Flutter | igor kavrakov

Low-order Structural Aerodynamics

Fri, 17 Mar 2023 00:00:00 +0000

Fluid-structure interaction is a complex phenomenon that involves two distinct mediums: a fluid and a solid. Simulating fluid-structure interaction can be accomplished through solving the coupled partial differential equations of the fluid and solid by means of numerical methods. These techniques require discretizing the fluid equations on many degrees of freedom, be that vortex particles, volume cells, or finite elements, and usually require long computational times. When we are interested in the effect of the fluid on the structure, low-order aerodynamic models are like a secret weapon since they model the forces as a function of the motion of a 2D rigid deck (3 degrees of freedom) and free-stream turbulent fluctuations, thereby reducing the degrees of freedom of the fluid and increasing computational efficiency. This provides deeper insight into the underlying physics, helping us understand the factors contributing to the modeled phenomena. Aerodynamic forces are also one of the most important aspects when designing slender civil structures, such as bridges, towers and masts.

Low-order aerodynamic forces acting on a 2D ridgid deck

Aerodynamic force models can be broadly classified into two categories: semi-analytical and data-driven. Semi-analytical models are white-box models based on physics and use aerodynamic coefficients to substitude the fluid behavior. On the other hand, data-driven models are black-box models that use machine learning methods to reverse-engineer the relationship between input (motion/gusts) and output (forces) by using Computational Fluid Dynamics (CFD) or wind tunnel data. We can use both models to efficiently simulate structural response and capture various aerodynamic phenomena such as buffeting, flutter, and vortex-induced vibrations.

Aeroelastic phenomena: structural response against wind speed

I studied aerodynamic force models from multiple perspectives, including their intrinsic mathematical properties, qualitative comparison, and development of new models. Part of my research work is summarised here, namely:

On the assesment of aerodynamic models using Category Theory
On the influence of the assumptions of aerodynamic models on the aeroelastic response
On the data-driven aerodynamic modelling using Gaussian Processes and Neural Nets

Aerodynamic Modelling via Category Theory

Initially, we employed and extended an abstract framework based on Category Theory to evaluate the complexity of aerodynamic models and establish a basis for comparison among semi-analytical models based on their mathematical constructions (i.e., assumptions). Category theory, sometimes referred to as “abstract nonsense” by some, is a mathematical theory that deals with structures such as categories, objects, arrows, and the rules that relate them. Using this theory, we constructed a diagram defining aerodynamic model complexity. This immediately allows for comparisons between models without the need for prior knowledge of the semi-analytical models. For example, we cannot compare models if there is no direct diagramic relation between them (see below). More information on the modeling framework, including the definition of model comparability, simplest and most complex models, as well as practical considerations, can be found in our Royal Society Open Science paper.

Influence of Model Assumptions on the Aeroelastic Response

In addition to quantitative comparison, we examined the direct impact of aerodynamic assumptions on the buffeting response and critical flutter velocity. To do this, we compared the semi-analytical models to each other and to a CFD model by conducting a one-to-one comparison using identical wind turbulence input. The video below illustrates a one-to-one comparison between the Linear Unsteady (LU) semi-analytical model and the CFD model.

It turns out that, the fluid memory assumption, captured by the Linear Unsteady (LU) model, plays a crucial role in the aerodynamic response. This can be seen from the figures below, which depict the buffeting response. The LU and the hybrid nonlinear model (HNL) demonstrate the closest match to the CFD model, which is used as the reference due to its higher complexity (as previously depicted in the complexity diagram above).

2D buffeting analysis of the Great Belt Bridge deck: instantaneous velocity field (top); sample of response time-histories (centre); root-mean-square of the response for a selected wind speed range and turbulence intensity (bottom)

Additional comparisons can be found in our two papers, one in Engineering, and the other in Fluids and Structures. The first paper is based on a modified model of the Mersey Gateway Bridge (UK) (see video below) in its construction stage and it also presents a simplified method for employing the aerodynamic admittance in the time-domain based on Fourier Transform. The second paper is, to the best of my knowledge, the first study that conducts a comprehensive one-to-one comparison between CFD and semi-analytical models for buffeting and flutter analyses.

Dealing with time-histories (response or forces), comparisons based on the root-mean-square and peak values are not exhaustive to quantify the differences between models. Thus, we introduced a set of eight comparison metrics to quantify the discrepancies between two signals by looking at salient signal features relevant for structural aerodynamics. Each metric ranges from 0 to 1, with 1 indicating a perfect match. We propose eight metrics, including: phase, peak, root-mean-square, magnitude, wavelet and normalized wavelet (time-frequency), and bispectrum (second-order harmonic). For instance, the bispectrum metric quantifies the difference in second-order nonlinearities that may occur in the self-excited aerodynamic forces at large angles of attack. The comparison metrics are explained in detail in our Engineering Mechanics paper (incl. code - Github link).

