CFD Analysis and Reduction
4.4 Compressible Euler
4.4.3 Reduction
Table 4.8: NumericalCL andCD data, with residue data for the FreeFem++ solver. The experimental AGARD data givesCL= 0.522 andCD= 0.0101.
Case CL CD ResCL ResCD
Coarse 0.212 0.0571 1.46e-4 3.13e-6 Normal 0.339 0.0579 1.88e-4 3.92e-6 Fine 0.440 0.0481 4.70e-5 8.22e-6 Very Fine 0.480 0.0357 2.35e-5 4.39e-6
(a) FreeFem++
Case CL CD
Coarse 0.605 0.0168 Normal 0.677 0.0067 Fine 0.722 0.0018
(b) SU(2)
same machine, the Fine mesh took 13 hours to compute with FreeFem++ whilst SU(2) took 7 minutes.
The Very Fine mesh took 2 days to compute in the same machine. This means that this model could not produce high-fidelity results as quickly as other solvers, and we are therefore inclined to label this model as a low to medium-fidelity model. The FreeFem++ model uses the characteristics method for treating the advection terms of the Euler equation, as stated in [101]. Since we are dealing with subsonic speeds and thus with problems elliptic in space, this method should handle the advection terms with low diffusion and dispersion, according to the authors of FreeFem++ (see [100] for a discussion linking the type of problem to be solved with the methods available in FreeFem++ and a study of the accuracy of the characteristics-method on a 2-dimensional linear advection problem). It could be that with higher-order elements or the use of parallel computing could speed-up both the convergence and the calculation of the solution, but when attempting to use the parallel MUMPS or Pastix solvers as specified in the manual [100] the results exploded, and using higher-order elements would require searching for a new Courant number that guarantees the stability of the model. We quickly step into the creation of high- fidelity models, which is outside of the scope of this thesis. Hence, given that this thesis concerns itself with model reduction and the deviations of ROMs relative to the models they reduce, we will still use the FreeFem++ solver as a test bed our model reduction methods, as it does seem to incorporate the physics present in compressible flows (albeit with error), and it is a time-marching non-linear model. Doing this will allow us to understand what sort of deviations to expect when using ROMs on higher-fidelity solvers that are also non-linear and time-marching.
0 0.2 0.4 0.6 0.8 1
−1 0 1 2
x/c
Cp
Coarse Normal Fine Very Fine
AGARD
Figure 4.63: Experimental [80] and numerical (FreeFem++) Cp data for the RAE-2822. Lead- ing edge is at x/c = 0.
0 0.2 0.4 0.6 0.8 1
−2
−1 0 1
x/c
Cp
Coarse Normal Fine AGARD
Figure 4.64: Experimental [80] and numeri- cal (SU2) Cp data for the RAE-2822. Leading edge is at x/c = 0.
Figure 4.65: Cp obtained with the FreeFem++ script for the Very Fine mesh.
Figure 4.66: Cp obtained with SU(2) for the Normal mesh.
Figure 4.67: Mach obtained with the FreeFem++ script for the Very Fine mesh.
Figure 4.68: Mach obtained with SU(2) for Normal mesh.
Figure 4.69: Vorticity obtained with the FreeFem++ script for the Very Fine mesh.
the outer boundary is a circle two and a half chords long. Though this penalizes fidelity, it was necessary to be able to do the sweep in a reasonable amount of time. The supersonic model was chosen to see what happens when ”high-frequency” data, such as shocks, is present. In other words, how does the ROM behave when the pressure at a point can change abruptly, even for small variations of a given parameter. This phenomena appears in the supersonic model due to the rotation of the shock and expansion fronts relative to the airfoil as it changes angle of attack, but is also present in transonic flows as the critical point is breached. As for the RAE model, it was chosen so as to have a physical model to reduce, with an airfoil similar to those encountered in real-life and in similar flight conditions.
The model as implemented will be of low fidelity, but this is a problem of the model itself, and not the reduction methods we are going to test. The latter should give us results close to the implemented HDM, regardless of how well the former describes reality. For the supersonic model, as mentioned, the original mesh was implemented. For the angle of attack the inbound Mach 1.5 flow was kept horizontal and the entire mesh was rotated. Only the sweep test was performed : 4 snapshots were taken for AoA at 1, 2, 3 and 4 degrees, and the ROM was analyzed and compared to the HDM for a sweep of the angle of attack going from 0 to 5 degrees, with a step of 0.1 degrees. For the Rae-2822 the same flight conditions as the validation tests were kept, and as for the supersonic airfoil the AoA was imposed by rotating the mesh.
