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

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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.

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Abbreviations

a N :

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

a T :

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

E a :

added enstrophy

E Y :

total or effective enstrophy

k 1, k 2 :

eigenvalues of the curvature tensor

k m :

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

l :

integral length scale

m :

mass of the non-material volume element, V

n i, j :

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

R e T :

turbulent Reynolds number

S d :

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

ᅟ:

the fluid

S i j :

flow strain rate tensor

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

added strain rate tensor

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

total or effective strain rate tensor

S c :

Schmidt number

T :

fluid temperature

T α :

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

T a α :

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

T Y α :

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

u ′ :

rms of the turbulent velocity fluctuations

W i j :

flow rotation rate tensor

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

added rotation rate tensor

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

total or effective rotation rate tensor

x i :

i th component of the spatial position vector x

x N :

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 )

i α , i β , i γ :

three eigenvectors of the flow strain rate tensor

n :

unit vector normal to the iso-scalar surface

n r :

unit vector in the direction of r

r :

infinitesimal non-material vector

v :

flow velocity

v Y :

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

S:

infinitesimal non-material surface area

V:

infinitesimal non-material volume element

1/τ η :

Kolmogorov strain rate

δ i j :

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

ω i :

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

τ Y :

flow mixing time due to molecular dissipation of scalar fluctuations

τ η :

Kolmogorov time micro-scale

ε :

turbulent kinetic energy dissipation rate

ε i j k :

Levi-Civitá alternating tensor

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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

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

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Authors and Affiliations

  1. School of Engineering and Architecture - Fluid Mechanics Area, University of Zaragoza / LIFTEC-CSIC, C/ Maria de Luna 3, Zaragoza, 50018, Spain

    Cesar Dopazo, Jesus Martin, Luis Cifuentes & Juan Hierro

  2. Institute for Combustion and Gasdynamics (IVG), Chair of Fluid Dynamics, University of Duisburg-Essen, Duisburg, 47048, Germany

    Luis Cifuentes

  3. Centro Universitario de la Defensa, AGM, Crta. de Huesca s/n, Zaragoza, 50090, Spain

    Juan Hierro

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  1. Cesar Dopazo
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  4. Juan Hierro
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Correspondence to Luis Cifuentes.

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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.

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Dopazo, C., Martin, J., Cifuentes, L. et al. Strain, Rotation and Curvature of Non-material Propagating Iso-scalar Surfaces in Homogeneous Turbulence. Flow Turbulence Combust 101, 1–32 (2018). https://doi.org/10.1007/s10494-017-9888-9

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