Inverse analysis of the time-dependent heat flux in stagnation point flow of incompressible fluid impinging on a cylinder with uniform surface suction-blowing using Levenberg–Marquardt method

Figures & data

Figure 1. Problem geometry and sensor position.

Table 1. Effect of number of nodes on local Nusselt number.

	Nu
τ	N = 50	N = 100	N = 150	N = 200	N = 250
0.1	13.437	13.293	13.243	13.218	13.216
0.2	18.969	18.849	18.803	18.779	18.771
0.3	25.421	25.486	25.496	25.498	25.499
0.4	32.041	32.529	32.678	32.750	32.758
0.5	37.930	39.045	39.413	39.597	39.604
0.6	42.571	44.395	45.022	45.339	45.345
0.7	46.052	48.561	49.446	49.899	49.912
0.8	48.855	52.014	53.148	53.732	53.784
0.9	51.460	55.275	56.660	57.376	57.395
1	53.960	58.431	60.067	60.916	61.008

Figure 2. Three-dimensional picture of axisymmetric stagnation flow on a cylinder.

Table 2. Comparison of numerical and analytical results of f"(1).

	f"(1)
b	Analytical results	Numerical Results
1.1	650.352	650.475
2	11.001	11.012
10	0.6674	0.6687

Table 3. Analytical and numerical results of Nusselt number for Re = 0.1.

	b
	1.1		2
Pr	Analytical result	Numerical result	Analytical result	Numerical result
0.7	42.40	42.57	6.27	6.32
7	46.26	46.48	10.08	10.29
70	78.88	79.23	22.36	22.47
700	181.82	182.16	48.52	48.74

Table 4. Maximum and minimum values of the $|J_{ij}|$.

	$\left|J_{ij}\right|$
aqw(τ)/2k	Maximum value	Minimum value
Exponential	0.1718	3.2 × 10^−5
Sinus–cosines	0.1717	2.89 × 10^−5
Triangular	0.0884	7.50 × 10^−6
Trapezoidal	0.1595	5.06 × 10^−5

Table 5. Temperature history at point η₁ = 1.06 for exponential function of wall heat flux and S = −0.1.

τ	Exact temperature(°C)	Estimated temperature(°C)
0.1	82.3264	82.3223
0.2	86.4909	86.4918
0.3	91.4101	91.4119
0.4	97.1456	97.1467
0.5	103.7834	103.7826
0.6	111.4330	111.4307
0.7	120.2297	120.2276
0.8	130.3368	130.3368
0.9	141.9502	141.9526
1	155.3036	155.3052
1.1	170.6751	170.6712

Figure 3. Variations of sensitivity coefficient for estimation of the heat flux in the form of exponential function.

Figure 4. Variations of sensitivity coefficient for estimation of the heat flux in the form of sinus–cosines function.

Figure 5. Variations of sensitivity coefficient for estimation of the heat flux in the form of triangular function.

Figure 6. Variations of sensitivity coefficient for estimation of the heat flux in the form of trapezoidal function.

Figure 7. Calculated Heat flux with Re = 100 and S = −0.6 vs. the exact heat flux in the form of an exponential function.

Figure 8. Temperature history at pointη₁ = 1.06 with Re = 100 and S = −0.6 for calculated heat flux vs. exact heat flux in the form of an exponential function.

Figure 9. Calculated Heat flux with Re = 100 and S = −0.3 vs. the exact heat flux in the form of an exponential function.

Figure 10. Temperature history at pointη₁ = 1.06 with Re = 100 and S = −0.3 for calculated heat flux vs. exact heat flux in the form of an exponential function.

Figure 11. Calculated Heat flux with Re = 100 and S = 0.5 vs. the exact heat flux in the form of an exponential function.

Figure 12. Temperature history at pointη₁ = 1.06 with Re = 100 and S = 0.5 for calculated heat flux vs. exact heat flux in the form of an exponential function.

Figure 13. Calculated Heat flux with Re = 300 and S = −0.1 vs. the exact heat flux in the form of a sinus–cosines function.

