Experiments in Fluids

, 46:499 | Cite as

Dissipation rate estimation from PIV in zero-mean isotropic turbulence

  • J. de Jong
  • L. Cao
  • S. H. Woodward
  • J. P. L. C. Salazar
  • L. R. Collins
  • H. Meng
Research Article

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 Reλ 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.

List of symbols

A

constant, from the scaling argument method

B

spatial spectral-filtering function

C1

first Kolmogorov constant

C2

second Kolmogorov constant

C3

third Kolmogorov constant

Cs

Smagorinsky constant

\(D_{L^n}\)

longitudinal velocity structure function of order n

DLL

second-order longitudinal velocity structure function

DLLL

third-order longitudinal velocity structure function

DNN

second-order transverse velocity structure function

E

three-dimensional energy spectrum

E11ι)

one-dimensional energy spectrum

L

large eddy length scale

L11

longitudinal integral length scale

N

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

Reλ

Taylor microscale Reynolds number

Te

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

ri

velocity vector separation in xi direction

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

unit vector in the r direction

rIR

inertial range velocity vector separation in xi direction

sij

fluctuating rate-of-strain tensor

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

filtered fluctuating rate-of-strain tensor

tη

Kolmogorov time scale

u

instantaneous velocity vector at a point

ui

instantaneous velocity vector at a point in the xi direction

u′

root mean square of the velocity fluctuations

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

root mean square of the velocity fluctuations in the xi direction

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

filtered root mean square of the velocity fluctuations in the xi 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

uη

Kolmogorov velocity scale

w

width of PIV interrogation volume

x

three-dimensional position vector

Δxi

separation between PIV vectors

Δ

“window” size over which velocity field is spatial averaged

Φij

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

κi

wavenumber in xi direction

κi, IR

inertial subrange wavenumbers in xi direction

Δκ1

separation between wavenumbers

ν

kinematic viscosity

τ

depth of PIV interrogation volume

τij

subgrid Reynolds stress

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • J. de Jong
    • 1
  • L. Cao
    • 1
  • S. H. Woodward
    • 1
  • J. P. L. C. Salazar
    • 2
  • L. R. Collins
    • 2
  • H. Meng
    • 1
  1. 1.Mechanical and Aerospace Engineering DepartmentUniversity of BuffaloBuffaloUSA
  2. 2.Sibley School of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA

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