Transition to chaos in a reduced-order model of a shear layer

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$ , leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters

Geometrically parametrized partial differential equations are currently widely used in many different fields, such as shape optimization processes or patient-specific surgery studies. The focus of this work is some advances on this topic, capable of increasing the accuracy with respect to previous approaches while relying on a high cost–benefit ratio performance. The main scope of this paper is the introduction of a new technique combining a classical Galerkin-projection approach together with a data-driven method to obtain a versatile and accurate algorithm for the resolution of geometrically parametrized incompressible turbulent Navier–Stokes problems. The effectiveness of this procedure is demonstrated on two different test cases: a classical academic back step problem and a shape deformation Ahmed body application. The results provide insight into details about the properties of the architecture we developed while exposing possible future perspectives for this work.

Model Order Reduction for a Piecewise Linear System Based on Dynamic Mode Decomposition

This paper presents a data-driven model order reduction strategy for nonlinear systems based on dynamic mode decomposition (DMD). First, the theory of DMD is briefly reviewed and its extension to model order reduction of nonlinear systems based on Galerkin projection is introduced. The proposed method utilizes impulse response of the nonlinear system to obtain snapshots of the state variables, and extracts dynamic modes that are then used for the projection basis vectors. The equations of motion of the system can then be projected onto the subspace spanned by the basis vectors, which produces the projected governing equations with much smaller number of degrees of freedom (DOFs). The method is applied to the construction of the reduced order model (ROM) of a finite element model (FEM) of a cantilevered beam subjected to a piecewise-linear boundary condition. First, impulse response analysis of the beam is conducted to obtain the snapshot matrix of the nodal displacements. The DMD is then applied to extract the DMD modes and eigenvalues. The extracted DMD mode shapes can be used to form a reduction basis for the Galerkin projection of the equation of motion. The obtained ROM has been used to conduct the forced response calculation of the beam subjected to the piecewise linear boundary condition. The results obtained by the ROM agree well with that obtained by the full-order FEM model.

A higher-order parametric nonlinear reduced-order model for imperfect structures using Neumann expansion

We present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

Down-Hole Directional Drilling Dynamics Modeling Based on a Hybrid Modeling Method With Model Order Reduction

This paper presents a dynamics model for the down-hole directional drilling system based on a hybrid modeling method with model order reduction. Due to the long dimension of the drill string, a drilling model purely based on numerical methods such as the finite element method (FEM) may require a large number of meshes, which induces high computational intensity. By using a hybrid method combining FEM and the transfer matrix method (TMM), the order of the model can be significantly reduced. To further reduce the modeling order, a proper orthogonal decomposition (POD)-Galerkin projection-based approach is applied, and a set of linear normal modes (LNMs) are identified to create a reduced-order projection subspace. To this end, simulation results are presented to prove that the method can effectively capture the dominant modes of the drilling dynamics, and a computationally efficient and high fidelity reduced-order hybrid model can be reached for real-time state estimation and control design.

Effect of Uncertainty in the Balancing Weights on the Vibration Response of a High-Speed Rotor

This work investigates how uncertainties in the balancing weights are propagating into the vibration response of a high-speed rotor. Balancing data are obtained from a 166-MW gas turbine rotor in a vacuum balancing tunnel. The influence coefficient method is then implemented to characterize the rotor system by a deterministic multi-speed and multi-plane matrix. To model the uncertainties, a non-sampling probabilistic method based on the generalized polynomial chaos expansion (gPCE) is employed. The uncertain parameters including the mass and angular positions of the balancing weights are then expressed by gPCE with deterministic coefficients. Assuming predefined probability distributions of the uncertain parameters, the stochastic Galerkin projection is applied to calculate the coefficients for the input parameters. Furthermore, the vibration amplitudes of the rotor response are represented by appropriate gPCE with unknown deterministic coefficients. These unknown coefficients are determined using the stochastic collocation method by evaluating the gPCE for the system response at a set of collocation points. The effects of individual and combined uncertain parameters from a single and multiple balancing planes on the rotor vibration response are examined. Results are compared with the Monte Carlo simulations, showing excellent agreement.