# 25. Computational Fluid Flow, Hydrodynamics

TLDRThis lecture delves into the complexities of computational fluid dynamics, focusing on hydrodynamics and its challenges. It introduces the fundamental equations, including the continuity equation and the Navier-Stokes equation, essential for describing fluid flow. The lecture simplifies the problem by using the stream function and vorticity, reducing the complexity of fluid dynamics to more manageable equations. It also discusses the importance of boundary conditions and provides an algorithm for solving these equations using relaxation techniques. The practical application involves using simulations to understand fluid behavior, particularly for environmental projects like creating habitats for salmon.

###### Takeaways

- π The lecture covers advanced topics in computational fluid dynamics (CFD) and computational hydrodynamics, emphasizing their complexity and the need for a dedicated class to fully understand them.
- π§ The speaker breaks down the lecture into two parts: basics and vorticity fields, highlighting the importance of understanding the equations and their forms before diving into simulations.
- π The practical application discussed involves the Oregon Department of Fish and Wildlife using CFD to restore rivers for salmon habitats, illustrating the real-world relevance of the subject.
- π‘ The importance of simulations is underscored, with the suggestion to use them to validate the equations by observing if the simulated fluid behaves as expected in the real world.
- π§ The lecture introduces two fundamental equations of fluid dynamics: the continuity equation and the Navier-Stokes equation, which describe the conservation of mass and motion of fluid, respectively.
- π The assumptions made in the models include incompressibility, steady-state conditions, and the effects of viscosity on fluid flow, which are crucial for setting up the equations.
- π The speaker explains the significance of boundary conditions in solving partial differential equations, noting that they are essential for obtaining a unique solution.
- π An analytical solution for a simplified two-plate problem is presented, demonstrating how the velocity of the fluid changes due to the presence of the plates.
- π The numerical solution process involves using relaxation techniques to solve the equations, with an emphasis on the challenges of dealing with non-linearities in fluid dynamics.
- π The concept of stream functions and vorticity is introduced to simplify the complex Navier-Stokes equations, transforming them into more manageable forms.
- π The Reynolds number is discussed as a measure of the non-linearity of the fluid flow, with its impact on whether the flow is laminar or turbulent.

###### Q & A

### What is the main topic discussed in the script?

-The main topic discussed in the script is Computational Fluid Dynamics (CFD), specifically focusing on the application of CFD in hydrodynamics, including the challenges and complexities of solving fluid dynamics equations.

### Why is the subject of computational fluid dynamics considered advanced?

-Computational fluid dynamics is considered advanced because it involves complex equations and concepts that require a deep understanding of fluid behavior, numerical methods for solving these equations, and the ability to interpret the results in a real-world context.

### What is the significance of the Navier-Stokes equation in fluid dynamics?

-The Navier-Stokes equation is significant in fluid dynamics as it describes the motion of fluid substances, including the effects of viscosity, pressure, and gravity. It is one of the fundamental equations in physics for understanding and predicting fluid flow behavior.

### What is the role of the continuity equation in fluid dynamics?

-The continuity equation in fluid dynamics ensures the conservation of mass, stating that the flow into and out of a region of space must be equal, which is crucial for solving fluid dynamics problems, especially when dealing with incompressible fluids.

### How does the script suggest approaching the learning of complex equations in fluid dynamics?

-The script suggests that learners should not expect to understand all the equations on the first pass. It encourages learners to get a feel for the form of the equations, how they are solved, and to use simulations to see the equations in action and to verify their appropriateness.

### What is the practical problem presented in the script related to the application of fluid dynamics?

-The practical problem presented is related to the Oregon Department of Fish and Wildlife's efforts to restore rivers for salmon habitats. The script discusses the use of computational fluid dynamics to determine the optimal size and placement of boulders or structures in rivers to provide resting places for salmon.

### What is the significance of the Reynolds number in the context of fluid dynamics?

-The Reynolds number is significant as it is a measure of the ratio of inertial forces to viscous forces and is used to predict the flow patterns in different fluid flow situations. It helps in determining whether the flow will be laminar or turbulent.

