Kelvin-Helmotz Instability

 

Introduction

This report is part of the General Purpose Fluid Mechanics Hands-on course at ENSEEIHT. The purpose of this study is to model an hydrodynamic instabilities with CFD codes like FLUENT. Hydrodynamic Instability is one of the central problem of fluid mechanics and can be often observed in Nature. We studied the Kelvin Helmholtz instabilities, that are produced by shear at the interface between two fluids with different physical properties (for example different temperature or different density). The purpose is to show the ability of CFD tools to predict those types of flow.
In this project, the first objective is to define the stability condition. Thanks to this condition, we can define the values of several parameters necessary to the development of the instability. After this theoretical study, we can simulate the Kelvin-Helmotz Instability with Fluent.

 

I.Physical Background

 

We consider the flow of 2 fluids in 2 horizontal parallel infinite streams with different densities and different velocities.The both fluids are incompressible,inviscible and unmiscible.



We define the difference of velocity between the upper fluid and the downer fluid:

We look for a solution using potentials and associated to velocities:



The velocities of fluids perpendicular to the interface must be equal for every fluid and equal to the velocity of the interface:

We consider we have a linear approximation which means magnitude of perturbations is little considering their wave length. Consequently, we can write:


Therefore, the velocity of both fluids perpendicular to the interface can be written:

Thus:

And finally:

But:

Then:

And finally, we obtain the 2 following equations:


Let us resolve the 2D Navier-Stokes equations in each fluid to obtain a third equation.


We suppose that both fluids are inviscid.



But:


Then:




And finally, we obtain the Bernoulli equation for an incompressible and inviscid fluid:


We can written for each fluid:



At the interface between the both fluids, the Bernoulli equations are equal. We obtain the equation(1):


At the interface, there is not equality of pressure because of the surface tension.


The equation(1) becomes the equation(2):


Let us consider the velocities of the 2 fluids:



At first order:


The equation(2) becomes the equation(3):


Finally, we obtain a system of three equations.


Let us resolve it. We look for solutions with the following form:


Let us develop the first equation:


Let us develop the second equation:


Let us develop the third equation:


The system of three equations becomes:


The determinant of the matrix of the system must be equal to zero:


Therefore:

Let us define:


Then:


We obtain 2 equations:


Therefore:


Finally, the flow is instable if:


The instability relation can be expressed by the following relation:


We represent the following function:


The fluid (1) is fuel-oil-liquid and the fluid (2) is water.
The values of parameters are:


Let us represent the stability diagramm:


 

II.Numerical simulation

1.Meshes and models


Geometry and meshes


At the beginning, we realize several studies in order to find the best method for compute this problem. this preliminary study highlights two important points. On the one hand, the Volume Of Fluids model, which is one of the multiphase models of fluent, is the best model to carry out our study. On the other hand, Kelvin Helmoltz instability can not appear without generating a perturbation at the interface of the two fluids. Hence, the boundary condition at the entry must be defined as an UDF in fluent.



We consider a rectangle of 2 meters lenght and 0.5 meters height. In fluent, the edges AB and CD are defined as wall, without friction. The edge BD is defined as a outflow outlet. Finally, the edge AC is defined as a velocity inlet.
For this study, we have defined two grids. the number of elements are the same for both: 200 elements on the x direction and 100 elements on the y direction. This discretisation provides a good resolution for the results.
The first one is regular whereas on the second one, the mesh in the y direction focus in the center. The explanation for this choice will be give in the other part.



The Volume Of Fluids model


We choose this model for study our 2 phases problem. The physical restrictions of the VOF model are the following: the flow must be incompressible, Heat transfer is not available, species mixing and reacting flow cannot be modeled and the LES turbulence model cannot be used.
Therefore, it seems to be the best model for our study because we consider two compressible flows, non miscible. Moreover, we choose the laminar solver in order to observe the development of the instability.

Physical parameters

Kelvin Helmoltz instability is difficult to vizualise therfore we choose two fluids with similar density. the parameters are given in the following table:



Taking into account gravity or surface tension depends on the simulation. It will be precised in the study.

Numerical parameters


The convergence criteria were set to the default values that is to say 0.001 for the residuals, X and Y velocity. Moreover, the parameters of the VOF model were set to the standard values as it is advised in Fluent help, that is to say:

Scheme: geo-reconstruct
courant number:0.25
Pressure velocity coupling:simple
Momentum:first order
Pressure:standard

The sinuso´dal oscillator


We can think that the instability cannot developed, even if the velocity difference is high, due to the numerical diffusion which stabilise the flow. Therefore, we have to include a perturbation.
Thus, The velocity inlet is defined as an UDF, wicht creates a beater at the interface of the fluids. It is represented in the following scheme (virtual case where there is not perturbation) :



At each time step, the position of the interface (at the velocity inlet boundary condition) is calculated with xo=Asin(wt). Then, to xo from Xmax, density and velocity take the values of the primary phase and it is the contrary for the secondary phase.





