Prediction of Transonic Flows Based on Proper Orthogonal Decomposition Method Benoit Malouin, Jean-Yves Tr´epanier and Martin Gari´epy ´ Department of Mechanical Engineering, Ecole Polytechnique de Montr´eal, Montr´eal, QC, H3T 2B2, Canada Email:
[email protected];
[email protected];
[email protected]
A BSTRACT A Proper Orthogonal Decomposition (POD) method is used to predict the flow around an airfoil. POD uses existing numerical simulations, called snapshots, to create eigenfunctions. These eigenfunctions are combined using weighting coefficients to create a new solution describing the same problem for different values of input parameters. In the present paper, these parameters are the Mach number and the angle of attack. Since POD methods are linear, their interpolation capabilities are quite limited when dealing with flow presenting non-linearities, such as shocks. As a potential remedy to this problem, a mapping method is proposed. The idea is to use POD to interpolate the difference between a CFD solution obtained on two different grids, a coarse and a fine one. Then, for any new value of the parameters, the POD interpolated difference is added to predict the general behaviour of the flow. Results for various Mach number and angle of attack are compared to full CFD results. The mapping method shows good improvements as opposed to the basic method.
1
I NTRODUCTION
Despite the incredible improvement in computer resources, the need to generate accurate low-cost reduced-order models (ROM) is always present in the conceptual design process. ROM are simple models representing the overal properties of a system. They are generally much less complex than the real model. As an example, in aerodynamics, the Euler equations can be seen as a ROM compared to the Navier-Stokes system. Even if the ROM are less accurate than the full complex model they are derived from, they have the advantage to be less expensive in terms of computer resources.
The particular interest of ROM resides in the early stages of a conceptual design study, where a large range of concepts needs to be surveyed over the design space with lesser accuracy to found only few specimens that will be investigated deeply. These models provide the capability of such preliminary studies, in addition of being quick and economical. They are many ways to produce ROM. However, in this paper, ROM are produced using a proper orthogonal decomposition (POD), which is a method to generate a series of functions (eigenfunctions) that describe the behavior of the system. These eigenfunctions are derived from existing, full-order solutions (snapshots) and they are combined, using weighting average coefficients, to produce a new solution describing the same problem but for different values of input parameters. Of course, the main idea behind this type of decomposition is to use a minimum set of existing solutions (the snapshots) to generate the new solutions, owing the principal target of ROM which is to reduce the computational time needed to analyse a large set of models. In this paper, an original combination of POD and mapping method is proposed to predict the transonic flow around an airfoil based on a set of snapshots generated with a CFD commercial solver. The varying input parameters are the Mach number and the angle-of-attack. Results show that this new method shows better performance in transonic flow compared to standard POD. The next section presents the state-of-the-art in POD used in the aerodynamic field. Following that, the POD and mapping method will be explained in de-
tails. Finally, results, based on the prediction of the flowfield around an RAE2822 airfoil, will be presented with the regular POD method and with the proposed new method.
2
S TATE - OF - THE - ART
POD has been used extensively in the field of aerodynamic. In particular, POD has been used to study steady subsonic flows [2, 5, 10, 12], transonic and supersonic flows [3, 11, 15, 23], and finally, for unsteady flows [1, 4, 6–9, 13, 14, 16, 17, 19–21]. The latter field has been the subject of intense research throughout the years owing the large computer resources needed for a transient solution. All the former studies were about unsteady flows where the time was the varying parameter. LeGresley et al. [11] were the first to use design variables such as Mach number instead of time evolution as the varying parameter. Over the years, modifications have been proposed to improve the accuracy. Maybe the most important contribution to POD applied to unsteady flow is what we call the splitting of the snapshots proposed by Cizmas et al. [19]. This consist of splitting the snapshots in two parts relative to the level of unsteadiness. The same idea was applied to steady aerodynamic by [18]. The snapshots were divided in two parts relative to the presence of shock. This method improved the accuracy but only for subsonic solution. However, the problem is still unresolved for transonic aerodynamic. Another method, the domain decomposition, was proposed by Lucia et al. citelucia2. They proposed to divide the flow domain in multiple sections to isolate the portion of the domain where POD have difficulties to interpolate. In their case, this was the shock zone of the flow. A full-order simulation was performed in the latter while POD was used for the remaining cells. This gives very accurate results for the 2-D blunt body problem. LeGresley et al. [11] used the same technique for airfoil analysis. Their POD based ROM combined with domain decomposition was able to precisely predict the position of the shock by using only 5 snapshots to construct their eigenfunctions (Mach number varying from 0.3 to 0.7 with increments of 0.1). Though, this method is not used here because it is not compatible with commercial softwares and the aim of this work is to avoid the use of full-order simulations.
ear based. The consequence is that the position and strength of a shock is unpredictable using the POD basic method described in section 3.1. This is why we pay special attention to transonic flow. The most recent method was developped by Taeibi-Rahni et al. [22]. It consists of filtering the snapshots in terms of low and high-wave modes. This approach has shown good improvement in the prediction of the shock. Furthermore, this is the only method that do not require the use of full-order simulation which was specifically developped to treat transonic flows. The aim of this paper is to present another approach to improve the fidelity in POD without complex use of full-order models.
