2.3. Modeling of Reynolds Shear Stress

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.

Reynolds shear stress, denoted as ρ in τij, is important for understanding how turbulent flow affects the average velocity and pressure in fluid dynamics. It represents the momentum transfer due to turbulence and needs to be modeled based on average flow characteristics to simplify complex turbulent behaviors.

Think of riding a bike on a windy day; the wind applies a force that affects your speed. Similarly, the Reynolds shear stress represents forces in a fluid that influence how it flows and the resulting speeds of various elements of that flow.

So, the best ways to put it in some form of an average value ... that particular problem is called a closure problem.

The closure problem arises because turbulence introduces fluctuations that complicate calculations of average quantities. The goal is to relate the Reynolds shear stress terms to known values that describe the mean flow, thereby facilitating easier calculations in fluid dynamics.

Consider trying to predict how a crowd moves in a busy market. You can't just look at one person; you need to average out their movements and how they interact. Similarly, in fluid dynamics, to understand the overall flow, we must average out the chaotic, turbulent fluctuations.

So, the first of the methods is called the k epsilon model. ... so, k of t can be written as capital K + small k.

The k-epsilon model is a widely used approach in fluid dynamics that helps calculate turbulent kinetic energy (k) and its dissipation rate (ε). It breaks down the energy in turbulence into two components: the average energy (capital K) and the fluctuation energy (small k). This separation helps simplify the turbulent flow modeling.

Imagine a sports team where both the star players (capital K) and the less consistent players (small k) contribute to the total performance. By analyzing both groups separately, coaches can develop better strategies to improve overall team performance — just like the k-epsilon model analyzes energy contributions in turbulent flow.

So, if you see there is a term called nu T. So, this nu t is the eddy viscosity ... turbulence there is no production or diffusion of turbulent kinetic energy.

The term νT represents turbulent eddy viscosity, which quantifies the effects of turbulence within the flow. It's derived from the relationship between kinetic energy (k) and its dissipation rate (ε). Understanding how turbulent viscous effects relate to energy dynamics helps in accurately modeling fluid flows.

Think about a blender mixing ingredients. The blades create turbulence, causing eddies that help mix everything uniformly. The eddy viscosity is like how strong those blades work to maintain a smooth mix; in turbulence modeling, it's crucial for understanding how energy is distributed.

The turbulent kinetic energy and the energy dissipation can be determined from the following equation. ... we still do not know so, the next step is finding out this turbulent kinetic energy k and epsilon.

To fully utilize the k-epsilon model, we need equations for calculating turbulent kinetic energy (k) and its dissipation rate (ε). These calculations are essential for determining how turbulent energy behaves and ultimately contributes to the fluid's motion.

It's like tuning an engine; you need to measure how much energy is used in acceleration (k) and how much energy is lost through heat and friction (ε). This balance is important to keep the engine running efficiently, similar to how we balance k and ε for fluid flows.

So, another method apart from Reynolds average Navier Stokes equation for solving the turbulence ... high order numerical accuracy, it is known as full turbulence simulation FTS.

Direct Numerical Simulation (DNS) represents a method that computes the full governing equations of turbulent fluid flow without approximations. Since DNS resolves all scales of turbulence, it provides highly accurate simulations but requires significant computational resources, making it applicable mainly in research and advanced applications.

Consider an artist painting a detailed mural. Instead of skimming over general shapes, the artist focuses on every single detail. This is like DNS, where every little fluctuation in turbulence is accounted for. However, just like painting a mural requires a lot of time and effort, DNS needs considerable computational power.

In this modeling Reynolds number is expressed as UL by nu ... characteristic timescale for viscous diffusion whereas L by U is the characteristics timescale for advection.

The Reynolds number (Re) is a key dimensionless quantity in fluid mechanics that helps predict flow patterns in different fluid flow situations. It is calculated using flow velocity (U), characteristic length (L), and kinematic viscosity (ν). A high Reynolds number indicates turbulent flow, whereas a low value indicates laminar flow.

Imagine a fast-moving river (high Reynolds number) versus a calm lake (low Reynolds number). The characteristics of flow in these two settings differ vastly, with the river displaying turbulent eddies and chaotic patterns and the lake exhibiting smooth, laminar flow.

So can we conclude that viscous effects are unimportant in turbulent flows ... turbulent energy is dissipated in form of heat by viscous effects?

Despite high Reynolds numbers indicating turbulence, viscous effects play a crucial role in energy dissipation within turbulent flows. Turbulent kinetic energy must be balanced by the viscosity-induced energy dissipation to maintain motion. This concept shows that even in chaos, the stabilizing effects of viscosity cannot be overlooked.

Think about how friction between car tires and the road surface allows vehicles to grip and move. Even though roads can be busy and chaotic, the friction (viscosity’s effect) is essential for controlling motion and preventing crashes. Similar dynamics occur in turbulent flows where viscous effects help manage energy dissipation.

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