Reduced modelling through identification on 2D incompressible laminar flows

Figures & data

Figure 1. The identification principle.

Figure 2. The computation grid with N = 15876 nodes and the boundary conditions.

Table 1. Maximal and mean velocity value of the identification data in the test case 1.

Velocity (ms^−1)	Maximal	Mean
u field	2.50 × 10^0	2.00 × 10^−1
v field	1.74 × 10^0	1.70 × 10^−1

Table 2. Influence of RM order.

Order n	J (m^2s^2)^−1	ΔMean (ms^−1)	ΔMax (ms^−1)
1	1.75 × 10^2	3.66 × 10^−2	2.05 × 10^−1
2	3.24 × 10^1	1.57 × 10^−2	1.18 × 10^−1
3	5.87 × 10^0	6.70 × 10^−3	4.71 × 10^−2
4	1.49 × 10^0	3.37 × 10^−3	2.13 × 10^−2
5	1.64 × 10^−1	1.12 × 10^−3	8.41 × 10^−3
6	1.68 × 10^−2	3.58 × 10^−4	2.83 × 10^−3
7	9.57 × 10^−4	8.54 × 10^−5	1.05 × 10^−3
8	9.78 × 10^−12	8.64 × 10^−9	3.33 × 10^−7
9	1.61 × 10^−11	1.11 × 10^−8	5.07 × 10^−7
10	1.62 × 10^−11	1.11 × 10^−8	6.01 × 10^−7

Figure 3. u velocity field of the DM (left) and the RM n = 8 (right) for Re = 1000. Available in colour online.

Figure 4. v velocity field of the DM (left) and the RM n = 8 (right) for Re = 1000. Available in colour online.

Figure 5. Stream function ψ fields of the DM (left) and the RM n = 8 (right) for Re = 1000. Available in colour online.

Figure 6. Vorticity ω fields of the DM (left) and the RM n = 8 (right) for Re = 1000. Available in colour online.

Table 3. Maximal and mean value of the DM velocity field at Re = 1000 and the corresponding errors of the RM n = 8.

	Maximal	Mean	ΔMax (ms^−1)	ΔMean (ms^−1)
u field	1.00 × 10^0	1.45 × 10^−1	1.29 × 10^−4	1.50 × 10^−5
v field	6.78 × 10^−1	1.21 × 10^−1	4.38 × 10^−4	1.06 × 10^−4

Figure 7. Comparison of DM and RM: velocity profiles of u component at x = 0.5d. Available in colour online.

Figure 8. Comparison of DM and RM velocity profiles of v component at y = 0.5d. Available in colour online.

Figure 9. Geometry and mesh of the pipe.

Figure 10. Centerline axial velocity (left) and axial outlet velocity profile (right) of the DM for Re = 100.

Table 4. Maximal and mean velocity value of the identification data in test case 2.

Velocity (ms^−1)	Maximal	Mean
u field	2.54 × 10^1	6.44 × 10^0
v field	4.9 × 10^−1	5.00 × 10^−3

Table 5. Influence of RM order.

Order n	J (m^2s^2)^−1	ΔMean (ms^−1)	ΔMax (ms^−1)
1	4.12 × 10^2	1.39 × 10^−1	8.11 × 10^−1
2	1.58 × 10^1	2.74 × 10^−2	3.04 × 10^−1
3	9.15 × 10^−1	6.59 × 10^−3	8.95 × 10^−2
4	5.80 × 10^2	1.66 × 10^−3	2.84 × 10^−2
5	1.90 × 10^−3	3.00 × 10^−4	4.00 × 10^−3
6	3.15 × 10^−9	3.87 × 10^−7	1.25 × 10^−5
7	1.18 × 10^−9	2.36 × 10^−7	1.14 × 10^−5
8	4.24 × 10^−11	4.48 × 10^−8	3.95 × 10^−6

Table 6. Maximal and mean value of the DM velocity field at Re = 400 and the corresponding errors of the RM n = 6.

	Maximal	Mean	ΔMax (ms^−1)	ΔMean (ms^−1)
u field	7.97 × 10^0	4.12 × 10^0	1.04 × 10^−2	1.69 × 10^−3
v field	2.31 × 10^−1	3.21 × 10^−3	4.90 × 10^−4	3.83 × 10^−5

Table 7. Maximal and mean value of the DM velocity field at Re = 800 and the corresponding errors of the RM n = 6.

	Maximal	Mean	ΔMax (ms^−1)	ΔMean (ms^−1)
u field	1.59 × 10^1	8.16 × 10^0	5.52 × 10^−3	9.63 × 10^−4
v field	3.67 × 10^−1	6.32 × 10^−3	3.37 × 10^−4	2.10 × 10^−5

Figure 11. Centerline axial velocity profiles of the DM and the RM of order n = 6 for Re = 400.

Figure 12. Outlet axial velocity profiles of the DM and the RM of order n = 6 for Re = 400.