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1. Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient

   https://doi.org/10.3390/fluids10040100  

   Ramirez, Miguel; Araya, Guillermo (April 2025, Fluids)

   A transient stability flow analysis is performed using the unsteady laminar boundary layer equations. The flow dynamics are studied via the Navier–Stokes equations. In the case of external spatially developing flow, the differential equations are reduced via Prandtl or boundary-layer assumptions, consisting of continuity and momentum conservation equations. Prescription of streamwise pressure gradients (decelerating and accelerating flows) is carried out by an impulsively started Falkner–Skan (FS) or wedge-flow similarity flow solution in the case of flat plate or a Blasius solution for particular zero-pressure gradient case. The obtained mean streamwise velocity and its derivatives from FS flows are then inserted into the well-known Orr–Sommerfeld equation of small disturbances at different dimensionless times (τ). Finally, the corresponding eigenvalues are dynamically computed for temporal stability analysis. A finite difference algorithm is effectively applied to solve the Orr–Sommerfeld equations. It is observed that flow acceleration or favorable pressure gradients (FPGs) lead to a significantly shorter transient period before reaching steady-state conditions, as the developed shear layer is notably thinner compared to cases with adverse pressure gradients (APGs). During the transient phase (i.e., for τ<1), the majority of the flow modifications are confined to the innermost 20–25% of the boundary layer, in proximity to the wall. In the context of temporal flow stability, the magnitude of the pressure gradient is pivotal in determining the streamwise extent of the Tollmien–Schlichting (TS) waves. In highly accelerated laminar flows, these waves experience considerable elongation. Conversely, under the influence of a strong adverse pressure gradient, the characteristic streamwise length of the smallest unstable wavelength, which is necessary for destabilization via TS waves, is significantly reduced. Furthermore, flows subjected to acceleration (β > 0) exhibit a higher propensity to transition towards a more stable state during the initial transient phase. For instance, the time response required to reach the steady-state critical Reynolds number was approximately 1τ for β = 0.18 (FPG) and τ = 6.8 for β = −0.18 (APG). 

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2. Surface Pressure Prediction for Bio-Inspired Unidirectional Canopies in Wall-jet Boundary Layers

   N. Hari, M. Szoke (August 2021, AIAA Aviation 2021)

   null (Ed.)

   An analytical approach has been developed to model the rapid term contribution to the unsteady surface pressure fluctuations in wall jet turbulent boundary layer flows. The formulation is based on solving Poisson’s equation for the turbulent wall pressure by integrating the source terms (Kraichnan, 1956). The inputs for the model are obtained from 2D time-resolved Particle Image Velocimetry measurements performed in a wall jet flow. The wall normal turbulence wavenumber two-point cross-spectra is determined using an extension of the von Kármán homogeneous turbulence spectrum. The model is applied to compare and understand the baseline flow in the wall jet and to study the attenuation in surface pressure fluctuations by unidirectional canopies (Gonzales et al, 2019). Different lengthscale formulations are tested and we observe that the wall jet flow boundary layer contributes to the surface pressure fluctuations from two distinct regions. The high frequency spectrum is captured well. However, the low frequency range of the spectrum is not entirely captured. This is because we have used PIV data only up to a height of 2.3𝜹, whereas the largest turbulent lengthscales in the wall jet are on the order of 𝒚𝟏/𝟐≈𝟔𝜹. Using the flow data obtained from PIV and Pitot probe measurements, the model predicts a reduction in the surface pressure due to canopy at low frequencies. 

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3. Assessment of Turbulence Models over a Curved Hill Flow with Passive Scalar Transport

   https://doi.org/10.3390/en15166013  

   Paeres, David; Lagares, Christian; Araya, Guillermo (August 2022, Energies)

