Nonlinear estimation in turbulent channel flows

Abstract We design a nonlinear estimator for channel flows at $$Re_{\tau }=180$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>180</mml:mn> </mml:mrow> </mml:math> and 590. The nonlinear estimator uses a linear estimator structure based on the linearised Navier–Stokes equations and explicitly calculates the nonlinear forcing from the estimated velocities in physical space. The goal is to use limited velocity measurements to predict the velocity field at other locations. We first use the velocities at one wall-normal height to estimate the velocities at other wall-normal heights. The estimation performance is compared among the nonlinear estimator, the linear estimator and the linear estimator augmented with eddy viscosity. At $$Re_{\tau }=180$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>180</mml:mn> </mml:mrow> </mml:math> , the nonlinear estimator and the linear estimator augmented with eddy viscosity outperform the linear estimator in terms of estimating the velocity magnitudes, structures and energy transfer (production, dissipation and turbulent transport) across the channel height. The limitations of using measurement data at one wall-normal height are discussed. At $$Re_{\tau }=590$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>590</mml:mn> </mml:mrow> </mml:math> , the nonlinear estimator does not work well with only one measurement plane, whereas the linear estimator augmented with eddy viscosity performs well. The performance of the nonlinear estimator and the linear estimator augmented with eddy viscosity at $$Re_{\tau }=590$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>590</mml:mn> </mml:mrow> </mml:math> is significantly enhanced by providing multiple measurement planes.