2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The aim is to determine the temporal displacement \(z(t)\) of the wall piston.

We consider the plane problem of this meridian model whose geometric, mechanical and fluid characteristics are described in figure [Figure 1.1-a]; the lateral piston is of length \(\mathrm{2L}\).

The two-dimensional problem is reduced to a one-dimensional problem by considering by approximation that the cross-flow velocities

induced by the movement of the piston are instantaneously transmitted in axial velocities in the channel.

In the control volume \(d\mathrm{\Omega }=\mathit{edx}\) with an area \(\mathit{dx}\) under the wall piston we can write:

(2.1)#\[ d (\ mathrm {\ delta} V) =\ frac {1} {2e}\ mathit {xx} d\ dot {z}\]

In the fluid, the variation in speed and the variation in pressure in adiabatic evolution are linked by:

(2.2)#\[ \ mathrm {\ delta} P= {\ mathrm {\ rho}}} _ {f} {f}\ mathrm {\ delta} V\]

The pressure at time \(t\) at a point on the abscissa \(x\) results from the superposition of the propagation of all the elementary sources distributed on the piston:

The coupling therefore consists in this: the movement of the acceleration piston \(\ddot{z}(t)\) induces in the channel a pressure field \(P(x,t)\) whose resulting force on the extent of the piston itself acts in return on the dynamics of the oscillator.

The geometric, mechanical, and fluid characteristics of the model are shown in figure [Figure 1.1-b].

It is first considered that the piston and the fluid are at rest and the oscillator is released at time \(t=0\) by imposing an initial speed on it.

The expression of pressure

At one point in the channel develops:

:math:`P(x,t)=frac{{mathrm{rho }}_{f}{c}_{f}}{2e}underset{0}{overset{t}{int }}left[underset{-L}{overset{x}{int }}ddot{z}left(mathrm{tau }-frac{

x-u

}{{c}_{f}}right)mathit{du}+underset{x}{overset{L}{int }}ddot{z}left(mathrm{tau }-frac{

u-x

}{{c}_{f}}right)mathit{du}dmathrm{tau }right]`

We have \(z(0)=0\) and \(z(t)=0\) for negative \(t\) and \(z\left(-\frac{L-x}{{c}_{f}}\right)=0\) since upstream of the piston (i.e. for \(x\) between \(-L\) and \(L\)) the quantity in brackets is always negative. Likewise, we have \(z\left(-\frac{L+x}{{c}_{f}}\right)=0\)

Finally it comes:

\[\]

: label: eq-4

P (x, t) =frac {{mathrm {rho}}} _ {f} {c} _ {f} ^ {2}} {2nd}left [2z (t) -zleft (t-frac {mathrm {rho}}}}left (t-frac {l+X}left (t-frac {L+x}left) (t-frac {L+x} {{c}} _ {c} _ {f}}right)right]

We integrate this expression on \(x\) in order to obtain the resultant of the pressure forces on the piston:

\[\]

: label: eq-5

R (t) =-Hunderset {-L} {overset {L} {overset {L} {L} {L} {overset {L} {overset {L} {overset {L} {int}} P (x, t)mathit {xx}}

Indeed \(P(x,t)\) is even in \(x\); it is therefore sufficient to integrate on half of the piston.

Hence the expression of the resultant of the pressure forces on the piston in the hypothesis of small movements:

(2.3)#\[ R (t) =-\ frac {H {\ mathrm {\ rho}}} _ {f} {c} _ {f} ^ {2}} {e}\ left [2\ mathit {Lz} (t) - {c} _ {f}} - {f}\ underset {t} {f}} (t) - {c} (t) - {c} (t) - {c} _ {f} (t) - {c} _ {f} (t) - {c} _ {f} (t) - {c} _ {f} (t) - {c} _ {f} (t) - {c} _ {f}\ int}} z (u)\ mathit {du}\ right]\]

The movement of the oscillator therefore obeys the equation:

(2.4)#\[ M\ ddot {z} +Kz+\ frac {2\ mathit {HL} {\ mathit {HL}} {\ mathrm {\ rho}}} {c} _ {f} ^ {2}} {e} z-\ frac {2} z-\ frac {2}} z-\ frac {2}} z-\ frac {2}} {e} {e} {c} _ {f}\ underset {t-\ frac {2L} {{c} _ {f}}} {\ overset {t} {\ int}}} z (u)\ mathit {du} =0\]

or even:

(2.5)#\[ M\ ddot {z} - {F} _ {\ text {int}}} - {F} _ {\ text {cpl}} =0\]

If we ask:

(2.6)#\[ {F} _ {\ text {int}} =-Kz-\ frac {2\ mathit {HL} {\ mathrm {\ rho}} _ {f} {c} _ {f} _ {f} ^ {2}} {e} z\]

We now consider the case of the propagation of a decompression wave with a steep front of amplitude \(\mathrm{\Delta }{P}_{0}\) along the duct. At instant \(t=0\), this wave attacks the wall piston still at rest, creating an excitation force on this piston such that:

(2.7)#\[\begin{split} {F} _ {\ text {exc}} =\ {\ begin {array} {ccc}} H {c} _ {f} t\ mathrm {\ Delta} {P} _ {0} &\ text {si}} & t<\ frac {2L} & t<\ frac {2L} {L} {L} {L}} {{c} _ {f}}}\\ 2HL\ mathrm {\ Delta} {P} {P} {si}} &\ text {si} &\ text {si} & t\ ge\ frac {2L} {{l}} {{c} _ {f}}\ end {array}\end{split}\]

The equation of motion is then written as:

(2.8)#\[ M\ ddot {z} = {F} _ {\ text {int}} + {\ text {int}}} + {\ text {cpl}} + {F} _ {\ text {exc}}\]

This equation is solved numerically for the presented characteristics of the meridian oscillator.

2.2. Benchmark result#

Displacement \(z(t)\) of the wall piston.

2.3. Uncertainty of the solution#

Analytical solution.

2.4. Bibliographical references#

1. STIFKENS: « Transient calculation in Code_Aster with vibro-acoustic elements ». Internal R&D note HP-51/97/026/A.
1. TEPHANY, A. HANIFI, C. LEHAUT: « Elements for the analysis of fluid-structure interaction in the annular cell-core shell space in the case of APRP » - Internal note SEPTEN ENTMS /94.057.