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After the filtering operation, the set of equation 3 and 4 can be cast into the following form, (5) (6) the subgrid stress tensor τij containing the information from the subgrid scales can be decomposed as (Leonard 1974) (7) Lij is the Leonard tensor and describes the interactions between the filtered scales, Cij is the crossing stress tensor and describes the interactions between the filtered scales and the subgrid ones and Rij is called the Reynolds tensor and describes the interaction between the subgrid scales. The introduction of τij involves the problem of modelling the tensor itself or its effects. Hence we need to build assumptions on the interaction between the resolved and the subgrid scales which preconceive the behaviour of the ones. Let us focus our attention on the modelisation. Equation 6 represents the evolution of the filtered quantity ū, in which the result of the interactions with the subgrid field u′ are translated through the term (8) Two ways exist to take into account this term (Sagaut 1995). Either the tensor τij is calculated explicitly (Love 1980, Lund 1992, Pope 1975) or a new term is introduced which has the similar effect on the resolved field ū as τij itself. In this paper, only the second approach is considered. The subgrid scales are assumed to have a Brownian motion superposed on the motion of the filtered scales. By analogy with the dissipation by molecular viscosity υ, a subgrid viscosity is introduced, denoted υSM. But there is an important difference between υ and υSM, υSM is not an intrinsic property of the fluid and hence depends heavily upon the flow itself. The subgrid viscosity has the advantage of being robust, that is to say a dissipative effect introduced through υSM which tends to stabilise the schemes. However, such a model is far from capable of describing every mechanism of the interactions between the subgrid scales and the filtered ones. Through the notion of subgrid viscosity, we assume implicitly that the root of the action of the subgrid scales on the filtered scales is energetic. In other words, we only need the energetic transfer balance between the two kinds of scales to describe the subgrid tensor effect. However, the energy drains from the filtered scales to the subgrid ones is not the only mechanism which describes the interaction between the two ranges of scales (Kraichnan 1976). Two other mechanisms can be distinguished that are first a reverse transfer of the energy from the scales to the filtered scales, such a mechanism is called backscattering (Chasnov 1991, Domaradzki 1997) and secondly a transfer of information related to the anisotropy from the greatest scales to the smallest ones. These last two mechanisms are ipso facto ignored by the subgrid viscosity approach, and it can be shown (Robinson 1991) that the backscattering and the anisotropy (Hartel 1994, Piomelli 1996) are the root of the whole dynamics of the turbulent boundary layer and hence cannot be neglected. In such area, a range of scales, as wide as possible, must be directly simulated. The subgrid is restricted to the smallest scales to the greatest extent possible with minimal effect on the filtered scales (Boris 1992). The number of discretisation meshes may be close to the one given by DNS criterion. The classical model used to calculate the subgrid viscosity is the Smagorinsky model (Smagorinsky 1963). An alternative is suggested by Boris et al (Boris 1992) and (Kawamura 1984). The numerical viscosity, resulting both from the finite volume discretization and from the numerical scheme is used as subgrid modelling. No subgrid viscosity is explicitly computed and hence this approach is called implicit modelisation. The relation between the numerical viscosity, and the subgrid viscosity is not supported by a clear mathematical theory. A drawback of this approach is that the numerical viscosity is utterly artificial, that is to say, not based on a comprehension of the simulated phenomena. A second drawback is the effect of the subgrid scales on the filtered ones is a strictly dissipative effect (Kawamura 1984, Sagaut 1995). However, the implicit simulation is quite manageable if we keep in mind that important mesh refinement is required in the areas where the viscosity appears explicitly (in boundaries layers, for example). Such areas present strong local anisotropy characteristics. In the areas where viscosity effects are not preponderant, larger meshes can be used and the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 303 numerical viscosity operates. In this paper, we intend to resolve the following set of Navier-Stokes equations, (9) (10) The numerical viscosity is introduced by a classical second-order accurate scheme with a finite volume discretization. The temporal resolution is performed by a first-order implicit Euler algorithm. These algorithms are carried through the commercial code FLUENT. Table 1 gives the main characteristics of the grid used for calculation. Table 1: Main characteristics of the mesh. NACA66 4 degrees BA 2 degrees BA 4 degrees BA 7 degrees Cells number 629069 828371 1322885 828371 Total span calculated 0.06 m 0.06 m 0.06 m 0.06 m Size of the axial mesh . 