Unsteady Fluid-structure Interaction Research Articles

Numerical methods for shape optimal design of fluid–structure interaction problems

We consider the method of mappings for performing shape optimization for unsteady fluid–structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on problems derived from an FSI benchmark to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Gust response of an elasto-flexible morphing wing using fluid–structure interaction simulations

Small and micro unmanned aircraft are the focus of scientific interest due to their wide range of applications. They often operate in a highly unstable flight environment where the application of new morphing wing technologies offers the opportunity to improve flight characteristics. The investigated concept comprises port and starboard adjustable wings, and an adaptive elasto-flexible membrane serves as the lifting surface. The focus is on the benefits of the deforming membrane during the impact of a one-minus-cosine type gust. At a low Reynolds number of Re = 264000, the morphing wing model is investigated numerically by unsteady fluid–structure interaction simulations. First, the numerical results are validated by experimental data from force and moment, flow field, and deformation measurements. Second, with the rigid wing as the baseline, the flexible case is investigated, focusing on the advantages of the elastic membrane. For all configurations studied, the maximum amplitude of the lift coefficient under gust load shows good agreement between the experimental and numerical results. During the decay of the gust, they differ more the higher the aspect ratio of the wing. When considering the flow field, the main differences are due to the separation behavior on the upper side of the wing. The flow reattaches earlier in the experiments than in the simulations, which explains the higher lift values observed in the former. Only at one intermediate configuration does the lift amplitude of the rigid configuration exceeds that of the flexible by about 12%, with the elastic membrane resulting in a smaller and more uniform peak load, which is also evident in the wing loading and hence in the root bending moment.

Numerical Investigation of Unsteady Characteristics of Gas Foil Journal Bearings with Fluid–Structure Interaction

Gas foil journal bearings (GFJBs) have been widely employed in high-speed rotating machinery in the aviation industry. However, the role of fluid–structure interaction in the unsteady aerodynamic character of the gas film and the dynamic response of the elastic foils have not yet been clarified. In this study, an unsteady shearing flow interacting with an exciting deformation of the top or bump foils was investigated by means of a large eddy simulation with bidirectional fluid–structure interaction (BFSI). The result shows that the main frequencies and amplitudes of stable fluctuations of different flow field parameters at different positions are different. The oscillating duration in the solid domain is much less than that in the fluid domain. The main positions for the interaction between the gas film pressure and the elastic foil are on both sides of θ = π. Compared with the case without FSI, the presence of the elastic foil flattens the distribution of the pressure of the gas film. As the rotational speed increases, the main frequency and the amplitude of pressure in the fluid domain continuously increase. With FSI, there is no interference frequency near the main frequency, which improves the stability of the shearing flow. However, an interference frequency appears near the main frequency of total displacement in the solid domain. The analysis in this paper lays the foundation for unsteady fluid–structure interaction research.

An interface and geometry preserving phase-field method for fully Eulerian fluid-structure interaction

We present an interface and geometry preserving (IGP) method for modeling fully Eulerian fluid-structure interaction via phase-field formulation. While the time-dependent mobility model preserves the hyperbolic tangent interface profile, the proposed method maintains the geometry of the solid-fluid interface by minimizing the volume-conserved mean curvature flow. To reduce the curvature flow, we construct a gradient-minimizing velocity field (GMV) for the convection of the order parameter. The constructed velocity field retains the solid velocity in the solid domain while extending the velocity in the normal direction throughout the diffuse interface region. With this treatment, the GMV reduces the normal velocity difference of the isosurfaces of the order parameter, alleviating the undesired thickening or thinning of the diffuse interface region due to the convection. As a result, the time-dependent mobility coefficient is substantially reduced and there is a lesser volume-conserved mean curvature flow. Furthermore, the GMV ensures that the diffuse interface region moves with the solid bulk despite the fluid flow, such that the fluid-solid interface conforms to the geometry of the solid. Using the unified momentum and mass conservation equations via the phase-dependent interpolation, we integrate the IGP method into a fully Eulerian fluid-structure interaction solver based on the incompressible viscous fluid and the neo-Hookean solid models. The kinematics of the solid in an Eulerian frame of reference is resolved by evolving the left Cauchy-Green tensor. We first demonstrate the ability of the phase-field-based IGP method for the convection of circular and square interfaces in a prescribed velocity field. The variational fluid-structure interaction framework with the IGP method is then examined on the deformation of a solid block under cavity flow. We next explore the geometry-preserving effect of the proposed framework when the fluid flow passes over a fixed deformable block in a channel. Finally, the vibration of a flexible plate attached behind a stationary cylinder is considered to demonstrate the ability of the framework in solving unsteady fluid-structure interaction problems.

