Abstract
This study investigates a nonlinear inverse convection problem for a laminar-forced convective flow between two parallel plates. The upper plate is exposed to unknown heat flux while the lower plate is insulated. The unknown heat flux is determined using temperature measured on the lower plate. The thermophysical properties of the fluid are temperature dependent, which renders the problem nonlinear. The sequential gradient method is applied to this nonlinear inverse problem in order to solve the problem efficiently. The function specification method is incorporated to stabilize the sequential estimation. The corresponding adjoint formalism is provided. Accuracy and stability have been examined for the proposed method with test cases. The tendency of deterministic error is investigated for several parameters. Stable solutions are achieved even with severely impaired measurement data.
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Abbreviations
- C :
-
Volumetric heat capacity
- f :
-
Unknown heat flux
- \(\bar f\) :
-
Unknown heat flux with function specification
- H :
-
Channel height
- J :
-
Residual norm
- k :
-
Thermal conductivity
- L :
-
Channel length
- m :
-
Number of future time steps
- M :
-
Number of time steps
- N :
-
Number of nodal points along duct
- p :
-
Parameter for iterative improvement
- Pe:
-
Peclet number
- R :
-
The total number of repeated estimations using the iterated final condition
- S:
-
Conjugate direction
- t :
-
Time
- t f :
-
Final time
- t i :
-
Final time for each sequence
- t o :
-
Initial time for each sequence
- T :
-
Temperature
- T o :
-
Initial temperature
- T in :
-
Inlet temperature
- u :
-
Velocity, or step function
- U :
-
Mean velocity
- v :
-
Perturbed temperature
- x :
-
Axial distance along channel
- y :
-
Transverse coordinate
- Y :
-
Measured temperature
- β:
-
Step size
- γ:
-
Conjugate coefficient
- Δt :
-
Time step
- Δx :
-
Distance between nodal points alongx− direction
- σ:
-
Standard deviation
- τ:
-
Time variable for solution of adjoint problem
- i :
-
Index
- o :
-
Nominal value
- +:
-
Dimensionless variable
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Kim, S.K., Lee, W.I. & Lee, J.S. Solving a nonlinear inverse convection problem using the sequential gradient method. KSME International Journal 16, 710–719 (2002). https://doi.org/10.1007/BF03184821
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DOI: https://doi.org/10.1007/BF03184821