0.0025 m 0.002 m 0.0025 m Size of the span mesh 0.0013 m 0.0011 m 0.0013 m Boundary layer mesh 0.002 m 0.001 m 0.002 m 6.6 106 6 106 3.6 106 3.6 106 Reynolds number For all cases, the mesh was tetrahedral hybrid non structured. Some difference appear in the way the boundary layer was meshed. For BA foil, it was hexahedral cells in the boundary layer and for NACA 66, it was tetrahedral cells. Figure 1 gives an idea of the calculation volume. Figure 1: Main dimensions of the calculation volume. EXPERIMENTAL SET-UP Two different tests were conducted in the small test section (1.14 m X 1.14 m X 6m) of the G.T.H. During the first one, we have studied a two-dimensional foil with a NACA66 modified section. It was equiped with 18 static pressure holes and was flush mounted on the wall using the 6 components balance. During the second test, a new two-dimensional foil (called BA foil) with a section specially defined was tested. This foil was also flush mounted with the balance and a fluctuating pressure transducer was installed in it. In order to reduce the lift of the two-dimensional foils, at high velocity, we have limited the span of the foil by using a big vertical plate (3.4 mX1.14mX0.048m) in the test section as describe in figure 2. Figure 2: Top view of the test section. The main characteristics of the foils are given in table 2. Table 2: Main characteristics of the foils. Section NACA 66 BA foil Plan form rectangular Rectangular Chord 0.6 m 0.6 m Span 0.45 m 0.45 m Maximum Thickness 6% 7% Maximum camber 2% 3.5% The tests were carried out with velocity range from 2 m/s to 13 m/s. So, the Reynolds number, based on the chord, ranged from 1.2 106 to 7.8 106 which is close to the full scale value for a section of a propeller. The the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 304 foil angle of attack was varied from −10 degrees to 14 degrees for the NACA 66 and from 0 degree to 8 degrees for BA foil. Table 3 presents physical parameters which were recorded on each foil. We note that forces and cavitation measurements are available for both foils but NACA66 will be used to validate stationary information and BA foil for non-stationary values. Table 3: Description of the physical quantities recorded. NACA 66 BA Static pressure Yes No Fluctuating pressure No Yes Boundary layer velocity No Yes Forces Yes Yes Cavitation Yes Yes COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS The first step of the validation consists of a comparison between measured and calculated mean stationary information like pressure on the foil. For that purpose, several control points were imposed in the numerical grid on the foil surface. For each control point and each time step, pressure was written in a file. The mean value for a chord position is then obtained by an average of all results. Figure 3 shows the comparison for Cp value versus chord position on the NACA66 at 4 degrees and 11 m/s. On this graph, 3 experimental flow velocities are given. For numerical results, we plot both LES results and Navier-Stokes stationary calculation with K-ε model and the same mesh as the one used for LES calculation. The agreement seems to be quite good, especially for the area close to the leading edge and flow velocity of 11 m/s. To continue with the validation of the numerical process, we decided to compare cavitation inception parameter σi measured during experiments (with nuclei injection) at cavitation inception with calculated −Cpmin or the value of −Cp at boundary layer detachment when calculation shows a detachment. Figure 4 presents the results for BA foil at several angles of attack. Once again, the agreement is very good at low angles of attack. At 4 degrees, the error between −Cpmin and σi is 10%. Figure 3: Pressure coefficients on NACA 66 foil: Comparison between calculation and experiments. Figure 4: Cavitation parameter on foil BA: Comparison between calculation and experiments. Now, as we consider that model gives good results for knowledge of stationary flow, we will look into the non- stationary informations obtained by LES model. Numerical flow description: the authoritative version for attribution. Several representations are issued from the numerical process. We can take a “photograph” of the flow at one time or we can “record” the temporal signal of pressure for a given control point. First we will consider some photographs of the flow. Figure 5 gives a view of the velocity field for BA foil at 7 degrees and 6 m/s. The field corresponds to a

