Lecture # 58
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Reynolds Shear Stress and the Closure Problem
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Welcome everyone! Today we're discussing the important concept of Reynolds shear stress, which plays a crucial role in understanding turbulence in fluid dynamics.
What exactly is Reynolds shear stress?
Good question! Reynolds shear stress refers to the stress exerted by turbulent fluctuations in the flow. It influences mean flow properties like velocity and pressure.
Why do we refer to the related issue as the 'closure problem'?
The closure problem arises because we need to express the complex turbulent shear stress in terms of simpler average quantities. It's essentially a challenge to remove fluctuations from our calculations.
Do we need to remember the entire equation?
No, focus on understanding the concept of shear stress and its impact rather than the mathematical complexity. This understanding will aid your grasp of turbulence models.
So how do we actually resolve this closure problem?
We can model tau_ij based on average flow using approaches like the k-epsilon turbulence model, which we'll cover next.
In summary, Reynolds shear stress is pivotal in analyzing turbulence, and the closure problem challenges us to link turbulent behavior to average flow metrics.
The k-epsilon Turbulence Model
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Now that we've discussed the closure problem, let's focus on the k-epsilon turbulence model, which is one effective approach to addressing it.
What does 'k-epsilon' signify?
'k' represents turbulent kinetic energy while 'epsilon' refers to its dissipation rate. This model establishes a relationship between these two important quantities.
So, how do we calculate turbulent viscosity in this model?
Great question! Turbulent viscosity (nu_T) can be modeled as a function of k and epsilon: nu_T = C_mu * k^2 / epsilon, where C_mu is a constant based on empirical data.
What are the governing equations in the k-epsilon model?
We primarily utilize the continuity equation and momentum equations to solve for k and epsilon, providing direct insights into energy dynamics and flow behavior.
Are there any notable constants we should remember?
Yes, remember the values of C_mu (approximately 0.09) and sigma constants for k and epsilon which are also important.
To summarize, the k-epsilon model is a powerful tool for resolving the closure problem, connecting turbulent energy dynamics with its dissipation.
Direct Numerical Simulation (DNS)
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Next, let's delve into Direct Numerical Simulation or DNS, an advanced approach contrasting the k-epsilon model.
How is DNS different from traditional turbulence models?
DNS solves the Navier-Stokes equations directly without turbulence approximations, providing a detailed flow analysis over scales.
Are there any considerable challenges with DNS?
Absolutely! The computational cost is high, especially at high Reynolds numbers, which require vast numbers of grid points for accurate simulation.
What are those Reynolds number implications I keep hearing about?
The Reynolds number describes the ratio of inertial to viscous forces. As we increase flow speed, inertial effects dominate, amplifying the complexity of simulations.
What's the significance of the Kolmogorov length scale?
The Kolmogorov length scale represents the scale at which energy dissipation occurs through viscous forces, influencing the design of our computational grids.
In conclusion, DNS serves as a powerful tool for understanding turbulence but requires careful consideration of computational resources and simulation scale.
Introduction & Overview
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Quick Overview
Standard
The lecture delves into the Reynolds shear stress equation, introducing the closure problem that arises when modeling average flow variables in turbulent flows. It explores the k-epsilon turbulence model, its constitutive equations, and introduces the concept of direct numerical simulation (DNS) as a method for solving turbulence without approximations.
Detailed
Detailed Summary
In this lecture, we explore Computational Fluid Dynamics (CFD) through the lens of turbulent flows, specifically examining the Reynolds shear stress equation and the pertinent closure problem.
Key Topics Discussed:
- Reynolds Shear Stress: This term indicates how turbulence affects mean flow properties, crucial for modeling. The shear stress is inherently connected to average flow velocities and pressures derived from Reynolds-Averaged Navier-Stokes (RANS) equations.
- Closure Problem: The key challenge here is relating the Reynolds shear stress (tau_ij) to average flow metrics without resorting to complex equations. This involves modeling tau_ij as a function of average flow, assisting in simplifying turbulent flow analyses.
- Turbulence Models: The k-epsilon model serves as an essential tool in the closure problem, where:
- Kinetic energy is partitioned into mean kinetic energy (K) and turbulent kinetic energy (k).
- Governing equations utilize the continuity equation and momentum equations to solve for variables such as turbulent viscosity (nu_T) and energy dissipation rates (epsilon).
- Direct Numerical Simulation (DNS): A detailed discussion on DNS highlights its capability to provide an exact solution by discretizing flow equations with high spatial and temporal resolution. It bypasses turbulence modeling altogether, demanding substantial computational resources, especially at high Reynolds numbers and their implications for grid requirements and computational cost.
In summary, this section underscores essential concepts in turbulent flow analysis and modeling strategies necessary to address turbulence in hydraulic engineering.
Audio Book
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Understanding Reynolds Shear Stress
Chapter 1 of 5
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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 Closure Problem
Chapter 2 of 5
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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...
Detailed Explanation
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.
Examples & Analogies
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.
K-epsilon Model Introduction
Chapter 3 of 5
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Turbulent Kinetic Energy and Its Equation
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Eddy Viscosity and Its Relation to Kinetic Energy
Chapter 5 of 5
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Chapter Content
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.
Detailed Explanation
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.
Examples & Analogies
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.
Key Concepts
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Reynolds shear stress: It highlights how turbulence affects the mean flow properties, crucial for modeling in CFD.
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Closure problem: The challenge of relating complex turbulence to simpler average values in flow analysis.
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k-epsilon model: A widely utilized turbulence model linking turbulent kinetic energy and dissipation.
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Direct Numerical Simulation (DNS): A high-fidelity simulation technique that directly solves fluid dynamics equations without turbulence modeling.
Examples & Applications
The k-epsilon model is employed in engineering applications like airfoil design to predict lift and drag forces in turbulent air.
In large industrial systems, DNS might be used to simulate the behavior of fluid in a condenser to ensure efficient heat transfer.
Memory Aids
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Rhymes
In turbulence, the key you'll find, is Reynolds stress that's intertwined. Closure's challenge, we must embrace, modeling flow makes us interlace.
Stories
Imagine a flowing river where swirling eddies emerge; they embody turbulence. To understand them, researchers create models like k-epsilon, telling the story of energy and its flight through the waters until it's smooth again.
Memory Tools
K-E-E-P: Kinetic energy, Epsilon for dissipation, Essential for turbulence, and Predicting flow behavior.
Acronyms
TURB
Turbulence Understanding Requires Balance - between kinetic energy and dissipation calculations in flow.
Flash Cards
Glossary
- Reynolds Shear Stress
The stress induced by turbulent fluctuations that affects mean flow properties in a fluid.
- Closure Problem
The challenge of connecting complex turbulent behaviors to simpler average flow metrics in fluid dynamics.
- kepsilon Model
A widely-used turbulence model that relates turbulent kinetic energy (k) to its dissipation rate (epsilon).
- Turbulent Kinetic Energy (k)
The energy associated with the chaotic and irregular motions of fluid flow.
- Dissipation Rate (epsilon)
The rate at which turbulent kinetic energy is converted into thermal energy.
- Direct Numerical Simulation (DNS)
A method of solving the Navier-Stokes equations directly without employing turbulence models for high-fidelity simulations.
- Kolmogorov Length Scale
A microscopic scale detailing the level at which energy is dissipated through viscosity in turbulent flows.
- Turbulent Viscosity (nu_T)
A measure of the momentum transfer due to turbulence, influenced by turbulent kinetic energy and dissipation.
Reference links
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