A Galerkin Method for the Simulation of Laminar Boundary Layers on Heated Walls
Abstract
:1. Introduction
2. Laminar Boundary-Layer Energy Equation
3. Solution Method
3.1. Galerkin Method with Respect to the y Variable
3.2. Finite Volume Method with Respect to the x Variable
- Solution initialization: in each cell, set: , where is the dynamic boundary layer thickness and , .
- End of the computation: Iterate until convergence to the steady state, i.e., until a given norm of the residual drops below a predefined threshold.
4. Results and Discussion
4.1. Test Case Definition
4.2. Convergence of the Method
4.2.1. Description of the Investigated Test Case
4.2.2. Grid Convergence
4.2.3. Convergence of the Galerkin Method
4.3. Influence of the Wall to Free-Stream Temperature Ratio
4.3.1. Influence on the Velocity Profile
4.3.2. Consequences on the Wall Heat Flux
4.4. Influence of the Wall Temperature Gradient
4.4.1. Uniform Temperature, m = 1
4.4.2. Uniform Temperature Gradient −400 K/m, m = 1
4.4.3. Uniform Temperature Gradient +400 K/m, m = 1
4.4.4. Uniform Temperature, m = −0.01961
4.4.5. Uniform Temperature Gradient −400 K/m, m = −0.01961
4.5. Discussion about the Computational Time
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Short Description of the Resolution of the Dynamic Boundary Layer in the Solver BLIM2D
Appendix B. Validation of the Model against an Empiric Relation
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Test-Case | L (m) | m | M | (Pa) | (K) | (K) | (K/m) |
---|---|---|---|---|---|---|---|
1 | 0.25 | 1 | 0.01 | 80,005 | 263.155 | 264.15 | 0 |
2 | 0.25 | −0.01961, 0, 1/3, 1 | 0.01 | 80,005 | 263.155 | 264.15, 273.15, 323.15, 373.15 | 0 |
3 | 0.25 | −0.01961, 0, 1/3, 1 | 0.25 | 83,555 | 266.439 | 323.15 | 0 |
4 | 0.25 | −0.01961, 0, 1/3, 1 | 0.25 | 83,555 | 266.439 | 373.15 | −400 |
5 | 0.25 | −0.01961, 0, 1/3, 1 | 0.25 | 83,555 | 266.439 | 273.15 | 400 |
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Radenac, E.; Harry, R.; Villedieu, P. A Galerkin Method for the Simulation of Laminar Boundary Layers on Heated Walls. Energies 2022, 15, 3267. https://doi.org/10.3390/en15093267
Radenac E, Harry R, Villedieu P. A Galerkin Method for the Simulation of Laminar Boundary Layers on Heated Walls. Energies. 2022; 15(9):3267. https://doi.org/10.3390/en15093267
Chicago/Turabian StyleRadenac, Emmanuel, Rémi Harry, and Philippe Villedieu. 2022. "A Galerkin Method for the Simulation of Laminar Boundary Layers on Heated Walls" Energies 15, no. 9: 3267. https://doi.org/10.3390/en15093267