Large-eddy estimate of the turbulent dissipation rate using PIV

Abstract

many particle images. The particle motion and thus the velocity field are averaged over the extent of the interrogation window. For a measurement of small-scale turbulence quantities, the velocity field must be resolved accordingly. For example, the dissipation rate ǫ = ν i,j (∂u i /∂x j ) 2 , where ν is the kinematic viscosity, involves the sum of squared derivatives of the components u i of the velocity field. It requires the resolution of the velocity field down to the Kolmogorov scale η, where the velocity field is smooth and where derivatives can be estimated from finite differences. Normally, the linear dimension L of the interrogation window is much larger than η, so that the magnitude of derivatives is underestimated. The obvious cure is to make the interrogation window as small as η, at the expense of a very limited view of the velocity field (Tanaka and Eaton 2010). On the other hand, when the size of the interrogation window is well within the inertial range, a now popular approach involves the assumption that the statistics of the resolved velocity field ū is universal, with a universal relation between the measured derivatives of the spatially averaged field ū and the true dissipation rate, ǫ LE (Sheng et al. 2000),For interrogation windows with inertialrange dimensions, finite velocity differences u scale as �(�u) 2 � ∝ ǫ 2/3 L 2/3 , so that ǫ LE becomes independent of the window size. This approach is attractive as it provides a means of estimating the dissipation rate from PIV measurements that still provide information about the large-scale structure of the velocity field.The commonly used value of the Smagorinsky constant is C Sm = 0.17 (Sheng et al. 2000;Lavoie et al. 2007), butAbstract The result of a particle image velocimetry (PIV) measurement is a velocity field averaged over interrogation windows. This severely affects the measurement of small-scale turbulence quantities when the interrogation window size is much larger than the smallest lengthscale in turbulence, the Kolmogorov length. In particular, a direct measurement of the dissipation rate demands the measurement of gradients of the velocity field, which are now underestimated because the small-scale motion is not resolved. A popular procedure is to relate the statistical properties of the measured, but underresolved gradients to those of the true ones, invoking a large-eddy argument (Sheng et al. in Chem Eng Sci 55(20):4423-4434, 2000). We argue that the used proportionality constant, the Smagorinsky constant, should depend on the window overlap, on the used elements of the strain tensor, and on the way in which derivatives are approximated. Using an analytic description, PIV measurements of velocity fields from a kinematic simulation and experiments in a synthetic jetdriven turbulent flow with zero mean velocity, we propose new values for this constant.

Experiments on breakup of bubbles in a turbulent flow

“…A series of tests, during which the size of interrogation area and vector spacing were changed, suggests that the most appropriate filter size Δ is the vector spacing (0.52 mm for present experiments). Note that we differ from other studies in this setting of Δ .…”

“…This rough estimate is based on following arguments: Integrated dissipation rate determined by LEPIV is approximately one half of the injected energy (Table ), although most of this difference can be caused by the uncaptured high dissipation rate close to the nozzles. Estimation of the dissipation rate from structure function (13) and from the scaling argument method (14) disagree with the LEPIV by up to 50%. We used a Smagorinsky's constant of 0.17, but smaller values (0.11 to 0.12) have been suggested A study by Bertens et al suggests that our method should have an error of approximately 30% (see their Table ; the present method is “2D CD” with “ α = 0.375”). …”

Scaling of an Atmospheric Model to Simulate Turbulence and Cloud Microphysics in the Pi Chamber

“…As the measurement has a limited resolution, the minimum PIV window size (5 cm) is 50 times larger than the Kolmogorov length (1 mm) of the small‐scale turbulent eddies. This results in an inherent filtering of the turbulent velocity field, which requires a correction of the resulting dissipation rate, detailed by Bertens et al (). This correction, based on the size of the PIV window to the Kolmogorov scale η , gives an uncertainty ≈25% in homogeneous, isotropic turbulence.…”

Resolving vorticity and dissipation in a turbulent boundary layer by tomographic PTV and VIC+

“…The underestimation of the dissipation rate by tomographic PIV is a problem recognized in recent literature (Tokgoz et al 2012) and sub-grid scale modelling approaches have been proposed to improve the estimation of the dissipation rate from PIV (Sheng et al 2000;Bertents et al 2015). Following the results in the previous section, an improved dissipation rate estimate is expected with VIC+, in comparison to tomographic PIV.…”