Resolvent-based design and experimental testing of porous materials for passive turbulence control

An extended version of the resolvent formulation is used to evaluate the use of anisotropic porous materials as passive flow control devices for turbulent channel flow. The effect of these porous substrates is introduced into the governing equations via a generalized version of Darcy’s law. Model predictions show that materials with high streamwise permeability and low wall-normal permeability (${\varphi }_{\mathit{xy}}=k_{\mathit{xx}}/k_{\mathit{yy}}\gg 1$) can suppress resolvent modes resembling the energetic near-wall cycle. Based on these predictions, two anisotropic porous substrates with ${\varphi }_{\mathit{xy}}>1$ and ${\varphi }_{\mathit{xy}}<1$ were designed and fabricated for experiments in a benchtop water channel experiment. Particle Image Velocimetry (PIV) measurements were used to compute mean turbulence statistics and to educe coherent structure via snapshot Proper Orthogonal Decomposition (POD). Friction velocity estimates based on the Reynolds shear stress profiles do not show evidence of discernible friction reduction (or increase) over the streamwise-preferential substrate with ${\varphi }_{\mathit{xy}}>1$ relative to a smooth wall flow at identical bulk Reynolds number. A significant increase in friction is observed over the substrate with ${\varphi }_{\mathit{xy}}<1$. This increase in friction is linked to the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices. Coherent structures extracted via POD analysis show qualitative agreement with model predictions.     

Computing mean fields with known Reynolds stresses at steady state

With the rising of modern data science, data-driven turbulence modeling with the aid of machine learning algorithms is becoming a new promising field. Many approaches are able to achieve better Reynolds stress prediction, with much lower modeling error( M), than traditional Reynolds-averaged Navier-Stokes(RANS) models, but they still suffer from numerical error and stability issues when the mean velocity fields are estimated by solving RANS equations with the predicted Reynolds stresses. This fact illustrates that the error of solving the RANS equations( P) is also very important for a RANS simulation. In the present work, the error P is studied separately by using the Reynolds stresses obtained from direct numerical simulation(DNS)/highly resolved large-eddy simulation to minimize the modeling error M, and the sources of P are derived mathematically. For the implementations with known Reynolds stresses solely, we suggest to run an auxiliary RANS simulation to make a first guess on ν*t and S 0 i j. With around 10 iterations, the error of the streamwise velocity component could be reduced by about one-order of magnitude in flow over periodic hills. The present work is not to develop a new RANS model, but to clarify the facts that obtaining mean field with known Reynolds stresses is nontrivial and that the nonlinear part of the Reynolds stresses is very important in flow problems with separations. The proposed approach to reduce P may be very useful for the a posteriori applications of the data-driven turbulence models.