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Dissipation rate estimation from PIV in zero-mean isotropic turbulence

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Abstract

Measuring the turbulent kinetic energy dissipation rate in an enclosed turbulence chamber that produces zero-mean flow is an experimental challenge. Traditional single-point dissipation rate measurement techniques are not applicable to flows with zero-mean velocity. Particle image velocimetry (PIV) affords calculation of the spatial derivative as well as the use of multi-point statistics to determine the dissipation rate. However, there is no consensus in the literature as to the best method to obtain dissipation rates from PIV measurements in such flows. We apply PIV in an enclosed zero-mean turbulent flow chamber and investigate five methods for dissipation rate estimation. We examine the influence of the PIV interrogation cell size on the performance of different dissipation rate estimation methods and evaluate correction factors that account for errors related to measurement uncertainty, finite spatial resolution, and low Reynolds number effects. We find the Re λ corrected, second-order, longitudinal velocity structure function method to be the most robust method to estimate the dissipation rate in our zero-mean, gaseous flow system.

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Abbreviations

A :

constant, from the scaling argument method

B :

spatial spectral-filtering function

C 1 :

first Kolmogorov constant

C 2 :

second Kolmogorov constant

C 3 :

third Kolmogorov constant

C s :

Smagorinsky constant

\(D_{L^n}\) :

longitudinal velocity structure function of order n

D LL :

second-order longitudinal velocity structure function

D LLL :

third-order longitudinal velocity structure function

D NN :

second-order transverse velocity structure function

E :

three-dimensional energy spectrum

E 11ι):

one-dimensional energy spectrum

L :

large eddy length scale

L 11 :

longitudinal integral length scale

N :

number of points for the Fourier transform of the velocity vector field

Re λ :

Taylor microscale Reynolds number

T e :

large eddy time scale

f(r):

longitudinal velocity spatial correlation function

h :

height of PIV interrogation volume

k :

turbulent kinetic energy

r :

three-dimensional direction vector

r i :

velocity vector separation in x i direction

\( \underline{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{r} }} \) :

unit vector in the r direction

r IR :

inertial range velocity vector separation in x i direction

s ij :

fluctuating rate-of-strain tensor

\( \tilde{s}_{ij} \) :

filtered fluctuating rate-of-strain tensor

t η :

Kolmogorov time scale

u :

instantaneous velocity vector at a point

u i :

instantaneous velocity vector at a point in the x i direction

u′ :

root mean square of the velocity fluctuations

\( u_{i}^{\prime } \) :

root mean square of the velocity fluctuations in the x i direction

\( \tilde{u}_{i} \) :

filtered root mean square of the velocity fluctuations in the x i direction

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u}_{j} \) :

Fourier transform of the velocity vector field

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u}_{j}^{*} \) :

complex conjugate Fourier transform of the velocity vector field

u η :

Kolmogorov velocity scale

w :

width of PIV interrogation volume

x :

three-dimensional position vector

Δx i :

separation between PIV vectors

Δ:

“window” size over which velocity field is spatial averaged

Φ ij :

three-dimensional velocity spectrum tensor

α :

proportionality constant for the Lin spectrum

ε :

turbulent kinetic energy dissipation rate

η :

Kolmogorov length scale

κ :

wavenumber vector

κ :

magnitude of wavenumber vector

κ i :

wavenumber in x i direction

κ i, IR :

inertial subrange wavenumbers in x i direction

Δκ 1 :

separation between wavenumbers

ν :

kinematic viscosity

τ :

depth of PIV interrogation volume

τ ij :

subgrid Reynolds stress

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Acknowledgments

This work was supported by the NASA Microgravity Fluid Physics Program grants NNCO5GA45G and NNCO5GA37G, by the National Science Foundation through grants CTS-0112514 and PHY-0554675, and by the New York State Office of Science, Technology and Academic Research (NYSTAR) under contract number 3538479. J.P.L.C. Salazar acknowledges support from the Brazilian Ministry of Education through the CAPES agency.

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de Jong, J., Cao, L., Woodward, S.H. et al. Dissipation rate estimation from PIV in zero-mean isotropic turbulence. Exp Fluids 46, 499–515 (2009). https://doi.org/10.1007/s00348-008-0576-3

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  • DOI: https://doi.org/10.1007/s00348-008-0576-3

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