LBE Exercise 1: Pressure-driven flows and Kármán vortex streets

Introduction

Pressure-driven flows are frequently found in all sorts of situations, most commonly when fluids are pumped through pipes. Any Navier-Stokes equation solver can be used to analyse these flows: LBE is capable of doing this, as well as being able to easily include obstacles in a flow by means of bounceback boundary conditions to apply no-slip conditions at their edges.

Placing an object in the path of a flowing fluid disrupts its flow. Above a particular Reynolds number, we can observe eddies being shed from each side of the object in turn, forming rows of vortices in its wake. This is referred to as a (von) Kármán vortex street and produces a distinctive regular pattern, as well as changes to the pressure distribution that create periodic sideways forces on the object. If the periodic forces correspond to the object’s natural resonating frequency, this can enhance the vibration and even cause structural damage. (One example of this happening was the collapse of three cooling towers at Ferrybridge Power Station in 1965 during high winds.)

Please refer to Theory to LBE Exercise 1: Pressure-driven flows and Kármán vortex streets if you would like more details about the theoretical background to this Exercise.

Aim

We want to check how well LBE can model pressure-driven flows, both unobstructed and with obstacles producing Kármán vortex streets. To do this, we will start with a simple two-dimensional system with two walls representing a channel and apply a constant force on the fluid to simulate a pressure drop across the channel length. We can vary the force (pressure drop) and the fluid viscosity in our LBE simulations to see what effect these have on the generated velocity profile. We can then add solid objects to the system and try to find conditions that can produce Kármán vortex streets.

Instructions

Run the DL_MESO Java GUI in your working directory (e.g. dl_meso/WORK), select LBE in the top menu and then Define LBE System at the side.

Use the fields available to create a system with the two-dimensional D2Q9 lattice scheme, dimensions of 250 by 50 grid points, a total of 60000 timesteps and a save span of 500 timesteps. Set the number of fluids to 1 and click on the set fluid parameters button: set both the initial density and relaxation time of the fluid to 1.0, and specify a constant body force on the fluid (say, 2.0E-7 for the \(x\)-component). All other values can be left as their defaults.

Once you have saved the system properties, click on Set LBE Space, select mid-grid bounce back for the top and bottom boundaries, leave the others as periodic and click Create. You will now have the lbin.sys and lbin.spa input files needed for a basic pressure-driven channel flow system.

Follow the instructions in Setting up DL_MESO (the section ‘Running DL_MESO’) to run the simulation and produce a series of VTK output files.

Open these up in Paraview and take a look at the velocity profile. (You can use the Plot Over Line filter, specifying the \(y\)-axis, to get a graph plot of the \(x\)-component of velocity.)

Tasks

  1. Run DL_MESO_LBE in your working directory with the input files you have just created. Open the output files in Paraview and take a look at the velocity profile. (You can use the Plot Over Line filter, specifying the y-axis, to get a graph plot of the x-component of velocity.)
    • Take a look at the velocity profile: what shape does it take? Where is its maximum velocity and what is it? What is the average (mean) velocity of the flow?
  2. Change the value of the body force in the lbin.sys file: you can either edit it in a text editor or open it back up in the GUI (under Define LBE System). Does the basic shape of the velocity profile change? How about the maximum velocity?
  3. Now try modifying the fluid’s kinematic viscosity by changing the relaxation time. How does the maximum velocity depend on relaxation time/viscosity?
  4. To add an obstacle in your channel flow simulation, reload the lbin.sys file in the GUI (under Define LBE System) and then select Set LBE Space. You will need to re-specify the boundary conditions at the top and bottom of the box, but you can also now add a rectangle or a circle with mid-grid bounceback.
    • Try a circle of radius 8 along the flow centreline not too far from the flow ‘entrance’ (say at x=49, y=24).
    • See how the obstacle affects the flow field, then try varying the force and relaxation time until you can achieve vortex shedding. (Hint: take a look at the input files in DEMO/DPD/2D_KarmanVortex for conditions that do produce vortex shedding.)
  5. Try the same again but add a vertical baffle instead of a circle (a e.g rectangle of width 1 and height 16). Can you still get a Kármán vortex street?
  6. To reduce vortex shedding and its destructive effects, we can reduce the area in which eddies can interact. Try adding an additional rectangle of height 1 and width 16 to the right of a circular object as a baffle. Does it suppress the vortex street?