Comparison metrics for time-histories of self-excited forces: small angles of attack (left); large angles of attack (right)

Data-driven Aerodynamic Models via Gaussian Processes and Neural Nets

Semi-analytical models are only as good as their mathematical properties, and it can be challenging to construct a white-box mathematical model that can capture memory-dependent nonlinearities and subsequently determine the corresponding aerodynamic coefficients (e.g., Volterra Series). To overcome this challenge, we employ data-driven techniques to reverse-engineer the model by learning the latent function that maps input (angle of attack) to output (aerodynamic force). We used both Gaussian Process (GP) Regression and Artificial Neural Networks (ANNs) as machine learning methods to construct the latent function; thereby, significantly increasing the mathematical capabilities of the model compared to semi-analytical models. The other key ingredient in constructing such model, which is fortunately abundant in wind engineering due to wind tunnel experiments, is data. Training the model requires appropriate data that reflects the use of the model. In practice, this means that the training signals should contain amplitudes and frequencies of the angle of attack that the bridge is expected to exhibit during flutter or buffeting. To address this, we developed a simple method based on Fourier transform that generates training signals for coupled motion.

Framework for data-driven modelling (top); Gaussian process model (bottom-left); artificial neural network model (bottom-right)

In our Journal of Wind Engineering and Industrial Aerodynamics paper (incl. code - Github link), we were able to train a GP model for self-excited forces, verify it using flat plate linear solutions, and predict second-order harmonics for Great Belt Bridge deck and Limit Cycle Oscillations (LCOs) for Tacoma-like deck. This is a promising development and we are continuing to explore this direction. However, we encountered an obstacle in predicting the LCO amplitude for coupled flutter, which we suspect is due to the non-stationary nature of the frequency content in the response. Specifically, at large amplitudes, the coupled frequency shifts towards torsional vibrations caused by large leading-edge separation. Similarly, we used standard ANNs in Computers and Structures paper and successfully predicted torsional flutter; however, we did not consider coupled excitation during the learning process in this study.

We are still exploring ways to use GPs as aerodynamic force models and incorporate physical principles into them. This research will eventually become a separate project - stay tuned!

Gaussian process aerodynamic force model: self-excited forces for large amplitude rotation of the Great Belt Bridge deck (top); limit cycle oscilation prediction of the rotation from free-vibration analysis of a Tacoma-like H-shaped deck (bottom)

Computational Wind Engineering

Sun, 15 Jan 2023 00:00:00 +0000

Computational Fluid Dynamics (CFD) has become an effective tool in the aerodynamic design of civil structures in the past few decades. These models can serve as a powerful tool in designer’s arsenal during the initial design phase, allowing them to explore various structural shapes and configurations. However, wind engineers may view CFD models with a dose of suspicion due to their inherent numerical uncertainty (maybe rightfully so), but that should not hinder us from harnessing their potentials. CFD models are commonly used to compute the aerodynamic coefficients and rarely the aeroelastic response (e.g. flutter or buffeting), enabling engineers to determine design forces or assess human comfort.

My work on CFD centers around the 2D Vortex Particle Methods. Unlike traditional Eulerian grid-based methods (e.g. Finite Volume or Finite Element), these Lagrangian methods discretise the vorticity field on point-like units called particles. These particles carry concentrated vorticity (circulation) enabling fluid kinematics to be solved through Biot-Savart’s relation (imagine fluid velocity as a magnetic field and vorticity as current). Advancing the fluid dynamics is straightforward with any explicit time-integration scheme, including an operator splitting technique to account for viscosity. Immersed bodies, like we usually have for line-like structures (bridges, towers), can be discretised using the Boundary Element Method on panels, allowing boundary conditions to be imposed inside the Biot-Savart relation and Kelvin’s circulation theorem. The scalar nature of vorticity in 2D makes the vortex methods computationally efficient. A schematic of the method is shown below.

Vortex Particle Method: mathematical model (left); numerical discretisation (right)

I worked on Vortex Methods from the perspective of aeroelastic analyses of structures. I used an in-house code, VXflow by Guido Morgenthal, and therefore, any extensions are related to this code. Next, part of the new methods and findings are summarised, in particular:

On a turbulent Pseudo-3D Vortex Method
On a numerical Active Turbulence Generator (ATG) for determining the complex aerodynamic admittance
On the nonlinear interaction between the motion- and gust-induced aerodynamic forces

