Advertisement

Flow, Turbulence and Combustion

, Volume 101, Issue 1, pp 1–32 | Cite as

Strain, Rotation and Curvature of Non-material Propagating Iso-scalar Surfaces in Homogeneous Turbulence

  • Cesar Dopazo
  • Jesus Martin
  • Luis Cifuentes
  • Juan Hierro
Article

Abstract

This research aims at gaining some physical insight into the problem of scalar mixing, following the time evolution of propagating iso-surfaces, Y (x, t) = constant, where Y (x, t) stands for any scalar field (e.g., species mass fraction or temperature). First, a rigorous kinematic analysis of non-material line, surface and volume elements, related to propagating iso-scalar surfaces, is presented; this formalism is valid for both constant and variable density flows. Time rates of change of the normal distance and volume between two adjacent iso-surfaces and of area elements, rotation rates of lines and surface elements and an evolution equation for the local mean curvature are obtained. Line and area stretch rates, which encompass additive contributions from the flow and the displacement speed (due to diffusion and reaction), are identified as total strain rates, normal and tangential to the iso-surfaces. Volumetric dilatation rates, addition of line plus area stretch rates, include the mass entrainment rate per unit mass into the non-material volume. Flow and added vorticities, the latter due to gradients of the displacement speed, yield the total vorticity, which provides the real angular velocity of lines and surface elements. A 5123 DNS database for the mixing of inert and reactive scalars in a box of forced statistically stationary and homogeneous turbulence of a constant-density fluid is then examined. A strongly segregated scalar field is prescribed as initial condition. A one-step reaction rate with a characteristic chemical time one order of magnitude greater than the Kolmogorov time micro-scale is used. Data are analyzed at 1.051 large-eddy turnover times after initialization of velocity and scalar fields. Mean negative normal (contractive) and positive tangential (stretching) flow strain rates occur over all mass fractions and scalar-gradient magnitudes. However, means of the total normal strain rate, conditional upon mass fraction, scalar-gradient and mean curvature, are positive everywhere and tend to destroy scalar-gradients for small times. Negative conditioned mean total tangential strain rates (area stretch factor) contract local areas, except for large values of scalar-gradients. Conditional averages of total and added enstrophies are almost identical, which implies a negligible contribution of the flow vorticity to the observed rotation of non-material line and surface elements. The added vorticity is exactly tangential to the iso-surfaces, whereas the flow and total ones are predominantly tangential. Flow sources/sinks of the mean curvature transport equation are much smaller than the added contributions; for this particular DNS database, the local mean curvature development is self-induced by spatial changes of the displacement speed.

Keywords

Iso-scalar surfaces Non-material line Surface and volume element kinematics Displacement speed Flow and added strain and rotation rate tensors Normal and tangential strain rates Mean curvature transport 

Nomenclature

aN

flow strain rate normal to an iso-scalar surface

\({a_{N}^{a}}\)

added strain rate normal to an iso-scalar surface

\({a_{N}^{Y}}\)

total or effective strain rate normal to an iso-scalar surface

aT

flow strain rate tangential to an iso-scalar surface

\({a_{T}^{a}}\)

added strain rate tangential to an iso-scalar surface

\({a_{T}^{Y}}\)

total or effective strain rate tangential to an iso-scalar surface

D

constant Fickian diffusion coefficient

E

flow enstrophy

Ea

added enstrophy

EY

total or effective enstrophy

k1, k2

eigenvalues of the curvature tensor

km

local mean curvature, (k m = ni, i/2)

l

integral length scale

m

mass of the non-material volume element, V

ni, j

ij component of the non-symmetric curvature tensor, (ni, j = n i /x j )

ReT

turbulent Reynolds number

Sd

normal displacement speed of point x on an iso-scalar surface relative to

the fluid

Sij

flow strain rate tensor

\(S_{ij}^{a}\)

added strain rate tensor

\(S_{ij}^{Y}\)

total or effective strain rate tensor

Sc

Schmidt number

T

fluid temperature

Tα

flow contributions to the mean curvature changes, (α = 1, 2, 3, 4 and 5)

Taα

added contributions to the mean curvature changes, (α = 1, 2, 3, 4 and 5)

