Chapter 3: Validation
At the second stage of the hierarchy, the difficulty of the problem increases. The study of the supercritical aerofoil designed means both compressible effects (compression and expansion) will be considered on a more complex shape. Pressure distribution computed with Fluent can be easily compared with the data computed with VGK.
At the third level of the hierarchy, the ONERA M6 wing enables to study a three dimensional wing in transonic flow. It is a benchmark case on which enough experiments have been carried out to ensure their accuracy. Three dimensional effects are introduced such as crossflow and wing tip vortices.
Finally, the last level corresponds to the complete problem with the wing designed either clean or with one of the wing tips. Some interactions between two flows are introduced, especially when adding the winglet.
3.3. Richardson extrapolation
The extrapolated data are obtained using Richardson extrapolation. It enables to work out the solution of the discretized equations that would have been obtained with an infinite number of cells using data obtained with three grids having a constant refinement ratio. The first step of this extrapolation is the computation of the order of convergence:
= ( f refers to the solution computed, r refers to the grid refinement ratio taken to be constant, subscripts 1 to 3 refer to the different grids, 1 being the fine grid and 3 the coarse one)
Chapter 3: Validation
Using the value obtained for the order of convergence, the continuum value, f 0, can be computed using the relation below:
The grid convergence index can then be estimated. It provides an error band on how far is the asymptotic numerical value from the data computed. Thus, it can be seen as an indicator to know whether further refinement of the grid is required. It is defined as:
)
where Fs is a safety factor (Fs=1.25 for comparisons over three or more grids) and ε is the relative error
Finally checking the convergence of the computations is required in order to ensure the data used to carry out the extrapolation is within the asymptotic range of convergence.
This can be done by checking the relation:
1
Richardson extrapolation can either be applied to some solutions at a grid point or to a solution functional. The f value can be seen as an estimation of the value that could0 be computed in the limit where the grid spacing tends toward zero.
This sort of procedure is relatively easy to do in 2D but it becomes computationally expensive in 3D. Indeed, a refinement ratio r=2 would multiply the number of cells in the grid by 8.
Chapter 3: Validation
3.4. Compression and expansion corner
The compression and expansion corners are some basic test cases involving compressible flows. Theory enables to compute rapidly from an angle of deviation and the incident Mach number the outlet Mach number and other physical data relating the inlet to the outlet.
In the case studied, the deviation angle was ±15° and the boundary conditions used were:
• Pressure far field: M=2.5, T=288K for both inlet and outlet
• Symmetry for the upper boundary
• Wall for the corner
Three structured grids were used to carry out the computations, enabling to establish a grid convergence using Richardson extrapolation. The coupled implicit solver was used to complete computations, with an inviscid model. Data computed are summed up in the tables below (tables 3, 4).
1/
(nbr cells) P2/P1 T2/T1 ρ2/ρ1 M2 angleµµµµ1 angle µµµµ2222
1.3E-02 0.328 0.748 0.439 3.169 25.0 18,0 6.6E-03 0.328 0.729 0.450 3.234 24.0 17,8 3.3E-03 0.328 0.728 0.451 3.235 23.5 17,8
order p 1.54 4.01 7.19 5.53
extrapolated value 0.328 0.727 0.451 3.235
error 2.03% 0.62% 1.47% 0.45%
asymptotic range of
convergence 0.999 0.972 1.027 1.021
Theoretical value 0.3212 0.723 0.444 3.25 23,623,623,623,6 17.9 Table 3: Data computed for the expansion corner with an incident Mach number M1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of nodes on the surface of the corner, µµµµrefers to the
Chapter 3: Validation
1/
(nbr cells) P2/P1 T2/T1 ρ2/ρ1 M2 angle θθθθ 1.8E-04 2.467 1.322 1.866 1.872 37.8 4.4E-05 2.466 1.321 1.867 1.874 37 1.1E-05 2.466 1.321 1.867 1.874 37
order p 1.57 2.56 2.95 4.35
extrapolated value 2.466 1.321 1.867 1.874
error 0.07% 0.22% 0.01% 0.22%
asymptotic range of
convergence 1.000 1.000 1.001 1.001
Theoretical value 2.468 1.324 1.867 1.87 37373737
Table 4 : Data computed for the compression corner with an incident Mach number M 1=2.5 and an angle of 15°.'nbr of cells ' refers to the number of connectors on the surface of the corner, θθθθ refers to the angle of the wave with the wall surface, the subscript 1 refers to the flow upstream of the corner and subscript 2 refers to the downstream flow
The angles measured and displayed in these tables are just presented to show that reasonably good predictions are obtained on this data. However, the measuring device used to obtain the data was not accurate enough to be able to obtain angles with a precision below a quarter of degree which is not enough as regards of the theoretical data. The error displayed in these tables is computed with comparing the computed data with the theoretical one and has nothing to do with the grid convergence index.
We can see on these data that the software is reliable in the computation of compressible flows and very few cells are required to obtain good predictions of the flow characteristics. Nevertheless, a good grid resolution is required in order to obtain good flow visualisations with a high degree of accuracy in the regions of discontinuities (shock wave and sudden change in angle). This change in angle should not appear that suddenly in the case of the wing. Besides, we can see a lower accuracy is obtained with the computation of the expansion corner than with the compression corner.
Chapter 3: Validation