2.Numerical study



2.1 Development of the instability


Parameters of the simulation:

Mesh : regular grid with 100 elements on the x axis and 200 elements on the y axis.
The velocity for fuel oil is 3ms.
The velocity for fuel oil is 1ms.


Presentation of the study and results:

In this first study, gravity and surface tension are not take into account. We use the regular grid because we do not know the height of the vortexes and we take a hight velocity difference because we only want to vizualise the instability. Below, you can see the evolution of the instability in the time. The first vortex which can be observed is not a Kelvin Helmholtz instability. Indeed, its form and height is not similar to following vortexes.Therefore, we can think that this first perturbation is linked to the VOF method.


Profile of density



Click here


Below, we have represented the static pressure profile. We can see the classical profile of a series of vortexes with alternance of hight pressure and low pressure.

Profile of global pressure




2.2 Stability study



After this first study, we decide to carry out a stability study. Firstly, we changed the difference of velocity inlet and secondly, we changed the frequency of the beater. Below, we present the result of these simulations. Then we try to explain the obsevations by using the stability curve which is represented in the Physical Background part.

2.2.1 Effects of the difference of velocity


In the following simulation, the frequency of the beater was fixed to 5 Hz. Moreover, the velocity of the secondary phase was fixed to 1m/s. The velocity of the primary phase was variable. We achieved several simulations in order to determine the value of the speed difference for which the instability could appear. We want to verify the stability curve which is drew in the physical background part.
Study 2.1:

Mesh : irregular grid with 100 elements on the x axis and 200 elements on the y axis

The velocity for fuel oil is 3ms
The velocity for fuel oil is 1ms

Gravity and surface tension are take into account.

The corresponding wave velocity is equal to the difference of velocity between the two fluids. As the frequency of the beater is known, the corresponding pulsation is :

Therefore, we know the theoritical wave number:


We can see on the following scheme that the instability is well developed.


We obseve 5 vortexes on this scheme therefore the wave lenght is 0.4 meters. therefore, the numerical wave number is:


According to the theory, for these values of the wave number and the difference of velocity for the two fluids, the flow is unstable.

Study 2.2:

Mesh : regular grid with 100 elements on the x axis and 200 elements on the y axis.

The velocity for fuel oil is 2ms.
The velocity for fuel oil is 1ms.

Gravity and surface tension are take into account.

Now, the difference of velocity is divided per 2 therefore the wave number too. So, the value of the wave number for this study is:



Therefore, the corresponding wave number is:







In theory, the flow is unstable and we should observe 10 vortexes on the domain. However, on the previous scheme, the vortexes are not well defined and we observe only 7 vortexes but it is difficult to observe.

Study 2.3:

Mesh : regular grid with 100 elements on the x axis and 200 elements on the y axis.
The velocity for fuel oil is 1.5ms.
The velocity for fuel oil is 1ms.

Gravity and surface tension are take into account.




We can see on the previous scheme that the flow seems to be stable. However, according to the theory, the stability limit is not reached.


We have see on this three cases that the theory is not verified by the numerical simulation.Indeed, the flow is stable whereas the difference of velocity between the two fluids is higher than the theory prevision. Nevertheless, we can think that the numerical diffusion stabilize the flow. Hence, we obtain higher values for stability criteria.

2.2.2 Effect of the perturbation frequency

Study 3.1:

f=1Hz


This study is not very realistic. Indeed, the result seems to be false because we have not a well instability. Maybe we can say that the frequency is too little.

Study 3.2:

f=2Hz


This study is more realistic. Indeed, we can observed 2 or 3 vortexes but it is difficult to determine.
According to the theory, the theoritical wave lenght for a frequency of 2 hz for the beater is 1 meter. So, the result between numerical simulation and theory is quite good in this case.

Study 3.3:

f=5Hz


This case have allready been treated in the previous part.

III.Examples of Kelvin-Helmotz Instability

We can observe the Kelvin-Helmotz Instability in natural phenomena, for example in clouds. A simulation of Stanford University represents this instability. The black region is above the cloud top and has no liquid water. In the pink region, there is about 0.8 grams of water per kilogram of dry air. Because of the cloud top entrainment instability, convective patterns develop which distord the interface.

The two following photographics are example of the Kelvin-Helmotz Instability:



 

Conclusion

Thanks to the VOF Model, Fluent enables to simulate the 2D Kelvin-Helmotz Instability. It is necessary to put a perturbation in order to observe the development of the Instability. The velocity inlet is defined as an UDF, which create a sinuso´dal perturbation. When gravity and surface tension are take into account, we observe the developement of the Instability for a critic value of the difference of velocity of the both fluids. This experimental result is similary to the theory of the first part.


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