CFD analysis of thermal fluid dynamic parameters inside an organ pipe: influence of air temperature variations

Organ pipes represent a fluid-dynamic challenge since the time of Bernoulli and Reynolds. A complete and reliable description of the fluid behaviour at the basement of the resonator named the “mouth” of the pipe is still absent. In the context of the design of an air-heating system for the attenuation of the inconsistencies between sounding frequencies of organ pipes, computing efforts were carried out focusing on the fluid-dynamic behaviour of the air at the mouth of the pipe at different temperature conditions. Describing the air dynamic and the heat transfer at the mouth level of the pipe, it would be possible to predict the heating conditions within the pipe resonator, which was the main subject of the project. Multiple points on the pipe organ air blowing scheme were considered for simulation results extraction. Temperature, pressure, and audio measurements were performed and no significant influence of the temperature on those parameters was raised from measurements, rising questions about the effective behaviour of the blowed air at the mouth of the pipe. Simulations were performed to predict both the fluid dynamic and the heat transfer. A finite volume approach in Fluent environment was chosen. The SST k-omega model was considered in reliability vs computation time balance. 3D simulations were performed based on a CAD reconstructed model of a real pipe. The results of the simulations confirmed experimental data, and they may prove the absence of a significant mass exchange between the in and outside of the mouth of the pipe. The oscillation of the air membrane at the mouth of the pipe may be cleared from instabilities and turbulence regime phenomena, as usually described in the literature.


Introduction
Pipe organs, with their rich and resonant tones, have enthralled audiences for centuries with their majestic and awe-inspiring sound.The intricate design and craftsmanship of these instruments involve a delicate balance of mechanics, acoustics, and materials.However, pipe organs also face thermal challenges that can significantly impact their performance and acoustic tuning [1,2].To address these challenges, the integration of Computational Fluid Dynamics (CFD) techniques has emerged as a stateof-the-art approach for understanding the physics behind sounding organ pipes [3][4][5][6].Pipe organs are complex musical instruments composed of a multitude of elements, including pipes, wind reservoirs, bellows, keyboards, and mechanisms for controlling airflow.The sound produced by a pipe organ is intricately connected to the physical properties of its pipes, which are designed to generate specific pitches and timbres.Achieving optimal acoustic tuning requires meticulous attention to factors such as pipe dimensions, materials, voicing techniques, and wind pressure regulation.Temperature and humidity fluctuations can significantly impact the performance, stability, and longevity of pipe organs.Recently, studies conducted report a shift of the 6% in the pipe tuning of a real pipe organ in a church in the Mediterranean area of Perugia, Italy, when the indoor temperature and relative humidity are in the 6 30 ° and 40 90% ranges, respectively [2].Understanding and managing these thermal challenges are crucial for maintaining the desired sound quality and overall functionality of pipe organs.In the literature, it is possible to find examples of heating system tested on church pipe organs.In St. Martin church in Oberesslingen, Germany, a fan coils heating system was experimented to fix beating effects between the different sections of the instrument [7].The researchers identified a threshold temperature difference of ∆ 0.5° between the sections of the instrument, resulting in emitted frequency difference of 0.3 , which generates a perceivable beating effect.Unfortunately, the designed system could limit the temperature variations only ∆ 1.4°.
In this work, a different heating method and system was tested on a test-bench pipe organ provided by Pinchi srl at the Environmental Control Lab at the University of Perugia.The heating system is directly embedded into the blowing system of the pipe organ itself, providing hot air at different temperature stages.The influence of the changing air conditions of the blowing air into the pipe-resonant system were measured in terms of temperature, acoustic pressure, and frequency.The absence of a significant influence on the resulting sounding frequency by the temperature change was investigated by a CFD model of the measured pipe.A Reynolds Average Navier-Stokes (RANS) SST k-omega model was used for the simulations and the hot-air flux was analyzed and compared to measurements.Interestingly, results may prove the absence of turbulence at the mouth level of the pipe, conversely to the hypothesis of Reinholds and Rosenhead [8,9], and the modal excitation of the resonance tube may be guided by an oscillating air flux where no turbulence phenomena happen.

Materials and method
To investigate the influence of temperature changes on the blowing air through the pipe organ system on the resulting frequency, a heating system was installed on the laboratory pipe organ provided by Pinchi srl organbuilder at the Environmental and Control Laboratory at the Engineering department of the University of Perugia.Italy.

