Reformulación de la regla de productividad de Hayek

The mesh for the numerical analysis of the flow along the NACA 0012 profile is shown in figure 4.1. The chord length of the profile is 600 mm. The domain is divided in three subregions. The first region is meshed with an O-grid around the profile. This gives good control of the quality of the boundary layer cells along the surface. A rectangular box is placed around the first region to make a transition from the O-grid to an H-grid. In this second region also an O-grid type of mesh is applied.

The third domain extends the numerical domain to either the tunnel walls in the experiments or to a distance to impose the far-field boundary conditions. At the interface between the second and third domain an arbitrary coupling method is employed, which allows non-matching cells at both sides of the interface.

The O-grid region closest to the profile surface can not be recognised well in the mesh plot due to the large number of mesh lines. The other two regions can be distinguished more clearly. In this approach the mesh around the profiles is identical for all calculated conditions. Regions 1 and 2 can be rotated to obtain the desired angle of attack for the profile.

Mesh dependency studies have been carried out to evaluate the variation in lift and drag prediction for an angle of attack of 4 degrees. The number of cells around the profile and in the direction perpendicular to the profile have been varied. The number of cells in the O-grid around the profile has been varied from 150 to 330 cells. In the normal direction the number of cells is increased from 32 to 48. The number of cell in normal direction in the first region is kept constant to keep a constant y+ value of about 110. This is in accordance with the requirements for the use of the wall functions.

At the upper and lower boundary of the domain two types of boundary conditions can be applied: (i) wall boundary conditions or (ii) constant pressure boundary conditions. The first type can be used if the experimental data is obtained from wind tunnel tests. The second type is suitable for an unbounded region. For sufficiently large numerical domains both types will

Chapter 4. Mathematical treatment

give comparable results. At the inlet boundary a uniform velocity distribution of 10 m/s is prescribed and the constant density of water is used. The Reynolds number for the calculations becomes . Turbulence intensity is set to 0.01% and the length scale is set to a small fraction of the tunnel height. All calculations have been carried out with the standard k-ε turbulence model and employing wall functions. Solution is based on a second order MARS (= Monotone Advection and Reconstruction Scheme) discretisation scheme for the momentum equations. This second order method is least sensitive to the mesh structure and skewness [9]. The k-ε model turbulence equations are discretised with a first order upwind differencing scheme. Convergence behaviour of one of the calculated conditions is shown in figure 4.2. The convergence criterion for all calculations is set to 10-4 for the momentum, mass and turbulent kinetic energy equations.

6 10⋅ 6

Figure 4.1 Plot of a part of the mesh as used in calculations of isolated NACA 0012 profile.

4.4 Two-dimensional test cases

Figure 4.3 shows a comparison of the calculated and measured lift and drag for the isolated NACA 0012 profiles. The experimental data is taken from Abbott & von Doenhoff [13], where the results of the measurements with smooth profiles at a Reynolds number of are used. Compressibility has been negligible for the tested conditions.

The numerical values for lift and drag are based on the integrated pressure and shear forces acting on the profile surface. The dimensionless lift and drag coefficients are defined as:

(4.35)

(4.36)

where ρ is the density, v the free-stream velocity and A the surface area of the wing, i.e. here the chord length times the width in span wise direction. L is the lift force and D the drag force.

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 0 50 100 150 200 250 U-velocity residu V-velocity residu Mass residu Turbulent energy residu Epsilon residu

Figure 4.2 Plot of convergence of a calculation of NACA 0012 profile at angle of attack of 4 degrees for mesh with 222 cells around profile and 40 cells in normal direction.

Iteration number mass Res id ual u velocity v velocity dissipation turbulent energy 6 10⋅ 6 c_l L 1 2 ---ρv2A --- = c_d D 1 2 ---_ρv2A --- =

Chapter 4. Mathematical treatment

Agreement is good for the lift up to an angle of attack of about 6 degrees. The comparison of calculated and measured drag shows a clear over-prediction. The relative error increases from 33% at 0 degrees angle of attack to 55% at an angle of attack of 4 degrees. The deviation between the measurements and the calculations continuously increases with larger angles of attack. The results of the mesh sensitivity study are shown in table 4.3 for the lift prediction and in table 4.4 for the drag prediction at an angle of attack of 4 degrees. The relative difference in lift coefficient between the minimum and maximum lift is about 3%. For the drag coefficient a variation of about 9% is found. The results of the finer meshes do not show a reduction of the deviation with the experimental data. The error in prediction of drag might be related to an error in the production term of the turbulence model at the stagnation point, as described by Moore & Moore [14].

0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 0 0.01 0.02 0.03 0.04 0.05 cl - EXP cl - CFD cd - EXP cd - CFD

Figure 4.3 Comparison of measured and calculated lift an drag for NACA 0012 profile. Reynolds number of experiments and

calculations is 6.0x106. Mesh is based on 222 cells around profile and 32 in normal direction

Section angle of attack [degrees]

cd [ - ] cl [ - ]

4.4 Two-dimensional test cases

The sensitivity of the turbulence model is evaluated by a variation of the turbulence intensity of the free stream flow at the inlet boundary condition. Calculations are carried out with a mesh with 222 cells around the profile and 32 in normal direction. The level of the turbulence intensity at the inlet boundary condition is increased from 0.01% to 1.0%. The results for the prediction of lift and drag coefficient are presented in table 4.5. Variation in lift coefficient is 4%, whereas the change in drag is about 42%.

The test calculations with the NACA 0012 profile show that the error in prediction of profile drag remains after some mesh refinement steps. Moreover the deviation between experimental data and calculations increases significantly when the level of turbulence intensity at the inlet boundary increases. It should be noted that the low turbulence levels as used in the experiments are not representative for the inflow to the waterjet pump. Table 4.3 Lift coefficient for mesh convergence study. Columns

show number of cells in normal direction and rows show cells in O-grid around the profile

c_l [-] 32 cells 40 cells 48 cells

150 cells 0.4418 0.4386 0.4407

222 cells 0.4384 0.4375 0.4366

330 cells 0.4291 0.4289 0.4302

Table 4.4 Drag coefficient for mesh convergence study. Columns show number of cells in normal direction and rows show cells in O-grid around the profile

c_d [-] 32 cells 40 cells 48 cells

150 cells 0.010754 0.011324 0.011493

222 cells 0.010522 0.010988 0.011332

330 cells 0.010784 0.011243 0.011495

Table 4.5 Lift and drag coefficients for calculations with variation of input values for turbulence intensity

Turbulence intensity [%] c_l [-] c_d [-]

0.01 % 0.4384 0.010522

0.10 % 0.4287 0.012713

0.50 % 0.4242 0.014147

Chapter 4. Mathematical treatment

It is concluded that with the currently used cell sizes, which is comparable with the sizes to be used in the three dimensional pump mesh, a significant deviation between calculated and measured profile drag will remain.