Phys. Fluids 25, 019901 (2013); https://doi.org/10.1063/1.4775381 25, 019901
© 2013 American Institute of Physics.
Erratum: “Investigations on the local
entrainment velocity in a turbulent
jet” [Phys. Fluids 24, 105110 (2012)]
Cite as: Phys. Fluids 25, 019901 (2013); https://doi.org/10.1063/1.4775381
Submitted: 12 December 2012 . Accepted: 18 December 2012 . Published Online: 09 January 2013 M. Wolf, B. Lüthi, M. Holzner, D. Krug, W. Kinzelbach, and A. Tsinober
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PHYSICS OF FLUIDS 25, 019901 (2013)
Erratum: “Investigations on the local entrainment velocity
in a turbulent jet” [Phys. Fluids 24, 105110 (2012)]
M. Wolf,1,a)B. L ¨uthi,1M. Holzner,1D. Krug,1W. Kinzelbach,1 and A. Tsinober2
1Institute of Environmental Engineering, ETH Zurich, 8093 Zurich, Switzerland 2School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University,
Tel Aviv 69978, Israel
(Received 12 December 2012; accepted 18 December 2012; published online 9 January 2013)
[http://dx.doi.org/10.1063/1.4775381]
There is a sign error in our article1 concerning the calculations of the mean curvature H. If
the normal vector is defined such that it points towards larger enstrophy values, as it is the case in Ref.1, Eq. (A4) should read K= 4ac − b2, H= −a − c. From this, it follows that Figures 11–14 in Ref.1show the wrong sign for the mean curvature H. Positive and negative mean curvature values must be exchanged. Figures 11–14 in the paper should be replaced by Figures1–4below.
−1000−6 −500 0 500 1000 −4 −2 0 2 4 6x 10 H K 4 4.5 5 5.5 6 6.5 7 7.5 8 −1000 0 1000 10− 10− 10− 10− H PDF −5 0 5 x 10 10− 10− 10− 10− K PDF
FIG. 1. Corrected Fig. 11 from Ref.1. Left: joint probability density function (PDF) of mean and Gaussian curvatures H and K; Right: PDFs of single components.
−1000−6 −500 0 500 1000 −4 −2 0 2 4 6x 10 5 H (a) (b) (c) K v n/<uk> −0.9 −0.6 −0.3 0 0.3 0.6 0.9 −1000−6 −500 0 500 1000 −4 −2 0 2 4 6x 10 5 H K vn inv /<uk> −0.9 −0.6 −0.3 0 0.3 0.6 0.9 −1000−6 −500 0 500 1000 −4 −2 0 2 4 6x 10 5 H K vn vis /<uk> −0.9 −0.6 −0.3 0 0.3 0.6 0.9
FIG. 2. Corrected Fig. 12 from Ref.1. Conditional average of local entrainment velocity,vn, and its components,vi nvn and
vvis
n conditioned on mean and Gaussian curvatures H and K.
a)wolf@ifu.baug.ethz.ch.
019901-2 Wolfet al. Phys. Fluids 25, 019901 (2013) −10001.5 −500 0 500 1000 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 H (a) (b) (c) ωi ωj sij −1000−3 −500 0 500 1000 −2 −1 0 1 2 H Λi /Λ 2 Λ1/Λ2 Λ3/Λ2 −1000 −500 0 500 1000 0.2 0.25 0.3 0.35 0.4 0.45 0.5 H cos 2(ω ,λi ) cos2(ω,λ1) cos2(ω,λ2) cos2(ω,λ 3)
FIG. 3. Corrected Fig. 13 from Ref.1.ωiωjsijand its components as shown in Ref.1, Eq. (12), conditioned on mean curvature
H. Ratios are shown for Eigenvaluesi.
First of all, we want to state that the main result of the article is not affected by the sign error. Before and after the correction, we find that the local entrainment velocityvnchanges considerably
depending on the local shape of the interface, and different small-scale mechanisms are dominating for different shapes of the interface. However, Fig.2(a)shows that local entrainment increases for convex (H> 0) and not for concave surface elements, looking from the turbulent region towards the convoluted boundary. Furthermore, viscous diffusion is dominating the local entrainment velocity for convex shapes, see Fig.2(c), and vortex stretching for concave shapes, see Fig.2(b). In other words, vortex stretching is the driving term to produce convolution while viscous diffusion is attempting to flatten or at least not to further convolute the interface.
By further decomposing the viscous component ofvn, it was concluded in Ref.1that the viscous
advancement of the interface into the non-turbulent region mostly depends on the enstrophy profile normal to the interface. This argument does not hold anymore. Fig. 2(c)shows that the viscous entrainment component vvisn decreases for positive H values. Hence, both the enstrophy profile
normal to the interface as well as the mean curvature of the interface can influencevvisn .
The resulting conclusion from Fig.3has not changed in comparison to Ref.1. We still see that the inviscid component of the local entrainment velocity,vni nv, is mainly governed by the alignment
of vorticity with the Eigenvectors of the rate of strain tensor.
−1000−6 −500 0 500 1000 −4 −2 0 2 4 6x 10 5 H K |u|/U c 0.1 0.11 0.12 0.13 0.14 0.15 0.16
FIG. 4. Corrected Fig. 14 from Ref.1. Magnitude of advection velocity|u| at the interface with respect to H and K, normalized by the mean centerline velocity Uc.
019901-3 Wolfet al. Phys. Fluids 25, 019901 (2013)
Figures2(a)and4illustrate that both the mean advection of the underlying fluid as well as the average local entrainment velocity counteract the convolution of the instantaneous interface. Hence, on average, the advection velocity together with a contribution ofvntend to flatten the interface.
In Fig. 1, it is shown that the joint distribution of mean and Gaussian curvature is skewed towards positive values indicating a slightly higher probability of convex shapes.
1M. Wolf, B. L¨uthi, M. Holzner, D. Krug, W. Kinzelbach, and A. Tsinober, “Investigations on the local entrainment velocity in a turbulent jet,”Phys. Fluids24, 105110 (2012).
