine the complete velocity gradient tensor in a turbulent boundary layer. Exp Fluids 39(2):222–231CrossRefGoogle Scholar
- Herpin S, Wong CY, Stanislas M, Soria J (2008) Stereoscopic piv measurements of a turbulent boundary layer with a large spatial dynamic range. Exp Fluids 45(4):745–763CrossRefGoogle Scholar
- Jiménez J, Wray AA, Saffman PG, Rogallo RS (1993) The structure of intense vorticity in isotropic turbulence. J Fluid Mech 255:65–90MathSciNetzbMATHCrossRefGoogle Scholar
- Kerr RM (1985) Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J Fluid Mech 153:31–58zbMATHCrossRefGoogle Scholar
- Kinzel M, Wolf M, Holzner M, Lüthi B, Tropea C, Kinzelbach W (2010) Simultaneous two-scale 3D-PTV measurements in turbulence under the influence of system rotation. Exp Fluids 1–8. 10.1007/s00348-010-1026-6Google Scholar
- Laizet S, Lamballais E (2009) High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy. J Comp Phys 228(16):5989–6015MathSciNetzbMATHCrossRefGoogle Scholar
- Laizet S, Lardeau S, Lamballais E (2010) Direct numerical simulation of a mixing layer downstream a thick splitter plate. Phys Fluids 22(015104):1–15Google Scholar
- Lavoie P, Avallone G, De Gregorio F, Romano GP, Antonia RA (2007) Spatial resolution of PIV for the measurement of turbulence. Exp Fluids 43(1):39–51CrossRefGoogle Scholar
- Lund TS, Rogers MM (1994) An improved measure of strain state probability in turbulent flows. Phys Fluids 6(5):1838–1847zbMATHCrossRefGoogle Scholar
- Lüthi B, Holzner M, Tsinober A (2009) Expanding the
*Q*−*R*space to three dimensions. J Fluid Mech 641:497–507zbMATHCrossRefGoogle Scholar - Lüthi B, Tsinober A, Kinzelbach W (2005) Lagrangian measurement of vorticity dynamics in turbulent flow. J Fluid Mech 528:87–118zbMATHCrossRefGoogle Scholar
- Mullin JA, Dahm WJA (2006) Dual-plane stereo particle image velocimetry measurements of velocity gradient tensor fields in turbulent shear flow II. Experimental results. Phys Fluids 18(035102):1–28Google Scholar
- Perry AE, Chong MS (1994) Topology of flow patterns in vortex motions and turbulence. Appl Sci Res 53:357–374zbMATHCrossRefGoogle Scholar
- Pirozzoli S, Grasso F (2004) Direct numerical simulation of isotropic compressible turbulence: influence of compressibility on dynamics and structures. Phys Fluids 16(12):4386–4407CrossRefGoogle Scholar
- Ruetsch GR, Maxey MR (1991) Small-scale features of vorticity and passive scalar fields in homogeneous isotropic turbulence. Phys Fluids A 3(6):1587–1597CrossRefGoogle Scholar
- Sreenivasan KR, Antonia RA (1997) The phenomenology of small-scale turbulence. Annu Rev Fluid Mech 29:435–472MathSciNetCrossRefGoogle Scholar
- Tao B, Katz J, Meneveau C (2000) Geometry and scale relationships in high Reynolds number turbulence determined from three-dimensional holographic velocimetry. Phys Fluids 12(5):941–944zbMATHCrossRefGoogle Scholar
- Tao B, Katz J, Meneveau C (2002) Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements. J Fluid Mech 457:35–78MathSciNetzbMATHCrossRefGoogle Scholar
- Taylor GI (1935) Statistical theory of turbulence. Proc R Soc Lond A 151(873):421–444zbMATHCrossRefGoogle Scholar
- Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, CambridgeGoogle Scholar
- Tsinober A (1998) Is concentrated vorticity that important?. Eur J Mech B/Fluids 17(4):421–449zbMATHCrossRefGoogle Scholar
- Tsinober A, Kit E, Dracos T (1992) Experimental investigation of the field of velocity gradients in turbulent flows. J Fluid Mech 242:169–192CrossRefGoogle Scholar
- van der Bos F, Tao B, Meneveau C, Katz J (2002) Effects of small-scale turbulent motions on the filtered velocity gradient tensor as deduced from holographic particle image velocimetry measurements. Phys Fluids 14(7):2456–2474MathSciNetCrossRefGoogle Scholar
- Vieillefosse P (1982) Local interaction between vorticity and shear in a perfect incompressible fluid. J Phys 43(6):837–842MathSciNetCrossRefGoogle Scholar
- Vincent A, Meneguzzi M (1994) The dynamics of vorticity tubes in homogeneous turbulence. J Fluid Mech 258:245–254zbMATHCrossRefGoogle Scholar
- Vukoslavčević P, Wallace JM, Balint JL (1991) The velocity and vorticity vector fields of a turbulent boundary layer. Part 1(Simultaneous measurement by hot-wire anemometry. J. Fluid Mech. 228):25–51Google Scholar
- Wallace JM, Vukoslavčević P (2010) Measurement of the velocity gradient tensor in turbulent flows. Annu Rev Fluid Mech 42:157–181CrossRefGoogle Scholar
- Westerweel J (2000) Theoretical analysis on the measurement precision in particle image velocimetry. Exp Fluids [Suppl] 29:S3–S12CrossRefGoogle Scholar
- Worth NA (2010) Tomographic-PIV Measurement of Coherent Dissipation Scale Structures. Ph.D. thesis, Cambridge UniversityGoogle Scholar
- Worth NA, Nickels TB, Swaminathan N (2010) A tomographic PIV resolution study based on homogeneous isotropic turbulence DNS data. Exp Fluids 49(3):637–656CrossRefGoogle Scholar