Feedback Flow Control using a Custom Boundary Condition in OpenFOAM

Before we begin, I would like to thank Michael Alletto for his wide variety of OpenFOAM resources. I would also like to thank Jozsef Nagzy and his excellent explanation of the use of overset meshes in OpenFOAM.

Introduction

Flow Induced Vibrations (FIV) is an important field of study that deals with the response of structures in various flow configurations with one or more degrees of freedom. It is a phenomena which is especially relevant in the areas of offshore drilling, energy harvesting and even vehicle aerodynamics.

On the other hand, Flow Control is a field that deals with the introduction of certain devices or the use of particular methods to regulate and, in theory, control the flow in a given test case. Flow control maybe done passively - attachment of splitter plates behind a bluff body to prevent vortex shedding - or it may be done actively - moving flaps in an aircraft.

The test case of an elastically mounted cylinder has been studied extensively by many reseachers in the field. The work of S.P. Singh and Mittal, Williamson and Govardhan etc. have been used to study the flow induced vibrations of such a system. For the laminar test case and zero mass damping, flow synchronization or lock-in is observed for a range of reduced velocities and the response can be divided into three branches - initial, lower and upper. Bourguet and Jacono conducted simulations with a similar cylinder which was rotating with a fixed rotational velocity. The results seemed to indicate a change in the amplitude of vibration for certain values of the non-dimensionalized rotational velocity. This indicates the potential use of addding rotation to a translating cylinder to cause a change in its vibrating response.

Finally, we arrive at the paper this project is based on. Vicente-Ludlam et al. made the use of the Lattice Boltzmann Method to simulate the test case of a rotating, elastically mounted cylinder with one degree of freedom. Their setup involved a constantly changing rotating velocity which was dependent on either the velocity or the acceleration of the cylinder in question. In this project we will be analyzing a similar rotation law to setup a feedback control system using a custom coded boundary condition in OpenFOAM

Setup

The cylinder is setup in a cross-flow configuration and only allowed to move in the transverse direction. It is attached to a spring and the rotation is given about its center of mass. The simulation is 2D in nature and involves zero mass damping. The following equations are valid for the non-dimensional parameters.

$$Re = \frac{U_{\inf}D}{\nu}$$ $$U^* = \frac{U_{\inf}}{f_ND}$$ $$m^* = \frac{m}{\frac{\pi}{4}\rho D^2 H}$$ $$k^* = \frac{k}{D}$$

The rotation-feedback law can be summed up as follows,

$$\omega = kU_y$$

It relates the angular velocity with the transverse velocity of the cylinder, this rotation affects the vortex formation and in turn the fluid forces acting on the body. Which ultimately causes the velocity and the amplitude to change, restarting the feedback loop.

For the following test case the value of the Reynold's Number and Mass Ratio is fixed at 100 and 10 respectively.

The Chimera Method - Mesh Setup

The domain size is 60DX40D with the cylinder situated at (20D,20D).The reduced velocity is varied by changing the spring constant in dynamicMeshDict. The mesh consists of two regions, the first is the cylinder mesh that will actually execute the motion and the second is the rectangular background mesh with two refinement zones. These two meshes are merged or connected using an overset region that interpolates information between the two meshes. This allows for the existences of two disconnected meshes that can interact with each other, this method is known as Chimera or the Overset Mesh Implementation. While it can simulate complex mesh motion and allow for a greater degree of freedom, it is prone to interpolation errors. Even so, overset meshes are a very powerful and widely used method for simulating moving meshes in CFD.

Image taken from D. Vicente-Ludlam, A. Barrero-Gil, A. Velazquez - https://doi.org/10.1016/j.jfluidstructs.2017.05.001 The equation of the 1DoF mass damper system can be given by

$$m\ddot{y} + c\dot{y} + ky = \text{Lift Force}$$

This equation is solved using the symplectic method provided in the 6D0F Motion Solver in OpenFOAM. Alletto's findings have shown that for low mass damping the Newmark solver is unstable while the symplectic one gives better results.

The circular mesh that represents the body and a little bit of the surrounding domain that actually oscillates

The stationary domain that the cylinder oscillates in

Mesh Convergence Study

A mesh convergence study was done using Shiels et al. as the reference paper - https://doi.org/10.1006/jfls.2000.0330. The test case simulated was that of mass ratio 5 and non-dimensionalized spring constant 4.74. Increasing refinement in each mesh case was employed. The finer mesh seems to have achieved convergence in results to the paper and is ready to be used for the simulation.

	Coarse Mesh	Fine Mesh	Finer Mesh	Shiels et al
$C_d$ Mean	2.283	1.5165	1.725	1.7
$C_L$ Amplitude	0.729	0.0515	0.038805	0.04
$St$	0.17175	0.15335	0.1565	0.156
$A^*$	0.7239	0.3683	0.421	0.46

Custom Boundary Condition

The implementation of the new boundary condition - "forcedRotation" - is needed as OpenFOAM does not support both free and forced motion at the same time. Our cylinder's translation is free motion while the rotation is dependent only on the velocity and is not affected by the fluid moments. To counter this, a novel boundary condition is used, the setup of this BC can be divided into two steps.