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 306 turbulent flow takes place. The spots seem to be periodically spaced. Near the trailing edge, we observe a separation longer than the one observed for 2 degrees. On figure 9 we present an experimental view of the trailing edge flow at 4 degrees and 6 m/s. Bubbles injected in the flow upstream enable to visualise the trailing edge separation. The accuracy between numerical photograph and experimental view is quite good. Figure 8: BA foil, 4 degrees and 6 m/s: axial vorticity. Figure 9: BA foil, 4 degrees and 6 m/s: photograph of trailing edge separation. For higher angles of attack (i.e. 7 degrees: figures 10 and 11), we can see the presence of the previously described (figure 5) separation bubble near the leading edge. Then the flow re-attaches and after the closure of the bubble we observe spots of high intensity of transverse vorticity which indicate the location of vortex. On the photograph corresponding to axial vorticity, we also notice, in front of the leading edge detachment, the presence of several lines of higher intensity regularly spaced in span. Figure 10: BA foil, 7 degrees and 6 m/s: axial vorticity. Figure 11: BA foil, 7 degrees and 6 m/s: transverse vorticity the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 307 Figure 12: NACA 66, 4 degrees and 11 m/s: axial vorticity. Figure 13: NACA 66, 4 degrees and 11 m/s: transverse vorticity. Concerning NACA66 foil, at 4 degrees, we observe, on the numerical photographs (figure 13), a little separation bubble near the leading edge then a reattachment of the boundary layer and spots of vorticity along the chord. Numerical pressure signal comparison: Let us go through the time dependent description of the flow for different given control points. We present from figure 14 to figure 15, the calculated pressure versus time for BA foil at 2, 4 and 7 degrees. The local fluctuating pressure is non-dimensionalised by flow dynamic pressure (i.e. 0.5 ρVref2). For BA foil at 2 degrees near the leading edge (figure 14), we notice that the three chosen locations give the same time dependant signal. The maximum fluctuation is ±1% of the dynamic pressure. Near the trailing edge, (figure 15), the fluctuation increase quickly between 93% and 97% of the chord. This correspond to the boundary layer separation near the trailing edge. In such a detachment, we had to notice that the pressure fluctuation can reach 30% of the reference dynamic pressure. Figure 14: calculated pressure signal for three control points. For BA foil at 4 degrees, we have chosen to illustrate the capability of the model to describe the evolution of the boundary layer. In figure 16, three control points are shown (i.e. 15%, 25% and 35% of the chord). At x/c=0.15, the time the authoritative version for attribution. dependant signal is similar to the one calculated for 2 degrees: low amplitude of fluctuation. At x/c=0.25 we can observe the existence of a well identified and regular frequency (near 100 Hz) in the time dependant pressure signal. It is a characteristic of Tollmien-Schlichting waves which will lead to turbulence. The pressure fluctuations are less than 5% of the reference value. Then at x/c=0.35, the flow is turbulent and the fluctuating pressure at that location is less than 1%.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 308 Figure 15: calculated pressure signal for three control Figure 16: calculated pressure signal for three control points. points. For BA foil at 7 degrees, the velocity map and flow vorticity both indicate the presence of a separation bubble near the leading edge. So, figure 17 shows the fluctuating pressure at three control points (x/c=0.03; x/c=0.07 and x/c=0.1). It appears that the fluctuation amplitude is less than 5% for x/c=0.03 and x/c=0.1. On the other hand, at the control point located near x/c=0.07, the recorded pressure fluctuation can achieve the reference pressure. This information corroborates the fact that this control point is situated in the closure area of the detachment bubble. Once again, for this angle of attack, the pressure fluctuations near the trailing edge (figure 18) are high and for a larger fraction of the chord than for 2 degrees. The detachment point is located further for 7 degrees than for 2 degrees. Figure 17: calculated pressure signal for three control Figure 18: calculated pressure signal for three control points. points. Numerical and experimental Spectrum The final step of our validation is the comparison between numerical and experimental frequency spectra of fluctuating pressure. For that purpose we use the data measured on BA foil at 4 degrees and 6 m/s. On Y-axis we use dB referenced to 1 µPa. The pressure transducer was a RESON transducer located at x/c=0.61. On figure 19, we present four numerical pressure spectra (i.e. x/c=0; x/c=0.21; x/c=0.61 and x/c=0.986) and one experimental spectrum (x/c=0.61). The following remarks must be written: - one numerical spectrum corresponds to an average of 12 spectra. Each spectrum corresponds to a time step of 5.10–4 s and a number of sample of 512. - the numerical spectrum is very accurate, compared to the experimental one, up to 150 Hz. For higher frequency, numerical model filters earlier than pressure transducer. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 309 - up to x/c=0.61, the pressure level is roughly 130 dB for low frequency. - we observe for x/c=0.21 the signature of the Tollmien-Schlichting waves near 100 Hz. - between x/c=0.21 and x/c=0.61 we note that energy switch from frequencies lower than 125 Hz to frequencies higher than 125 Hz. - near the trailing edge, the pressure fluctuation is very high and the spectrum can reach 30 dB more than at mid chord. Figure 19: pressure spectrum on Foil BA; 4 degrees; V=6 m/s. Comparison between calculations and experiments. Figure 20: pressure spectrum on foil BA. Comparison between calculation, experiments and empirical laws. The main purpose of this development is to be able to calculate the boundary layer excitation on a foil in order to calculate the foil vibro-acoustic response. On figure 20, we illustrate the usefulness of our model. So, up to now, propeller designer had empirical models based on boundary layer characteristics. These models were developed from flat plate experiments. In that way, the adverse pressure coefficient distribution can't be well represented. During tests in G.T.H, we measured velocity profiles in the boundary layer at the pressure transducer location. Hence, we had a measure of the boundary layer characteristics corresponding to the pressure measurements. Using empirical models and measured boundary layer characteristics, we obtain two of the curves presented on figure 20. It is clear that it is far from (more than 10 dB) both the reality (experimental curve) and LES spectrum. With LES calculation, we are able to know the exact low frequency level. Then, since the model, at the moment, filters to early, we can use this low frequency model to calibrate empirical models. EFFECT OF REYNOLDS NUMBER ON CAVITATION Another purpose of this project is to have a better understanding of the flow on a foil in order to predict more accurately the type and the development of cavitation patterns. Hence, with the photographs shown here after, it is clear that the boundary layer state and its characteristics are very important for the development of cavitation. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 310 Figure 21: effect of Reynolds number on cavitation. Flow with nuclei injection. For small angles of attack (i.e. BA foil at 2 degrees; figure 21) we note a strong effect of Reynolds number on the cavitation pattern. For velocity lower than 8 m/s and σ=0.85, we did not observe any cavitation. Then from 8 m/s to 13 m/ s, cavitation appears like little fingers attached to the foil surface. As the velocity increases, their number and length increase too and their attachment location moves near the leading edge. At this angle of attack, the LES calculation at 6 m/s indicates that the boundary layer is attached to the foil. Some lines of axial vorticity are also visible along the span. The evolution of cavitation with Reynolds number must be linked to the ratio between boundary layer and roughness height. For high angles of attack (i.e. BA foil at 7 degrees; figure 22) photographs show sheet cavitation related to separation bubble which was observed on numerical photographs and with numerical pressure fluctuation versus time. When the velocity increases at a fixed σ value, the length of the sheet seems to be constant. But we observe some little cavitating fingers attached to the leading edge upstream the cavity detachment line. Their number increases with Reynolds number. Figure 22: effect of Reynolds number on cavitation. Flow with nuclei injection. For NACA 66 foil, as illustrated on figure 23, the sheet cavitation appears also near the leading edge in correlation with calculated separation bubble (figure 12). Figure 23: sheet cavitation developed on NACA 66 foil. Top view of the foil. The cavitating unsteady flow is illustrated on figure 24 a) and b). These correspond to the same flow condition at two different times. We can observe cavitating structures carried along the chord by the flow. This type of cavitating flow was already observed on small foils by Kawanami (Kawanami and al. 1998). Their spatial repartition seems to be the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 311 periodic. The time frequency between two consecutive structures is 50 Hz. Figure 24: sheet cavitation developed on NACA 66 foil. Top view of the foil. CONCLUSION This paper presents results of an evaluation of a new numerical tool, based on LES method and developed by Bassin d'essais des carènes. This evaluation was conducted by comparison with experimental results obtained at high Reynolds number in the G.T.H. for non-cavitating and cavitating flows. The accuracy of numerical calculation had been obtained after many mesh tests. The numerical tool provides the knowledge of nonstationary quantities for non cavitating flow. Moreover, the LES calculation gives pressure coefficient repartition more accurately than standard Navier-Stokes calculation. The boundary layer development on the foil is well represented including transition or Tollmien-Schlichting waves. Pressure fluctuations acting everywhere on a foil can be quantified which is important to minimise radiated noise at the design stage of a