An Explicit Threshold for the Appearance of Lift on the Deck of a Bridge

Gazzola, Filippo; Patriarca, Carlo

doi:10.1007/s00021-021-00643-6

Cited by 14 publications

(9 citation statements)

References 20 publications

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“…The main goal of this work regards the quantification of the smallness assumption on the inlet velocity that ensures the well-posedness of problem (1.3): an explicit upper bound on the size of the boundary datum h, in terms of the geometric constraints of the obstacle K and of the domain Ω, is given in Section 3, see Theorem 3.3 and Corollary 3.1, which then also yield an explicit upper bound for the solution of (1.3). Similar results were obtained in [20,22] regarding the unique solvability of (1.3) 1 under non-homogeneous Dirichlet boundary conditions on ∂Q. By following closely the proofs given by Fursikov & Rannacher in [15], one discovers that, in order to yield such explicit threshold, several tools of functional analysis must be carefully studied.…”

Section: Introductionsupporting

confidence: 69%

“…Therefore, in Section (2.1) we firstly provide lower bounds for the Sobolev constant of the embedding H 1 (Ω) ⊂ L p (Ω), p ∈ [2,6], involving functions that vanish only on Γ W or on Γ I ∪ Γ W . Thus, symmetrization techniques as in [20,21,22] can only be applied after properly reflecting Ω with respect to the planes x = ±L, and performing a suitable even extension of the functions considered, as in [32,Chapter 2]. Secondly, in Section 2.2 we build an estimable solenoidal extension of the boundary datum h, namely, we look for a vector field v 0 ∈ H 1 (Ω) such that…”

Section: Introductionmentioning

confidence: 99%

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On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

Sperone¹

2020

Preprint

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We analyze the steady motion of a viscous incompressible fluid in a three-dimensional channel containing an obstacle through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet, together with the standard no-slip assumption on the obstacle and on the remaining walls of the domain. Explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity through the Bogovskii formula. A quantitative analysis of the forces exterted by the fluid over the obstacle constitutes the main application of our results: by deriving a volume integral formula for the drag and lift, explicit upper bounds on these forces are given in terms of the geometrical constraints of the domain.

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Section: Introductionsupporting

confidence: 69%

Section: Introductionmentioning

confidence: 99%

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

Sperone¹

2020

Preprint

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“…Here •, • ∂K denotes the duality pairing between W − 2 3 , 3 2 (∂K) and W 2 3 ,3 (∂K), while the minus sign is due to the fact that the outward unit normal n to Ω is directed towards the interior of K. In the case of suspension bridges, the boundary conditions should model an horizontal inflow on the (2d or 3d) face x = −L of Q, as in conditions (3.2) and (3.4), see also [19]. Then, the most relevant component of the force (2.9), leading to structural instability, is the lift force L K (u, p) which is oriented vertically and, in our generalized context, can be computed as…”

Section: Main Results and Applications To The Navier-stokes Equationsmentioning

confidence: 99%

Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier-Stokes equations

Fragalà¹,

Gazzola²,

Sperone³

2020

Preprint

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We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow-outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier-Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow.

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“…To this end, it is easy to show from (10.4) that, in the case at hand, we have Consequently, the bifurcation due to these solutions is either subcritical or supercritical, a two-sided bifurcation being excluded. (18)…”

Section: Formulation Of Assumption (H4)mentioning

confidence: 99%

“…The main objective in [4] is rather basic and regards the existence of possible equilibrium configurations of the structure, at least for "small" data. Successively, several other papers have been dedicated to the investigation of further relevant properties of this model, such as well-posedness of the relevant initial-boundary value problem [22], large-time behavior and existence of a global attractor [19], and sharp threshold for uniqueness of the equilibrium configuration [18].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of Flow-Induced Oscillations of a Spring-Mounted Body in a Navier-Stokes Liquid

Galdi¹

2022

Preprint

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We study the motion of a rigid body B subject to an undamped elastic restoring force, in the stream of a viscous liquid L . The motion of the coupled system B-L ≡ S is driven by a uniform flow of L at spatial infinity, characterized by a given, constant dimensionless velocity λ e 1 , λ > 0. We show that as long as λ ∈ (0, λ c ), with λ c a distinct positive number, there is a uniquely determined time-independent state of S where B is in a (locally) stable equilibrium and the flow of L is steady. Moreover, in that range of λ, no oscillatory flow may occur. Successively we prove that if certain suitable spectral properties of the relevant linearized operator are met, there exists a λ 0 > λ c > 0 where an oscillatory regime for S sets in. More precisely, a bifurcating time-periodic branch stems out of the time-independent solution. The significant feature of this result is that no restriction is imposed on the frequency, ω, of the bifurcating solution, which may thus coincide with the natural structural frequency, ω n , of B, or any multiple of it. This implies that a dramatic structural failure cannot take place due to resonance effects. However, our analysis also shows that when ω becomes sufficiently close to ω n the amplitude of oscillations can become very large in the limit when the density of L becomes negligible compared to that of B.

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An Explicit Threshold for the Appearance of Lift on the Deck of a Bridge

Cited by 14 publications

References 20 publications

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier-Stokes equations

Mathematical Analysis of Flow-Induced Oscillations of a Spring-Mounted Body in a Navier-Stokes Liquid

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