3. Reminders of the equations of the problem

3.1. Equations in the fluid

It is assumed that \(K\) vibrating structures are immersed in a perfect fluid domain (non-viscous), incompressible and at rest (without flow). The effect of gravity is overlooked. We can therefore write the Euler equations associated with the fluctuations in the fluid without flow:

`

• mass conservation:

\(\frac{\partial {\rho }_{f}}{\partial t}+\text{div}({\rho }_{f}\mathrm{v})=0\) eq 3.1-1

• conservation of momentum (in a linearized hypothesis):

\(\frac{\partial \mathrm{v}}{\partial t}+(\mathrm{v}\mathrm{.}\mathrm{grad})\mathrm{v}+\frac{1}{{\rho }_{f}}\mathrm{grad}p=0\) eq 3.1-2

Because of the incompressibility of the fluid, equation [éq 3.1-1] becomes:

\(\text{div}v=0\) eq 3.1-3

In volume \({\mathrm{\Omega }}_{f}\) of the fluid, the convection induced by the supposed low-amplitude movement of the structure is overlooked. So the equation [éq 3.1-2] becomes:

\(\frac{\partial \mathrm{v}}{\partial t}+\frac{1}{{\rho }_{f}}\mathrm{grad}p=0\) eq 3.1-4

By differentiating [éq 3.1-3] with respect to time and transferring the expression for \(\frac{\partial v}{\partial t}\) as a function of pressure into this equation, we get:

\(\text{div}\mathrm{grad}p=0\)

either:

\(\Deltap =0\) in \({\mathrm{\Omega }}_{f}\)

which is the Laplace equation in a fluid without flow.

At the fluid/structure interface, which is assumed to be airtight, it can be written that the normal acceleration of the wall of the structure is equal to the normal acceleration of the fluid (continuity of normal accelerations—impermeability condition of the structure). Here we use the following convention for the normal: it is the normal exterior to the structure, oriented from the structure to the fluid.

\(\frac{\partial \mathrm{v}}{\partial t}\mathrm{.}\mathrm{n}={\ddot{\mathrm{x}}}_{{S}_{l}}\mathrm{.}\mathrm{n}\)

With equation [éq 3.1-4], we get:

\(\text{grad}p\mathrm{.}\mathrm{n}\text{=}-{\rho }_{f}\frac{\partial \mathrm{v}}{\partial t}\mathrm{.}\mathrm{n}\text{=}-{\rho }_{f}{\ddot{\mathrm{x}}}_{{S}_{l}}\mathrm{.}\mathrm{n}\)

That is:

\((\frac{\partial p}{\partial \mathrm{n}}{)}_{{\gamma }_{l}^{\mathit{fs}}}\text{=}-{\rho }_{f}\ddot{{\mathrm{x}}_{{S}_{l}}}\mathrm{.}\mathrm{n}\) out of \({\gamma }_{l}^{\mathit{fs}}\), fluid interface/structure of the structure indexed by \(l\).

In summary, the fluid problem consists in solving a Laplace equation in the fluid domain with von Neumann-type boundary conditions on mobile walls \({\mathrm{\Gamma }}_{\mathit{fs}}\) and fixed walls \({\mathrm{\Gamma }}_{\mathit{fix}}\) and Dirichlet conditions on the free surface \({\mathrm{\Gamma }}_{\mathit{lib}}\):

\(\{\begin{array}{c}\mathrm{\Delta }p=0\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ (\frac{\partial p}{\partial \mathrm{n}}{)}_{{\mathrm{\Gamma }}_{\mathit{fs}}}\text{=}-{\rho }_{f}\ddot{{\mathrm{x}}_{s}}\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{\mathit{fs}},\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{\mathit{fs}}=\text{}\underset{l=1,K}{\cup }{\mathrm{\gamma }}_{l}\\ (\frac{\partial p}{\partial \mathrm{n}}{)}_{{\mathrm{\Gamma }}_{\mathit{imp}}}=0\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{2},\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{2}=\partial {\mathrm{\Omega }}_{f}-{\mathrm{\Gamma }}_{\mathit{fs}}\end{array}\) eq 3.1-5

