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Reduced-Order Modeling of Steady Flows Subject to Aerodynamic Constraints

Ralf Zimmermann, Alexander Vendl, Stefan Görtz

2014
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AIAA Journal
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A novel reduced-order modeling method based on proper orthogonal decomposition for predicting steady, turbulent flows subject to aerodynamic constraints is introduced. Model-order reduction is achieved by replacing the governing equations of computational fluid dynamics with a nonlinear weighted least-squares optimization problem, which aims at finding the flow solution restricted to the low-order proper orthogonal decomposition subspace that features the smallest possible computational fluid
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... namics residual. As a second and new ingredient, aerodynamic constraints are added to the nonlinear least-squares problem. It is demonstrated that the constrained nonlinear leastsquares problem can be solved almost as efficiently as its unconstrained counterpart and outperforms all alternative approaches known to the authors. The method is applied to data fusion, seeking to combine the use of computational fluid dynamics with wind-tunnel or flight testing to improve the prediction of aerodynamic loads. It is also demonstrated that it can be used to compute aerodynamic loads for a given aerodynamic configuration subject to aerodynamic design or performance targets. Exemplary results considering both applications are computed for the NACA 64A010 airfoil and the DLR-F15 high-lift configuration. Nomenclature A = mean-flow state vector of given m flow solution snapshots; element of R n t a = vector of proper orthogonal decomposition coefficients; element of Rm C L , C D , C Mx , C My , C Mz , C p = lift, drag, moment, and pressure coefficients d = number of model parameters; element of N d k = search direction vector in Gauss-Newton method; element of Rm E = total energy, kg · m 2 s −2 I = identity matrix; element of R n×n m = number of flow solution snapshots; element of Ñ m = dimension of proper orthogonal decomposition subspace; element of N M ∞ = freestream Mach number; element of R n c = number of constraints imposed on reduced-order modeling problem; element of N n g = number of grid cells; element of N n t = total dimension of discretized flow problem; n g n v ; element of N n v = number of flow variables; element of N p = vector of model parameters; element of R d RICm = relative information content of firstm proper orthogonal decomposition eigenmodes res = discretized flux residual; R n t → R n t r j = relative information content of jth proper orthogonal decomposition mode; element of R s k = step length in Gauss-Newton method; element of R u x , u y , u z = flow velocity vectors, ms −1 U j = jth proper orthogonal decomposition mode; element of R n t V j = jth eigenvector of Y T ΩY; element of R m W = flow solution state vector; element of R n t x, y, z = Cartesian spatial coordinates; element of R Y = snapshot matrix; element of R n t ×m α = angle of attack, deg λ j = jth eigenvalue of Y T ΩY; element of R μ t = eddy viscosity, kg·ms −1 ν = Spallart-Allmaras viscosity, kg·ms −1 ρ = density, kg·ms −3 Ω = diagonal matrix of grid cell volumes; element of R n t ×n t h·; ·i L 2 = L 2 scalar product k · k L 2 = L 2 norm; h·; ·i L 2 p

doi:10.2514/1.j052208
fatcat:uyj36dtsajbrdc6zvsvmwgb5oa