Shape optimization of viscous flow domain considering unsteady fluid structure interaction

Shape optimization of viscous flow domain considering unsteady fluid structure interaction

Projection Based Semi-Implicit Partitioned Reduced Basis Method for Fluid-Structure Interaction Problems

In this manuscript a POD-Galerkin based Reduced Order Model for unsteady Fluid-Structure Interaction problems is presented. The model is based on a partitioned algorithm, with semi-implicit treatment of the coupling conditions. A Chorin–Temam projection scheme is applied to the incompressible Navier–Stokes problem, and a Robin coupling condition is used for the coupling between the fluid and the solid. The coupled problem is based on an Arbitrary Lagrangian Eulerian formulation, and the Proper Orthogonal Decomposition procedure is used for the generation of the reduced basis. We extend existing works on a segregated Reduced Order Model for Fluid-Structure Interaction to unsteady problems that couple an incompressible, Newtonian fluid with a linear elastic solid, in two spatial dimensions. We consider three test cases to assess the overall capabilities of the method: an unsteady, non-parametrized problem, a problem that presents a geometrical parametrization of the solid domain, and finally, a problem where a parametrization of the solid’s shear modulus is taken into account.

Data-driven nonlinear reduced-order modeling of unsteady fluid–structure interactions

A novel data-driven nonlinear reduced-order modeling framework is proposed for unsteady fluid–structure interactions (FSIs). In the proposed framework, a convolutional variational autoencoder model is developed to determine the coordinate transformation from a high-dimensional physical field into a reduced space. This enables the efficient extraction of nonlinear low-dimensional manifolds from the high-dimensional unsteady flow field of the FSIs. The sparse identification of a nonlinear dynamics (SINDy) algorithm is then used to identify the dynamical governing equations of the reduced space and the vibration responses. To investigate and validate the effectiveness of the proposed framework for modeling and predicting unsteady flow fields in FSI problems, the two-dimensional laminar vortex shedding of a fixed cylinder is considered. Furthermore, the proposed data-driven nonlinear reduced-order modeling framework is applied to the three-dimensional vortex-induced vibration of a flexible cylinder. Using the SINDy model to analyze the vibration responses, the dynamics of the flexible cylinder are found to be correlated with the flow wake patterns, revealing the underlying FSI mechanism. The present work is a significant step toward the establishment of machine learning-based nonlinear reduced-order models for complex flow phenomena, the discovery of underlying unsteady FSI physics, and real-time flow control.

Aeroelastic mode decomposition framework and mode selection mechanism in fluid–membrane interaction

In this study, we present a global Fourier mode decomposition framework for unsteady fluid–structure interaction. We apply the framework to isolate and extract the aeroelastic modes arising from a coupled three-dimensional fluid–membrane system. We investigate the frequency synchronization between the vortex shedding and the structural vibration via mode decomposition analysis. We explore the role of flexibility in the aeroelastic mode selection and perform a systematic comparison of flow features among a rigid flat wing, a rigid cambered wing and a flexible membrane. The camber effect can enlarge the pressure suction area on the membrane surface and suppress the turbulent intensity, compared to the rigid flat wing counterpart. With the aid of our mode decomposition technique, we find that the dominant structural mode exhibits a chordwise second and spanwise first mode at different angles of attack. The structural natural frequency corresponding to this mode is estimated using an approximate analytical formula. By examining the dominant frequency of the coupled system, we show that the dominant membrane vibration mode is selected via the frequency lock-in between the dominant vortex shedding frequency and the structural natural frequency. From the fluid modes and the mode energy spectra at α=20∘ and 25°, the aeroelastic modes corresponding to the non-integer frequency components lower than the dominant frequency are observed, which are associated with the bluff body vortex shedding instability. The non-periodic aeroelastic behaviors observed at higher angles of attack are related to the interaction between aeroelastic modes caused by the frequency lock-in and the bluff-body-like vortex shedding. Using the mode decomposition analysis, we suggest a feedback cycle for flexible membrane wings undergoing synchronized self-sustained vibration. This feedback cycle reveals that the dominant aeroelastic modes are selected through the mode and frequency synchronization during fluid–membrane interaction to exhibit similar modal shapes in the membrane vibration and the pressure pulsation.

