3. The elasto-acoustic beam model

3.1. Assumptions

The vibrations of elastic, linear, homogeneous and isotropic pipes coupled to a compressible fluid are studied at low frequency.

The effects due to viscosity and fluid flow are overlooked.

Pipes are elongated bodies. In fact, their transverse dimensions are much less than their length: \(D\mathrm{\ll }L\), and the thicknesses are such that the methods of swelling and ovalization of the pipe can be overlooked. A beam model can be used.

At low frequency, the acoustic wavelengths associated with the problems studied are large compared to the transverse dimensions and small compared to the longitudinal dimension of the circuit: \(\omega \mathrm{.}L/c>1\) and \(\omega \mathrm{.}D/c\ll 1\). In fact, compressibility acts mainly on longitudinal movements. Transversely, it is considered that the fluid moves as an undeformable solid, that is, it acts as an added mass. Since the pressure in a straight section of the pipe is then constant, it is said that the acoustic wave is plane.

3.2. Functional of the coupled problem

The variational formulation of the problem of fluid-filled pipes can be written from the equations of equilibrium and behavior of the fluid and the pipe as well as from the boundary conditions. From the general functional of the three-dimensional coupled problem ([bib1], [bib2]), we can write the functional applied to the particular case of beams.

The variational formulation of the 3D problem amounts to minimizing the functional:

\(F(u,p,\Phi )=\frac{1}{2}\{{\int }_{{\Omega }_{s}}\left[\sigma (u):\varepsilon (u)-{\mathrm{\rho }}_{s}{\omega }^{2}{u}^{2}\right]\text{dV}\)

\(+{\int }_{{\Omega }_{f}}\left[\frac{{P}^{2}}{{\mathrm{\rho }}_{0}{c}^{2}}(\frac{\mathrm{2P}\Phi }{{\mathrm{\rho }}_{0}{c}^{2}}-{(\text{grad}\Phi )}^{2})\right]\text{dV}\}-{\mathrm{\rho }}_{0}{w}^{2}{\int }_{S}\Phiu \text{.}n\text{dA}\)

with:

\(\Omegas\), the field of structure

\(\Omegaf\), the fluid domain

\(\Sigma\), the fluid interaction surface - structure.

3.2.1. Contribution of pipework

The beam model used is that of Timoshenko with shear force deformations and rotational inertia of the cross section. It corresponds to modeling POU_D_T, whose elementary calculations it uses. We do not take into account the effects of ovalization [bib3].

The terms associated with piping in the variational formulation are then written:

\({\int }_{L}\left[\sigma (u):\varepsilon (u)-{\mathrm{\rho }}_{s}{\omega }^{2}{u}^{2}\right]{S}_{s}\text{ds}\)

with: \(L\), the middle fiber of the pipe and \({S}_{s}\), the pipe section at abscissa \(s\) (cf. [Figure3.2.1-A]).

Figure 3.2.1-a: pipe geometry

3.2.2. Fluid contribution

In this paragraph, we are only interested in the fluid part of the functional, that is to say, coupling term aside, in the term that is written in 3D:

\({\int }_{{\Omega }_{f}}\left[\frac{{P}^{2}}{{\mathrm{\rho }}_{0}{c}^{2}}-{\mathrm{\rho }}_{0}{\omega }^{2}(\frac{\mathrm{2P}\Phi }{{\mathrm{\rho }}_{0}{c}^{2}}-{(\text{grad}\Phi )}^{2})\right]\text{dV}\) eq 3.2.2-1

It is assumed that the pressure is broken down into two terms:

\(P(M(s),t)=p(s,t)+\tilde{p}(M(s),t)\)

where \(p\) is the value averaged over a straight section of the pressure:

\(p(s,t)=\frac{1}{{S}_{f}(s)}{\int }_{{S}_{f}(s)}P(M(s),t)\text{dM}\)

and \(\tilde{p}\) is a term of fluctuating pressure that corresponds to the contribution of transversal modes.

According to the assumptions in paragraph [§1], \(p\) verifies the 1-D Helmholtz equation and \(\tilde{p}\) the Laplace equation (incompressible). So the integral [éq 3.2.2-1] is broken down into two terms.

3.2.2.1. Term corresponding to the contribution of \(\tilde{p}\)

In movements perpendicular to the axis of the pipe, it is considered that the fluid intervenes only through its added mass [bib4], the term linked to \(\tilde{p}\) is therefore a term of inertia:

\({\int }_{L}{\mathrm{\rho }}_{0}{\omega }^{2}{({u}_{t})}^{2}{S}_{f}\text{ds}\)

\({u}_{t}\) being the transverse components of the displacement vector of the structure and \({S}_{f}\) being the cross section of the fluid at the abscissa \(s\).