3.2. Equations in structures

Consider \(K\) elastic structures bathed by a fluid medium. The equation of their motion in the presence of fluid is written, having noted \(\mathrm{U}\) their degrees of freedom:

\(\{\begin{array}{c}\forall l\phantom{\rule{2em}{0ex}}\text{indice de structure},l\in \text{{}0,\mathrm{...},K\text{}},\phantom{\rule{2em}{0ex}}{\mathrm{M}}_{l}\mathrm{.}{\ddot{\mathrm{U}}}_{l}+{\mathrm{K}}_{l}\mathrm{.}{\mathrm{U}}_{l}={\mathrm{F}}_{l}\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{{S}_{l}},\text{volume de la structure}\phantom{\rule{2em}{0ex}}l\\ \forall l,\mathrm{\sigma }\mathrm{.}\mathrm{n}\text{=}-p\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}{\mathrm{\gamma }}_{l}^{\mathit{fs}},\text{contour de la structure}\phantom{\rule{2em}{0ex}}l\end{array}\)

\({M}_{l}\) is the mass matrix of the structure, \({\mathrm{K}}_{l}\) its stiffness matrix. The boundary condition on the outline of the structures reflects the continuity of the normal stress at the fluid/structure interface (the fluid stress tensor being reduced to its non-deviatory part, the fluid being perfect). By integrating this normal stress into the outline of each structure, we obtain a force \({\mathrm{F}}_{l}\) resulting from the pressure forces of the fluid at the fluid/structure interface. This force is the integral of the pressure field on contour \({\gamma }_{l}^{\mathit{fs}}\) of each structure:

\(\forall l\phantom{\rule{2em}{0ex}}\text{indice}\text{de}\text{structure},l\in \text{{}0,\mathrm{...},K\text{}},\phantom{\rule{2em}{0ex}}{\mathrm{F}}_{l}\text{=}-\underset{{\mathrm{\gamma }}_{l}^{\mathit{fs}}}{\int }p\mathrm{n}\mathit{dS}\)

The pressure field verifies the problem [éq 3.1-5].

3.3. Coupled problem equations - highlighting the added mass matrix

In short, the fluid/structure problem can be written as:

\(\{\begin{array}{c}\mathrm{\Delta }p=\text{0}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ \forall l\in \text{{}0,\mathrm{...},K\text{}},{\left(\frac{\partial p}{\partial \mathrm{n}}\right)}_{{\mathrm{\gamma }}_{l}^{\mathit{fs}}}\text{=}-{\rho }_{f}{\ddot{\mathrm{x}}}_{{S}_{l}}\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\gamma }}_{l}^{\mathit{fs}}\\ \forall l\in \text{{}0,\mathrm{...},K\text{}},\phantom{\rule{2em}{0ex}}{\mathrm{M}}_{l}\mathrm{.}{\ddot{\mathrm{U}}}_{l}\text{+}{\mathrm{K}}_{l}\mathrm{.}{\mathrm{U}}_{l}\text{=}{\mathrm{F}}_{l}\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{{S}_{l}}\\ \forall l\in \text{{}0,\mathrm{...},K\text{}},\phantom{\rule{2em}{0ex}}{\mathrm{F}}_{l}\text{=}-\underset{{\mathrm{\gamma }}_{l}}{\int }p\mathrm{.}\mathrm{n}\mathit{dS}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\gamma }}_{l}^{\mathit{fs}}\end{array}\) eq 3.3-1

We will now show that the effort that submerged structures experience is proportional to their acceleration. A good way to demonstrate this is to place yourself in the modal base of structures in a vacuum. We can thus break down the acceleration on this basis (which is in fact the combination of the modal bases of each of the structures). So:

\({{x}_{{S}_{l}}}_{}(r,t)\text{=}\sum _{\text{i=1}}^{\infty }{a}_{\mathrm{il}}(t){X}_{\mathrm{il}}(r)\)

By putting this expression into the second equation of the system [éq 3.3-1], we have to search for the pressure field in the form:

