Appendix I Evaluation of the Integral Length Scale
Appendix I
Evaluation of the Integral Length Scale
During the discussion of this work, has been widely introduced the notion of integral length scale (ILS). In this Appendix, the evaluation of the ILS is proposed through the study of the autocorrelation function proposed in chapter 2.
Before exposing and explaining the results a brief introduction of the meaning of the integral length scale is proposed. Previously the ILS was adopted in the evaluation of the energy dissipation with the dimensional analysis, approximating the integral length scale (L), which appears in the empirical relationshipε = Au /3 L.
The autocorrelation function from the definition is the integral from - to + , but in this case, the double integral limits have been chosen to be of finite dimension, because of the finite dimensions of the system under investigation. The integral of the autocorrelation function should be done until the first zero crossing value. This value is variable though the flow field and in our case varies from 10mm to 20mm, thus the ILS has been investigated over a square of constant dimension over the whole flow field with the side of 15mm.
∞ ∞
Figure A.1: Integral length scale evaluated over ux’
Appendix I Evaluation of the Integral Length Scale
Figure A.2: Integral length scale evaluated over uy’
The border of flow field results inevitably extended, reducing the size of the analyzed flow field area. The normalization in the autocorrelation was done evaluating the mean value over the velocity fluctuation over the integration area.
Comparing the integral length scale evaluated with the local energy dissipation rates, estimated with the dimensional analysis, which results for the same angle and configuration are reported in Figure 4.14; the areas of the flow field where lower is the energy dissipation, correspond to the areas in which the integral length scale is higher. Reaching in the discharge stream, values of 0.45 times the values used previously (D/10) and in the further areas 1.1 times D/10.
In the next figures is shown the result of the application of the correction to the previously calculated energy dissipation rates.
Appendix I Evaluation of the Integral Length Scale
Figure A.3:Energy comparison between the dimensional analysis with the variable ILS in the first image and in the second image the sub-grid-scale model reported from chapter 4.
In Figure A.3, is shown the comparison of the energy dissipation rate of the dimensional analysis method with variable ILS, with the sub-grid-scale model. The results in the discharge jet present similar values. For the dimensional analysis, the maximum value for the coordinates r/R=0.5 and z/H=0.13 is 8.11; while for the sub-grid-scale model at the same coordinates, the value of Esgs/N3D2 is 8.45.
Appendix I Evaluation of the Integral Length Scale The value obtained with a constant ILS (D/10), reported in paragraph 4.3.3 and illustrated in
Figure 4.14, is for the coordinates described above, higher than 250 Ewp/N3D2.
Thus the evaluation of the ILS with the autocorrelation described in the previous chapters increased significantly the precision of the method.
