Acoustics of a Pipe System with3D Bend and Junction

Acoustics of a Pipe System with
3D Bend and Junction

The propagation and response of acoustic signals in pipe systems is used to detect leaks and deformations in the pipes. This is, for example, relevant in the oil and gas industry or in water delivery piping systems. This model uses the pipe acoustics interfaces to model the propagation of acoustic waves in a pipe system using a 1D model. The 1D model is coupled to a detailed 3D model of a junction and a bend. In the 3D domains, a pressure acoustics interface is used. The 3D domains could represent any complex geometry like valves, reservoirs, or pipe expansions where the 1D model is not valid. The tutorial sets up both a transient and a frequency-domain version of the model. A Pressure Acoustics, Boundary Mode interface is used to compute and analyze the propagating modes in the pipe cross section, as only plane propagating modes are supported in the 1D pipe formulation.

The application requires the Acoustics Module.

Model Definition

Pipe acoustics is used to model the propagation and response of acoustic signals in the long pipe segments using a 1D model. This interface is only valid for the analysis of plane wave modes. The 3D structures are modeled using pressure acoustics. To ensure correct coupling to the 1D model, the Acoustics-Pipe Acoustics Connection multiphysics coupling is used. The coupling requires a straight pipe segment connected to the 3D structure and it is only valid when the background fluid velocity speed is zero.

For a given pipe system in quiescent conditions (u0 = 0), constant background pressure, and completely rigid walls, the governing equations of the pipe system become:

(1)

(2)

where ρ is the fluid density, A is the cross-sectional area of the pipe, u is the area-averaged acoustic velocity, which is defined in the tangential direction u = uet, where et is the tangential vector, ∇t is the tangential derivative, τw is the tangential wall drag force, Z is the inner perimeter of the pipe, F is a volume force applied to the pipe, and c is the effective speed of sound in the pipe (in a rigid pipe this has the same value as the isentropic bulk speed of sound).

The pipe acoustics domain (pipe domain) is connected to the pressure acoustics domain (acoustic domain) through the Acoustics-Pipe Acoustics Connection multiphysics coupling. This coupling guarantees continuity of pressure and velocity between both domains through the equations:

(3)

(4)

Here, ppipe is the pipe pressure at the connection point, A is the total area of the connected acoustic boundary, pt is the total pressure in the acoustic domain and S is the connected acoustic boundary, n is the vector normal to the boundary, and upipe is the pipe velocity at the connection point. The possible dipole domain source, qd, is zero in this model.

The geometry of the pipe system, including both 1D and 3D domains, is shown in Figure 1.

Figure 1: Model geometry.

The model has two separated 3D domains, shown in Figure 2, which will be included in the model using the Pressure Acoustics interface.

Figure 2: T-junction and bend.

The pipe ends, not connected to the Pressure Acoustics domains, have an end impedance condition of an infinite pipe, allowing the acoustic waves to exit the system. In our case, this simply corresponds to the characteristic specific impedance.

(5)

The points where this impedance condition is imposed can be seen in Figure 3.

Figure 3: The pipe end impedance conditions are applied to the open ends of the pipe system.

The tutorial sets up both a transient and a frequency domain version of the model. Under the transient analysis, a wave train is created through the imposition of a volume force on one end of the pipe system. The force is defined through an analytical function created under the node. Figure 4 shows a plot of the analytical function used in the model to generate the wave train.

Figure 4: Imposed volume force as direct output of the plot functionality when defining an analytic function.

At last, a Pressure Acoustics, Boundary Mode interface is used to solve for the propagation modes on the pipe cross section at the driving frequency to check if higher order modes can propagate. This is easy and quick to check by solving an eigenvalue problem. The eigenvalue solver computes a specified number of solutions {pj, λj} to the equation

(6)

with

(7)

where pt is the total pressure, keq is the in-plane wave number, ω is the angular frequency, kn is the out-of-plane wave number, and λ = −ikn is the eigenvalue.

Results and Discussion

Figure 5 shows the transient pressure distribution in the pipe system at various times. The figure shows the continuity in the pressure at the Acoustics-Pipe Acoustics Connection points.

Figure 5: Evolution of the Pressure in the pipe system at various times.

The initial wave train travels through the pipe system producing reflections as the wave finds discontinuities at the T-junction and the bend. Figure 6 shows the pressure at different points of the pipe system. The reflected wave from the T-junction can be observed in Point A after 0.09 s and the reflected wave coming from the pipe bend can be seen after 0.13 s. By updating the parameter rbend, that controls the bending radius of the bend, to 160 cm this reflection can be greatly reduced.