The heating system and the measurement campaign
The heating system design followed the scheme reported in Fig. 1 and consisted in an electric heater with two temperature set points (333 and 383 ), which influence the air boosted to the windchest of the instrument by the fan-bellows system (Fig. 2).
A F1 sounding pipe was chosen for measurements because its dimensions (reported in Table.1) are average for a pipe of an organ.Acoustic pressure and temperature were measured during the whole process, and the following measurement set-up was involved: -8 thermocouples managed by a central device-manager SpecView (± 0.02°C); -2 Tinytags TGU-1500 probes for room T and RH (± 0.2°C from 0°C to 50°C); -2 Endevco pressure transducers class 8507 C-1 / C-2 -2 mm Ø (±50 mV/psi); -2 Bruel&Kjaer pressure transducer types 4938 and 4939 (± 2 dB in 4 Hz to 70 kHz range); -NI-9234 cards for transducer acquisition with a  51.2 .
-Fluke 922 airflow meter for air velocity data acquisition (±1 m/s).Temperature and pressure probes' locations are reported in Fig. 3.The Endevco pressure probes were located at the rank-chest and the inner part of the mouth of the pipe, together with the thermocouples (Fig. 4), whereas the Bruel&Kjaer pressure probes (bigger in size) were outside the mouth of the pipe

The CFD model
The Computational Fluid-Dynamic CFD simulation was done in Fluent Ansys environment.The pipe was modeled together with the volume of the inner air in a 3D CAD file.The 3D model (Fig. 6) was modelled based on metric measurement and photo relief, and represents a simplified geometrical model of the pipe especially at the mouth level, where craftmanship modifications to the geometry of the pipe made it difficult to be realistically reported into the virtual model.The 3D model was imported into Ansys mesher module and a detailed mesh of 1003107 nodes and 5349166 elements were generated (element size 1.2e-003 m [max 2.4e-003m, defeature 6e-006m], target Skewness=0.9).Further refinement was reserved to the area of the mouth, with an increase of the volume number of 6 times (Fig. 7).. Then inlet and outlet surfaces were defined (Fig. 6), and the 3D model was imported into the Fluent software environment for CFD calculation, where gravity acceleration was also considered in the y-vertical-direction.The fluid boundaries were set up with wall boundary condition with roughness constant of 0.5.Default Fluent converge criteria were used, 1e-03 for continuity, momentum and 1e-06 for energy equations.Although it would be possible to analyse only half of the 3D Geometry of the pipe due to its symmetry around the y-longitudinal axis, the model was completely considered to study turbulence within the pipe.RANS analysis was considered because both the k − ϵ and k − ω models have complementary pros and cons in the simulation of a pipe, where the turbulent regime is not well defined, and the presence of the wall is essential, especially at the mouth level.It was chosen to perform a transient analysis using the SST k-ω model.The model is based on the differential equation system as available in the literature [10,11].
SST k-ω model was chosen among the possible RANS model because: -it combines the high reliability in predicting flows present in boundary layers due to gradients of contrary pressure and flow separation of the k − ω model, and the high accuracy of k − ϵ model solving turbulent flows; -compared with the BSL model, it considers the shear stress transport due to the presence of turbulence.This may be important in the dynamic of the fluid at the mouth to evaluate the cause of the oscillation of the "air membrane", as introduced by the work of Elder [9].In green, the wall boundary condition layer.

Results and discussion
The results of the measurements on F1 pipe, at different working stages of the heater, are reported in Fig. 8-13 and Table 2. Fig. 6-9 show the temperature trend over functioning time of the heater at the measurement points reported in Fig. 3, plus all the stages of the air blowing system.In particular, the three stages are highlighted by the temperature trends at the upstream measurement, from the off stage, the first stage appear between 1000 and 1500 , while the second stage between 1500 and 2000 .Fig. 3 highlights the significant thermal dissipation at the heating windchest stage, where more than 80  of temperature decrease are reported.Furthermore, temperature trends focused on the sounding pipe system show important thermal dissipation estimated in a 3 4  decrease between the mouth and the top of the pipe.However, although there is a thermal dissipation between the mouth and the top of the pipe, the most significant thermal dissipation is measured at the previous windchest-pipe foot stage, where temperature decreases by about 20 .The thermal dissipation at that stage, and the consequent absence of a significant influence of the heating system influence on the air-flow temperature, can explain why variations on the pressure are visible only at the pipe-foot inlet level (Fig. 10, Table 2), and there are variations at the mouth and top of the pipe (Fig. 11-12, Table 2).Pressure magnitudes from measurements reported in Table 2 are trustworthy because the higher pressure is expected to be at the inlet of the pipe foot, where the section of the pipe is the smaller and the air velocity is the higher, while pressure at the mouth and the top of the pipe (the two extremities of the resonator) are according to the physics of the resonator of the pipe.Indeed, the closer to the extremities of the pipe resonator, the closer to zero is the average pressure by the oscillations.However, differences in terms of pressure in function of the temperature are reported only at the inlet stage.There, a higher temperature corresponds to a higher average pressure with comparable oscillations (Standard deviations St.d).No differences are reported at the other stages.Consistently to the absence of influence of temperature on the pressure, the sound of the pipe remained the same during the measurements, and the spectral content of its sound reports no significant differences (Fig. 11).
The absence of influence of the temperature changes on the resonator acoustic features, pressure patterns, justifiable by the temperature drops between the foot and the basement of the resonator where the temperature probe was located, suggests that the air flow at the mouth of the pipe direct the warmer air outside the pipe rather than splitting in and out from the pipe as hypothesized in the literature [8,9].To visualize the path of the airflow from the foot of the pipe to the mouth, and then the resonator, the RANS SST k − ω model in Fluent was used to investigate the thermal dissipation.The air velocity measurement at the inlet of the pipe (air velocity 20 1   ⁄ and the temperature measurements as reported in Fig. 9 were used as input into the CFD model.Fig. 14-15 graphically represent the results in function of time and not a specific frame of the simulation. The results of the simulations (Fig. 14-15) show the vector direction at the mouth level as outgoing the mouth of the pipe.In particular, the dominant direction is toward the external upper side of the mouth, where the largest magnitude is reported (Fig. 14).In the same figure it is possible to observe the preserve of other vector directions, smaller in magnitude, which can be due to interactions between the airflow, the low and up borders of the mouth of the pipe.However, velocity vectors in different directions than the largest one report magnitude of several order smaller, and they may be neglected.To investigate the behaviour of the airflow at the mouth of the pipe and evaluate if the warmer air flow moved toward the resonator of the pipe or externally, the velocity vector component along the perpendicular direction to the plane of the mouth of pipe were isolated (Fig. 15).There, it is possible to observe the largest magnitude of the vector outcoming the mouth of the pipe, only a small number of vectors is represented as incoming the mouth of the pipe and their magnitude is at least smaller for one order.
This may demonstrate that the airflow at the mouth of the pipe generates a driving oscillation which moves the air membrane at the mouth, but there is no significant mass exchange from outside the mouth as frequently hypnotized in the literature [8,9].

