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ABSTRACT
Current reduced-order models (ROM) by the proper orthogonal decomposition (POD) for fluid flow are on a fixed domain. We expect to combine the POD model reduction and the body-fitted coordinate methods so that the POD reduced-order model for fast calculations of fluid flow can be expanded to domains with different shapes. In this paper, the principle is studied for obtaining basis functions of body-fitted POD model for fluid flow. Three methods to obtain basis functions are discussed theoretically. It is pointed out that contravariant components must be utilized to construct the sample matrix for the fast calculation of steady laminar flow in different-shape domains. Taking this principle into account, a standard POD ROM for steady laminar flow problems is established. It is found that the model appears instable when the geometric parameters vary a lot. To overcome this problem, supplementary matrix is introduced to establish a calibrated model with much better accuracy and stability. The test cases show that the calibrated POD reduced-order model, with higher accuracy and stability than the standard one, can fulfill the fast calculations of steady laminar flow on different-shape domains.
Nomenclature
| ak | = | spectrum coefficient for the kth basis function |
| ae | = | exact spectral coefficient |
| bj, cj | = | coefficients of the jth sampling velocity |
| C | = | supplementary matrix |
| = | ||
| eaver | = | average error of reduced-order model |
| ek | = | residual of Eq. (15) after including the kth basis function |
| h | = | eccentricity between the inner and the outer circles (m) |
| p | = | Pressure (Pa) |
| r2 | = | radius of inner circle (m) |
| r1 | = | radius of outer circle (m) |
| SU, SV | = | source terms of momentum equation |
| u, v | = | Cartesian components of velocity |
| = | ||
| U, V | = | contravariant components of velocity |
| = | ||
| x, y | = | Cartesian coordinate |
| (xη)j, (yη)j | = | xη, yη in the jth sampling condition |
| α, β, γ, J | = | parameters related with grids |
| ε | = | residual of reduced-order model |
| ϕu, ϕv | = | Cartesian components of basis functions, corresponding to u, v |
| = | ||
| ω | = | rotational speed of inner wall (rad/s) |
| θ | = | angle of parallelogram cavity |
| ξ, η | = | body-fitted coordinate |
| ψu, ψv | = | contravariant components of basis functions, corresponding to U, V |
| = | ||
| ∇ | = | Hamilton operator, |
| ⋄ | = | author defined operator, |
| (,) | = | Hilbert inner product |
| Subscript | = | |
| ()ξ, ()η | = |
Nomenclature
| ak | = | spectrum coefficient for the kth basis function |
| ae | = | exact spectral coefficient |
| bj, cj | = | coefficients of the jth sampling velocity |
| C | = | supplementary matrix |
| = | ||
| eaver | = | average error of reduced-order model |
| ek | = | residual of Eq. (15) after including the kth basis function |
| h | = | eccentricity between the inner and the outer circles (m) |
| p | = | Pressure (Pa) |
| r2 | = | radius of inner circle (m) |
| r1 | = | radius of outer circle (m) |
| SU, SV | = | source terms of momentum equation |
| u, v | = | Cartesian components of velocity |
| = | ||
| U, V | = | contravariant components of velocity |
| = | ||
| x, y | = | Cartesian coordinate |
| (xη)j, (yη)j | = | xη, yη in the jth sampling condition |
| α, β, γ, J | = | parameters related with grids |
| ε | = | residual of reduced-order model |
| ϕu, ϕv | = | Cartesian components of basis functions, corresponding to u, v |
| = | ||
| ω | = | rotational speed of inner wall (rad/s) |
| θ | = | angle of parallelogram cavity |
| ξ, η | = | body-fitted coordinate |
| ψu, ψv | = | contravariant components of basis functions, corresponding to U, V |
| = | ||
| ∇ | = | Hamilton operator, |
| ⋄ | = | author defined operator, |
| (,) | = | Hilbert inner product |
| Subscript | = | |
| ()ξ, ()η | = |