TYα

total contributions to the mean curvature changes, (α = 1, 2, 3, 4 and 5)

u

rms of the turbulent velocity fluctuations

Wij

flow rotation rate tensor

\(W_{ij}^{a}\)

added rotation rate tensor

\(W_{ij}^{Y}\)

total or effective rotation rate tensor

xi

i th component of the spatial position vector x

xN

local spatial coordinate normal to an iso-scalar surface

Y

species mass fraction

Y, i

i th component of the scalar gradient ∇Y, (Y, i = Y/x i )

iα, iβ, iγ

three eigenvectors of the flow strain rate tensor

n

unit vector normal to the iso-scalar surface

nr

unit vector in the direction of r

r

infinitesimal non-material vector

v

flow velocity

vY

absolute velocity of a point x on an iso-scalar surface

S

infinitesimal non-material surface area

V

infinitesimal non-material volume element

Greek symbols

1/τη

Kolmogorov strain rate

δij

ij component of the Kronecker delta tensor

\(\dot {\omega }_{Y}\)

chemical rate of conversion of Y, (\(\dot {\omega }_{Y}=\dot {\omega }/\rho \))

η

Kolmogorov length micro-scale

Γ

iso-scalar surface value

ν

kinematic viscosity

ωi

i th component of the flow vorticity vector

\({\omega _{i}^{a}}\)

i th component of the added vorticity vector

\({\omega _{i}^{Y}}\)

i th component of the total or effective vorticity vector

ρ

constant fluid density

τY

flow mixing time due to molecular dissipation of scalar fluctuations

τη

Kolmogorov time micro-scale

ε

turbulent kinetic energy dissipation rate

εijk

Levi-Civitá alternating tensor

Notes

Acknowledgments

The authors gratefully acknowledge the support of this research by the Spanish Ministry of Economy and Competitiveness, under the CONSOLIDER-INGENIO Program, Project CS D2010-00011-SCORE.

Funding Information

This study was funded by the Spanish Ministry of Economy and Competitiveness (Project CS D2010-00011-SCORE CONSOLIDER-INGENIO Program).

Compliance with Ethical Standards

This manuscript has not been submitted or published elsewhere. The results presented and discussed in the manuscript are originals. No data have been fabricated or manipulated to support the conclusions. No data, text, or theories by others are presented as if they were the authors’ own.

Conflict of interests

The authors declare that they have no conflict of interest.

