An Exact Navier‐Stokes Solution for Three‐Dimensional, Spanwise‐Homogeneous Boundary Layers

John, Michael O.; Obrist, Dominik; Kleiser, Leonhard

doi:10.1002/pamm.201210227

“…With κ denoting its non-dimensional strength, wall suction is known to increase both the linear critical Reynolds number Re crit (κ) (Hall et al 1984) and the transition Reynolds number Re tr (κ) (Spalart 1988;Arnal et al 1997). This behaviour was linked to the linear stability theory of a broader class of homogeneous flat-plate boundary layers by John, Obrist & Kleiser (2012, 2014a. In particular, the SHBL with suction turns into the highly stable asymptotic effective in leading to transition, as demonstrated for Tollmien-Schlichting (TS) eigenmodes interacting with cross-flow disturbances (Bippes 1989;Meyer & Kleiser 1989;Wintergerste & Kleiser 1995;Bippes 1999;Wintergerste 2002), for travelling cross-flow vortices (Wassermann & Kloker 2003) or for two streaks interacting with each other (Brandt & de Lange 2008).…”

mentioning

confidence: 97%

Secondary instability and subcritical transition of the leading-edge boundary layer

John

¹

,

Obrist

²

,

Kleiser

³

2016

Self Cite

View full text Add to dashboard Cite

The leading-edge boundary layer (LEBL) in the front part of swept airplane wings is prone to three-dimensional subcritical instability, which may lead to bypass transition. The resulting increase of airplane drag and fuel consumption implies a negative environmental impact. In the present paper, we present a temporal biglobal secondary stability analysis (SSA) and direct numerical simulations (DNS) of this flow to investigate a subcritical transition mechanism. The LEBL is modelled by the swept Hiemenz boundary layer (SHBL), with and without wall suction. We introduce a pair of steady, counter-rotating, streamwise vortices next to the attachment line as a generic primary disturbance. This generates a high-speed streak, which evolves slowly in the streamwise direction. The SSA predicts that this flow is unstable to secondary, time-dependent perturbations. We report the upper branch of the secondary neutral curve and describe numerous eigenmodes located inside the shear layers surrounding the primary high-speed streak and the vortices. We find secondary flow instability at Reynolds numbers as low as Re ≈ 175, i.e. far below the linear critical Reynolds number Re crit ≈ 583 of the SHBL. This secondary modal instability is confirmed by our three-dimensional DNS. Furthermore, these simulations show that the modes may grow until nonlinear processes lead to breakdown to turbulent flow for Reynolds numbers above Re tr ≈ 250. The three-dimensional mode shapes, growth rates, and the frequency dependence of the secondary eigenmodes found by SSA and the DNS results are in close agreement with each other. The transition Reynolds number Re tr ≈ 250 at zero suction and its increase with wall suction closely coincide with experimental and numerical results from the literature. We conclude that the secondary instability and the transition scenario presented in this paper may serve as a possible explanation for the well-known subcritical transition observed in the leading-edge boundary layer.

show abstract

“…With κ denoting its non-dimensional strength, wall suction is known to increase both the linear critical Reynolds number Re crit (κ) (Hall et al 1984) and the transition Reynolds number Re tr (κ) (Spalart 1988;Arnal et al 1997). This behaviour was linked to the linear stability theory of a broader class of homogeneous flat-plate boundary layers by John, Obrist & Kleiser (2012, 2014a. In particular, the SHBL with suction turns into the highly stable asymptotic suction boundary layer (ASBL) in the limit of vanishing chordwise flow, e.g.…”

Section: Introductionmentioning

confidence: 99%

Secondary Instability and Subcritical Transition in the Leading-Edge Boundary Layer

John¹,

Obrist²,

Kleiser³

2017

Direct and Large-Eddy Simulation X

View full text Add to dashboard Cite

The leading-edge boundary layer (LEBL) in the front part of swept airplane wings is prone to three-dimensional subcritical instability, which may lead to bypass transition. The resulting increase of airplane drag and fuel consumption implies a negative environmental impact. In the present paper, we present a temporal biglobal secondary stability analysis (SSA) and direct numerical simulations (DNS) of this flow to investigate a subcritical transition mechanism. The LEBL is modelled by the swept Hiemenz boundary layer (SHBL), with and without wall suction. We introduce a pair of steady, counter-rotating, streamwise vortices next to the attachment line as a generic primary disturbance. This generates a high-speed streak, which evolves slowly in the streamwise direction. The SSA predicts that this flow is unstable to secondary, time-dependent perturbations. We report the upper branch of the secondary neutral curve and describe numerous eigenmodes located inside the shear layers surrounding the primary high-speed streak and the vortices. We find secondary flow instability at Reynolds numbers as low as Re ≈ 175, i.e. far below the linear critical Reynolds number Re crit ≈ 583 of the SHBL. This secondary modal instability is confirmed by our three-dimensional DNS. Furthermore, these simulations show that the modes may grow until nonlinear processes lead to breakdown to turbulent flow for Reynolds numbers above Re tr ≈ 250. The three-dimensional mode shapes, growth rates, and the frequency dependence of the secondary eigenmodes found by SSA and the DNS results are in close agreement with each other. The transition Reynolds number Re tr ≈ 250 at zero suction and its increase with wall suction closely coincide with experimental and numerical results from the literature. We conclude that the secondary instability and the transition scenario presented in this paper may serve as a possible explanation for the well-known subcritical transition observed in the leading-edge boundary layer.

show abstract

Stabilizing a Leading-edge Boundary Layer Subject to Wall Suction by Increasing the Reynolds Number

John

¹

,

Obrist

²

,

Kleiser

³

2015

Procedia IUTAM

0

View full text Add to dashboard Cite

Laminar flow in the boundary layer at the leading edge of swept airplane wings typically becomes transitional and turbulent shortly downstream of the attachment line. Flow control techniques to maintain the flow laminar, such as suction into the wall, therefore must focus on this instability, which otherwise leads to turbulent flow and thus contaminates the flow over the entire wing chord. The present paper presents new results on how the linear leading-edge boundary layer (LEBL) instability of swept-cylinders flow, which models swept-wing flow, may be avoided. The classical Reynolds number definition is employed, which is based on the far-field velocity Q ∞ , the cylinder radius R * , and the sweep angle Λ. It is demonstrated that the flow can be stabilized by increasing the Reynolds number at constant wall suction through an increase of R * or Λ, but not of Q ∞ . The stability analysis is carried out for the swept Hiemenz boundary layer (SHBL), a widely used flat-plate approximation of the swept-cylinder LEBL. As demonstrated recently ? , the SHBL with suction becomes similar to the two-dimensional asymptotic suction boundary layer (ASBL) when increasing the classical Reynolds number Re SH to large values. In the limit of Re SH → ∞, the SHBL with suction becomes identical to the highly stable ASBL, and hence inherits its linear stability properties. The transformation of these recent findings concerning the linear stability of the SHBL with suction to the swept-cylinder LEBL unveils that stabilization of flow with constant suction can be observed by increasing Re S H .

show abstract

An Exact Navier‐Stokes Solution for Three‐Dimensional, Spanwise‐Homogeneous Boundary Layers

Cited by 5 publications

References 5 publications

Secondary instability and subcritical transition of the leading-edge boundary layer

Secondary instability and subcritical transition of the leading-edge boundary layer

Secondary Instability and Subcritical Transition in the Leading-Edge Boundary Layer

Stabilizing a Leading-edge Boundary Layer Subject to Wall Suction by Increasing the Reynolds Number

Contact Info

Product

Resources

About