Holding Force and Vertical Vibration of Emergency Gate in the Closing Process: Physical and Numerical Modelling

A two-dimensional unsteady fluid–structure interaction numerical model was established, based on the physical model test, to investigate the influence of vertical vibration on the holding force of an emergency gate in the closing process. Gate motion was controlled by the user-defined function in Fluent. Attention was paid to the relationship between the vertical vibration, hydrodynamic loads and flow discharge. The experiment results show that holding force has three typical forms in the closing process and it is related to the service gate height. The numerical model can reflect the gate vertical vibration and the gate-closing displacement in the form of steps. Gate vertical vibration in the closing process is a motion-induced vibration caused by gate active falling. Moreover, the transition from full-flow to open-flow behind the emergency gate has a great influence on the gate vertical vibration. With a small gate opening, gate vertical vibration makes the flow discharge fluctuation increase. Furthermore, flow discharge has an influence on the gate body loads, which is mainly concentrated in the upstream plate and gate bottom. Finally, the lift force coefficient at the gate bottom is different from the standard and is mainly controlled by the outflow boundary condition. The simulation result is in good agreement with the experiment and the relative error meets engineering requirements, suggesting that the numerical model can successfully simulate the gate fluid–structure interaction and reproduce the characteristics of physical quantities in the closing process.

Insights on the unsteadiness and stall effects on the characteristics and responses of continuous wing-based systems

The effects of the aerodynamic loads on continuous wing-based systems can be represented using a quasi-steady or a fully unsteady reduced-order modeling. Frequently, the quasi-steady formulation is used to model the aerodynamic loads at small reduced frequency values. This approximation, unlike the unsteady formulation, does not account for any lag between the unsteady oscillations and its effect on the aerodynamic loads which might lead to over or under predicting some crucial information related to the system studied, particularly concerning the onset of flutter or type of instability. To this end, in this effort, a comparison of the effects of different aerodynamic formulations on a wing-based system is conducted. A linear analysis considering the influence of each noncirculatory term in both formulations is inspected individually, and the results generated from both aerodynamic formulations are compared while depicting the effects of the structural damping on the linear flutter speed and frequency. A nonlinear analysis is then performed to investigate the stall effect on the system’s response as well as the impacts of different nonlinearities including inertial and geometric. The system’s behavior is examined using higher modes in the Galerkin discretization to make sure that it is accurately modeled and that the number of modes selected does not fail to correctly portray its aeroelastic response. The results of this study show the importance of using the unsteady fluid–structure interaction with stall modeling for continuous wing systems even for small angles of attack and reduced frequency values.

Aeroelastic analysis of wind turbines under turbulent inflow conditions

Abstract. The aeroelastic response of a 2 MW NM80 turbine with a rotor diameter of 80 m and interaction phenomena are investigated by the use of a high-fidelity model. A time-accurate unsteady fluid–structure interaction (FSI) coupling is used between a computational fluid dynamics (CFD) code for the aerodynamic response and a multi-body simulation (MBS) code for the structural response. Different CFD models of the same turbine with increasing complexity and technical details are coupled to the same MBS model in order to identify the impact of the different modeling approaches. The influence of the blade and tower flexibility and of the inflow turbulence is analyzed starting from a specific case of the DANAERO experiment, where a comparison with experimental data is given. A wider range of uniform inflow velocities are investigated by the use of a blade element momentum (BEM) aerodynamic model. Lastly a fatigue analysis is performed from load signals in order to identify the most damaging load cycles and the fatigue ratio between the different models, showing that a highly turbulent inflow has a larger impact than flexibility, when low inflow velocities are considered. The results without the injection of turbulence are also discussed and compared to the ones provided by the BEM code AeroDyn.