3.2.2.2. Term corresponding to the contribution of \(p\)

\({\int }_{L}\left[\frac{{P}^{2}}{{\mathrm{\rho }}_{0}{c}^{2}}-\mathrm{\rho 0}{\omega }^{2}(\frac{\mathrm{2P}\Phi }{{\mathrm{\rho }}_{0}{c}^{2}}-{(\frac{\partial \Phi }{\partial s})}^{2})\right]{S}_{f}\text{ds}\)

3.2.3. Coupling term

3.2.3.1. Current section

From the references [bib4] and [bib5], it is shown that the coupling term \(C\):

\(C\text{=-}{\mathrm{\int }}_{S}{\rho }_{0}\Phi \mathrm{u}\text{.}\mathrm{n}\text{dA}\mathrm{=}{\mathrm{\int }}_{L}\mathrm{-}\frac{{\rho }_{0}\Phi }{R}\mathrm{u}\text{.}\mathrm{j}{S}_{f}\text{ds}+{\mathrm{\int }}_{L}{\rho }_{0}\Phi \mathrm{u}\text{.}\mathrm{i}\frac{{\text{dS}}_{f}}{\text{ds}}\text{ds}\)

The structural equilibrium equations and the plane wave propagation equation (Helmholtz) in the fluid are therefore coupled at the level of the bent parts and the straight parts where there is a change in the cross section of the pipe. In the case of a pipe with a constant straight cross section:

\(R\to \mathrm{\infty }\) and \(\frac{{\text{dS}}_{f}}{\text{ds}}\mathrm{=}0\) so \(C\mathrm{=}0\)

There is therefore no coupling between the movements of the beam of the structure and the longitudinal movements of the fluid in the right parts of the circuit. In this case, the fluid is characterized solely by its added mass associated with transverse displacements.

3.2.3.2. Pipe bottom

In the case of a pipe bottom, we note \(\Sigmat =\Sigma +{S}_{f}\), the total area of interaction between the fluid and the pipe.

In the case of a straight pipe with a closed constant cross section, the coupling term \(C\) is then equal to:

\(C\text{=-}{\int }_{S+{S}_{f}}{\mathrm{\rho }}_{0}\Phiu \text{.}n\text{dA}\text{=-}{\int }_{{S}_{f}}{\mathrm{\rho }}_{0}\Phiu \text{.}n\text{dA}\)

That is the background effect.

This term is added to the coupling term of a current section. Thus, a free node that achieves a zero flow condition across the [bib6] section realizes an acoustic background condition. In fact, an incident plane wave is completely reflected on the background: the acoustic pressure in the duct obeys the Helmoltz equation with zero normal pressure gradient (fluid and solid displacements being zero).

\(\{\begin{array}{c}\frac{{\partial }^{2}p}{\partial {x}^{2}}+{k}^{2}p=0\\ (\frac{\partial p}{\partial x}{)}_{{S}_{f}}=0\end{array}\)

We are looking for the solution in the form: \(p=A\text{cos}(\omegat -\text{kx})+B\text{cos}(\omegat +\text{kx})\)

i.e. in the form of a linear combination of an incident plane acoustic wave and a reflected wave.

The condition of zero gradient on the background, verified at all times, imposes:

\(A=B\)

The reflected wave is therefore « equal » to the incident wave (reflection coefficient equal to unity).

3.2.4. Functionality of the coupled system in the case of pipes

In the particular case that we are dealing with non-bent pipes, with a constant cross section, the functional of the coupled problem is therefore written in the following form:

\(\begin{array}{cc}F(u,p,\Phi )=& \frac{1}{2}\{{\int }_{L}\sigma (u):\varepsilon (u){S}_{s}\text{ds}-{\omega }^{2}{\int }_{L}\left[{\rho }_{s}{S}_{s}{u}^{2}+{\rho }_{0}{S}_{f}{({u}_{t})}^{2}\right]\text{ds}\\ & +{\int }_{L}\left[\frac{{P}^{2}}{{\rho }_{0}{c}^{2}}-{\rho }_{0}{\omega }^{2}(\frac{\mathrm{2P}\Phi }{{\rho }_{0}{c}^{2}}-{(\frac{\partial \Phi }{\partial \sigma })}^{2})\right]{S}_{f}\text{ds}\}-{\omega }^{2}{\int }_{{S}_{f}}{\rho }_{0}\Phi u\text{.}n\text{dA}\end{array}\)

3.3. Finite element discretization

The \((u,p,\Phi )\) solution sought minimizes functional \(F\). The finite element approximation of the complete problem then leads to the symmetric system:

\(\left[\begin{array}{ccc}K& 0& 0\\ 0& \frac{{K}_{f}}{{\mathrm{\rho }}_{0}\text{.}{c}^{2}}& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}u\\ p\\ \Phi \end{array}\right]-{\omega }^{2}\left[\begin{array}{ccc}M+{M}_{f}& 0& {M}_{\Sigma }\\ 0& 0& \frac{{M}_{\text{fl}}}{{c}^{2}}\\ {M}_{\Sigma }^{T}& \frac{{M}_{\text{fl}}^{T}}{{c}^{2}}& {\mathrm{\rho }}_{0}\text{.}H\end{array}\right]\left[\begin{array}{c}u\\ p\\ F\end{array}\right]=0\)