3
POD
In this section, we will first present the theory of POD, as described in [11]. Then, we will describe the proposed improvements by giving the theory of mapping and how it can be combined to POD.
3.1
Theory
The POD method is an algorithm that can be described as follows: 1. Obtain first M solutions, called snapshots, by varying the inlet Mach number and angle of attack using a given CFD solver. 2. Construct the correlation matrix R as follows: R = UTU
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where U is a matrix representing the collection of snapshots for a state variable (u, v, ρ, P, H, T ) of the flow simulation. This matrix has the size N ×M where N is the number of cells in the mesh. Note that R is symetric and must be obtained for each state variable. 3. Calculate eigenvalues λ and eigenvectors V of R by solving RV = λV (2) 4. Using λ and V , eigenvalues are sorted in descending order and M eigenfunctions can be constructed for a given state variable as: φ = UV
(3)
5. Eigenfunctions are normalized as follows: Instead of all progress made in that field, there is still a lack of development when dealing with transonic flow. The shock is a non-linear phenomena and POD are lin-
φNormalized =
φ ||φ||2
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Once these steps are completed, a new solution, K, can be computed and expressed in terms of the eigenfunctions as follows:
An interpolated solution of a given variable β can be computed as follows: Coarse βFine Interpolated = βCFD + ∆βPOD
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K(x, y) = ∑ ηi φiK (x, y)
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i=1
where η is a weighting coefficient chosen to construct the new solution, K is any state variable and φK are the associated eigenfunctions. In the present work, the method called Interpolated POD [3] was used. The idea is to first reconstruct each snapshot in function of the eigenfunctions: M
K n (x, y) − ∑ ηni φiK (x, y) = 0
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For each parameter, angle of attack and Mach number, two simulations are run. One on the fine grid and a second on the coarse grid. In this work, two coarse grids were tested containing 5876 and 13600 cells compared to 52650 cells for the fine one. The difference (∆β) is then computed and interpolated with POD. The whole procedure described in Section 3.1 is applied on ∆β instead of β. However, to compute the difference, all results must be on the same mesh. To do so, an interpolation must be performed. One could do an interpolation with cubic spline or linear, in our case, a nearest neighbor interpolation was used.
i=1
where n corresponds to the nth snapshot. The weighting coefficients, η, who respect Eq. 6, are computed as follows:
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ηni = φiK K n φiK
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Kn
where and are both vectors of equal length. Thus, the multiplication returns a scalar. φiK is the eigenvector i associated with the state variable K when K n is the nth snapshot of K. Once each solution K n (x, y) is reconstructed, all coefficients ηni are known. Therefore, the method interpolates the coefficients ηi to obtain new solutions. In our case, for each eigenfunction, the weighting coefficients ηi are functions of the flow parameters (Mach number and angle of attack). Weighting coefficients are interpolated with a cubic spline in the two dimensional parameter space.
3.2
R ESULTS AND VALIDATION
Description of the problem
The flow in transonic regime around an RAE2822 (Fig. 1) airfoil is studied. Two parameters varies to generate the 9 snapshots, which are the angle of attack and the Mach number. The snapshots, generated with Fluent 12.1.4, are summarized in Fig. 2. The square dots represent where a new solution is wanted while the circles represent the snapshots.
RAE 2822 ai rfoi l
Figure 1: RAE2822
Proposed Improvement
Since POD is not able to predict the position of the shock, it is possible to find a way to get this information to help POD interpolate the solution. One could use an inviscid solver, however, this idea was tested in [18] and results were inconclusive because of the variation in shock’s position in both solvers. Therefore, one should be able to precisely predict the position of the shock. The solution is to run a simulation on a coarser mesh and interpolate the results on the finer mesh: the mapping. This is the first time that such approach is proposed to deal with transonic flows and this is the core of this paper.
Figure 2: Lattice of snapshots
Three grids were used, containing 5876, 13600 and 52650 cells. For all computations, y+ was less than 5. The Spalart-Allmaras turbulence model was used with 4% of turbulence viscosity ratio at boundaries. The density-based solver combined with the Roe implicit scheme was selected. Second order upwind resolution associated with the Green-Gauss cell center algorithm was used. Boundary conditions were set to pressure-far-field. All simulations were ran on a Linux 64 bits, with 4 processors.
