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1. Accurate near-wall measurements in wall bounded flows with optical flow velocimetry via an explicit no-slip boundary condition

   https://doi.org/10.1088/1361-6501/acf872  

   Jassal, Gauresh Raj; Schmidt, Bryan E. (September 2023, Measurement Science and Technology)

   Abstract High fidelity near-wall velocity measurements in wall bounded fluid flows continue to pose a challenge and the resulting limitations on available experimental data cloud our understanding of the near-wall velocity behavior in turbulent boundary layers. One of the challenges is the spatial averaging and limited spatial resolution inherent to cross-correlation-based particle image velocimetry (PIV) methods. To circumvent this difficulty, we implement an explicit no-slip boundary condition in a wavelet-based optical flow velocimetry (wOFV) method. It is found that the no-slip boundary condition on the velocity field can be implemented in wOFV by transforming the constraint to the wavelet domain through a series of algebraic linear transformations, which are formulated in terms of the known wavelet filter matrices, and then satisfying the resulting constraint on the wavelet coefficients using constrained optimization for the optical flow functional minimization. The developed method is then used to study the classical problem of a turbulent channel flow using synthetic data from a direct numerical simulation (DNS) and experimental particle image data from a zero pressure gradient, high Reynolds number turbulent boundary layer. The results obtained by successfully implementing the no-slip boundary condition are compared to velocity measurements from wOFV without the no-slip condition and to a commercial PIV code, using the velocity from the DNS as ground truth. It is found that wOFV with the no-slip condition successfully resolves the near-wall profile with enhanced accuracy compared to the other velocimetry methods, as well as other derived quantities such as wall shear and turbulent intensity, without sacrificing accuracy away from the wall, leading to state of the art measurements in the $y^{+}<1$region of the turbulent boundary layer when applied to experimental particle images. 

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2. Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

   https://doi.org/10.1142/S021820251950057X  

   Borthagaray, Juan Pablo; Nochetto, Ricardo H.; Salgado, Abner J. (December 2019, Mathematical Models and Methods in Applied Sciences)

   We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condition. The weight is a power of the distance to the boundary [Formula: see text] of [Formula: see text] that accounts for the singular boundary behavior of the solution for any [Formula: see text]. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension [Formula: see text], which is optimal for [Formula: see text]. 

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3. Rotating horizontal convection

   https://doi.org/10.1017/jfm.2013.136  

   Barkan, Roy; Winters, Kraig B.; Llewellyn Smith, Stefan G. (May 2013, Journal of Fluid Mechanics)

   null (Ed.)

   Abstract ‘Horizontal convection’ (HC) is the generic name for the flow resulting from a buoyancy variation imposed along a horizontal boundary of a fluid. We study the effects of rotation on three-dimensional HC numerically in two stages: first, when baroclinic instability is suppressed and, second, when it ensues and baroclinic eddies are formed. We concentrate on changes to the thickness of the near-surface boundary layer, the stratification at depth, the overturning circulation and the flow energetics during each of these stages. Our results show that, for moderate flux Rayleigh numbers ( $$O(1{0}^{11} )$$ ), rapid rotation greatly alters the steady-state solution of HC. When the flow is constrained to be uniform in the transverse direction, rapidly rotating solutions do not support a boundary layer, exhibit weaker overturning circulation and greater stratification at all depths. In this case, diffusion is the dominant mechanism for lateral buoyancy flux and the consequent buildup of available potential energy leads to baroclinically unstable solutions. When these rapidly rotating flows are perturbed, baroclinic instability develops and baroclinic eddies dominate both the lateral and vertical buoyancy fluxes. The resulting statistically steady solution supports a boundary layer, larger values of deep stratification and multiple overturning cells compared with non-rotating HC. A transformed Eulerian-mean approach shows that the residual circulation is dominated by the quasi-geostrophic eddy streamfunction and that the eddy buoyancy flux has a non-negligible interior diabatic component. The kinetic and available potential energies are greater than in the non-rotating case and the mixing efficiency drops from $${\sim }0. 7$$ to $${\sim }0. 17$$ . The eddies play an important role in the formation of the thermal boundary layer and, together with the negatively buoyant plume, help establish deep stratification. These baroclinically active solutions have characteristics of geostrophic turbulence. 

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4. Linear instability of Z-pinch in plasma: Inviscid case

   https://doi.org/10.1142/S0218202521500093  

   Bian, Dongfen; Guo, Yan; Tice, Ian (February 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   The [Formula: see text]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [Formula: see text] direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [Formula: see text]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system. 

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5. Flow in oscillatory boundary layers over permeable beds

   https://doi.org/10.1063/5.0104305  

   Meza-Valle, Claudio; Pujara, Nimish (September 2022, Physics of Fluids)

   In fluid dynamics applications that involve flow adjacent to a porous medium, there exists some ambiguity in how to model the interface. Despite different developments, there is no agreed upon boundary condition that should be applied at the interface. We present a new analytical solution for laminar boundary layers over permeable beds driven by oscillatory free stream motion where flow in the permeable region follows Darcy's law. We study the fluid boundary layer for two different boundary conditions at the interface between the fluid and a permeable bed that was first introduced in the context of steady flows: a mixed boundary condition proposed by Beavers and Joseph [“Boundary conditions at a naturally permeable bed,” J. Fluid Mech. 30, 197–207 (1967)] and the velocity continuity condition proposed by Le Bars and Worster [“Interfacial conditions between a pure fluid and a porous medium: Implications for binary alloy solidification,” J. Fluid Mech. 550, 149–173 (2006)]. Our analytical solution based on the velocity continuity condition agrees very well with numerical results using the mixed boundary condition, suggesting that the simpler velocity boundary condition is able to accurately capture the flow physics near the interface. Furthermore, we compare our solution against experimental data in an oscillatory boundary layer generated by water waves propagating over a permeable bed and find good agreement. Our results show the existence of a transition zone below the interface, where the boundary layer flow still dominates. The depth of this transition zone scales with the grain diameter of the porous medium and is proportional to an empirical parameter that we fit to the available data. 

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