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The logarithmic law of the wall does not capture the mean flow when a boundary layer is subjected to a strong pressure gradient. In such a boundary layer, the mean flow is affected by the spatio-temporal history of the imposed pressure gradient; and accounting for history effects remains a challenge. This work aims to develop a universal mean flow scaling for boundary layers subjected to arbitrary adverse or/and favourable pressure gradients. We derive from the Navier–Stokes equation a velocity transformation that accounts for the history effects and maps the mean flow to the canonical law of the wall. The transformation is tested against channel flows with a suddenly imposed adverse or favourable pressure gradient, boundary layer flows subjected to an adverse pressure gradient, and Couette–Poiseuille flows with a streamwise pressure gradient. It is found that the transformed velocity profiles follow closely the equilibrium law of the wall.
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This content will become publicly available on April 1, 2026
Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient
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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- Award ID(s):
- 2314303
- PAR ID:
- 10596985
- Publisher / Repository:
- Fluids
- Date Published:
- Journal Name:
- Fluids
- Volume:
- 10
- Issue:
- 4
- ISSN:
- 2311-5521
- Page Range / eLocation ID:
- 100
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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