References

  1. 1.
    Pope, S.B.: The evolution of surfaces in turbulence. Int. J. Eng. Sci. 26, 445–469 (1988)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Girimaji, S.S., Pope, S.B.: Propagating surfaces in isotropic turbulence. J. Fluid Mech. 234, 247–277 (1992)CrossRefMATHGoogle Scholar
  3. 3.
    Yoda, M., Hesselink, L., Mungal, M.G.: Instantaneous three-dimensional concentration measurements in the self-similar region of a round high-schmidt-number jet. J. Fluid Mech. 279, 313–350 (1994)CrossRefGoogle Scholar
  4. 4.
    Chong, M.S., Perry, A.E., Cantwell, B.J.: A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dopazo, C., Calvo, P., Petriz, F.: A geometric/kinematic interpretation of scalar mixing. Phys. Fluids 11, 2952–2956 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dopazo, C., Martin, J., Hierro, J.: Local geometry of isoscalar surfaces. Phys. Rev. E 76, 056316 (2007)CrossRefGoogle Scholar
  7. 7.
    Batchelor, G.K.: The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349–366 (1952)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Batchelor, G.K.: Small-scale variation of convected quantities like temperature in turbulent fluid.i. general discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133 (1959)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gibson, C.H.: Fine structure of scalar fields mixed by turbulence. i. zero gradient points and minimal gradient surfaces. Phys. Fluids 11, 2305–2315 (1968)CrossRefMATHGoogle Scholar
  10. 10.
    Cocke, W.J.: Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12, 2488 (1969)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Corrsin, S.: Random geometric problems suggested by turbulence, in statistical models and turbulence. Lect. Notes Phys., Springer-Verlag 12, 300–316 (1972)CrossRefMATHGoogle Scholar
  12. 12.
    Kraichnan, R.H.: Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762 (1974)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Drummond, I.T., Münch, W.: Turbulent stretching of line and surface elements. J. Fluid Mech. 215, 45–59 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Drummond, I.T., Münch, W.: Distortion of line and surface elements in model turbulent flows. J. Fluid Mech. 225, 529–543 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Moffat, H.K.: In: Lesieur, M., Yaglom, A., Davies, F. (eds.) The Topology of Turbulence, in New Trends in Turbulence, pp. 319–340. Springer, Berlin (2001)Google Scholar
  16. 16.
    Moffat, H.K.: In: Kambe, T. et al. (eds.) The Topology of Scalar Fields in 2D and 3D Turbulence, in Iutam Symposium on Geometry and Statistics of Turbulence, pp. 13–22. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  17. 17.
    Pelcé, P.: Dynamics of Curved Fronts. Academic Press Inc., Cambridge (1988)Google Scholar
  18. 18.
    Candel, S.M., Poinsot, T.J.: Flame stretch and the balance equation for the flame area. Combust. Sci Flame Technol. 70, 1–15 (1990)CrossRefGoogle Scholar
  19. 19.
    Chung, S.H., Law, C.K.: An invariant derivation of flame stretch. Combust. Flame 55, 123–125 (1984)CrossRefGoogle Scholar
  20. 20.
    Vervisch, L., Bidaux, E., Bray, K.N.C., Kollmann, W.: Surface density function in premixed turbulent combustion modeling. similarities between probability density function and flame surface approaches. Phys. Fluids 7, 2496–2503 (1995)CrossRefMATHGoogle Scholar
  21. 21.
    Kollmann, W., Chen, J.H.s: Pocket formation and the flame surface density equation. Proc Pocket Combust. Inst. 27, 927–934 (1998)CrossRefGoogle Scholar
  22. 22.
    Sankaran, R., Hawkes, E.R., Chen, J.H., Lu, T., Law, C.K.: Structure of a spatially developing turbulent lean methane-air bunsen flame. Proc. Combust. Inst. 31, 1291–1298 (2007)CrossRefGoogle Scholar
  23. 23.
    Echekki, T., Chen, J.H.: Analysis of the contributions of curvature to premixed flame propagation. Combust. Flame 118, 308–311 (1999)CrossRefGoogle Scholar
  24. 24.
    Chakraborty, N., Cant, S.: Unsteady effects of strain rate and curvature on turbulent premixed flames in inlet-outlet configuration. Combust. Flame 137, 129–147 (2004)CrossRefGoogle Scholar
  25. 25.
    Chakraborty, N., Cant, S.: Influence of Lewis number on curvature effects in turbulent premixed flame propagation in the thin reaction zones regime. Phys. Fluids 17, 105105 (2004)CrossRefMATHGoogle Scholar
  26. 26.
    Chakraborty, N.: Comparison of displacement speed statistics of turbulent premixed flames in the regimes representing combustion in corrugated flamelets and thin reaction zones. Phys. Fluids 19, 105109 (2007)CrossRefMATHGoogle Scholar
  27. 27.
    Uranakara, H.A., Chaudhuri, S., Dave, H.L., Arias, P.G., Im, H.G.: A flame particle tracking analysis of turbulence-chemistry interaction in hydrogen-air premixed flames. Combust. Flame 163, 220–240 (2016)CrossRefGoogle Scholar
  28. 28.
    Lee, T.-W., North, G.L., Santavicca, D.A.: Curvature and orientation statistics of turbulent premixed flame fronts. Combust. Sci. Technol. 84(1-6), 121–132 (1992)CrossRefGoogle Scholar
  29. 29.
