2.1. Reynolds Shear Stress

Hydraulic Engineering Prof. Mohammad Saud Afzal Department of Civil Engineering Indian Institute of Technology-Kharagpur Lecture # 58 Computational Fluid Dynamics (Contd.) Welcome back to the last lecture of this module computational fluid dynamics and the last lecture we ended at a point where we saw the Reynolds shear stress equation. We discussed about the different terms in it you do not need to worry about the mathematical equation of those terms, but some common terms like Reynolds shear stress, those things are recommended that you remember them, but not the complex terms.

This introduction sets the context for the discussion on Reynolds shear stress within the larger topic of computational fluid dynamics. The emphasis is on understanding basic concepts rather than complex mathematical equations.

Think of Reynolds shear stress as the friction between layers of water flowing in a river, where some layers move faster than others. Just like how we don’t always need to know the physics behind every bump in the road, here, we focus on understanding the basics without getting lost in equations.

The effect of rho in tau i j on the mean flow is like that of a stress term. 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.

This chunk explains how Reynolds shear stress, denoted as  ho in tau i j, functions similarly to a stress term in fluid dynamics. It relates to how the mean flow interacts with the shear stress to derive average flow velocities (u i) and pressures.

Imagine a busy highway where cars are changing lanes. The friction between these moving vehicles is like the Reynolds shear stress; it affects how quickly and easily cars can move (or flow) under normal conditions.

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

The closure problem arises when trying to model complex fluctuations in turbulent flow. By relating the shear stress to average flow measurements, we simplify calculations and make them more manageable.

Think of it like trying to predict the average wait time in a busy restaurant. Instead of considering how long each individual group waits—each with their own unique circumstances—we use an average to simplify our prediction.

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 effects the turbulent kinetic energy, k stands for kinetic energy.

The k-epsilon model is a commonly used method to model turbulence. Here, k represents the turbulent kinetic energy, explaining how energy is distributed and dissipated in a flow. This model helps in understanding the overall behavior of turbulent flows.

Consider stirring a pot of soup. As you stir, some areas of the soup whip around faster (high kinetic energy) while others remain still (low kinetic energy). The k-epsilon model helps predict how this mixing process works on a larger scale.

So, this nu t is the eddy viscosity. So, this actually is not should better be called as turbulent eddy viscosity, nu T.

Eddy viscosity (nu T) quantifies the turbulent effects within fluid flow. It's a concept used to relate turbulence to its dissipative forces. Understanding this is crucial for enabling more accurate models of fluid dynamics.

Think of eddy viscosity as the way a blender works. When blending, the eddies mix the ingredients together unevenly, just as eddies in the fluid create turbulence, affecting how energy flows and dissipates in the mixture.

The turbulent kinetic energy and the energy dissipation can be determined from the following equation.

This content highlights how turbulent kinetic energy (k) and dissipation rate (epsilon) can be calculated, providing foundational insight into turbulent flows. These two concepts are essential for modeling turbulence accurately.

Imagine running water in a river. The energy of the water (turbulence) comes from the slopes (kinetic energy), while boulders and bends dissipate this energy by slowing it down or spreading it out.

This for just writing it down solve for k and epsilon then nu T is related to k square by epsilon.

This chunk discusses how the turbulent kinetic energy and dissipation rates are linked together and used to find turbulent eddy viscosity. This relationship highlights the complexity involved in modeling turbulence in fluid dynamics.

It's like trying to solve a puzzle. Each piece (k, epsilon, nu T) fits together to show the complete picture of how turbulence works, but finding the right connections takes knowledge and practice.

Most of the turbulence model in the world follow k and epsilon, there is one other turbulence model for k omega.

This chunk summarizes the prevalence of k-epsilon models in fluid dynamics while also acknowledging the existence of the k-omega model. Both have their respective advantages and applications.

Think of k-epsilon and k-omega as different brands of shoes. Each brand features unique designs (models) but ultimately serves the same purpose (modeling turbulence) depending on user preference and needs.

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