Piecewise Linear Dimension Reduction as a Regularization Strategy in Data Assimilation for RANS Simulations

Data Assimilation can reduce the model-form errors of RANS simulations. A spatially distributed corrective parameter field can be introduced to the model, whose optimal values can be efficiently found by an adjoint method and a gradient-based optimization. For assimilation of experimental data, which in most cases are sparsely distributed or are based on a low-resolution grid, the inverse problem will be severely ill-posed. A regularization strategy is needed to reduce the number of local minima associated with unphysical solutions. Common regularization methods such as Tikhonov and Sobolev gradient are capable of reproducing smooth and physically reasonable internal velocity fields, however, if measurements are located close to walls and an accurate wall shear stress profile is sought, they cannot prevent over-fitting at these locations and will result in unphysical jagged profiles. We propose a new regularization strategy based on a projection of the parameter field which prevents over-fitting by constraining the parameter field into piecewise linear subdomains. We show that the method provides accurate velocity and wall shear stress profiles. In addition, it introduces no hyper-parameters and also leads to faster minimization convergence.