Unsteady fluid-structure interactions in a soft-walled microchannel: A one-dimensional lubrication model for finite Reynolds number

We develop a one-dimensional model for the unsteady fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it. A beam equation, which accounts for both transverse bending rigidity and nonlinear axial tension, is coupled to a one-dimensional fluid model obtained from depth-averaging the two-dimensional incompressible Navier--Stokes equations across the channel height. Specifically, the Navier--Stokes equations are scaled in the viscous lubrication limit relevant to microfluidics. The resulting set of coupled nonlinear partial differential equations is solved numerically through a segregated approach employing fully-implicit time stepping. We explore both the static and dynamic FSI behavior of this example microchannel system by varying a reduced Reynolds number $Re$, which necessarily changes the Strouhal number $St$, while we keep the geometry and a modified dimensionless Young's modulus $\Sigma$ fixed. At steady state, an order-of-magnitude analysis (balancing argument) shows that the axially-averaged pressure in the flow, $\langle P\rangle$, exhibits two different scaling regimes, while the maximum deformation of the top wall of the channel, $H_{\mathrm{max}}$, can fall into four different regimes, depending on the magnitudes of $Re$ and $\Sigma$. These regimes are physically explained as resulting from the competition between the inertial and viscous forces in the fluid flow as well as the bending resistance and tension in the elastic wall. Finally, the linear stability of the steady inflated microchannel shape is assessed via a modal analysis, showing the existence of many highly oscillatory but stable modes, which further highlights the computational challenge of simulating unsteady FSIs.

Analysis of shape optimization problems for unsteady fluid-structure interaction

Shape optimization via the method of mappings is investigated for unsteady fluid-structure interaction (FSI) problems that couple the Navier–Stokes equations and the Lamé system. Building on recent existence and regularity theory we prove Fréchet differentiability results for the state with respect to domain variations. These results form an analytical foundation for optimization und inverse problems governed by FSI systems. Our analysis develops a general framework for deriving local-in-time continuity and differentiability results for parameter dependent nonlinear systems of partial differential equations. The main part of the paper is devoted to conducting this analysis for the FSI problem, transformed to a shape reference domain. The underlying shape transformation—actually we work with the corresponding shape displacement instead—represents the shape and the main result proves the Fréchet differentiability of the solution of the FSI system with respect to the shape transformation.

A combined digital image correlation/particle image velocimetry study of water-backed impact

Several aeronautical and naval structures are routinely exposed to impulsive loading conditions that elicit complex fluid-structure interactions. An archetypal problem that has attracted considerable attention is the impact on a compliant plate, fixed on a water surface. Despite significant progress in mathematical modeling of the impact, several technical questions remain open due to the lack of a simultaneous experimental characterization of the structural response and fluid flow. Here, we seek to fill this gap of knowledge through the integration of digital image correlation (DIC) and particle image velocimetry (PIV). We employ DIC to measure the apparent in-plane displacement of the plate, from which we reconstruct its out-of-plane deflection. On the other hand, PIV is utilized to measure the velocity field of the water, from which we infer the pressure field in the fluid and the hydrodynamic loading on the plate. We examine a number of aggregated measures, including the mid-span deflection, strain energy stored in bending and stretching of the plate, modal contribution factors, hydrodynamic loading, and added mass coefficient. Just as our approach constitutes a significant methodological step forward in the study of unsteady fluid-structure interactions, our experimental results contribute new evidence for an improved understanding of water-backed impact.

Experimental and numerical investigation of blade resonance in a centrifugal compressor for varying gas properties

The blades of centrifugal compressors are exposed to unsteady forces during operation which can result in resonance response conditions and failures due to high cycle fatigue. A typical source of excitation is the unsteady fluid structure interaction between the impeller blades and the downstream vaned diffuser. Centrifugal compressors are operated with various working fluids with a wide range of applications in the power and process industry. Understanding the excitation mechanisms for different working fluids will help to design aerodynamically efficient compressors, while ensuring mechanical integrity and reducing the number of experimental design validations. A variation in working fluid properties allows investigation of the contribution of blade forcing and damping while the modal response remains unchanged. Experiments have been conducted at ETH Zurich’s radial compressor facility with a state of the art industrial compressor design. Dynamic strain gauge measurements on the impeller blades were used to determine the amplitude response, damping properties and forcing at a defined resonance condition. Two working fluids have been investigated to vary compressor flow settings while the modal response remains unchanged. Unsteady flow simulations and harmonic FSI simulations were used to complement the experiments and to investigate the local blade forcing distribution, which then were linked to flow effects. Experiments showed a change in resonance amplitude up to a factor of 4 due to a change in the applied working fluid. Estimation of the damping ratio with a single degree of freedom model found the exciting force to be the main contributor to the differences in resonant response. The unsteady flow simulations were able to identify the locations on the blade surface which are responsible for the change in forcing. It was found that the forcing depends on wave propagation effects in the flow channel and on how the pressure field matches the mode shape.

