1.4. Lecture # 58

So I am going to write here rho in tau i j is actually Reynolds shear stress. So it is like shear stress to obtain u i and pressure. So, u i is the average flow velocities and pressure from RANS equation, we need to model this shear stress rho in tau i j as a function of the average flow so if you look at the equations.

Reynolds shear stress, denoted as rho in tau ij, represents a form of stress that affects how fluid flows. In fluid mechanics, shear stress is crucial because it helps determine the average flow velocity (u_i) and pressure. The Reynolds-averaged Navier-Stokes (RANS) equations are a set of equations attributed to how we analyze fluid flow. By relating the Reynolds shear stress to average flow velocities, we can simplify complex flow dynamics, emphasizing the need to model this shear stress effectively.

Think of Reynolds shear stress like the friction experienced when sliding a book across a table. The book is the fluid moving over a surface. The amount of force you need to apply (equivalent to shear stress) to keep the book moving consistently relates to how smooth or rough the table surface is (analogous to average flow conditions).

The best way to put it in some form of an average value and to do that, that particular problem is called a closure problem. So, we will go back to the closure problem...

The closure problem arises when studying fluid dynamics because there are terms in the equations that are unknowns. The goal is to model these unknown terms, like the Reynolds shear stress, in a way that allows us to compute the average flow properties. Essentially, we want closure (a complete description of the flow) by expressing these stresses in terms of quantities we can compute from the average flow. This process reduces the complexity of analyzing turbulent flows.

Imagine trying to predict the amount of traffic on a highway. Instead of knowing exactly how many cars are on the road at every moment, we can use the average number of cars observed during rush hour. Solving the closure problem in fluid dynamics is similar: we must find a way to accurately predict the behavior of average conditions without needing to track every individual fluctuation.

There are elegant ways the first of the methods is called the k epsilon model. In this technique, the model focuses on the mechanism that affects the turbulent kinetic energy, k stands for kinetic energy.

The k-epsilon model is a widely used approach in computational fluid dynamics to simulate turbulence. It emphasizes two key components: turbulent kinetic energy (k) and its dissipation rate (epsilon). The model assumes that the behavior of turbulent flows can be accurately captured using these two factors, which allows for more straightforward calculations that represent complex flow phenomena effectively.

Picture boiling water: as the water heats up, bubbles form and rise. The energy of these bubbles can be thought of as kinetic energy (k). As the bubbles burst on the surface, that energy dissipates, similar to the dissipation modeled by epsilon. The k-epsilon model helps predict how the boiling water will behave by understanding both the energy of the bubbles and how that energy is lost.

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.

In turbulent flows, the kinetic energy can be divided into two components: the mean kinetic energy (K) and the turbulent kinetic energy (k). The mean kinetic energy represents a large-scale flow pattern, while the turbulent kinetic energy accounts for the small-scale fluctuations and disturbances in the flow. This division allows engineers and scientists to analyze turbulent flows more systematically and comprehensively.

Consider a crowded concert where people are moving around. The average position of people swaying to the music represents the mean kinetic energy (K). However, individual movements, like someone jumping up and down or waving their arms, represent turbulent kinetic energy (k). Analyzing these two components helps us understand crowd behavior more accurately.

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 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 (nu_t) is a concept used to describe how turbulent flows resist movement. It quantifies the turbulence's effect on momentum transfer and is linked to both the turbulence's kinetic energy (k) and its rate of dissipation (epsilon). The relationship expressed as nu_t = c_mu * (k^2 / epsilon) translates these factors into a meaningful way to model fluid dynamics, with c_mu being a constant obtained from empirical data.

Think of eddy viscosity like butter melting in a pan. Just as the melted butter allows the food to move and slide more easily, eddy viscosity allows turbulent energy to spread through the fluid. By understanding how these elements interact, we can better predict how liquids behave under different conditions.

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