4.3
Results
4.3.1
Basic Method
As mentionned previously, POD has difficulties to interpolate when there is a shock. Analyse of Fig. 4 shows the reason:
Snapshots were taken only in transonic regime because shocks cause a lot of difficulties to the POD. Recall that the standard POD method gives good results in subsonic flow and do not need further attention [18]. 1 POD 0.8
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A way to validate the algorithm is to reconstruct the snapshots. The eigenfunctions are able to perfectly reconstruct any snapshots used to derive them. Analyse of Fig. 3 shows that both solutions, the one computed with POD and the one computed with CFD, overlap perfectly, leading to the conclusion that the computer program is working fine.
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Figure 4: Contours of pressure at Mach=0.7125 and A.O.A.=3.375
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Figure 3: Reconstruction at Mach=0.725 and A.O.A.=3.25
It is possible to notice that there are two shocks on the contours. Instead of interpolating the position of the shock, POD creates two smaller shocks. From a mathematical point of view, it makes sense because it is an average of the shocks of all snapshots. However, from a physic point of view, it is unrealistic. We can explain this behaviour by taking a look at the first eigenfunction in Fig. 5. The first eigenfunction is always the one associated with the higest weighting coefficient, thus, the one having the biggest impact on the interpolated solution. We notice that this eigenfunction presents two shocks. The same behaviour is expected on the interpolated solution as seen on Fig. 4.
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Figure 5: Contours of the first eigenfunction
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Figure 6: Contours of the first eigenfunction for mapping with 5875 cells
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Proposed Method
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Figure 7: Contours of the first eigenfunction for mapping with 13600 cells POD is not able to predict the position of the shock, to solve this problem, a simulation on a coarser mesh is run. The role of the latter is to get these informations. The difference between the variables on both grids are expected to be more predictable and, thus, are interpolated with POD. The coarse grid gives us the position of the shock while the POD predicts the rest.
It is possible to notice, on Fig. 6, that there is only one shock with some perturbations upwind and downwind. The results is more obvious when using more cells (see Fig. 7). Since the eigenfunctions represent the flow behaviour in a more physical way, the final results must be improved.
As an explanation to the improvements of our method, we may begin by plotting the contours of the first eigenfunction, which has the biggest impact on the solution.
The following tables show the impact of the method. Table 1 contains the error generated by the POD method alone while Table 2 and Table 3 represents the improvement when using the mapping method. Val-
1
ues in Tables 2-3 are calculated using Eq. 9. We notice the improvement. For example, the error on all cells associated with a solution at M0.7175 and 3.125◦ is decreased by 67% for the first grid and by 79% for the second grid. The improvement is considerable.
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The error is calculated by considering two different zones of the flow field. In the first place, the error is calculated over all cells of the mesh. The second zone considers the error only in the cells around the airfoil, the wall cells. All the aerodynamic coefficients are calculated with those last cells. This is why it is important to know if the error was reduced not only in the farfield region, but mainly in the wall region where the non-linear phenomenons are predominant.
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E-Global 2373 2623 1842 1588
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Table 1: Least-square error on pressure (in Pa) while using the basic method
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Table 2: Least-square error reduction on pressure with mapping and 5876 cells
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E-Global 79% 74% 53% 32%
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Table 3: Least-square error reduction on pressure with mapping and 13600 cells
The pressure contours reinforce the results presented in the previous tables.
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Figure 8: Contours of pressure at Mach=0.7375 and A.O.A.=3.125
We can notice on Fig. 8(b) the presence of two shocks, before and after the real position of the shock shown in Fig. 8(a). However, by using the mapping method, the position of the shock is more accurate. Fig. 8(c) shows that the contour lines are moving toward the real position. The result is more obvious on Fig. 8(d).
simulation, though, they give a fast approximation of the solution and could be really helpful in conceptual design process. Thus, further development and attention should be granted to this field.
The more fine is the coarse mesh, the more accurate the position of the shock will be. Though, finer is the coarse mesh, the more computational time a new solution will require. Fig. 9 shows the evolution of the error with the increase in computational time. It is possible to see that the optimal number of cells to use in the coarse mesh is around 30000 cells, half the number of cells contained in the fine mesh.
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Results for the three other test points are presented at the end of this paper and lead to the exact same conclusion: the mapping method helps POD interpolate in transonic flow and the finer the coarse mesh is, the better the results are.
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Error Time
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5
C ONCLUSION
A method of improving the accuracy of POD based ROM in transonic flow was developped. This approach has shown good results compared to the basic POD method. However, the interpolation was nearest type. It is believed that the error could be further decreased by using another method, such as cubic spline or krigging, to transpose the CFD results on the fine mesh. These approaches are not as accurate as a full order
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Figure 10: Contours of pressure at Mach=0.7125 and A.O.A.=3.125
Figure 11: Contours of pressure at Mach=0.7125 and A.O.A.=3.375
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Figure 12: Contours of pressure at Mach=0.7375 and A.O.A.=3.375