propeller or appendages. The accuracy of the numerical model applied to real flows (with adverse pressure gradient) is better than that of empirical models for the prediction of pressure fluctuation spectrum. Moreover, the knowledge of the boundary layer development help for cavitation patterns. This type of calculation is still time consuming, however, we are looking to include it into a more complete design process. REFERENCES Boris J.P., Grinstein F.F., Oran E.S., Kolbe R.L., “New insights into large eddy simulation”, Fluid Dynamic research, vol. 10, pp 199–228, 1992. Chasnov J.R., “Simulation of the Kolmogorov inertial subrange using an improved subgrid model”, Physic of Fluids A, vol. 3, pp 188–200, 1991. Domaradzki J.A., Saiki E.M., “Backscatter models for large eddy simulations”, Theoretical and Computational Fluid Dynamics, vol. 9, pp 73–83, 1997. Hartel C., Klieser L., Unger F., “Subgrid scale energy transfer in the near wall region of turbulent flows”, Physics of Fluids A, vol. 6, pp 3130–3143, 1994. Jordan S., “Large-Eddy simulation of the Vortical motion resulting from Flow over Bluff Bodies”, 21st ONR Symposium on naval Hydrodynamics, Trondheim, 1996. Kawamura T., Kuwahara K., “Computation of high Reynolds number flow around circular cylinder with surface roughness”, AIAA, paper 84–0340, 1984. Kawanami Y., Kato H., Yamaguchi H., “Three-dimensional characteristics of the cavities formed on a two-dimensional hydrofoil”, Third International Symposium on Cavitation, Grenoble, 1998. the authoritative version for attribution. Kraichnan R.H., “Eddy viscosity in two and three dimensions”, Journal of Atmospheric Science, vol. 33, pp 1521–1536, 1976. Leonard A., “Energy cascade in large eddy simulations of turbulent fluid flows”, Advanced in Geophysics A, vol. 18, pp 237–248, 1974. Love M.D., “Subgrid modelling studies with Burger's equation”, Journal of Fluid Mechanics, vol. 100, pp 87–110, 1980.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 312 Piomelli U., Yu Y., Adrian R.J., “Subgrid scale energy transfer and near wall turbulent structure”, Physics of Fluids A, vol; 8, pp 215–224, 1996. Pope S.J., “A more general effective viscosity hypothesis”, Journal of Fluid mechanics, vol. 70, pp 331–340, 1975. Robinson S.K., “Coherent motions in the turbulent boundary layer”, annual Review of Fluids Mechanics, vol. 23, pp 601–639, 1991. Sagaut P., “Simulations numériques d'écoulements décollés avec des modèles de sous-maille”, PhD thesis, université de Paris VI, Juin 1995. Smagorinsky J., “General circulation experiment with the primitive equations. I) the basic experiment”, Monthly Weather Review, vol. 91, pp 99–165, 1963. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line UNSTEADY FLOW QUANTITIES ON TWO-DIMENSIONAL FOILS: EXPERIMENTAL AND NUMERICAL RESULTS 313 DISCUSSION I.Celik West Virginia University, USA It is surprising on one hand, and encouraging on the other hand, to see that the authors attempted to perform LES using a commercial code, FLUENT. The time accuracy is first order. The use of unstructured and non-uniform grids will reduce the spatial accuracy to almost first order. Hence, the numerical viscosity will be large acting as a filtering mechanism. The only way to obtain reasonable results is to use very small time step and very small grid sites. How many grid points were located in the boundary layers? What was the time step used? Is the time step used related to the resolution limit of 100 Hz in the power spectra of pressure fluctuations? Another issue is the dependence of the calculations on Reynolds number without using a sub-grid scale model. Could the simulations predict dependence of separation point on the Reynolds number? As for validation of unsteady calculations the only results shown concern the power spectra of pressure fluctuations. But there anything beyond 100 Hz seems to be numerical noise. It is necessary to compare the unsteady pressure fluctuations directly to measurement to see if the amplitude and the frequency can be predicted. AUTHOR'S REPLY The cell size in the boundary layer is typically 1.1 mm following the chord and 1.1 mm following the span. The first grid point is located at y+=1 and we mesh in altitude up to y+=100-200 using a geometrical ratio of 1.5 with hexahedrons. 200000 grid points are located in this zone (16% of the total amount of the cells number in the domain.). The time step used for the simulation is 0.0005 seconds and it is not a priori related to the resolution limits of 100hz. No simulation has been performed to evaluate the dependence of separation point on the Reynolds number. Perhaps we could pay some attention on this subject in the future. Concerning the numerical noise, it is very dangerous to perform comparison between numerical and experimental results. The numerical noise is the result of a diffusion of the rounding error and as a result it is not at all representative of physical noise. The objective is to make it as low as possible so that it can be assumed to be negligible. the authoritative version for attribution.