Ifthe upper lip is removed, the jet is considered free. In this caseshear layersare present on the free jet both sides of the jet as depicted in the right hand side of Figure 9.1. The vortices inside the shear
layer form aKármán vortex street. From the aeroacoustic point of view the pressure fluctuations of the free jet flow inside the shear layer are quadrupole sources with a broadband noise spectrum.
When edge tone
generation the upper lip is present in the model, the air jet hits it and starts a quasi-periodic motion
around the edge. The pressure fluctuations on the sides of the lip have opposite phases in opera-tion, and hence the lip acts as a dipole sound source. Due to the oscillation of the air jet, vortices are generated near the edge of the upper lip as depicted in the right hand side of Figure 9.1.
In a complete pipe model, the acoustic feedback from the resonator has a very strong effect on the flow in the steady state and the oscillation frequency of the jet are synchornized with the acoustic oscillations in the resonator in this configuration. From an aeroacoustic viewpoint, the presence of the resonator also means the appearance of a monopole sound source. Nevertheless, our discussion here is limited to the free jet and edge tone generartion cases. Thus, the resonator is not a part of the computational model and the acoustic feedback effect is not considered here.
9.3 The numerical framework
The numerical model was implemented inside theOpenFOAM 2.1framework [118] based on the three-dimensionalfinite volume method. The governing equations of the model are the con-servation of mass (3.2) and the concon-servation of momentum (3.5). By assuming constant density and taking both pressure and shear forces into account in the momentum conservation equation, the incompressible set of Navier – Stokes equationsare obtained. For anewtonian fluid the set of equations to be solved over the numerical grid is:
Navier –
withudenoting the velocity,pthe pressure,ρ0the density, andνthe kinematic viscosity of the fluid. The notation·symbolizes grid-filtered variables. The rate of strain tensorSis defined as Sij =12(∂ui/∂xj+∂uj/∂xi). Einstein notation is applied here and subsequently. The grid-filtered advection termuiujcan be expressed as
uiuj =τijr +uiuj, (9.2)
with theresidual stress tensorτijr containing all unclosed terms. When no turbulence model is used (i.e. in the laminar approximation), the residual stress tensorτijr is simply omitted.
9.3.1 Dynamic Smagorinsky LES turbulence model
In order to arrive at a more sophisticated approximation of the grid-filtered advection termuiuj compared to the simple laminar model, different turbulence models can be applied. Apply-ing turbulence models can enhance the reliability of the CFD simulations without the need of increasing the spatial resolution of the computational domain and hence keeping the compu-tational costs affordable. One of the popular choices of turbulence models is the Smagorinsky model thanks to its simplicity. In the Smagorinsky turbulence model [132] the residual stress tensorτijr is approximated by introducing the artificial kinematic viscosity (also known as eddy viscosity) denoted byνT as follows:
artificial
whereδij is the Kronecker symbol,∆ is the grid size andC is the Smagorinsky constant. The shorthand notation
S
= (2SklSkl)1/2is applied.
In the LES framework filtering by a so-called test filter, denoted by b· is introduced. By ap-plying the test filter to the Navier – Stokes equations (9.1), an approximation of the subtest-scale stress tensorTijcan be attained similar to (9.3)
Tij−1
where∆b is the test filter width. By expressing the resovled turbulent stress tensorLij the Ger-mano identity is obtained:
Lij =Tij−τbijr =udiuj−buiubj. (9.5) The dynamic Smagorinsky LES model proposed by Lilly [94] finds the locally dependent value ofC by substituting (9.4) into (9.5) and using a least-squares error method to satisfy the resulting overdetermined tensor equation. The optimal solution is found as
dynamic The model described by equations (9.3) and (9.6) was incorporated into the simulations with the choice ∆/∆ = 2b and using a simple top-hat filter as the test filter. The grid-size ∆ was approximated as∆ =Vcell1/3withVcelldenoting the cell volume.
9.3.2 Mesh generation
Because of the complexity of the geometry setting up a structured mesh is not straightforward.
Hence, unstructured triangular meshes were built based on the geometry data of Außerlechner’s pipe foot model, presented in Section 9.2.
