1.2. Department of Civil Engineering

Hydraulic Engineering Prof. Mohammad Saud Afzal Department of Civil Engineering Indian Institute of Technology-Kharagpur Lecture # 58 Computational Fluid Dynamics (Contd.,)

In this lecture, we are continuing our module on Computational Fluid Dynamics (CFD). The focus is on understanding turbulent flow and the models used for its simulation, including the K-epsilon model. This section serves as an introduction to the complexities involved in modeling turbulent flows and sets the stage for the detailed discussions that follow.

Think of turbulent flow like a busy street filled with cars in different lanes, some moving quickly and others slowly. The way the cars interact with each other represents the complex dynamics of turbulent flow, just as CFD models help us predict those interactions in fluid dynamics.

We are going to study the closure problem. The effect of the Reynolds shear stress rho in tau i j on the mean flow is like that of a stress term. So, rho in tau i j is actually Reynolds shear stress. So it is like shear stress to obtain u i and pressure.

The chapter introduces the concept of Reynolds shear stress, which is a critical part of modeling the mean flow in turbulent fluid dynamics. The 'closure problem' refers to the challenge of calculating this stress term accurately. Understanding how shear stress affects mean flow is essential for predicting fluid behaviors.

Imagine trying to predict how a crowd of people will move through a narrow hallway. Just as the interactions (shear) between individuals impact the overall flow, Reynolds shear stress quantifies how turbulence affects the average flow of a fluid.

The term 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 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.

The k-epsilon model is a widely used approach in turbulence modeling. It defines turbulent kinetic energy (k) as the total energy of the flow, composed of both average (mean) and fluctuating components. This helps modelers understand and predict how energy is distributed and dissipated in turbulent flows.

Think of it like a baseball game where the overall score (mean energy) includes both the runs scored by the team and the errors made on the field (turbulent fluctuations). The combination gives a complete picture of the game’s outcome.

The governing equations for this are one we have a continuity equation and the other is now modeling of the Reynolds shear stress as the momentum equation.

In turbulence modeling, two essential equations govern fluid flow: the continuity equation, which ensures mass conservation, and the momentum equation, which governs the flow's motion (including Reynolds shear stress). The relationship between these equations is fundamental in CFD analysis.

Consider a water pipe where you need to conserve the amount of water flowing in and out (continuity), while also accounting for how fast the water is moving and how it interacts with the pipe walls (momentum). These principles guide engineers in designing efficient fluid systems.

This nu T is the eddy viscosity. It can be related to kinetic energy k and kinetic energy dissipation rate epsilon through nu T is written as c mu into k square by epsilon.

Eddy viscosity, denoted as nu T, is a measure of how turbulence affects momentum transfer in fluid flow. It is defined in relation to turbulent kinetic energy and its dissipation rate. This relationship is vital for accurately capturing how turbulent energy dissipates as heat.

Imagine a stirred cup of coffee. The stirring creates eddies (vortices) that mix the coffee. Just as these eddies help mix the coffee and transfer heat, eddy viscosity helps model how turbulence mixes and dissipates energy in fluids.

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

The discussion continues with equations that help model turbulent kinetic energy (k) and its dissipation (epsilon). These are crucial for deriving turbulent eddy viscosity and establishing boundary conditions used in CFD simulations.

Think of cooking where the heat energy (like turbulent kinetic energy) needs to be balanced with the energy that dissipates during cooking. You control the heat (dissipation) to maintain the right cooking temperature; similarly, these equations maintain energy balance in fluid flows.

To sustain the fluctuations, the supply of kinetic energy must be balanced by these dissipation of turbulent energy...

In turbulent flow, it's crucial that the energy supplied to the system is matched by energy losses due to turbulence dissipation. This principle helps create realistic simulations of fluid dynamics by maintaining energy balance throughout the flow.

Consider balancing your bank account. Just as you need to ensure your expenses (dissipation) do not exceed your income (supply), in turbulent flows, the energy supplied needs to equal the energy dissipated to ensure stability in the system.

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