\(p\text{=}\sum _{l=\mathrm{1,}\mathrm{...},K}\sum _{i=\mathrm{1,}\mathrm{...},\infty }{\ddot{a}}_{\mathrm{il}}(t){p}_{\mathrm{il}}(r)\)

By bringing these expressions into the problem [éq 3.3-1], we have to solve in the fluid as many Laplace problems as we have chosen modes for each of the structures. This results in:

\(\forall l\in \text{{}1,\mathrm{...},K\text{}},\phantom{\rule{2em}{0ex}}\forall i\text{{}1,\mathrm{...},\mathrm{\infty }\text{}},\phantom{\rule{4em}{0ex}}\{\begin{array}{c}\mathrm{\Delta }{p}_{\mathit{il}}=\text{0}\phantom{\rule{2em}{0ex}}\text{dans}\text{}{\mathrm{\Omega }}_{f}\\ {\left(\frac{\partial {p}_{\mathit{il}}}{\partial \mathrm{n}}\right)}_{{\mathrm{\gamma }}_{l}^{\mathit{fs}}}\text{=}-{\rho }_{f}{\mathrm{x}}_{\mathit{il}}\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\gamma }}_{l}^{\mathit{fs}}\\ \left[\begin{array}{c}{\mathrm{m}}_{\mathit{il}}\end{array}\right]({\ddot{\mathrm{a}}}_{l})\text{+}\left[\begin{array}{c}{\mathrm{k}}_{\mathit{il}}\end{array}\right]({\mathrm{a}}_{l})\text{=}({\mathrm{f}}_{\mathrm{il}})\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{l}\end{array}\)

The generalized mass and stiffness « matrices » written in these bases are diagonal.

Each of the components of the resulting pressure force projected on a modal basis is written as:

\(\forall i\text{{}1,\mathrm{....},\mathrm{\infty }\text{}},\forall l\text{{}1,\mathrm{....},K\text{}},\phantom{\rule{2em}{0ex}}\left({f}_{\mathit{il}}\right)\text{=}-\sum _{k=1}^{K}\sum _{j=1}^{\mathrm{\infty }}{\ddot{a}}_{\text{jk}}\underset{{\mathrm{\gamma }}_{l}^{\mathit{fs}}}{\int }{p}_{\text{jk}}{X}_{\mathit{il}}\mathrm{.}\mathrm{n}\mathrm{.}{N}_{j}\mathit{dS}\)

We can then write the vector of the generalized pressure force on a submerged structure in matrix form:

\(({\mathrm{f}}_{\mathit{il}})=-\left[\begin{array}{c}{\mathrm{m}}_{\mathit{il}\text{jk}}\end{array}\right]\mathrm{.}{\ddot{a}}_{\text{jk}}\phantom{\rule{2em}{0ex}}\text{avec}\phantom{\rule{2em}{0ex}}{m}_{\mathit{il}\text{jk}}=\underset{{\mathrm{\gamma }}_{l}^{\mathit{fs}}}{\int }{p}_{\text{jk}}{X}_{\mathit{il}}\mathrm{.}\mathrm{n}\mathit{dS}\)

Here, \(l\) is fixed: the matrix consisting of the components \(\left[{\mathrm{m}}_{\mathit{il}\text{jk}}\right]\) is called mass matrix added of the fluid on the contour structure \({\gamma }_{l}^{\mathit{fs}}\). When considering the modal basis of all \(K\) structures, we generalize the notation of the added mass matrix of \(\left[{\mathrm{m}}_{\mathit{il}\text{jk}}\right]\) components on a modal basis in a vacuum, \(l\) varying from \(1\) to \(K\). This matrix is generally non-diagonal.

3.4. Some definitions

3.4.1. Definition 1

When \(l\mathrm{=}k\) (same structure) and \(i=j\) (same mode order), the coefficient \({m}_{\mathrm{ilil}}\) is the auto‑mass added of the mode \(i\) of the \(l\) structure. This is the additional inertia due to the fluid displaced by the mode of order \(i\) of the structure, taking into account the geometric confinements induced in the fluid by the presence of the other structures that are supposed to be fixed.