Figure 14. Temperature history at pointη₁ = 1.0225 with Re = 300 and S = −0.1 for calculated heat flux vs. exact heat flux in the form of a sinus–cosines function.

Figure 15. Calculated Heat flux with Re = 300 and S = 0.1 vs. the exact heat flux in the form of a sinus–cosines function.

Figure 16. Temperature history at pointη₁ = 1.015 with Re = 300 and S = 0.1 for calculated heat flux vs. exact heat flux in the form of a sinus–cosines function.

Figure 17. Calculated Heat flux with Re = 300 and S = 0.3 vs. the exact heat flux in the form of a sinus–cosines function.

Figure 18. Temperature history at pointη₁ = 1.015 with Re = 300 and S = 0.3 for calculated heat flux vs. exact heat flux in the form of a sinus–cosines function.

Figure 19. Calculated heat flux with Re = 200 and S = −0.1 vs. the exact heat flux in the form of a triangular function.

Figure 20. Temperature history at pointη₁ = 1.005 with Re = 200 and S = −0.1 for calculated heat flux vs. exact heat flux in the form of a triangular function.

Figure 21. Calculated Heat flux with Re = 200 and S = 0.5 vs. the exact heat flux in the form of a triangular function.

Figure 22. Temperature history at pointη₁ = 1.005 with Re = 200 and S = 0.5 for calculated heat flux vs. exact heat flux in the form of a triangular function.

Figure 23. Calculated Heat flux with Re = 200 and S = 1 vs. the exact heat flux in the form of a triangular function.

Figure 24. Temperature history at pointη₁ = 1.005 with Re = 200 and S = 1 for calculated heat flux vs. exact heat flux in the form of a triangular function.

Figure 25. Calculated heat flux with Re = 150 and S = −0.1 vs. the exact heat flux in the form of a trapezoidal function.

Figure 26. Temperature history at pointη₁ = 1.03 with Re = 150 and S = −0.1 for calculated heat flux vs. exact heat flux in the form of a trapezoidal function.

Figure 27. Calculated Heat flux with Re = 150 and S = 0.5 vs. the exact heat flux in the form of a trapezoidal function.

Figure 28. Temperature history at pointη₁ = 1.005 with Re = 150 and S = 0.5 for calculated heat flux vs. exact heat flux in the form of a trapezoidal function.

Figure 29. Calculated Heat flux with Re = 150 and S = 1 vs. the exact heat flux in the form of a trapezoidal function.

Figure 30. Temperature history at pointη₁ = 1.005 with Re = 150 and S = 1 for calculated heat flux vs. exact heat flux in the form of a trapezoidal function.

Figure 31. Calculated heat flux with Re = 100 and S = −0.6 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of an exponential function.

Figure 32. Calculated heat flux with Re = 100 and S = −0.6 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of an exponential function.

Figure 33. Calculated heat flux with Re = 100 and S = −0.3 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of an exponential function.

Figure 34. Calculated heat flux with Re = 100 and S = −0.3 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of an exponential function.

Figure 35. Calculated heat flux with Re = 100 and S = 0.5 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of an exponential function.

Figure 36. Calculated heat flux with Re = 100 and S = 0.5 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of an exponential function.

Figure 37. Calculated heat flux with Re = 300 and S = −0.1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 38. Calculated heat flux with Re = 300 and S = −0.1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 39. Calculated heat flux with Re = 300 and S = 0.1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 40. Calculated heat flux with Re = 300 and S = 0.1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 41. Calculated heat flux with Re = 300 and S = 0.3 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 42. Calculated heat flux with Re = 300 and S = 0.3 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a sinus–cosines function.

Figure 43. Calculated heat flux with Re = 200 and S = −0.1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a triangular function.

Figure 44. Calculated heat flux with Re = 200 and S = −0.1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a triangular function.

Figure 45. Calculated heat flux with Re = 200 and S = 0.5 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a triangular function.

