RD-V: 0400 Mok FSI Benchmark

Convergent channel with internal flexible plate is experiencing a parabolic profile inlet.

Figure 1. Model overview


In this analysis, a fluid flow through a convergent channel with an internal flexible plate and the deflection of that plate are examined.

The geometric parameters of this problem are shown in Figure 1.

Fluid domain is modeled with /MAT/LAW6 for the main domain and /MAT/LAW11 for inlet and outlet boundaries, while the plate is modeled with /MAT/LAW2.

The Fluid-Structure Interaction (FSI) is modeled using two different approaches:
  • Arbitrary Lagrangian Eulerian (ALE) formulation using the Displacement Method for the ALE mesh formulation, in which the nodes at the fluid and solid interfaces belong to both fluid and solid parts.

    For the ALE formulation, three different interfaces are used to simulate the interface forces.

  • Coupled Eulerian Lagrangian (CEL) formulation, where two separate meshes are created for each one of the fluid and solid domains and these domains can move independently, with a correction force applied between them.

The results of the analysis are compared with the results proposed by Mok (2001), based on two control points in the pressure surface of the plate.

Options and Keywords

The following keywords are used in the models:

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

Model Description

Units: kg, s, m, Pa, N

To model the fluid domain, /BRICK elements (/HEXA8N) are used and /MAT/LAW6 (HYDRO or HYD_VISC) is used. For the boundaries (inlet and outlet), /MAT/LAW11 (BOUND) is used. The elements at the interface between the fluid domain and the inlet and outlet must have the same nodes. An Ityp=2 formulation is used for both inlet and outlet.

To calculate the pressure in the fluid domain, a linear equation of state (/EOS/LINEAR) is used in /MAT/LAW6 and calibrated accordingly so that the fluid exhibits incompressible behavior.

To model the solid plate, /BRICK elements (/HEXA8N) are again used and assigned with an isotropic elastic-plastic Johnson-Cook material law (/MAT/LAW2).

The velocity inlet profile is inserted using the /IMPVEL load, with a different scale factor for each node, depending on its z coordinate. The velocity inlet is additionally a function of time. The equations describing the inlet are:

u y z,t =4vbarz 1z   m s MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaWgaaWcbaWdbiaadMhaa8aabeaak8qadaqadaWdaeaa peGaamOEaiaacYcacaWG0baacaGLOaGaayzkaaGaeyypa0JaaGinai abgwSixlaadAhacaWGIbGaamyyaiaadkhacqGHflY1caWG6bWaaeWa a8aabaWdbiaaigdacqGHsislcaWG6baacaGLOaGaayzkaaGaaiiOam aadmaapaqaa8qadaWccaWdaeaapeGaamyBaaWdaeaapeGaam4Caaaa aiaawUfacaGLDbaaaaa@519D@
vbar=  0.06067 2 1cos πt 10 ,   0t10 0.06067,   10<t25 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamODaiaadkgacaWGHbGaamOCaiabg2da9iaacckadaGabaWdaeaa faqabeGabaaabaWdbmaalaaapaqaa8qacaaIWaGaaiOlaiaaicdaca aI2aGaaGimaiaaiAdacaaI3aaapaqaa8qacaaIYaaaamaabmaapaqa a8qacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbWaaSaaa8aabaWdbi abec8aWjaadshaa8aabaWdbiaaigdacaaIWaaaaaGaayjkaiaawMca aiaacYcacaGGGcGaaiiOaiaacckacaaIWaGaeyizImQaamiDaiabgs MiJkaaigdacaaIWaaapaqaa8qacaaIWaGaaiOlaiaaicdacaaI2aGa aGimaiaaiAdacaaI3aGaaiilaiaacckacaGGGcGaaiiOaiaaigdaca aIWaGaeyipaWJaamiDaiabgsMiJkaaikdacaaI1aaaaaGaay5Eaaaa aa@68AF@

For the fast reproduction of different /IMPVEL loads for every node, two Altair Compose scripts are used and the 0000.rad is modified directly.

The different formulations for the fluid-structure interaction are presented below.
  • ALE formulation
    • Coincident Mesh (no interface established)
    • Slip Interface - different velocity tangential to the interface are allowed (/INTER/TYPE1)
    • Non-slip (tied) Interface – interface nodes have exactly the same normal and tangential velocity (/INTER/TYPE2)
  • CEL formulation
    • The solid mesh (Lagrangian) can move within the fluid mesh (Eulerian) (/INTER/TYPE18)

Due to its high accuracy and low runtime, the CEL model is further pushed with a multi-domain method to speed up the simulation. More precisely, the fluid and solid domains are solved with a different time step and connect in points defined by the higher time step. With this formulation, the CEL model is able to run in about an hour on a conventional laptop, achieving 10-times better performance than without this method.

Results

Figure 2 shows the horizontal velocity contour to display not only the plate deformation but also the fluid behavior.
Figure 2. Horizontal velocity contour


In Figure 2, the flow phenomena are simulated properly with the vertices located behind the plate. Additionally, despite the slow response caused by the deformation of the plate, the flow finally speeds up due to the convergent channel geometry and reaches a steady-state condition.

Figure 3 shows the streamlines which were captured in different time frames and illustrates the flow phenomena once again.
Figure 3. Streamlines in different time frames


In Figure 4, the pressure contour is exhibited.
Figure 4. Pressure contour


In Figure 4, the pressure is initially raised by the large resistance created by the neck the neck generated by the plate. This increase is responsible for the strong deformation of the plate in the first 10 seconds of the simulation. Then the systems reach a steady-state and the fluid pressure increases while the deformation of the plate reaches a constant value. In this contour, warm colors are used for positive pressure values and cool ones for negative pressure to clearly represent the phenomena.

The control points A (at the top of the pressure side of the plate) and B (at the middle of the pressure side of the plate) are used to investigate the simulation performance.

Figure 5 show an analysis of the influence of mesh size for the coincident mesh formulation, and it is clear that a denser mesh can lead to better results.
Figure 5. Mesh size influence analysis


In Figure 6, the results with the different formulations and the same mesh size (70k elements) are compared.
Figure 6. Comparison between different formulations


The different methods can lead to competing results, with the CEL method being the most accurate of all.

Figure 7 shows a sensitivity analysis for the values of the bulk modulus in /EOS/LINEAR. At low values of the bulk modulus, compressibility phenomena occur, thus affecting the deformation of the plate, as part of the energy is consumed to compress the fluid instead of deforming the plate. This behavior is clearly shown in Figure 7 by the oscillating response of the plate and the response delay at t=2.5 seconds when the plate starts to deform.
Figure 7. Effect of Bulk Modulus (/EOS/LINEAR)


In Figure 8, different values of /INTER/TYPE18 Stiffness are displayed, and the different results are compared. The automatically calculated stiffness value can provide good results, especially when capturing the permanent deformation of the plate. Different values of stiffness calculated with respect to the automatically calculated value can lead to a better agreement throughout the simulation time. In general, this value can be calibrated to maximize accuracy.
Figure 8. Effect of Interface Stiffness (/INTER/TYPE18)