2.7. Production of Turbulent Kinetic Energy

So, this in this technique the model focuses on the mechanism that affects the turbulent kinetic energy, k stands for kinetic energy. So, the instantaneous kinetic energy k as a function of time k t of a turbulent flow is the sum of the mean kinetic energy and the turbulent kinetic energy k. So, k of t can be written as capital K + small k. So, this is small k and capital K is written is simply half u square + v squared + w squared actually u bar v bar and w bar and small k is written is half of u dash squared + v dash squared + w dash squared.

In fluid dynamics, turbulent kinetic energy (TKE) is the energy contained in the swirling motions of a fluid. The total kinetic energy of a turbulent flow at a given instant can be thought of as being made up of two parts: the mean kinetic energy (K) and the turbulent kinetic energy (k). The mean kinetic energy is derived from the average velocities of the flow (represented by u, v, and w), while turbulent kinetic energy accounts for the fluctuations around these average velocities. Consequently, they combine to form the instantaneous kinetic energy represented mathematically as k(t) = K + k, where K is the mean part and k is the turbulent part.

Think of a river. The water flowing steadily represents the mean kinetic energy (K), while the swirls and eddies that form and dissipate are akin to the turbulent kinetic energy (k). Just as the river has both calm areas and turbulent sections, flows in different scenarios can be described similarly.

The governing equations for this are one we have continuity equation and the other is so, the way we write is So, D D t of tau i j that is what we are modeling is -1 by rho del p del x i + del x j into 2 nu + nu t into del i j. How does equations have been derived we are not going to discuss but these are the 2 equations that we use for momentum.

The behavior of turbulent flows is modeled using governing equations. In this context, we have the continuity equation and a momentum equation, which expresses how the shear stress (τij) evolves over time under various influences, such as pressure gradients and viscous effects. The term 'nu' represents the kinematic viscosity of the fluid, while 'nu_t' indicates the turbulent eddy viscosity, essential in turbulence modeling.

Imagine driving a car. The continuity equation is like ensuring you have enough fuel to continue driving, while the momentum equation relates to how fast and in which direction you accelerate or decelerate based on the pressure you apply to the gas pedal or brakes.

So, if you see there is a term called nu T. So, this nu t is the eddy viscosity. So, this actually is not should better be called as turbulent eddy viscosity, nu T, it can be actually dimensionally related to kinetic energy k kinetic energy dissipation rate epsilon through nu T is written as c mu into k square by epsilon.

Eddy viscosity (ν_T) refers to the turbulent expansion of fluid momentum within a fluid flow, helping to quantify the effects of turbulence. It connects to the turbulent kinetic energy and its dissipation rate, making it essential for modeling turbulent flows in engineering. The equation ν_T = cμ * (k^2 / ε) shows that eddy viscosity is influenced by the square of turbulent kinetic energy divided by its dissipation rate, with cμ being a constant determined experimentally.

Think of a busy highway. The eddy viscosity is like the traffic flow around obstacles (like slow-moving vehicles or construction) that cause changes to usual traffic speed. Vehicles can maintain their speed but will also find ways around slower cars, showing how energy and movement are transferred and dissipated in a fluid-like manner.

The turbulent kinetic energy and the energy dissipation can be determined from the following equation. So, now, we said that this is the equation which we are going to use for determining tau i j and that we can use in our average equation Reynolds average Navier stokes equation... And use it in the average equation of Reynolds Navier Stokes equation.

To computationally approach the challenge of modeling turbulence, we need equations that can help us ascertain not just the turbulent kinetic energy (k) but also the dissipation rate (ε). The key relationship is that these values must be solved alongside the Reynolds-averaged Navier-Stokes equations, which together handle averages in fluid dynamics. Thus, we find specific equations tailored for k and ε that ultimately help us model the turbulent behavior effectively.

Consider the weather. Meteorologists analyze not only current temperatures (like k) but also how fast they change (like ε) in their models to predict weather patterns. This goes to show how key values are essential for accurate modeling, akin to what we do with turbulent flows.

So, this for just writing it down solve for k and epsilon then nu T is related to k square by epsilon. Nu T, put this in Reynolds shear stress equation and use that in Reynolds average Navier stokes equations. It is quite a complex way, but it gives good results.

Modeling turbulence is inherently complex because it involves iteratively solving equations for k, ε, and ν_T, before ultimately substituting back into the shear stress equations for Reynolds-averaged equations. While this may seem convoluted, this process has proven reliable and accurate for predicting turbulent flows effectively within various engineering contexts.

Think about baking a cake. You need to follow a series of steps—mixing ingredients, checking the oven temperature, timing—the complexity ensures the final product is delicious. In turbulence modeling, every step must be carefully considered to achieve precise outcomes.

Most of the turbulence model in the world follow k and epsilon, there is one other turbulence model for k omega. So, k is the same kinetic energy omega is somehow related to dissipation epsilon, but that is also outside the scope, but it is better to remember the name.

The two prevalent models for turbulence in engineering are the k-epsilon and k-omega models. They both focus on kinetic energy, but they handle the dissipation differently. The k-epsilon model works well for free shear flows and is widely used due to its reliability, while the k-omega model may offer advantages in near-wall flows. Understanding both will give engineers flexibility based on the flow conditions they are evaluating.

Choosing between k-epsilon and k-omega models is like selecting a specific recipe based on the occasion. For a quick dessert, a no-bake recipe (k-epsilon) is more straightforward, while for a complex dish (k-omega), one might need deeper techniques and expertise.

So, another method apart from Reynolds average Navier Stokes equation for solving the turbulence, for solving the computational fluid dynamics Navier Stokes equation is called direct numerical simulation.

Direct Numerical Simulation (DNS) is a technique that solves the Navier-Stokes equations without using turbulence models, capturing all scales of motion. DNS requires very fine spatial resolution and numerical accuracy, making it computationally expensive but allowing for the most accurate simulations of turbulence.

Imagine you were to film a live performance instead of just getting a summary of highlights. DNS is like that detailed film—capturing every nuance and movement. Though it requires a lot of resources and effort, the richness of the data is invaluable.

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