The meshes used are based on those of the validation tests: we used a Coarse and Normal mesh with the same element size at the airfoil as in 4.7, with the element size at the farfield boundary being 10 times bigger. A sweep test was done on the Coarse mesh, where 5 snapshots were taken at 1,2,3,4 and 5 degrees of AoA, and the ROM was analyzed for an interval ranging from 0 to 6 degrees of AoA. After, a scalability test was done, where ROMs were built for the Normal and Coarse meshes by computing the snapshots in the same points as before. We then computed the ROM response at AoA = 6 degrees and compared its performance with the HDM. For these tests we expect as before that the ROM exactly
”interpolates” the HDM at the snapshots: we should have very small deviations at the snapshots. We also expect the ROM be significantly faster than the HDM when a finer mesh is used. Besides this, we will pay attention to the following problems that may arise: ROM stability and error due to shocks. As presented in the bibliographical section, the created ROM may not be stable (which is why we chose to test MOR on a time marching method), and the POD may have difficulty in capturing abrupt variations, resulting in a wrong pressure field. For this reason, besides the RMS error for the velocity and pressure fields, we will monitor:
eL=|LROM−LHDM|
|LHDM| , eD= |DROM−DHDM|
|DHDM| (4.20)
rpCauchy= qR
Ω(pROMi −pROMi−1 )2 qR
Ω(pHDMi −pHDMi−1 )2
(4.21)
HereeL is the relative error in liftL,eD relative error in dragD,rCauchyp ratio between the last value of the Cauchy convergence criterion in pressure of the HDM and the ROM. This last measure is specific to the supersonic airfoil, to check if the convergence characteristics between the ROM and HDM are too different : if the ratio is far superior to one, the ROM may be diverging; if it is close or inferior to one, the
Table 4.9: Scalability test results for Rae-2822 for AoA = 6 degrees. The solution time for the ROM is given in seconds and as a percentage of the HDM solution time. In the last two columns the Cauchy Residue for the Lift and Drag coefficients computed by the ROM are given.
Mesh tHDM tSV D tROM N DOF eu ep ResCLROM ResCDROM Coarse 157.69 s 0.52 s 134.33 (85%) 7 000 0.34 % 0.064 % 6.31e-5 1.20e-5 Normal 2520.88 6.99 s 1626.7 (65 %) 25 808 0.71 % 0.54 % 2.36e-4 4.41e-5
ROM is converging at the same speed or even faster than the HDM. For the Rae-2822 case we supply the Cauchy residue of CL andCD for the scalability tests in table 4.9. The results for the sweep tests and the obtained aerodynamic coefficients can be seen in figure 4.70 through 4.73. As it can be seen we obtain errors below 1 % for most of the sweep for the RAE-2822 airfoil, with near-perfect fit of the aerodynamic coefficients. We do not seem to obtain the expect ”dip” in error at the snapshots, except for the relative velocity error in some points. Unlike for the the NSI case, for this time-dependent problem we were forced to add another level of discretization that is not accounted in the error estimates of the POD+Galerkin Projection: the time derivative must be discretized. The absence of perfect interpolation might be due to the error in time-discretization dominating over error of model reduction. Still, the error seems quite acceptable by industry standards, except for the lift for very low angles of attack where the near-zero value obtained naturally exacerbates the error. For the sweep of the supersonic airfoil, the
”dips” in the RMS relative errors are visible, indicating the interpolation properties of the ROM, but unlike what was observed for the NSI reduction, the error seems have a lower bound stopping them from going to machine 0. Again, we suspect this to be due to time-discretization error. With the supersonic airfoil we seem to have higher relative deviations than with the RAE-2822 when we are outside of the original interval at which the snapshots were taken. It might be therefore wise when constructing a basis to take snapshots at the extremes of the intervals we wish to simulate whenever abrupt variations in data (like shocks) may be present. Finally, for either sweep test, even with the presence of shocks, the ROM showed excellent stability : the Cauchy residues in pressure and aerodynamic coefficients tended to be lower for the ROM and for all models they decreased monotonically. This is exemplified in figure 4.74, where it is shown that throughout the sweep the Cauchy residue in pressure for the ROM is smaller than that of the HDM. Small residues for the RAE-2822 case are also shown in table 4.9. The scalability test results are shown in table 4.9, where we can see that for the RAE-2822 case, with104NDOF, the ROM is almost two times faster than the HDM, all the while maintaining relative deviations below 1%. This confirms the trend observed with NSI reduction, where with finer meshes the ROM proved to be more and more agile than the HDM, but in this case the deviations over the pressure are far more acceptable (even with shocks present, as observed for the sweeps). This might be due to the fact that there are no pressure singularities as observed in the lid-driven cavity problem.
Finally it seems that for non-linear CFD codes, the SVD of the snapshots is the least costly step in MOR, which was not the case in the benchmarks evaluated in the previous chapter.
0 1 2 3 4 5 6 10−4
10−3 10−2 10−1 100
AoA (deg)
adim
Velocity Pressure
Lift Drag
Figure 4.70: Relative error results for the sweep the RAE-2822 model.
0 1 2 3 4 5
10−3 10−2 10−1 100 101
AoA (deg)
adim
Velocity Pressure
Lift Drag
Figure 4.71: Relative error results for the sweep the original supersonic model in [101].
0 1 2 3 4 5 6
0 1 2 3 4
·10−2
AoA (deg)
Coefficients
CDHDM CLHDM CDROM CLROM
Figure 4.72: Aerodynamic coefficients for the sweep the RAE-2822 model.
0 1 2 3 4 5
0 0.2 0.4
AoA (deg)
Coefficients
CDROM CLROM CDHDM CLHDM
Figure 4.73: Aerodynamic coefficients for the sweep the original supersonic model in [101].
0 1 2 3 4 5
10−2 10−1
AoA (deg) rCauchy p
Figure 4.74: Ratio of the Cauchy residues in pressure for the supersonic airfoil in [101].
Figure 4.75: Pressure (Pa) distribution for the RAE-2822 HDM with the Normal mesh for an AoA of 6 degrees.
Figure 4.76: Pressure (Pa) distribution for the RAE-2822 ROM with the Normal mesh for an AoA of 6 degrees.