3.4.2. Definition 2

When \(l\mathrm{=}k\) (same structure) and \(i\mathrm{\ne }j\) (different order of mode), the coefficient \({m}_{\mathrm{iljl}}\) is the mass added of coupling between the modes of order \(i\) and \(j\) of the structure \(l\). In air, these extra-diagonal mass terms are zero, because the modes are orthogonal to each other. Given the general expression of the coefficient \({m}_{\mathrm{il}\text{jk}}\), the modes \(i\) and \(j\) can be coupled en masse, because the pressure field \({p}_{\mathit{jl}}\) created by the mode \(j\) of the structure \(l\) is not necessarily orthogonal to the mode of order \(i\) of this same structure. It is sufficient for this structure to be immersed in an environment that does not include a geometric symmetry for this coefficient to be non-zero. In a symmetric environment, on the other hand, the orthogonality of the pressure field with the mode is observed.

3.4.3. Definition 3

When \(l\mathrm{\ne }k\) (different structures) and \(i\ne j\) (different order of mode), the coefficient \({m}_{\mathrm{il}\text{jk}}\) is the mass added of coupling between the modes of order \(i\) and \(j\) respectively of the structures \(l\) and \(k\). This coefficient reflects the inertial force that the structure \(k\) vibrating in its mode of order \(j\) subjects to the structure \(l\) vibrating in its mode \(i\).

3.5. Properties of the added mass matrix

3.5.1. Theorem 1: the added mass matrix is symmetric

To simplify the demonstration, we will consider a single structure immersed in a perfect, incompressible and non-viscous fluid. We break down the movement of the structure on its modal basis (truncated to \(n\) modes), but the result can be demonstrated just as well on a « physical » basis (i.e. the basis of nodal interpolation functions). Finally, the result is generalized in the case of \(K\) structures immersed in the same fluid.

We need to show that: \({m}_{\text{ij}}=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}{\mathrm{X}}_{j}\mathrm{.}\mathrm{n}\mathit{dS}={m}_{\text{ji}}=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{j}{\mathrm{X}}_{i}\mathrm{.}\mathrm{n}\mathit{dS}\)

• \({p}_{i}\) (respectively \({p}_{j}\)) represents the pressure field created in the fluid and at the interface with the structure by the mode of order \(i\) (respectively of order \(j\)) of the structure,

• \({\mathrm{X}}_{j}\) (respectively \({\mathrm{X}}_{i}\)) represents the modal deformation of the mode of order \(j\) (respectively of order \(i\)).

Gold:

\(\{\begin{array}{c}\mathrm{\Delta }{p}_{i}=0\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ \frac{\partial {p}_{i}}{\partial \mathrm{n}}\text{=}-{\rho }_{f}{\mathrm{X}}_{i}\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}\mathrm{\Gamma }\end{array}\) and \(\{\begin{array}{c}\mathrm{\Delta }{p}_{j}=0\phantom{\rule{2em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ \frac{\partial {p}_{j}}{\partial \mathrm{n}}\text{=}-{\rho }_{f}{\mathrm{X}}_{\mathrm{j}}\mathrm{.}\mathrm{n}\phantom{\rule{2em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}\mathrm{\Gamma }\end{array}\)

Hence, using Green’s formula with a normal oriented from the structure to the fluid and the harmonicity of \({p}_{i}\) and \({p}_{j}\):

