In acoustic fluid-structure interaction (FSI) problems, the structural dynamics equation must be considered along with the Navier-Stokes equations of fluid momentum and the flow continuity equation.

From the law of conservation of mass the flow continuity equation is:

(8–1)

where:

= the velocity vector in the x-, y-, and z-directions

= density

= mass source (kg/m3t)

= time

From the law of conservation of momentum, the Navier-Stokes equation is:

(8–2)

where:

= viscous stress tensor

= pressure

= body force

The discretized structural dynamics equation can be formulated using the structural elements as shown in Equation 15–5. The fluid momentum (Navier-Stokes) equations and continuity equations are simplified to get the acoustic wave equation using the following assumptions:

The linearized continuity equation is:

(8–3)

The linearized Navier-Stokes equation is:

(8–4)

where:

= acoustic velocity

= acoustic pressure

The acoustic wave equation is given by:

(8–5)

where:

c = speed of sound in fluid medium (input as SONC on the MP command)

ρo = mean fluid density (input as DENS via MP)

K = bulk modulus of fluid

μ = dynamic viscosity (input as VISC via MP)

p = acoustic pressure (=p(x, y, z, t))

Q = mass source in the continuity equation (input as MASS via BF)

t = time

Since the viscous dissipation has been taken in account using the Stokes hypothesis, Equation 8–5 is referred to as the lossy wave equation for propagation of sound in fluids. The discretized structural Equation 15–5 and the lossy wave Equation 8–5 must be considered simultaneously in FSI problems. The wave equation will be discretized in the next subsection, followed by the derivation of the damping matrix to account for the dissipation at the FSI interface. The acoustic pressure exerting on the structure at the FSI interface will be considered in Derivation of Acoustic Matrices to form the coupling stiffness matrix.

Harmonically varying pressure is given by:

(8–6)

where:

p = amplitude of the pressure

ω = 2πf

f = frequency of oscillations of the pressure

Equation 8–5 is reduced to the following inhomogeneous Helmholtz equation:

(8–7)

Note that some theories may not be applicable for 2D acoustic elements using Equation 8–7. See FLUID29 and FLUID129 for available details.

The finite element formulation is obtained by testing wave Equation 8–5 using the Galerkin procedure (Bathe [2]). Equation 8–5 is multiplied by testing function w and integrated over the volume of the domain (Zienkiewicz [87]) with some manipulation to yield the following:

(8–8)

where:

dv = volume differential of acoustic domain ΩF

ds = surface differential of acoustic domain boundary ΓF

From the equation of momentum conservation, the normal velocity on the boundary of the acoustic domain is given by:

(8–9)

Substituting Equation 8–9 into Equation 8–8 yields the “weak” form of Equation 8–5, given by:

(8–10)

The normal acceleration of the fluid particle can be presented using the normal displacement of the fluid particle, given by:

(8–11)

where:

= the displacement of the fluid particle

After using Equation 8–11, Equation 8–10 is expressed as:

(8–12)

The fluid momentum (Navier-Stokes) equations and continuity equations are simplified to arrive at the convective wave equation using the following assumptions:

The linearized continuity equation (Equation 8–1) is:

(8–13)

The linearized Navier-Stokes equation (Equation 8–2) is:

(8–14)

where:

= mean flow velocity

= body force

With the introduction of velocity potential and force potential,

(8–15)

(8–16)

where:

= the velocity potential

= the force potential

The second order convective wave equation is given by:

(8–17)

where:

The acoustic pressure is derived by:

(8–18)

As an alternative to the pressure formulation in a transient analysis, the velocity potential formulation (KEYOPT(1) = 4 on the acoustic element) is cast by setting the mean flow to zero in Equation 8–17 and multiplying by . Either particle velocity or mass source can be applied as the excitation source for the velocity potential formulation (rather than the acceleration or mass source rate used for the pressure formulation).

The finite element formulation is obtained by Equation 8–17 using the Galerkin procedure. Equation 8–17 is multiplied by testing function w and integrated over the volume of the domain with some manipulation to yield the following:

(8–19)

Harmonically varying velocity potential is given by:

(8–20)

where:

The static mean flow can be obtained with the assumption of irrotational and incompressible flow. The Laplacian equation, in regard to the velocity potential, is given by:

(8–21)

The known mean flow values on the domain exterior surface are used as the non-zero Neumann boundary for the Laplacian equation. The zero Neumann boundary is applied to the rest of the domain exterior surface, which implies that the mean flow is tangential to the surface. The reference potential φ = 0 is specified by the program for the pure Neumann boundary value problem.

With the assumptions that the mean flow is tangential to the wall and the convective wave equation is solved in the frequency domain, the boundary condition on the moving boundary is given by:

(8–22)

where:

= outward normal unit vector of the boundary

un = the normal displacement of the boundary

The boundary integral on the moving boundary in Equation 8–19 is cast by:

(8–23)

If the normal velocity v_n on the boundary is known, that is:

(8–24)

the boundary integral in Equation 8–19 is given by:

(8–25)

If the impedance boundary condition exists on the moving boundary, that is:

(8–26)

where:

The boundary integral in Equation 8–19 is given by:

(8–27)

On an FSI interface, the coupled matrix equation is obtained by substituting Equation 8–18 into Equation 8–150 and using the moving boundary condition given by Equation 8–23.

For an axisymmetric configuration, the Helmholtz equation is utilized in cylindrical coordinates. The rotation direction, , contribution is approximated as in the following equation for a variable in the Helmholtz equation.

(8–28)

Additionally, the testing function is given by

(8–29)

Based upon the given equilibrium, the gradients of p, Q, and w are obtained. After substituting the reformed variables into Equation 8–12, the weak formulation for an axially-symmetric element is expressed as:

(8–30)

where:

= mode number

r = distance to rotation axis