\(K\) and \(M\) being respectively the stiffness and mass matrices of the structure,

\({K}_{f}\), \({M}_{\text{fl}}\), \(H\) being the fluid matrices, respectively obtained from quadratic forms:

\({\int }_{L}{p}^{2}{S}_{f}\text{ds},{\int }_{L}p\Phi {S}_{f}\text{ds},{\int }_{L}{(\frac{\partial \Phi }{\partial s})}^{2}{S}_{f}\text{ds}\)

\({M}_{f}\) being the fluid matrix obtained from the quadratic form: \({\int }_{L}{p}_{0}{S}_{f}{({u}_{t})}^{2}\text{ds}\)

\({M}_{\Sigma }\) being the coupling matrix obtained from the bilinear form: \({\int }_{{S}_{f}}{p}_{0}\Phiun \text{dA}\).

By discretizing \(p\) and \(\Phi\) linearly, we therefore have:

\(p={p}_{1}\frac{L-x}{L}+{p}_{2}\frac{x}{L}\text{et}\Phi ={\Phi }_{1}\frac{L-x}{L}+{\Phi }_{2}\frac{x}{L}\), \(L\) being the length of the element in question.

In this case, the elementary fluid stiffness matrix is written as:

\({K}_{f}=\frac{{S}_{f}L}{3}\left\{{p}_{1}{p}_{2}\right\}\left[\begin{array}{cc}1& 1/2\\ 1/2& 1\end{array}\right]\left\{\begin{array}{c}{p}_{1}\\ {p}_{2}\end{array}\right\}\)

The elementary coupling matrix is written as:

\({M}_{\Sigma }={r}_{0}{S}_{f}\left\{{\Phi }_{1}{\Phi }_{2}\right\}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left\{\begin{array}{}{u}_{1}\\ {u}_{2}\end{array}\right\}\)

The various elementary fluid mass matrices can be written as:

\({M}_{\Sigma }={r}_{0}{S}_{f}\left\{{\Phi }_{1}{\Phi }_{2}\right\}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left\{\begin{array}{}{p}_{1}\\ {p}_{2}\end{array}\right\}\)

\({M}_{\Sigma }=-\frac{{S}_{f}}{L}\left\{{\Phi }_{1}{\Phi }_{2}\right\}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left\{\begin{array}{}{\Phi }_{1}\\ {\Phi }_{2}\end{array}\right\}\)

3.4. Implementation in Code_Aster

Based on the principles we have just described, a vibro-acoustic beam element, by Timoshenko for the pipe part, straight with a constant or variable cross section (in this case, only circular sections are allowed), has been implemented in*Code_Aster*. It belongs to the “FLUI_STRU” modeling of the “MECANIQUE” phenomenon.

This element has 8 degrees of freedom per node: the movements and rotations of the pipe, the pressure and the potential for fluid movement (see [Figure 3.4-a]). The formulation is written for local movements in the coordinate system local to the element consisting of the neutral fiber (axis \(X\)) and the main inertia axes (axis \(Y\), axis \(Z\)) of the section. The two scalars \(p\) and \(\Phi\) (pressure and potential of fluid movements) are invariant by change of frame of reference.

On each node of this element, Dirichlet-type boundary conditions can be imposed in terms of pressure, potential for fluid movements and displacements (translation or rotation).

Figure 3.4-a: fluid-filled beam element

To date, this element only makes it possible to calculate the natural modes of a straight pipe filled with fluid and to calculate the harmonic response. The effects of curvature or sudden widening of the section are not taken into account for the moment, but these fluid-structure effects, when dealing with low-density fluids such as steam from an intake pipe, do not seem to have a decisive importance on the calculation of the first modes: the correct mechanical representation of the elbow (flexibility coefficient) seems sufficient to calculate these frequencies [bib7].

In modal analysis, we can cite the case of a straight pipe filled with fluid with a free end:

Figure 3.4-b: embedded fluid-filled beam - free

The natural frequency of the tension-compression mode of this coupled fluid/structure system is given by the relationship:

\(\text{tg}(\frac{\omegal }{c})=\sqrt{\frac{{S}_{s}}{{S}_{f}}\frac{E}{{\mathrm{\rho }}_{0}{c}^{2}}}\)

We denote by:

\(E\): Young’s modulus of solid material

\({S}_{S}\): solid section

\({S}_{f}\): fluid section

Here we assume that the speed of speed of sound in the fluid is equal to the speed of sound in solid \({c}_{S}=\sqrt{\frac{E}{{\mathrm{\rho }}_{S}}}\) [bib7].

The transient response calculation for this type of finite element \((u,p,\phi )\) is not yet available in*Code_Aster*.