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The influence of probe resolution on the measurement of a passive scalar and its derivatives

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Abstract

The influence of probe resolution on the statistical measurement of a passive scalar is reported. A spectral method is employed to simulate degradation of the spatial resolution of a probe on the measured variances of a fluctuating scalar and its streamwise derivative by low-pass filtering a time-series of data at different cutoff frequencies. Direct measurements are also employed by varying probe sensor separation. The far field of a circular jet and the near wake of a circular cylinder are both investigated using air as the working fluid. The use of this low-Schmidt number working fluid and relatively low turbulence Reynolds numbers allows for good resolution of small scales of scalar fluctuations. By comparison, the same level of resolution is much more difficult to achieve when utilising a high-Schmidt number working fluid. A small temperature differential above ambient is used to mark the passive scalar, which is measured using a cold-wire anemometer. Taylor's hypothesis is employed to determine length scales. The present results are in good agreement with previous direct measurements using both optical techniques and cold-wire probes. It is found that the spatial resolution required for accurate measurement of the scalar dissipation rate is well described by the characteristic smallest scale of the scalar fluctuation, i.e. 'the Batchelor scale'. However, an order of magnitude less resolution is required for the scalar variance. The effect of degrading resolution on the variance measurements is more significant in the near wake than the far-field jet, suggesting that these requirements may be flow-dependent.

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Abbreviations

C :

empirical constant in Eq. (4)

C 1 :

empirical constant in Eqs. (5), (6) and (7), defined by C 1=C −1/4

D :

scalar (molecular) diffusivity

d j :

jet initial diameter

d c :

wake cylinder diameter

f :

frequency

f B :

Batchelor frequency

f c :

filtering cutoff frequency

f s :

sampling frequency

f K :

Kolmogorov frequency

k :

wavenumber vector whose magnitude is k=(k 1 2+k 2 2+k 3 2)1/2

k 1, k 2, k 3, k 2, k 3 :

wavenumber components in the x-, y- and z-directions, respectively

L :

local characteristic length scale

Re L :

Reynolds number defined by Re L ≡UL/ν

Re λ :

turbulent Reynolds numbers defined by Re λ ≡u′λ T/ν

r 1/2 :

mean velocity half-width of a jet at which <U>=0.5U c

Sc :

molecular Schmidt number defined by Sc≡ν/D

t :

time

u′:

root mean-squared streamwise velocity fluctuation

U :

local characteristic velocity scale

U c :

local centreline mean velocity of a jet

U o :

jet exit bulk velocity

U ∞ :

free-stream velocity

x, y, z :

streamwise, lateral and spanwise coordinates, respectively

<β 2>:

variance or mean square of fluctuating quantity β

δ :

local jet width based on the mean scalar field

<ε>:

mean energy dissipation rate

Φ β :

power spectral density (or 'spectrum' for simplicity) of fluctuating quantity β defined by ∫Φ β (f)df=<β 2>

Γ :

three-dimensional scalar spectrum

G w :

cold-wire effective length

λ B :

Batchelor length scale defined by λ B≡(vD 2/<ε>)1/4

λ D :

smallest strain-limited diffusion length scale

λ K :

Kolmogorov length scale defined by λ K≡(v 3/<ε>)1/4

λ r :

spatial resolution scale

λ T :

Taylor's microscale defined by \(\lambda _{\rm{T}} \equiv {{\sqrt { < u^2 > } } \mathord{\left/ {\vphantom {{\sqrt { < u^2 > } } {\sqrt { < \left( {\partial u/\partial x} \right)^2 > } }}} \right. \kern-\nulldelimiterspace} {\sqrt { < \left( {\partial u/\partial x} \right)^2 > } }}\)

θ :

scalar (e.g. concentration, temperature) fluctuation

<θ 2>:

scalar variance

ν :

kinematic viscosity of the fluid

<χ>:

>mean scalar dissipation rate defined by <χ>≡D[<(∂θ/∂x)2>+<(∂θ/∂y)2>+<(∂θ/∂z)2>]

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Acknowledgements

The support of the Australian Research Council is gratefully acknowledged. Partial support for the authors' positions was provided by FCT combustion, which is also greatly acknowledged.

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  1. Department of Mechanical Engineering, The University of Adelaide, 5005, South Australia, Australia

    J. Mi & G. J. Nathan

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  1. J. Mi
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  2. G. J. Nathan
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Correspondence to J. Mi.

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Mi, J., Nathan, G.J. The influence of probe resolution on the measurement of a passive scalar and its derivatives. Exp Fluids 34, 687–696 (2003). https://doi.org/10.1007/s00348-003-0603-3

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