Figure 46. Calculated heat flux with Re = 200 and S = 0.5 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a triangular function.

Figure 47. Calculated heat flux with Re = 200 and S = 1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a triangular function.

Figure 48. Calculated heat flux with Re = 200 and S = 1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a triangular function.

Figure 49. Calculated heat flux with Re = 150 and S = −0.1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a trapezoidal function.

Figure 50. Calculated heat flux with Re = 150 and S = −0.1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a trapezoidal function.

Figure 51. Calculated heat flux with Re = 150 and S = 0.5 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a trapezoidal function.

Figure 52. Calculated heat flux with Re = 150 and S = 0.5 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a trapezoidal function.

Figure 53. Calculated heat flux with Re = 150 and S = 1 with noisy data (σ = 0.01T_max) vs. the exact heat flux in the form of a trapezoidal function.

Figure 54. Calculated heat flux with Re = 150 and S = 1 with noisy data (σ = 0.03T_max) vs. the exact heat flux in the form of a trapezoidal function.

Table 6. Temperature history at point η₁ = 1.0225 for sinus–cosines function of wall heat flux and S=−0.1.

τ	Exact temperature (°C)	Estimated temperature (°C)
0.1	46.5274	46.7134
0.2	48.5330	48.5686
0.3	50.5463	50.4659
0.4	52.4117	52.2968
0.5	54.0286	53.9595
0.6	55.3476	55.3673
0.7	56.3649	56.4595
0.8	57.1211	57.2164
0.9	57.7163	57.7290
1	58.3223	58.2908
1.1	59.1481	59.2397

Table 7. Temperature history at point η₁ = 1.005 for triangular function of wall heat flux and S = −0.1.

τ	Exact temperature (°C)	Estimated temperature (°C)
0.1	49.3113	49.2427
0.2	52.3592	52.3006
0.3	55.7201	55.7634
0.4	59.3232	59.4951
0.5	63.1318	63.3086
0.6	67.2927	67.9537
0.7	70.3718	70.1495
0.8	72.5738	72.6426
0.9	74.1141	74.2854
1	75.0867	75.1158
1.1	75.5584	75.4294

Table 8. Temperature history at point η₁ = 1.03 for trapezoidal function of wall heat flux and S = −0.1.

τ	Exact temperature (°C)	Estimated temperature (°C)
0.1	50.9891	50.5880
0.2	52.3944	52.6362
0.3	54.3783	54.9982
0.4	57.2096	57.5645
0.5	60.2797	60.1976
0.6	63.0932	62.7306
0.7	65.4690	64.9810
0.8	66.8693	66.7747
0.9	67.9637	67.4946
1	68.2741	68.4681
1.1	69.1343	68.7815

Table 9. RMS error for exponential function of wall heat flux.

	eRMS
Dimensionless transpiration	σ = 0	σ = 0.01Tmax	σ = 0.03Tmax
S = −0.6	0.0102	0.142	0.289
S = −0.3	0.0057	0.138	0.281
S = 0.5	0.0024	0.102	0.248

Table 10. RMS error for sinus–cosines function of wall heat flux.

	eRMS
Dimensionless transpiration	σ = 0	σ = 0.01Tmax	σ = 0.03Tmax
S = −0.1	0.209	0.393	0.452
S = 0.1	0.091	0.112	0.241
S = 0.3	0.0043	0.185	0.293

Table 11. RMS error for triangular function of wall heat flux.

	eRMS
Dimensionless transpiration	σ = 0	σ = 0.01Tmax	σ = 0.03Tmax
S = −0.1	0.231	0.287	0.403
S = 0.5	0.218	0.304	0.448
S = 1	0.205	0.310	0.458

Table 12. RMS error for trapezoidal function of wall heat flux.

	eRMS
Dimensionless transpiration	σ = 0	σ = 0.01Tmax	σ = 0.03Tmax
S = −0.1	0.401	0.451	0.562
S = 0.5	0.346	0.387	0.492
S = 1	0.338	0.376	0.483