Week 5 - Rayleigh Taylor Instability

Rayleigh Taylor instability

Aim: To visualize the interface of the Water and Air using the Volume of Fluid method in ANSYS FLUENT. 

Objectives:

1. Understand the Rayleigh Taylor instability and Atwood number(A).

2. Understand in detail about the Volume of fluid(VOF) method and its applications. 

2. Explain why a steady state approach is not suitable for this simulation.

Theory:

Rayleigh Taylor instability : This is the instability at the interface of two fluids of different densities which happens when the lighter fluid tries to push the denser fluid.

One of the general phenomenon of the Rayleigh Taylor instability is the water suspended on the top of the lighter fluid, kerosene oil. When the equilibrium is subjected to any disturbance such as a volume of the heavier fluid seeping into the lighter fluid displaces equal amount of the lighter fluid upwards. As the Rayleigh Taylor instability developes, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards.

                                                          Fig.1 : Rayleigh Taylor instability

Practical CFD models based on the Rayleigh Taylor instability waves :

1. Kevin Helmholtz instability: A Kevin Helmholtz instability typically arises when there is a velocity shear in a single continuous fluid or there is a velocity difference across the interface of two different fluids. The shera flow induces vorticity at the interface as a result of which the interface become an unstable vortex sheet that rolls up into a spiral.

2. Richtmyer-Meshkov instability: This instability occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. 

Stages of development and eventual evolution into turbulent mixing :

                                                               Fig.2 : Development of the turbulent mixing

Stage.1 (fig.a): In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. After the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). 

Stage.2(fig.b): The growth of the mushroom like structures continue resulting in a growth rate that is approximately constant in time. 

Stage.3(fig.c): The spikes and bubbles begin to interact wiith one another. Bubble merging, where the smaller bubbles combine to form larger bubbles takes place. 

Stage.4(fig.d): Development of turbulent mixing region.

Atwood number(A) :

This is the ratio of the differences of the densities of the fluids to the sum of the densities of the fluids. 

`A = (rho_2 ~ rho1)/(rho_1+rho_2)`.

Significance : For an Atwood number close to 0; RT instability flow takes the form of symmetry fingers of fluid.

For an Atwood number close to 1; the much lighter fluid below the heavier fluid takes the form of large bubble like plums.

Calculation of Atwood number :

Case. 1: Fluid 1 as water and Fluid 2 as air

`A = (1000 - 1.225)/(1000+1.225) = 0.9975`

Case. 2 : Fluid 1 as water and fluid 2 is User defined

`A = (1000 - 400)/(1000+400) = 0.428`

Voume of Fluid (VOF) method :

                                              Fig. 2(a) : Pictorial representation of Volume of Fluid method

In CFD, the Volume of Fluid is a free surface modelling technique for tracking and locating the free surface.  It belongs to the class of Eulerian methods which are characterized by a mesh that is either stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface.

This method is based on the scalar function called Fraction function 'C'. C is defined as the integral of the characteristic function in the control volume. 

cases :

1. If C is 0, there is no trace of fluid inside the cell.

2. If C is 1, when the cell is fully filled with the fluid.

3. 0 < C>

CFD simulation:

In the workbench of ANSYS, an ANSYS FLUENT analysis system is created. 

1. Geometry modelling :

1. Launch the SpaceClaim component of the analysis system. 
2. Change the view to the top view and draw a rectangle of 10mm x 20mm. On top of it draw another rectangle with the same dimesions.
3. Using the pull tool, which is used to create the surfaces from sketches, select the rectangles individually and create two surfaces.
4. Name the top surface as water and the lower surface as air.
5. In the workbench tab of SpaceClaim, click on the share toplogy feature which is essential as it distinguishes the interface.

                                                                   Fig. 3 : Geometry modelling in the SpaceClaim

Meshing :

1. Launch the ANSYS Mesher. 
2. Define the element size in the Mesh sub window of the Meshing interface. For three different mesh sizes - Coarse, Fine and Superfine; the element sizes selected are 0.5 mm, 0.1 mm and 0.08mm.
3. Click on the Generate Mesh present on the ribbon bar of the interface.