    Lee, T.-W., North, G.L., Santavicca D.A.: Surface properties of turbulent premixed propane/air flames at various lewis numbers. Combust. Flame 93(4), 445–456 (1993)CrossRefGoogle Scholar
  30. 30.
    Shepherd, IG, Ashurst, WmT.: Flame front geometry in premixed turbulent flames. Proc. Combust. Inst. 24(1), 485–491 (1992)CrossRefGoogle Scholar
  31. 31.
    Ashurst, WmT., Shepherd, I.G.: Flame Front curvature distributions in a turbulent premixed flame zone. Proc. Combust. Inst. 124(1-6), 115–144 (1997)Google Scholar
  32. 32.
    Dunstan, T.D., Swaminathan, N., Bray, K.N.C., Kingsbury, N.G.: Flame interactions in turbulent premixed twin v-flames. Combust. Sci. Tech. 185, 134–159 (2013)CrossRefGoogle Scholar
  33. 33.
    Dunstan, T.D., Swaminathan, N., Bray, K.N.C., Kingsbury, N.G.: The effects of non-unity Lewis numbers on turbulent premixed flame interactions in a twin v-flame configuration. Combust. Sci. Tech. 185, 874–897 (2013)CrossRefGoogle Scholar
  34. 34.
    Minamoto, Y., Swaminathan, N., Cant, S.R., Leung, T.: Morphological and statistical features of reaction zones in mild and premixed combustion. Combust. Flame 161, 2801–2814 (2014)CrossRefGoogle Scholar
  35. 35.
    Griffiths, R.A.C., Chen, J.H., Kolla, H., Cant, R.S., Kollmann, W.: Three-dimensional topology of turbulent premixed flame interaction. Proc. Combust. Inst. 35, 1341–1348 (2014)CrossRefGoogle Scholar
  36. 36.
    Wang, H., Hawkes, E.R., Chen, J.H.: Turbulence-flame interactions in dns of a laboratory high karlovitz premixed turbulent jet flame. Phys. Fluids 28, 095107 (2016)CrossRefGoogle Scholar
  37. 37.
    Dopazo, C., Cifuentes, L., Martin, J., Jimenez, C.: Strain rates normal to approaching iso-scalar surfaces in a turbulent premixed flame. Combust. Flame 162, 1729–1736 (2015)CrossRefGoogle Scholar
  38. 38.
    Dopazo, C., Cifuentes, L., Hierro, J., Martin, J.: Micro-scale mixing in turbulent constant density reacting flows and premixed combustion. Flow Turbul. Combust. 96, 547–571 (2015)CrossRefGoogle Scholar
  39. 39.
    Kim, S.H., Pitsch, H.: Scalar gradient and small-scale structure in turbulent premixed combustion. Phys. Scalar Fluids 19, 115104 (2007)CrossRefMATHGoogle Scholar
  40. 40.
    Chakraborty, N., Klein, M., Swaminathan, N.: Effects of Lewis number on the reactive scalar gradient alignment with local strain rate in turbulent premixed flames. Proc. Combust. Inst. 32, 1409–1417 (2009)CrossRefGoogle Scholar
  41. 41.
    Dopazo, C., Cifuentes, L.: The physics of scalar gradients in turbulent premixed combustion and its relevance to modeling. Combust. Sci. Tech. 188, 1376–1397 (2016)CrossRefGoogle Scholar
  42. 42.
    Weatherburn, C.E.: Differential Geometry of Three Dimensions, vol. 1. Cambridge University Press, Cambridge (2016)MATHGoogle Scholar
  43. 43.
    Dopazo, C., Valino, L., Martin, J.: Stochastic Modelling of a Scalar Field and Its Gradient Undergoing Turbulent Mixing and Chemical Reaction. Joint Meeting of the Italian and Spanish Sections of the Combustion Institute (1993)Google Scholar
  44. 44.
    Cerutti, S, Meneveau, C: Statistics of filtered velocity in grid and wake turbulence. Phys. Fluids 12(5), 1143–1165 (2000)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Cerutti, S., Meneveau, C., Knio, O.M.: Spectral and hyper eddy viscosity in high-reynolds-number turbulence. J. Fluid Mech. 421, 307–338 (2000)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Alvelius, K: Random forcing of three-dimensional homogeneous turbulence. Phys. Fluids 11(7), 1880–1889 (1999)CrossRefMATHGoogle Scholar
  47. 47.
    Eswaran, V., Pope, S.B.: Direct numerical simulations of the turbulent mixing of a passive scalar. Phys. Fluids 31, 506–520 (1988)CrossRefGoogle Scholar
  48. 48.
    Eswaran, V., Pope, S.B.: An examination of forcing in direct numerical simulations of turbulence. Comp. Fluids 16, 257–278 (1988)CrossRefMATHGoogle Scholar
  49. 49.
    Chandra, R., Dagun, L., Kohr, D., Maydan, J., Mcdonald, D., Menon, R.: Parallel Programming in Openmp. Morgan Kaufmann Publishers, Burlington (2001)Google Scholar
  50. 50.
    Chapman, B., Jost, G., Van, R.: Usingopenmp: Portable Shared Memory Parallel Programming. The MIT Press Library of Congress Cataloging-in-Publication Data (2008)Google Scholar
  51. 51.
    Peyret, R.: Spectral methods for incompressible viscous flow. In: Applied Mathematical Sciences, 148 (2000)Google Scholar
  52. 52.
    Orszag, S., Patterson, G.: Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 12, 76–79 (1972)CrossRefGoogle Scholar
  53. 53.
    Rogallo, R.S.: Numerical Experiments in Homogeneous Turbulence. NASA TM, 81315, NASA Ames Research Center, CA (1981)Google Scholar
  54. 54.
    Ashurst, W., Kerstein, A., Kerr, R., Gibson, C.: Alignment of vorticity and scalar gradient in simulated navier-stokes turbulence. Phys. Fluids 30, 2343–2353 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Engineering and Architecture - Fluid Mechanics AreaUniversity of Zaragoza / LIFTEC-CSICZaragozaSpain
  2. 2.Institute for Combustion and Gasdynamics (IVG), Chair of Fluid DynamicsUniversity of Duisburg-EssenDuisburgGermany
  3. 3.Centro Universitario de la Defensa, AGM, Crta. de Huesca s/nZaragozaSpain

Personalised recommendations