### How are the concepts of stream function and vorticity introduced in the script?

-The stream function is introduced as a scalar potential that helps describe the direction of the flow, and the vorticity is introduced as a vector potential that measures the rotation or 'curl' in the fluid flow. Both are used to simplify the solution of the Navier-Stokes equations.

### What numerical method is suggested for solving the fluid dynamics equations presented in the script?

-The script suggests using a relaxation technique, specifically the successive over-relaxation (SOR) method, to solve the fluid dynamics equations numerically.

### How does the script relate the study of fluid dynamics to the study of electromagnetism (EM)?

-The script relates fluid dynamics to electromagnetism by drawing parallels between the equations and concepts used in both fields, such as the use of potentials (stream function and vector potential in fluid dynamics compared to scalar and vector potentials in EM) and the application of vector identities.

###### Outlines

##### π Introduction to Computational Fluid Dynamics

The speaker introduces the topic of computational fluid dynamics (CFD), emphasizing its complexity and importance. CFD is crucial for simulating the behavior of fluids, and the lecture is divided into two parts: basics and vorticity fields. The speaker suggests that viewers should not expect to grasp everything in one go and recommends using provided code to experiment with simulations and observe fluid behavior. The practical application discussed involves the Oregon Department of Fish and Wildlife using boulders to create habitats for salmon, highlighting the real-world relevance of CFD.

##### π Theoretical Foundations of Fluid Dynamics

This paragraph delves into the theoretical aspects of fluid dynamics, starting with the continuity equation, which is fundamental in physics and essential for describing incompressible fluids. The speaker explains the concept of fluid viscosity and the steady-state condition, assuming no change in velocity over time. The Navier-Stokes equation is introduced as a complex but vital equation in fluid dynamics, accounting for various forces acting on a fluid, including pressure gradients and friction. The lecture aims to help viewers understand these equations and their significance in simulating fluid flow.

##### π Navier-Stokes Equation and Fluid Dynamics Challenges

The Navier-Stokes equation is further elaborated upon, highlighting its role as a 'grand challenge' in science and its complexity due to the nonlinearities it introduces. The speaker discusses the equation's relation to Newton's second law, describing how it represents the change in momentum of a fluid due to various forces. The equation is simplified for the context of the lecture, assuming steady-state flow and incompressibility, leading to a system of partial differential equations that need to be solved simultaneously. The importance of boundary conditions in obtaining a unique solution is also emphasized.

##### π οΈ Setting Up the Computational Problem

The speaker outlines the computational problem setup, focusing on the geometry of the problem and the assumptions made, such as a deep, fast-flowing, and wide river. The problem is simplified by considering only the middle part of the river, ignoring edge effects. The speaker introduces the boundary conditions necessary for solving the Navier-Stokes equation, including constant stream velocity, no-slip conditions on the plates, and symmetry considerations. The goal is to find an appropriate solution that describes the flow around an obstacle like a boulder or a beam in the river.

##### π Analytical and Numerical Solutions for Fluid Flow

The lecture presents an analytical solution for the velocity in the x-direction around thin plates in a stream, derived from the simplified Navier-Stokes equation. The solution accounts for the Bernoulli effect, showing how pressure changes as fluid moves around an obstacle. The speaker then transitions to discussing numerical methods for solving these equations, mentioning relaxation techniques and successive over-relaxation to improve convergence. The importance of the grid system and finite differences in numerical solutions is highlighted.

##### π Exploring Vorticity and Stream Functions

The second part of the lecture begins with an exploration of vorticity fields and stream functions, which simplify the solution of fluid dynamics problems. The stream function is introduced as a way to represent the velocity field without directly solving for velocity components, automatically satisfying the continuity equation. The vorticity, a measure of the rotation in the flow, is defined as the curl of the velocity. The speaker discusses the significance of these concepts in understanding and simulating fluid flow, including the transition from laminar to turbulent flow.

##### π Mathematical Representation of Vorticity

This paragraph focuses on the mathematical representation of vorticity, establishing it as a vector quantity that quantifies the rotation within a fluid flow. The vorticity is derived from the curl of the velocity, and its relationship with the stream function is explored. The speaker simplifies the Navier-Stokes equation using these new potentials, leading to Poisson's equation for the stream function, with vorticity acting as a source term. The importance of the right-hand rule in determining the direction of vorticity is mentioned.

