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, co...
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2025-04-01
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| description | 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>). 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>), 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> > 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 1<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> = 0.18 (FPG) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> = 6.8 for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> = −0.18 (APG). |
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| spelling | doaj-art-b45a5f8f58b04837b3a2a26e2def8bd42025-08-20T02:17:14ZengMDPI AGFluids2311-55212025-04-0110410010.3390/fluids10040100Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure GradientMiguel Ramirez0Guillermo Araya1Department of Mechanical Engineering, University of Puerto Rico, Mayaguez, PR 00682, USAComputational Turbulence and Visualization Laboratory, Department of Mechanical, Aerospace, and Industrial Engineering, University of Texas at San Antonio, San Antonio, TX 78249, USAA 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>). 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>), 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> > 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 1<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> = 0.18 (FPG) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> = 6.8 for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> = −0.18 (APG).https://www.mdpi.com/2311-5521/10/4/100Falkner–Skanlaminar boundary layerOrr–Sommerfeldstreamwise pressure gradientstability analysistransitional flow |
| spellingShingle | Miguel Ramirez Guillermo Araya Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient Fluids Falkner–Skan laminar boundary layer Orr–Sommerfeld streamwise pressure gradient stability analysis transitional flow |
| title | Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient |
| title_full | Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient |
| title_fullStr | Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient |
| title_full_unstemmed | Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient |
| title_short | Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient |
| title_sort | stability analysis of unsteady laminar boundary layers subject to streamwise pressure gradient |
| topic | Falkner–Skan laminar boundary layer Orr–Sommerfeld streamwise pressure gradient stability analysis transitional flow |
| url | https://www.mdpi.com/2311-5521/10/4/100 |
| work_keys_str_mv | AT miguelramirez stabilityanalysisofunsteadylaminarboundarylayerssubjecttostreamwisepressuregradient AT guillermoaraya stabilityanalysisofunsteadylaminarboundarylayerssubjecttostreamwisepressuregradient |