1. Getting the velocity of the overset mesh, this is the free translational velocity that is influenced by fluid forces.
2. Adding a rotational velocity on the boundary walls of the cylinder to account for the rotation.

Vector addition of both these velocities will give us the required result. The implementation is as follows

Solution Setup

For running the test case follow the given commands.

1.Initialize the 0 directory

cd backgroundMesh
restore0Dir

2. Cylinder Mesh

cd cylinderMesh
blockMesh

3. Background Mesh

cd backgroundMesh
blockMesh
topoSet
refineMesh -dict system/refineMeshDictR1 -overwrite
refineMesh -dict system/refineMeshDict -overwrite
mergeMeshes . ../cylinderMesh -overwrite
checkMesh
setFields
tansformPoints -scale 0.01

4. Starting the solution

potentialFoam -pName p -writephi
decomposePar
mpirun -np 8 renumberMesh -overwrite -parallel
pyFoamPlotRunner.py mpirun -np 8 overPimpleDyMFoam -parallel 

5. Extracting the displacement and velocity

cd backgroundMesh
./makeFiles

The solver used is overPimpleDymFoam available in the ESI version of OpenFOAM, it utilizes a merged PISO-SIMPLE algorithm. The solution is initialized by potentialFoam and the divergence scheme is that of Gauss QUICK. A large amount of outer corectors are utilized along with a Max Courant Number of 0.5. The interpolation scheme for the overset meshes is the standard inverseDistance. Mass flux is solved for along with displacement, velocity and pressue and the use of both momentum predictors and adjustment of overset phi are used.

Results and Inference

Velocity Contours and Wake Structures

For the case of reduced velocity 5.5 and a fixed $k^* = 2$, we have the following velocity contours. The wake structure is also analyzed by the plots of the Z component of the vorticity. We can clearly see a 2P mode for very low vibrational amplitude and a coalesced 2P mode for higher amplitude vibrations, even for the rotating case.

VID_20230808134418.mp4

Response Curve and Amplitude Branches

The maximum amplitude of the cylinder across the ranges of the reduced velocity are as follows. Clearly the existence of the initial and lower branches can be seen as stated by Khalak and Williamson. The rotating case transitions earlier than the standard one, and clear reduction in the maximum amplitude can be observed. Hence the rotation law helps in curbing the flow induced vibrations of the cylinder.

Vibration Inducing Phenomena and Synchronization

To find out the cause of the reduction we take a look at the ratio of the vortex-shedding frequency and the natural frequency of the system for the case of the rotating cylinder. Here, one can clearly see that the region defined by the synchronization of the frequencies is same as that of the high amplitude vibrations. This tells us that the vibrations are caused by wake-body resonance and the reduction in amplitude is due to the decrease in the vortex power by the motion of the cylinder.

References

1. S.P. Singh, S. Mittal, Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes, Journal of Fluids and Structures, Volume 20, Issue 8, 2005, Pages 1085-1104, ISSN 0889-9746, https://doi.org/10.1016/j.jfluidstructs.2005.05.011.
2. A KHALAK, C.H.K WILLIAMSON, MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING, Journal of Fluids and Structures, Volume 13, Issues 7–8, 1999, Pages 813-851, ISSN 0889-9746, https://doi.org/10.1006/jfls.1999.0236.
3. Bourguet, R., & Lo Jacono, D. (2014). Flow-induced vibrations of a rotating cylinder. Journal of Fluid Mechanics, 740, 342-380. https://doi.org/10.1017/jfm.2013.665
4. GOVARDHAN, R., & WILLIAMSON, C. (2000). Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics, 420, 85-130. https://doi.org/10.1017/S0022112000001233
5. D. Vicente-Ludlam, A. Barrero-Gil, A. Velazquez, Flow-Induced Vibration of a rotating circular cylinder using position and velocity feedback, Journal of Fluids and Structures, Volume 72, 2017, Pages 127-151, ISSN 0889-9746, https://doi.org/10.1016/j.jfluidstructs.2017.05.001.
6. D. SHIELS, A. LEONARD, A. ROSHKO, FLOW-INDUCED VIBRATION OF A CIRCULAR CYLINDER AT LIMITING STRUCTURAL PARAMETERS, Journal of Fluids and Structures, Volume 15, Issue 1, 2001, Pages 3-21, ISSN 0889-9746, https://doi.org/10.1006/jfls.2000.0330.
7. Alletto, M. (2022). Comparison of Overset Mesh with Morphing Mesh: Flow Over a Forced Oscillating and Freely Oscillating 2D Cylinder. OpenFOAM® Journal, 2, 13–30. https://doi.org/10.51560/ofj.v2.47
8. Jozsef Nagzy's explanation on the use of overset meshes - https://youtu.be/nCIsS0VqypA