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 CifuentesEmail author
  • Juan Hierro


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.


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 



flow strain rate normal to an iso-scalar surface


added strain rate normal to an iso-scalar surface


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


flow strain rate tangential to an iso-scalar surface


added strain rate tangential to an iso-scalar surface


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


constant Fickian diffusion coefficient


flow enstrophy


added enstrophy


total or effective enstrophy

k1, k2

eigenvalues of the curvature tensor


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


integral length scale


mass of the non-material volume element, V

ni, j

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


turbulent Reynolds number


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

the fluid


flow strain rate tensor


added strain rate tensor


total or effective strain rate tensor


Schmidt number


fluid temperature


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


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


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


rms of the turbulent velocity fluctuations


flow rotation rate tensor


added rotation rate tensor


total or effective rotation rate tensor


i th component of the spatial position vector x


local spatial coordinate normal to an iso-scalar surface


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


unit vector normal to the iso-scalar surface


unit vector in the direction of r


infinitesimal non-material vector


flow velocity


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


infinitesimal non-material surface area


infinitesimal non-material volume element

Greek symbols


Kolmogorov strain rate


ij component of the Kronecker delta tensor

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

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


Kolmogorov length micro-scale


iso-scalar surface value


kinematic viscosity


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


constant fluid density


flow mixing time due to molecular dissipation of scalar fluctuations


Kolmogorov time micro-scale


turbulent kinetic energy dissipation rate


Levi-Civitá alternating tensor



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.


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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

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