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“Turbulence” is the word that has puzzled physicists for so many decades. Physics and mathematics are complex to work on and still don’t describe the problems to the fullest. Today’s article is all about the basics of turbulence and its modeling. We will also brush up on some required fluid mechanics before jumping into the sea of turbulence modeling. So let’s start with some basic ideas in Fluid Mechanics.
Index
The idea of motion: Fluid mechanics is the branch of physics that deals with fluids. Fluids are everything that can flow and some things which can’t flow may be also considered as fluids as they will flow after some threshold value of the shear force. There are some new ideas coming out recently on states of matter which focus on how not to classify matter as fluids and solids and focus on the spectrum of it. For more on this, you can read this article https://www.mechnflow.com/post/states-of-matter
Coming back to fluid mechanics, as we said it deals with fluids and our basic aim is to analyze the fluid flows that we come across. To analyze we need to have a mathematical model of the flows so that we can analyze it. Here we have realized that for theoretical fluid mechanics we need to understand a lot of mathematics as our job is very clear i.e. to analyze the flows. This analysis is done using the equations of motion which are partial coupled differential equations and also using the boundary + initial conditions which are having physical significance.
One of the sets of equations that describe the Newtonian fluid behavior compressible or incompressible in the presence of pressure as well as viscous stresses is called “Navier-Stokes equations.”
The Navier Stokes equations are written using Einstein’s summation notations as follows. If you don’t follow the summation notations then don’t worry it's quite simple and any standard textbook available on fluid mechanics will be good for you to understand it.
For our simplicity, we are going to assume the Newtonian incompressible fluid flow only in this article and work towards turbulence. The first equation is the statement of conservation of mass and the second statement is for the conservation of the momentum. There are four unknowns u,v, w, and p, and four equations. The only issue is about pressure-velocity coupling and how to deal with it. So these equations are mathematically closed and can be solved. But valid for Newtonian fluids in a laminar regime with incompressibility.
The second regime of the flow where these equations are not valid is nothing the turbulent regime. The flow must be having one more regime called the Transition regime. This article is now going to be focused on the turbulent regime and other things can be discussed in later articles.
Turbulence involves a variety of scales in space as well as time. This variety has always puzzled everybody but is also the driving force behind the beautiful flows which we can see.
Before we look at the modeling part, we should know where it all started. Osborne Reynolds was one of the pioneers studying turbulence back in the 19th century. He had an idea of decomposing the various fields into mean and fluctuating quantities. For example velocity in x-direction say u. This u at any position and at any instant of time can be written as mean quantity and fluctuation about it. Mean also known as an Average can be of different types. Time average, ensemble average, etc. Time average is very easy to understand. The average over the time period would be time average. What is an ensemble average? Let’s look at it.
Suppose we have a coin. Let’s assume if we toss a coin and we get head we have 1 and for tails, we have 0. Now as I continue doing experiments I will start getting 1’s and 0’s. Ideally, the average taken of all entries should come out to be 0.5 (which is a very ideal case) though we don’t get it often. This 0.5 is an ensemble average. So it doesn’t depend on time as well as it doesn’t depend on space. It is purely a function of the no. of experiments done. So for now in this article whenever we talk about the average we are referring to the ensemble average. Using Reynold’s decomposition and ensemble averaging technique we land up with Reynold’s Average Navier-Stokes equation which looks like the following equation in Einstein’s summation notations form.
Where the capital letters such as U denote that these are the mean or average values and small letters such as u or v denote the fluctuations. If we compare our previous Navier – Stokes equations and this we can see that it is quite similar with just one extra term. As the equations have the same form it is called Reynold’s Average Navier Stokes equations i.e. RANS. As you can see that there is one extra term and this extra term is what leads to something called Turbulence Modeling. All the efforts we put in are just to write this extra term in terms of some mean quantities.
There are different ways to model Reynold’s stress terms i.e. we can approximate the stress terms using various approaches. Eddy Viscosity is one of the approaches or classes which is very famous and can be understood easily. Using this class we can write the extra stress terms as follows.
Since the velocity gradients are known the only term remaining is nu t which is famously known as the turbulent viscosity. There are various ways to model this nu t term and classification of all the models is also done. The most famous models are two-equation models which we all must have heard about for example K-ɛ model.
The classification can be done on the basis of the no. of extra equations we get through the modeling of turbulent viscosity. So we can classify models as zero equation models, one equation models, two-equation models, etc. The zero equation model is nothing but the explicit equation for turbulent viscosity. This is the easiest way to model the stress terms but the results we get are not up to the accuracy level one would like to have. In this article, we will look at this model only and we will look at different models in the coming up blogs.
The zero equation model is famously known as “Prandtl’s mixing length model”. The mixing length model uses the analogy of atoms and the collisions at the atomic levels. The distance traveled by a molecule before it colloids with other molecules is the “mean free path” similar to this in turbulent flows we have eddies that travel and colloid. So the distance traveled by the eddies before the collision is called mixing length by Prandtl. Since the collision leads to better mixing the term mixing length is used. The mixing length has also an explicit equation stating its value in terms of y. The mixing length is nothing but the Von Karman constant times y which is the direction of travel of eddy. Von Karman constant is given by (K) which is 0.41. This constant is found using the experimental data. So the set of equations will look as follows.
With this, our equations are now closed and we can solve for the unknown values of Ui’s and the pressure. This model is quite basic and not used anywhere in the commercial software but it is important for us to understand the basics so we have studied this model. In the next article, we will discuss more of the one equation models for turbulent viscosity.
Summary :
Reynold's Averaged Navier Stokes equations lead to the extra stress term called Reynold's stress. Turbulence modeling starts with modeling of this term.
There are various types of models Eddy Viscocity is one of the famous models. Inside the eddy viscosity models, we have 0 equation model, 1 equation model, 2 equation model, etc.
Prandtl's mixing length model is a zero equation model which explicitly calculates the turbulent viscosity in the eddy viscosity model.
Mixing length theory depends on the mean free path between molecular collisions.