The unstructured meshes were created using Delaunay triangularization with a goal function over the geometry on the element sizele. The smallest elements withle≈0.04 mmwere located in the windway and around the lower edge of the upper lip, where the smallest coherent flow structures are expected. It was ensured that the edge length of neighboring elements do not differ from each other more than5%, which results in a smooth transition of the size of the elements over the whole mesh. Contrary to the model of Vaik & Paál [140], the complete pipe foot was not a part of the mesh, only a smaller pressure tank was incorporated to imitate the pipe foot. The latter reduces the number of elements in the mesh, while at the same time it can also be expected not to influence the simulation results outside of the pipe foot.
Two mesh
geometry
different mesh configurations were created. In the “Jet” configuration the upper lip was not a part of the model, whereas in case of the “Edge” configuration the upper lip was also included. The meshes were generated using one of the settings also reported in Refs. [18, 19, 140] withd = 1.3 mm, α = 45◦, h = 10 mmand l = 4.05 mm. The width of the walls was set as1 mm, the width of the languid was4 mmat the jet side and2 mmon the other side. The pressure tank representing the pipe foot had a size of7 mm×7 mm. The size of the computational domain was chosen as−40 mm ≤ x≤ 60 mmand−20 mm ≤ y ≤90 mmin the “Jet” case and
−30 mm≤x≤45 mmand−20 mm≤y≤70 mmin the “Edge” case. Close-up views of the two generated meshes with the element edge sizeleare displayed in Figure 9.2.
Both 2D and 3D
meshes
for the “Jet” and “Edge” configurations two- and three-dimensional meshes were cre-ated. Since two-dimensional meshes are not explicitly handled inOpenFOAM 2.1, the 2D meshes were also three-dimensional, with only one element along thez-axis. The 3D meshes were cre-ated by extruding the 2D mesh in thezdirection to a distance ofdusing 16 elements along the z-axis. The key properties of the resulting meshes are summarized in Table 9.1.
9.3. THE NUMERICAL FRAMEWORK 117
(a) “Jet” mesh (b) “Edge” mesh
Figure 9.2.Close-up view of the generated triangular meshes
Mesh name Jet2D Edge2D Jet3D Edge3D
Cells 27 625 38 111 442 000 609 766
All faces 96 864 133 801 1 135 449 1 569 151
Internal faces 41 261 56 754 1 074 551 1 479 729
Processor cores 4 4 32 48
Average # cells / core 6 906 9 528 13 813 12 704 Average # shared faces / core 156 126 1 169 1 044
Table 9.1.Mesh statistics and decomposition properties
Due to the large number of cells and faces (> 106 faces in case of 3D meshes) paralleliza-tion was an essential part of handling the problem. The 2D problems were run using 4 processor cores, whereas 32 and 48 cores were used in 3D simulations in the “Jet” and “Edge” cases, respec-tively. The parallelization was performed using the Scotch decomposition method [123], which minimizes the communication (i.e. the number of shared faces) between the cores. The results of the decomposition are also shown in Table 9.1.
9.3.3 Boundary conditions
Inlet At the left and bottom boundaries of the pressure tank a fixed total pressure ofpfoot was set.
Free flow Free flow boundary conditions were provided by prescribing a dynamic inlet/outlet condition depending on the current pressure. In case of overpressure this boundary condi-tion acts as an outlet condicondi-tion withuj evaluated from the flux. In case of underpressure the boundary functions as an inlet with zero pressure gradient.
Walls No-slip boundary condition withuj= 0and∂p/∂n= 0.
Front & back In case of 2D models anemptyboundary condition was imposed on the front and back walls, prescribing no flow through these surfaces. Whereas, for 3D models cyclic boundary conditions were set up providing continuity of the front and back surfaces of the mesh.
−6 −4 −2 0 2 4
y [mm] Velocity magnitude |u| [m/s]
0
y [mm] Velocity magnitude |u| [m/s]
0
(b) Larger structures of the flow field
Figure 9.3.Visualization of the flow in the “Jet” case (2D laminar model,t= 5.0 ms)
9.3.4 Time stepping
The simulations were performed using ∆t = 5·10−7s time steps, which ensured a Courant number always less than 1. The 2D simulations were run up toTmax = 0.2 s, whereas the 3D configurations were run settingTmax = 0.1 s. This resulted in 4·105 and 2·105 time steps in the 2D and 3D cases, respectively. The complete flow field was saved every 100thtime step for subsequent visualization.