💨Mathematical Fluid Dynamics Unit 11 – Fluid-Structure Interaction & Flow Control

Key Concepts and Definitions

• Fluid-structure interaction (FSI) the mutual interaction between a deformable or movable structure and a surrounding or internal fluid flow
• Coupling the interaction between the fluid and the structure, which can be one-way (fluid affects structure) or two-way (fluid and structure affect each other)
• Fluid dynamics the study of fluid flow and its interactions with solid boundaries, including concepts such as velocity, pressure, and viscosity
  • Incompressible flow a type of fluid flow where the density remains constant throughout the flow field
  • Compressible flow a type of fluid flow where the density varies significantly due to changes in pressure or temperature
• Structural dynamics the study of the behavior of structures under dynamic loading, including concepts such as vibration, deformation, and stress
  • Elastic deformation reversible deformation of a structure that returns to its original shape after the load is removed
  • Plastic deformation permanent deformation of a structure that remains even after the load is removed
• Boundary conditions the conditions specified at the boundaries of a fluid or structural domain, such as velocity, pressure, or displacement
• Turbulence the chaotic and irregular motion of fluids characterized by the presence of eddies and fluctuations in velocity and pressure
• Vortex a region in a fluid where the flow revolves around an axis, creating a swirling motion (whirlpools, tornadoes)

Governing Equations

• Conservation of mass (continuity equation) states that the rate of change of mass in a control volume is equal to the net mass flux through its boundaries
  • Incompressible flow: $\nabla \cdot \mathbf{u} = 0$
  • Compressible flow: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$
• Conservation of momentum (Navier-Stokes equations) describes the balance of forces acting on a fluid element, including pressure, viscous, and body forces
  • Incompressible flow: $\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$
  • Compressible flow: additional terms for compressibility effects and energy equation
• Structural equations of motion describe the dynamic behavior of a structure under external loads and fluid forces
  • Elastic structures: $\mathbf{M}\ddot{\mathbf{d}} + \mathbf{C}\dot{\mathbf{d}} + \mathbf{K}\mathbf{d} = \mathbf{F}$
  • Plate and shell theories for thin-walled structures (Kirchhoff-Love, Mindlin-Reissner)
• Fluid-structure coupling equations enforce the compatibility of displacements and tractions at the fluid-structure interface
  • Kinematic condition: continuity of displacements and velocities
  • Dynamic condition: balance of forces and stresses
• Constitutive equations relate the stress and strain in a material, describing its mechanical behavior (Hooke's law for linear elasticity, hyperelastic models for large deformations)

Fluid-Structure Interaction Fundamentals

• Types of FSI problems
  • External flow around flexible structures (aircraft wings, wind turbines, bridges)
  • Internal flow in deformable channels or pipes (blood flow in arteries, flow in collapsible tubes)
  • Flow-induced vibrations and instabilities (vortex-induced vibrations, flutter)
• Coupling schemes for FSI
  • Monolithic approach solves the fluid and structural equations simultaneously as a single system
  • Partitioned approach solves the fluid and structural equations separately and exchanges information at the interface
    • Explicit coupling updates the fluid and structure sequentially without iteration
    • Implicit coupling iterates between the fluid and structure solvers until convergence
• Arbitrary Lagrangian-Eulerian (ALE) formulation a method for handling moving and deforming domains in FSI problems by introducing a referential coordinate system
• Immersed boundary methods treat the structure as a collection of Lagrangian points or markers immersed in the Eulerian fluid domain, applying forces to the fluid to represent the structure
• Stability and convergence issues in FSI
  • Added-mass effect the inertial effect of the fluid on the structure, which can cause instabilities in explicit coupling schemes
  • Geometric conservation law ensures that the numerical scheme preserves the geometry of the moving domain
• Fluid-structure interface conditions
  • No-slip condition the fluid velocity matches the structural velocity at the interface
  • Traction continuity the fluid and structural stresses balance at the interface

Numerical Methods and Simulations

• Finite element method (FEM) a numerical technique for solving partial differential equations by discretizing the domain into elements and approximating the solution with basis functions
  • Weak formulation transforms the governing equations into an integral form, which is then discretized using the finite element approximation
  • Isoparametric elements elements that use the same shape functions for both geometry and solution approximation
• Finite volume method (FVM) a numerical method that discretizes the domain into control volumes and enforces conservation laws on each volume
  • Flux calculation schemes (upwind, central differencing, high-resolution schemes)
  • Pressure-velocity coupling algorithms (SIMPLE, PISO, COUPLED)
• Temporal discretization schemes
  • Explicit schemes (Forward Euler, Runge-Kutta) calculate the solution at the next time step using only information from the current time step
  • Implicit schemes (Backward Euler, Crank-Nicolson) solve a system of equations involving both the current and next time steps
• Mesh generation and adaptivity
  • Structured meshes regular grid-like meshes with a fixed topology (quadrilateral, hexahedral)
  • Unstructured meshes irregular meshes with a flexible topology (triangular, tetrahedral)
  • Adaptive mesh refinement (AMR) dynamically refines or coarsens the mesh based on solution features or error estimates
• Verification and validation
  • Verification assesses the accuracy of the numerical implementation by comparing it to analytical solutions or benchmarks
  • Validation assesses the accuracy of the mathematical model by comparing it to experimental data or real-world observations