Mesh details:

Type of Mesh	Number of elements	Element size
Coarse mesh	3200	0.5 mm
Fine mesh	80,000	0.1 mm
Superfine mesh	1,24,991	0.08 mm

In order to check the quality of the mesh, click on the quality option in the Mesh details sub window and select the optionAspect ratio from the available options.

                                                 Fig.4 : Superfine Mesh corresponding to the geometry and the Aspect ratio chart.  

From the above bar graph, it is evident that 1,24,926 elements out of the 1,24,991 elements have the AR 1. This is an indication of the highest quality of the elements.

Case setup in FLUENT :

                                                  Fig.5 : Initial steps for setting up the case in ANSYS Fluent.

1. Launch the FLUENT solver.
2. Under the Setup > Models, set the enable the Multiphase option. Also use the implicit formulation. 
3. Under the viscous models, select the Laminar option.
4. Right click on the materials tab and select the new option. (This is done to create a user defined fluid)

                                                       Fig.6 : Creating a User defined fluid in the FLUENT. 

         e. A new pop up window titled create/edit materials appears. Set the name of the fluid as User defined fluid and change the values of                       density and absolute viscosity to 400 `(kg)/m^3` and 0.001 `(kg)/(m-s)` respectively and click on change/create button.

                                                      Fig.7 : Defining the fluid properties of the User defined fluid in the FLUENT. 

      f. In our analysis since we have water as the high density fluid, it has to be listed under the materials. In order to list water navigate to                  materials > air > Fluent database > water and click on the copy button. 

                                  Fig.8 : Copying the water from the FLUENT database to the materials section. 

      g. Now as the basic setup is done, initialize the solution using the Standard initialization option (Solution > Initialization > standard                  initialization). 

      h. An important step is to patch the top surface as water and the bottom surface as air in the Multiphysics window. Navigate as follows :

         Initialization > Patch.

       Set the volume fraction of the water to 1 and air to 0.

                                              Fig.9 : patching the water surface with the volume fraction to 1. 

   i. Click on the run calculation option after defining a time step value of 0.02 and iteration count to 2000.

Results :

Case. 1: RT instability analysis for Water and air case

a. Coarse Mesh (3200 elements).

      Residual plot :

b. fine Mesh (80,000 elements.)

Animation files : 

1. Basic case setup (Coarse mesh):

<iframe src="//www.youtube.com/embed/JsfS6msZYNk" allowfullscreen="allowfullscreen"></iframe>

2. Fine Mesh (80,000 elements)

<iframe src="//www.youtube.com/embed/cmedmpT8lsg" allowfullscreen="allowfullscreen"></iframe>

3. Super fine mesh (124,991 elements)

<iframe src="//www.youtube.com/embed/HTJOsE7DAEY" allowfullscreen="allowfullscreen"></iframe>

4. Animation for water and User defined fluid.

<iframe src="//www.youtube.com/embed/DGmUg4I_Y-Q" allowfullscreen="allowfullscreen"></iframe>

Analysis:

Effect of mesh size on the simulation results :

A mesh with more number of elements should be constructed in order to resolve the interface accurately. The number of elements depend on the grid independence study.

1. Case 1 : Water and air:

As the Atwood number is close to 1, the heavier fluid i.e, water takes the form of bubble like plumes.

2. Case 2 : Water and UDF:

As the Atwood number is some what close to 0, the RT instability flows take the form of symmetric "fingers" of fluid.

References:

1. Rayleigh taylor instability : https://en.wikipedia.org/wiki/Rayleigh–Taylor_instability

2. Richtmyer-Meshkov instability :https://en.wikipedia.org/wiki/Richtmyer–Meshkov_instability

3. Kevin Helmholtz instability :https://en.wikipedia.org/wiki/Kelvin–Helmholtz_instability

4. Volume of Fluid method : https://en.wikipedia.org/wiki/Volume_of_fluid_method