Although my background is in earthquake engineering, I’ve recently been fascinated by wave-structure interaction problems. From a structural perspective, we are often focused on deformations due to external loading on the structure. In contrast, fluid mechanics is often focused on the effects of boundary conditions on the fluid flow.

In some situations, the solid and fluid mechanics cannot be decoupled, resulting in fluid-structure interaction, i.e., the solid is deformed by the fluid and (because it is now deformed) causes the fluid flow and loading to change. If I were to draw parallels to earthquake engineering, this interaction effect would correspond to a change in the input ground motion (or potential soil-structure interaction effects) due to the motion of the structure.

Fluid-structure interaction spans both a solid and fluid domain.

**Fluid domain: **The Navier-Stokes equations can be used to define the fluid domain (herein, I will assume incompressible flow):

*mass balance:* $latex \nabla \cdot \bold{v} = 0$

*momentum balance:* $latex \rho \frac{D\bold{v}}{Dt} = – \nabla p + \mu \nabla^2 \bold{v} + \rho \bold{g}$

where $latex \bold{v} = $ fluid velocity, $latex p = $ fluid pressure, $latex \rho = $ fluid density, and $latex \mu = $ fluid viscosity. As written, the mass balance equation represents the conservation of volume (the constraint of incompressibility). Momentum balance can be interpreted in terms of Newton’s second law, where the right-hand side represents the the sum of the forces and the left-hand side represents inertial effects (i.e., mass $latex \times$ acceleration).

From left to right, the different terms correspond to the inertial forces, pressure gradient (flow movement from high-to-low pressure), viscous forces (shear effect), and external forces acting on the fluid (such as gravity). The Navier-Stokes equations are nonlinear in $latex \bold{v}$ because of the convective acceleration term, which results from expansion of the acceleration using the chain rule:

$latex \frac{D\bold{v}}{Dt} = \frac{\partial \bold{v}}{\partial t} + \bold{v} \cdot \nabla \bold{v}$

where $latex \frac{\partial \bold{v}}{\partial t} = $ change of velocity with time and $latex \bold{v} \cdot \nabla \bold{v} = $ convective term describing the change of velocity with position. The resulting nonlinear partial differential equations are expanded below in 2D Cartesian coordinates $latex (x,y)$:

*mass balance:* $latex \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} = 0$

*momentum balance in $latex x$:* $latex \rho (\frac{\partial v_x}{\partial t} + v_x\frac{\partial v_x}{\partial x} + v_y\frac{\partial v_x}{\partial y}) = – \frac{\partial p}{\partial x} + \mu (\frac{\partial^2 v_x}{\partial x^2}+\frac{\partial^2 v_x}{\partial y^2}) + \rho g_x$

*momentum balance in $latex y$:* $latex \rho (\frac{\partial v_y}{\partial t} + v_x\frac{\partial v_y}{\partial x} + v_y\frac{\partial v_y}{\partial y}) = – \frac{\partial p}{\partial y} + \mu (\frac{\partial^2 v_y}{\partial x^2}+\frac{\partial^2 v_y}{\partial y^2}) + \rho g_y$

Both the velocities and pressures are unknown. Numerical solution methods for the incompressible Navier-Stokes equations often iterate between a predictive step for the flow velocity, $latex \bold{v}$, and a subsequent corrective step for the pressures, $latex p$.

**Structural Domain:** Structural dynamics (or alternatively a solid mechanics approach) can be used to define the solid domain. Below is a discretized form of the equations of motion (assuming the mass is lumped at the structure’s degrees-of-freedom):

*equation of motion*: $latex \bold{m} \ddot{\bold{u}} + \bold{k} \bold{u} = \bold{F}(t)$ (undamped and assuming elastic response)

where $latex \bold{u} = $ displacements, $latex \bold{m} = $ mass matrix, $latex \bold{k} = $ stiffness matrix, and $latex \bold{F}(t) = $ applied forces. From left to right, the terms correspond to the inertial forces, resisting forces internal to the structure, and applied loads. This results in $latex N$ coupled second-order ordinary differential equations, where $latex N$ is the number of degrees-of-freedom associated with the mass.

Similar to the Navier-Stokes equations, structural dynamics can be interpreted in terms of Newton’s second law, where the relevant forces sum, including inertial effects. Because the equations are undamped, the structure will exhibit oscillatory motions corresponding to the natural frequencies of the structure. This motion is often described in terms of the displacements, $latex \bold{u}$, which are unknown and result from the solution to the equations of motion.

**Interface:** At the interface between the solid and fluid, we would like to satisfy a kinematic condition, e.g., $latex \bold{v} = \dot{\bold{u}}$ (neglecting mixing or sliding along the interface), and equilibrium between the forces along the solid-fluid interface, $latex \bold{F}(t) =$ fluid forces derived from the viscous forces, pressure gradient, etc. In addition, each domain must satisfy its own fluid and solid boundary conditions.

This set of equations can be solved using a partitioned approach (i.e., solving for the solid and fluid domains separately and iterating until the interface conditions are satisfied). Alternatively, the equations can be rewritten and solved using a monolithic approach (i.e., simultaneously solving a set of coupled equations including the fluid, solid, interface, and boundary conditions).

The extent to which you need to consider fluid-structure interaction effects then depends on the density and deformation of the solid relative to the fluid, defined by the relative magnitude of the dimensionless mass, reduced velocity, and Cauchy numbers (more on these later).

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