2. Closure Problem

The effect of 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.

In fluid dynamics, Reynolds shear stress is an important concept that modifies the way we predict the behavior of fluid flow. It introduces a correction factor to the average flow calculations. The term 'tau i j' denotes the stress in the fluid, influenced by turbulence, while 'rho' represents fluid density. Here, 'u i' indicates the average velocity of the fluid.

To accurately describe turbulent flows, we want to relate this shear stress to measurable quantities, specifically to the average flow parameters. This is vital for practical applications since turbulence can significantly impact fluid motion and the resultant forces on structures.

Imagine driving through a crowded street. The average speed of cars (u i) fluctuates due to traffic jams and different driving behaviors, much like how turbulent flows behave in a fluid. Just as you need to understand these averages to estimate your arrival time, engineers must understand Reynolds shear stress to predict how fluids will behave under different conditions.

This particular problem is called a closure problem. To obtain u i and pressure from Reynolds equation, we need to model this Reynolds's shear stress given by rho into tau i j as a function of average flow, this will remove the fluctuations.

The closure problem arises from the need to express turbulent stresses (like Reynolds shear stress) in terms of mean quantities that can be more easily estimated or calculated. In turbulence modeling, we treat the known average flow parameters as the starting point and seek to express the unknown turbulent effects (the closures) in terms of these averaged values. By successfully modeling these stresses, engineers can predict fluid behavior without needing to capture every little fluctuation in the flow.

Consider a classroom with students varying in their heights – if you want to estimate the average height of the class (closure), you measure each student but need to account for the outliers (very short or very tall students) that could skew your average. Similarly, in fluid dynamics, while we can measure the average flow, we also need a good model to account for the unpredictable turbulence.

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 turbulence model in computational fluid dynamics. Here, 'k' represents the turbulent kinetic energy of a flow, while 'epsilon' indicates the rate at which this energy is dissipated. This model helps bridge the gap between observable mean flows and complex turbulence by relating the average kinetic energy and its dissipation, making it easier to compute fluid behavior under various conditions.

Think about how a group of friends plays around in a park (turbulence). Some friends are more energetic (k) and will eventually tire out (epsilon). If we want to know how much energy they exert over time, we consider both their current activity (energy) and how quickly they tire (dissipation), similar to how engineers use the k-epsilon model to predict turbulent flows.

This nu t is the eddy viscosity. This actually is better 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 measure of the effective viscosity in turbulent flows. It accounts for the mixing and momentum transfer due to eddies (or swirling motions) within the fluid. The relationship nu T = c mu * (k^2/epsilon) links eddy viscosity to turbulent kinetic energy and its dissipation rate, helping to quantify how turbulence affects flow characteristics.

Consider stirring a thick soup. As you stir, you create swirling motions that mix the ingredients more effectively than if you just let them sit. The 'thickness' of the soup can be thought of as viscosity. Just as stirring (eddy viscosity) mixes the soup (momentum transfer), turbulent eddies mix a fluid, making flow predictions more complex yet more accurate.

So, the next step is finding out this turbulent kinetic energy k and epsilon that that is how this equation gets in the model gets its name. And to be able to do that we have two other equations - one is in terms of kinetic energy and one is in terms of epsilon.

To solve the closure problem, we need to derive equations to calculate both turbulent kinetic energy (k) and the dissipation rate (epsilon). These equations use variables derived from the flow characteristics. Solving these equations allows us to estimate how energy is produced and dissipated in the turbulence, thereby enabling better predictions of flow behavior.

Imagine trying to predict how fast a campfire will burn out using knowledge of how much wood (energy) is available (k) and how quickly it turns to ash (epsilon) once consumed. By understanding both how flames burn and how they weaken, you can effectively manage the fire, similar to managing turbulence in fluid flows.

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