Conclusions and future perspectives
In the present work the influence of temperature changes on the airflow driving-resonant of a pipe organ is introduced and investigated.To evaluate this phenomena, temperature, pressure, and acoustic frequency measurements were carried out on a modified test-bench pipe organ provided by Pinchi srl organ builder at the Environmental Control Laboratory at the Department of Engineering, University of Perugia.The pipe organ was provided with an electrical heating system installed directly between the bellows and the windchest of the instrument.The measurement campaign highlighted significant thermal dissipation between the heater and the windchest, as well as between the foot of the pipe and the resonator, rising questions about the effective direction of the air flow from the foot, mainly hypothesized as equally distributed inside and outside the mouth of the pipe because of instabilities and turbulence phenomena.The absence of a significant change in the temperature of the air at the top of the pipe suggested the use of a CFD model in Fluent environment to confirm and better investigate the directions of the velocity vector of the air incoming from the foot of the pipe.RANS SST k − ω model was used to consider the transport of the share stress in the air flow.Results from simulations showed the outgoing direction as the dominant one, and the absence of contribution of the warmer air into the resonator of the pipe was proved by the absence of a significant magnitude of the air vector velocity inside the mouth of the pipe from the external part.Further investigations are required to improve the precision of the RANS simulation.For example, DNS simulation may provide data useful for the RANS approach, improving the numerical setup, and justify and prove the behaviour of the air at the mouth level.

Figure 1 .
Figure 1.Scheme of the pipe organ heating system.Figure 2. The heating system installed before the windchest stage of the laboratory organ at the Environmental and Control Lab at the University of Perugia.

Figure 2 .
Figure 1.Scheme of the pipe organ heating system.Figure 2. The heating system installed before the windchest stage of the laboratory organ at the Environmental and Control Lab at the University of Perugia.

Figure 3 -
Figure 3-a.Positions of the probes during measurements inside the sounding pipe.

Figure 3 -
Figure 3-b.Positions of the probes during measurements inside pipe organ system.

Figure 4 .
Figure 4. Pressure and temperature probes at the mouth of the pipe.

Figure 5 .
Figure 5. Pressure and temperature probes at the top end of the pipe.

Figure 6 .
Figure 6.The 3d model of the pipe (on the left) and the Inlet-outlet configuration (on the right).In green, the wall boundary condition layer.

Figure 7 .
Figure 7. Detail of the 3D with the faces object of the mesh refinement process (in blue, on the left), and mesh representation (on the right).

Figure 8 .
Figure 8. Temperature measurements results from the sensors at the different stages of the blowing system of the instrument.

Figure 9 .
Figure 9. Temperature measurements results fromthe sensors at the three points of the pipe (as reported in Fig.3).

Figure 10 .
Figure 10.Pressure measurements at the foot of the pipe at the three different temperature stages of the heating system.

Figure 11 .
Figure 11.Pressure measurements at the mouth of the pipe at the three different temperature stages of the heating system.

Figure 12 .
Figure 12.Pressure measurements at the top of the pipe at the three different temperature stages of the heating system.

Figure 13 .
Figure 13.FFT of the sound recordings of the pipe at the three different temperature stages of the heating system (yellow-red-blue).

Figure 14 .
Figure 14.Vector velocity magnitude of the airflow from CFD simulations.

Figure 15 .
Figure 15.x-component of the vector velocity magnitude of the airflow from CFD simulations.

Table 1 .
Geometrical characteristics of the F1 sounding pipe.

Table 2 .
Pressure means and standard deviations values at each heating step, at each measurement point, for both pipes.