Flow Control Techniques

• Passive flow control techniques that do not require external energy input
  • Vortex generators small protrusions that create vortices to delay flow separation and enhance mixing (dimples on golf balls, winglets on aircraft)
  • Riblets small streamwise grooves on a surface that reduce turbulent skin friction drag
  • Compliant surfaces surfaces that deform in response to fluid forces to reduce drag or delay transition
• Active flow control techniques that require external energy input
  • Boundary layer suction removes the low-momentum fluid near the wall to delay separation or reduce drag
  • Boundary layer blowing injects high-momentum fluid near the wall to energize the boundary layer and prevent separation
  • Plasma actuators use electric fields to ionize the air and create a body force that can manipulate the flow
  • Synthetic jets zero-net-mass-flux jets that create vortices to enhance mixing or delay separation
• Feedback control strategies
  • Proportional-integral-derivative (PID) control a simple feedback control algorithm that adjusts the control input based on the error between the desired and actual output
  • Model predictive control (MPC) an optimization-based control strategy that predicts the future behavior of the system and determines the optimal control input
  • Adaptive control a control strategy that adjusts the controller parameters based on the changing dynamics of the system
• Flow sensing and estimation
  • Hot-wire anemometry measures fluid velocity by detecting changes in the resistance of a heated wire exposed to the flow
  • Particle image velocimetry (PIV) measures the velocity field by tracking the motion of tracer particles in the flow
  • Kalman filtering a recursive algorithm for estimating the state of a dynamic system from noisy measurements

Applications in Engineering

• Aerospace engineering
  • Aircraft wings and control surfaces (ailerons, flaps, rudders)
  • Helicopter rotor blades and fuselages
  • Spacecraft and satellite structures (solar panels, antennas)
• Automotive engineering
  • Vehicle aerodynamics and drag reduction (spoilers, underbody diffusers)
  • Engine components (valves, pistons, turbochargers)
  • Suspension systems and tires
• Biomedical engineering
  • Cardiovascular systems (heart valves, arteries, veins)
  • Respiratory systems (airways, lungs)
  • Prosthetic devices and implants (stents, artificial joints)
• Civil and structural engineering
  • Bridges and cable-stayed structures
  • Wind-excited buildings and towers
  • Offshore structures (oil platforms, wind turbines)
• Energy and power systems
  • Wind turbines and hydroelectric turbines
  • Heat exchangers and cooling systems
  • Fuel cells and batteries

Advanced Topics and Current Research

• Multiphase and multicomponent FSI
  • Fluid-structure-acoustics interaction the coupled interaction between fluid flow, structural deformation, and acoustic waves (underwater explosions, sonar systems)
  • Fluid-structure-porous media interaction the interaction between fluid flow, deformable structures, and porous materials (biological tissues, geomechanics)
• Multiscale and multiphysics modeling
  • Atomistic-to-continuum methods bridging the gap between molecular dynamics simulations and continuum mechanics models (nanofluidics, biomolecular systems)
  • Multiscale homogenization deriving effective macroscopic properties from microscopic simulations (composite materials, porous media)
• Uncertainty quantification and stochastic FSI
  • Polynomial chaos expansions representing random variables and fields using a set of orthogonal polynomials (Hermite, Legendre)
  • Bayesian inference updating the probability distribution of model parameters based on observed data (Markov Chain Monte Carlo, Kalman filtering)
• Machine learning and data-driven methods
  • Neural networks and deep learning using artificial neural networks to learn complex relationships from data (convolutional neural networks, recurrent neural networks)
  • Reduced-order modeling constructing low-dimensional models that capture the essential dynamics of high-dimensional systems (proper orthogonal decomposition, dynamic mode decomposition)
• Optimization and inverse problems
  • Shape optimization finding the optimal shape of a structure or fluid domain to minimize an objective function (drag reduction, structural compliance)
  • Parameter identification estimating unknown model parameters from experimental data or observations (material properties, boundary conditions)

Problem-Solving Strategies

• Identify the type of FSI problem (external flow, internal flow, flow-induced vibrations) and the relevant physical phenomena (incompressible/compressible flow, elastic/plastic deformation)
• Determine the appropriate governing equations for the fluid (Navier-Stokes) and the structure (equations of motion, constitutive relations) based on the problem characteristics
• Select a suitable numerical method (FEM, FVM) and discretization scheme (explicit, implicit) for solving the coupled FSI problem, considering factors such as accuracy, stability, and computational cost
  • Monolithic approach for strongly coupled problems with significant fluid-structure interactions
  • Partitioned approach for weakly coupled problems or when using existing fluid and structural solvers
• Generate a computational mesh that accurately represents the geometry and provides adequate resolution in regions of interest (boundary layers, interfaces, regions of high gradients)
• Specify the appropriate boundary conditions and initial conditions for the fluid and structural domains, ensuring compatibility at the fluid-structure interface
• Implement the numerical scheme and coupling algorithm, paying attention to issues such as convergence, stability, and conservation properties
  • Under-relaxation or stabilization techniques for explicit coupling schemes to mitigate instabilities
  • Convergence acceleration techniques (Aitken's relaxation, quasi-Newton methods) for implicit coupling schemes
• Verify the implementation using benchmark problems or analytical solutions, and validate the model using experimental data or real-world observations
• Analyze the results, extracting relevant quantities of interest (forces, displacements, velocities, pressures) and visualizing the fluid and structural fields
• Interpret the findings in the context of the application, identifying key physical insights and potential design improvements or optimizations
• Communicate the results effectively through clear and concise presentations, reports, or publications, highlighting the key conclusions and their significance for the field of study