\(\begin{array}{c}{m}_{\text{ij}}=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}{\mathrm{X}}_{j}\mathrm{.}\mathrm{n}d\mathrm{\Gamma }\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}\frac{\partial {p}_{j}}{\partial \mathrm{n}}\mathrm{.}\mathrm{n}dS\\ \phantom{\rule{6em}{0ex}}\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}(\underset{0}{\underset{⏟}{\underset{\mathrm{\Omega }}{\overset{}{\int }}{p}_{i}\mathrm{\Delta }{p}_{j}d\mathrm{\Omega }}}-\underset{\mathrm{\Omega }}{\overset{}{\int }}\mathrm{grad}{p}_{i}\mathrm{.}\mathrm{grad}{p}_{j}d\mathrm{\Omega })\\ \phantom{\rule{6em}{0ex}}\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}(\underset{0}{\underset{⏟}{\underset{\mathrm{\Omega }}{\overset{}{\int }}{p}_{j}\mathrm{\Delta }{p}_{i}d\mathrm{\Omega }}}-\underset{\mathrm{\Omega }}{\overset{}{\int }}\mathrm{grad}{p}_{j}\mathrm{.}\mathrm{grad}{p}_{i}d\mathrm{\Omega })\\ \phantom{\rule{6em}{0ex}}\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{j}\frac{\partial {p}_{i}}{\partial \mathrm{n}}\mathrm{.}\mathrm{n}dS=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{j}{\mathrm{X}}_{i}\mathrm{.}\mathrm{n}dS\\ \phantom{\rule{6em}{0ex}}={m}_{\text{ji}}\end{array}\)

3.5.2. Theorem 2: the added mass matrix is positive definite

Refer to reference [bib1] for the complete demonstration.

3.5.3. Theorem 3

Suppose we have \(K\) structures that have identical linear elasticity properties and that are immersed in the same fluid. In addition, these structures allow two degrees of freedom of movement in plane \(\mathit{Oxy}\) (cf. diagram in § 3.1). Each of these structures admits the same spectrum \({f}_{1},\mathrm{...},{f}_{n},\mathrm{...}\) of natural frequencies in a vacuum.

For any natural frequency \({f}_{n}\), there are \(2K\) natural frequencies \(\left\{{\omega }_{1},\mathrm{...},{\omega }_{\mathrm{2K}}\right\}\) of the fluid/structure coupled system verifying \(\mathrm{\forall }i\mathrm{\in }\left\{\mathrm{1,}\mathrm{...}\mathrm{,2}K\right\},{\omega }_{i}\mathrm{\le }{f}_{n}\)

Refer to reference [bib1] for the complete demonstration.

3.5.4. Other properties

The added auto-mass coefficients are always positive.

We always assume that we have a single structure immersed in a perfect, incompressible and flowless fluid. The demonstration is easily generalized to \(K\) submerged structures.

It must be shown that:

\(\forall i\text{indice}\text{de}\text{mode}\phantom{\rule{2em}{0ex}}\in \left\{1,\mathrm{...},n\right\},\phantom{\rule{2em}{0ex}}{m}_{\text{ii}}=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}{\mathrm{X}}_{i}\mathrm{.}\mathrm{n}dS\phantom{\rule{2em}{0ex}}\ge \phantom{\rule{2em}{0ex}}0\)

Gold:

\(\begin{array}{c}{m}_{\text{ii}}=\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}{\mathrm{X}}_{i}\mathrm{.}\mathrm{n}dS\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}\underset{\mathrm{\Gamma }}{\overset{}{\int }}{p}_{i}\frac{\partial {p}_{i}}{\partial \mathrm{n}}dS\\ \phantom{\rule{4em}{0ex}}\text{=}-\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}(\underset{0}{\underset{⏟}{\underset{{\mathrm{\Omega }}_{f}}{\overset{}{\int }}{p}_{i}\mathrm{\Delta }{p}_{i}d\mathrm{\Omega }}}-\underset{\mathrm{\Omega }}{\overset{}{\int }}\mathrm{grad}{p}_{i}\mathrm{.}\mathrm{grad}{p}_{i}d\mathrm{\Omega })\\ \phantom{\rule{4em}{0ex}}=\frac{\phantom{\rule{2em}{0ex}}1}{{\rho }_{f}}\underset{{\mathrm{\Omega }}_{f}}{\overset{}{\int }}(\mathrm{grad}{p}_{i}{)}^{2}d\mathrm{\Omega }\\ \phantom{\rule{4em}{0ex}}\ge 0\end{array}\)

Suppose we have \(K\) structures immersed in the same fluid. It is assumed that they have only one degree of translational freedom following \(\mathit{Ox}\). So the sum of all the added mass coefficients of this matrix gives the self-mass added over all of the \(K\) structures all moving in the same sinusoidal rectilinear motion.

Refer to reference [bib2] for the complete demonstration.