Turbulent Pseudo-3D Vortex Method

Computing the response of a structure subjected to free-stream turbulence (buffeting analysis) in CFD is notoriously computationally intensive. The 3D nature of free-stream turbulence requires a very fine discretisation to accurately capture turbulent scales and turbulent boundary layers in the fluid-structure interaction. To address this challenge, we developed a Turbulent Pseudo-3D Vortex method that combines the Laminar Pseudo-3D Vortex method (Morgenthal and McRobie 2002, Wind Struct. 5) with velocity-based free-stream random turbulence generation (Prendergast 2007, PhD Thesis, Univ. of Cambridge). The idea is to position 2D fluid planes (slices) along a line-like structure (e.g. bridge or tower) and introduce correlated free-stream turbulence upwind of the structure that preserves some key atmospheric statistical properties such as point spectra and coherence. The beam-like structure couples with the fluid slices through multiple vibration modes and we needed to demonstrate that the generated upstream turbulence remains correlated between slices. In other words, we needed to ensure that the wind felt by two people standing apart longitudinally on the bridge is not vastly different, as is the case in nature. Our Engineering Structures Paper shows this analytically for Vickery’s coherence with a certain loss due to a couple of assumptions. Later, in my PhD thesis, we can combined Davenport’s and Vickery’s coherence to eliminate the assumptions, resulting in improved coherence between slices.

Pseudo-3D Vortex Method (top). Lateral (bottom-left) and vertical (bottom-right) coherence (Vickery - VC; Modified - MC) between six fluid slices

We used this method to conduct buffeting analysis of the Great Belt Bridge and then compared the structural displacements to the semi-analytical linear unsteady (LU) method, which represents common practice in bridge aerodynamics. The figure below shows a snapshot of the CFD simulation and response:

Pseudo-3D buffeting analysis of the Great Belt Bridge (top); root-mean-square of the response for the CFD and LU models (bottom)

Further, we computed the critical flutter limit for the first time with high spatial discretisation (50 slices), and compared to previous experimental results:

Critical flutter velocity of the Great Belt Bridge in Pseudo-3D

Determination of Complex Aerodynamic Admittance using a Numerical Active Turbulence Generator

Simulating sinusoidal deterministic free-stream gusts can be quite useful in practice, particularly when determining aerodynamic admittance. In wind tunnel testing, this is commonly done using an apparatus known as an Active Turbulence Generator (ATG). An ATG is essentially a set of vertically arranged airfoils that pitch sinusoidally. When fluid flows through them, a sinusoidal gust is generated downstream. We propose a numerical ATG model that involves injecting vortex particles at two locations to mimic the wake of two pitching airfoils that are located upstream of the section (see figure below). By adjusting the circulation amplitude, we were able to generate a sinusoidal gust downstream with prescribed intensity. If the ‘virtual’ airfoils are in phase, the gust is vertical; if they’re out of phase, the gust is longitudinal (pulsating). We derived an analytical relation between the gust and circulation amplitudes, building upon an existing model (Stapountzis 1978, J. Phys. E. Sci. Instrum. 15).

Concept of numerical active turbulence generator (top); predicted circulation using a proposed closed-form solution (bottom)

We used this method to determine the complex aerodynamic admittance, which represents a complex transfer function between the free-stream gusts and buffeting forces. This is particularly useful in practice since aerodynamic admittance is a key ingredient when performing buffeting analysis using reduced-order models. We verified the accuracy of our numerical admittance results by comparing them to the analytical Sears admittance for a flat plate, and further validated them in the Politecnico di Milano Wind Tunnel using the Third Bosphorus Bridge deck as a test case. For all the nitty-gritty details, check out our Journal of Wind Engineering and Industrial Aerodynamics paper.

Aerodynamic admittance of the Third Bosphorus Bridge decek: testing at the Politecnico di Milano Boundary Layer Wind Tunnel - courtesy of T. Argentini, D. Rocchi, S. Omarini (top); instantaneous particle map of the section under vertical sinusoidal gust (centre); aerodynamic admittance of the lift force (bottom)

Nonlinear Interaction between the Motion- and Gust-induced Aerodynamic Forces

Another interesting application for which we employed the numerical ATG is to study the interaction between the gust- and motion-induced forces acting on bridge decks. To do so, we first made sure that the individual force (gust and motion) remain linear for a given amplitude of the resultant angle of attack. Then, the bridge deck was subjected to both, motion and gust, and the resultant force was compared with the Linear Unsteady (LU) model, which simply superimposes the individual components. This was done in a parametric way for various angle of attack amplitudes and phase between the individual forces. Interestingly, we found out that for bluff decks at large angles of attacks, the difference is prominent, yielding the conclusion that the interaction causes nonlinear behaviour. This is quite important as the LU model is typically the go-to model when it comes to practical applications. The study was done primarely by Samuel Tesfaye, a PhD student with whom I co-advise as a postdoc, and the findings are summarized in our Fluids and Structures paper.

Concept of studying nonlinear interaction between gust- and motion-induced forces (top); pressure and velocity fields of two individual excitation cases and one combined (centre); deviation of the moment due to combined gust and moment interaction with respect to the linear semi-analytical solution (bottom)