   An incoming canonical spatially developing turbulent boundary layer (SDTBL) over a 2-D curved hill is numerically investigated via the Reynolds-averaged Navier–Stokes (RANS) equations plus two eddy-viscosity models: the K−ω SST (henceforth SST) and the Spalart–Allmaras (henceforth SA) turbulence models. A spatially evolving thermal boundary layer has also been included, assuming temperature as a passive scalar (Pr = 0.71) and a turbulent Prandtl number, Prt, of 0.90 for wall-normal turbulent heat flux modeling. The complex flow with a combined strong adverse/favorable streamline curvature-driven pressure gradient caused by concave/convex surface curvatures has been replicated from wind-tunnel experiments from the literature, and the measured velocity and pressure fields have been used for validation purposes (the thermal field was not experimentally measured). Furthermore, direct numerical simulation (DNS) databases from the literature were also employed for the incoming turbulent flow assessment. Concerning first-order statistics, the SA model demonstrated a better agreement with experiments where the turbulent boundary layer remained attached, for instance, in Cp, Cf, and Us predictions. Conversely, the SST model has shown a slightly better match with experiments over the flow separation zone (in terms of Cp and Cf) and in Us profiles just upstream of the bubble. The Reynolds analogy, based on the St/(Cf/2) ratio, holds in zero-pressure gradient (ZPG) zones; however, it is significantly deteriorated by the presence of streamline curvature-driven pressure gradient, particularly due to concave wall curvature or adverse-pressure gradient (APG). In terms of second-order statistics, the SST model has better captured the positively correlated characteristics of u′ and v′ or positive Reynolds shear stresses ( > 0) inside the recirculating zone. Very strong APG induced outer secondary peaks in and turbulence production as well as an evident negative slope on the constant shear layer. 

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4. Exact solutions for ground effect

   https://doi.org/10.1017/jfm.2020.149  

   Baddoo, Peter J.; Kurt, Melike; Ayton, Lorna J.; Moored, Keith W. (May 2020, Journal of Fluid Mechanics)

   ‘Ground effect’ refers to the enhanced performance enjoyed by fliers or swimmers operating close to the ground. We derive a number of exact solutions for this phenomenon, thereby elucidating the underlying physical mechanisms involved in ground effect. Unlike previous analytic studies, our solutions are not restricted to particular parameter regimes, such as ‘weak’ or ‘extreme’ ground effect, and do not even require thin aerofoil theory. Moreover, the solutions are valid for a hitherto intractable range of flow phenomena, including point vortices, uniform and straining flows, unsteady motions of the wing, and the Kutta condition. We model the ground effect as the potential flow past a wing inclined above a flat wall. The solution of the model requires two steps: firstly, a coordinate transformation between the physical domain and a concentric annulus; and secondly, the solution of the potential flow problem inside the annulus. We show that both steps can be solved by introducing a new special function which is straightforward to compute. Moreover, the ensuing solutions are simple to express and offer new insight into the mathematical structure of ground effect. In order to identify the missing physics in our potential flow model, we compare our solutions against new experimental data. The experiments show that boundary layer separation on the wing and wall occurs at small angles of attack, and we suggest ways in which our model could be extended to account for these effects. 

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5. On the Kármán momentum-integral approach and the Pohlhausen paradox

   https://doi.org/10.1063/5.0036786  

   Majdalani, Joseph; Xuan, Li-Jun (December 2020, Physics of Fluids)

   This work explores simple relations that follow from the momentum-integral equation absent a pressure gradient. The resulting expressions enable us to relate the boundary-layer characteristics of a velocity profile, u(y), to an assumed flow function and its wall derivative relative to the wall-normal coordinate, y. Consequently, disturbance, displacement, and momentum thicknesses, as well as skin friction and drag coefficients, which are typically evaluated and tabulated in classical monographs, can be readily determined for a given profile, F(ξ) = u/U. Here, ξ = y/δ denotes the boundary-layer coordinate. These expressions are then employed to provide a rational explanation for the 1921 Pohlhausen polynomial paradox, namely, the reason why a quartic representation of the velocity leads to less accurate predictions of the disturbance, displacement, and momentum thicknesses than using cubic or quadratic polynomials. Not only do we identify the factors underlying this behavior but also we proceed to outline a procedure to overcome its manifestation at any order. This enables us to derive optimal piecewise approximations that do not suffer from the particular limitations affecting Pohlhausen’s F = 2ξ − 2ξ3 + ξ4. For example, our alternative profile, F = (5ξ − 3ξ3 + ξ4)/3, leads to an order-of-magnitude improvement in precision when incorporated into the Kármán–Pohlhausen approach in both viscous and thermal analyses. Then, noting the significance of the Blasius constant, s¯≈1.630 398, this approach is extended to construct a set of uniformly valid solutions, including F=1−exp[−s¯ξ(1+12s¯ξ+ξ2)], which continues to hold beyond the boundary-layer edge as y → ∞. Given its substantially reduced error, the latter is shown, through comparisons to other models, to be practically equivalent to the Blasius solution. 

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