Numerical method for unsteady fluid structure interaction problem

Numerical method for unsteady fluid structure interaction problem

Experimental Determination of Mechanical Stress Induced by Rotating Stall in Unshrouded Impellers of Centrifugal Compressors

Unshrouded centrifugal compressor impellers typically operate at high rotational speeds and volume flow rates. The resulting high mean stress levels leave little margin for dynamic excitations that can cause high-cycle fatigue. In addition to the well-established high-frequency impeller blade excitations of centrifugal compressors caused by the stationary parts, such as vaned diffusers or inlet guide vanes (IGVs), the presented study addresses an unsteady rotating flow feature (rotating stall) which should be taken into account when addressing the high-cycle fatigue during the design phase. The unsteady fluid–structure interaction between rotating stall and unshrouded impellers was experimentally described and quantified during two different measurement campaigns with two full-size compression units operating under real conditions. In both campaigns, dynamic strain gauges and pressure transducers were mounted at various locations on the impeller of the first compression stage. The casing was also equipped with a set of dynamic pressure transducers to complement the study. Rotating pressure fluctuations were found to form an additional impeller excitation at a frequency that is not a multiple of the shaft speed. The measurements show that the excitation amplitude and frequency caused by the rotating pressure fluctuations depend on the operating conditions and are therefore challenging to predict and consider during the design phase. Furthermore, the excitation mechanism presented was found to cause resonant impeller blade response under specific operating conditions. For the experimentally investigated impeller geometries, a rotating pressure fluctuation caused approximately 1.5 MPa of additional dynamic stress in the structure per 1 mbar of dynamic pressure amplitude when exciting the first bending mode of the impeller. The induced dynamic mechanical stresses due to rotating stall are in the order of 10% of the endurance limit of the material for the tested impeller geometries; therefore, they are not critical and confirm a robust and reliable design.

Method of successive approximations for a unsteady fluid structure interaction problem

Method of successive approximations for a unsteady fluid structure interaction problem

On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid

We study the existence of weak solution for unsteady fluid-structure interaction problem for shear-thickening flow. The time dependent domain has at one part a flexible elastic wall. The evolution of fluid domain is governed by the generalized string equation with action of the fluid forces. The power-law viscosity model is applied to describe shear-dependent non-Newtonian fluids.

Partitioned coupling strategies for fluid-structure interaction with large displacement: Explicit, implicit and semi-implicit schemes

In this paper the unsteady fluid-structure interaction (FSI) problems with large structural displacement are solved by partitioned solution approaches in the arbitrary Lagrangian-Eulerian finite element framework. The incompressible Navier-Stokes equations are solved by the characteristic-based split (CBS) scheme. Both a rigid body and a geometrically nonlinear solid are considered as the structural models. The latter is solved by Newton-Raphson procedure. The equation governing the structural motion is advanced by Newmark-<TEX>${\beta}$</TEX> method in time. The dynamic mesh is updated by using moving submesh approach that cooperates with the ortho-semi-torsional spring analogy method. A mass source term (MST) is introduced into the CBS scheme to satisfy geometric conservation law. Three partitioned coupling strategies are developed to take FSI into account, involving the explicit, implicit and semi-implicit schemes. The semi-implicit scheme is a mixture of the explicit and implicit coupling schemes due to the fluid projection splitting. In this scheme MST is renewed for interfacial elements. Fixed-point algorithm with Aitken's <TEX>${\Delta}^2$</TEX> method is carried out to couple different solvers within the implicit and semi-implicit schemes. Flow-induced vibrations of a bridge deck and a flexible cantilever behind an obstacle are analyzed to test the performance of the proposed methods. The overall numerical results agree well with the existing data, demonstrating the validity and applicability of the present approaches.