##### π Solving the Simplified Fluid Dynamics Equations

The speaker presents the simplified form of the Navier-Stokes equation in terms of the stream function and vorticity, resulting in a set of coupled equations that are easier to solve than the original system. The equations are elliptic in nature, and the speaker discusses the use of relaxation techniques to solve them numerically. The role of the Reynolds number in determining the flow characteristics, such as laminar or turbulent flow, is highlighted. The lecture also touches on the challenges of implementing boundary conditions for these equations.

##### ποΈ Applying Boundary Conditions and Simulation

The final paragraph of the lecture discusses the application of boundary conditions for the simplified fluid dynamics equations. The speaker provides guidance on setting boundary conditions for different regions of the flow, such as the inlet, outlet, and surfaces of obstacles like beams. The challenge of accurately implementing these conditions in simulations is acknowledged. The speaker encourages viewers to experiment with the provided code to gain a deeper understanding of fluid dynamics and to explore the effects of different parameters and obstacle placements on flow behavior.

##### π£ Conclusion and Practical Application

In conclusion, the speaker reiterates the importance of understanding the physical representation of the simulation, particularly in the context of the salmon habitat restoration project introduced earlier. The lecture wraps up with an invitation for the viewers to engage with the simulation, manipulate parameters, and observe the effects on fluid flow and potential resting places for salmon. The goal is to apply the theoretical and computational knowledge gained to solve a real-world problem and to deepen the understanding of fluid dynamics.

###### Mindmap

###### Keywords

##### π‘Computational Fluid Dynamics (CFD)

##### π‘Hydrodynamic Equations

##### π‘Navier-Stokes Equation

##### π‘Continuity Equation

##### π‘Vorticity

##### π‘Stream Function

##### π‘Laminar Flow

##### π‘Incompressible Fluid

##### π‘Boundary Conditions

##### π‘Successive Over-Relaxation (SOR)

##### π‘Reynolds Number

###### Highlights

Computational fluid dynamics (CFD) is a complex and advanced topic that typically requires a dedicated term of study to fully understand.

CFD involves solving serious equations that may not be fully understood on the first pass, suggesting the need for repeated study and practical application.

The lecture is divided into two parts: basics of CFD and vorticity fields, with a recommendation to take a break between for better comprehension.

The practical application of CFD is demonstrated through a real-world problem of restoring rivers for salmon habitats, introducing the importance of boulder placement.

The theoretical foundation of CFD includes the continuity equation, a standard physics equation derived from quantum mechanics, indicating the fluid's incompressible nature.

The Navier-Stokes equation is introduced as the fundamental equation of hydrodynamics, describing the motion of fluid substances.

The significance of the Navier-Stokes equation is emphasized as one of the most challenging and important in physics, even featured on a mouse pad of grand challenge equations.

The introduction of the stream function and vorticity as additional quantities to simplify the solution of Navier-Stokes equations, showing a deeper layer of mathematical modeling in fluid dynamics.

The stream function is defined as the curl of a vector function, providing a way to describe the direction of the stream flow without directly solving for the velocity field.

Vorticity is introduced as a measure of the rotation in a fluid flow, crucial for understanding turbulence and other complex flow behaviors.

The relationship between the stream function and vorticity is established through the curl operation, showing how they are mathematically linked in fluid dynamics.

The simplification of the Navier-Stokes equation using the stream function and vorticity results in a form similar to Poisson's equation from electromagnetism, highlighting the connection between different fields of physics.

The numerical solution of the simplified Navier-Stokes equation is discussed, involving the use of relaxation techniques and finite difference methods.

The importance of boundary conditions in defining the solution to CFD problems is emphasized, noting that they must be adequately specified to avoid over-specification.

The practical implementation of the algorithm for solving the vorticity form of the Navier-Stokes equation is outlined, including the use of successive over-relaxation for faster convergence.

The Reynolds number is introduced as a measure of the non-linearity of the fluid flow equations, with significant implications for the behavior of the fluid, such as the transition from laminar to turbulent flow.

The final application of the CFD lecture is to answer the practical question of whether a salmon can find a resting place behind a beam in a stream, tying the theoretical discussion back to the initial real-world problem.

###### Transcripts

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