13.021 Marine Hydrodynamics, Fall 2003

Lecture 2

Copyright © 2003 MIT - Department of Ocean Engineering, All rights reserved.

13.021 - Marine Hydrodynamics
Lecture 2

Chapter 1 - Basic Equations

1.1 - Description of a Flow.

Flows are often defined either by an Eulerian description or a Lagrangian description.

• Eulerian description: This is a field description that is often easy to apply. The velocities of the flow are given at fixed points in space as time varies. Imagine a case where both the measuring device and the frame of reference are fixed.

  The velocity, pressure, density, ...can be mathematically represented as follows:

  $\vec{v}(\vec{x},t), p(\vec{x},t), \rho(\vec{x},t), \ldots$

• Lagrangian description: This description is easier to understand but harder to apply. Here the quantities of the flow are given for a particular moving particle at varying times. The velocity, pressure, density, ...can be mathematically represented as follows:

  $\vec{v}_{p}(t), p_{p}(t), \rho_{p}(t), \ldots$

Useful terms for flow description

• Streamline: A line everywhere tangent to the fluid velocity $\vec{v}$ at a given time. In an Eulerian description, it would be a `snapshot' of the flow (See tutorial ).

• Streakline: Instantaneous locus of all particles that pass a given point. In an Eulerian description, it would be a `snapshot' of certain particles (See tutorial ).

• Pathline: The trajectory of a given particle P in time. The photograph analogy would be a long time exposure of a given particle (See tutorial ).

Some Eulerian Quantities of Interest

• Scalars: have magnitude only.

  Pressure: $p\left( \vec{x},t \right)$ ; Density: $\rho \left( \vec{x},t \right)$

• Vectors: have magnitude and direction

  $$\vec{v} = u\hat{i}+v\hat{j}+w\hat{k}$$

  $$=u_{1}\hat{x_{1}}+u_{2}\hat{x_{2}}+u_{3}\hat{x_{3}}$$

  $$=\sum_{i}^{3} u_{i}\vec{x_{i}}$$

  $$= u_{i} \vec{x_{i}}$$

  
  
  Einstein Notation: Repeated indices are summed by implication over all values of the index i. In this example, the summation is over i =1, 2, 3.
  

  Note that if the equation looks like this: $\left( u_{i} \right) \left( \vec{x}_{i} \right)$ , the indices are not summed.

Concept of Continuous Flow

For a fluid flow to be continuous, we require that the velocity $\vec{v}(\vec{x},t)$ be a finite and continuous function of $\vec{x}$ and t. i.e. $\nabla \cdot \vec{v}$ and $\frac{\partial \vec{v} }{ \partial t}$ are finite but not necessarily continuous. Since $\nabla \cdot \vec{v}$ and $\frac{\partial \vec{v} }{ \partial t}$ < $\infty$, there is no infinite acceleration, which is physically consistent.

Consequences of Continuous Flow

• Material volume remains material. No segment of fluid can be joined or broken apart.

• Material surface remains material. The interface between two material volumes always exists.

• Material line remains material. The interface of two material surfaces always exists.
  

• Material neighbors remain neighbors. To prove this mathematically, we must prove that, given two particles, the distance between them at time t is small, and the distance between them at time t + $\delta$t is still small.

Given: Two particles with initial position $\vec{x}$ and $\vec{x}+\delta\vec{x}$, initial time t and the fluid velocity v.

Proof:

Where $\vec{v}\delta t$ is the distance travelled by the particle. The difference in position between the two particles is:

$$\left[\vec{x}(t)+\delta\vec{x}(t)\right]+\left(\vec{v}+\delta\vec... ...\vec{v}\right)\delta t-\vec{x}(t)-\vec{v}\delta t = \delta\vec{x}(t+\delta t)$$ (1)

$$\Rightarrow \delta\vec{x}(t+\delta t) = \delta\vec{x}(t)+\delta\vec{x}(t)\cdot\nabla\vec{v}\delta t \propto \delta\vec{x}(t)$$

Therefore $\delta\vec{x}(t+\delta t) \propto \delta \vec{x}(t)$ since $\nabla \vec{v}$ is finite due to the continuous flow assumption. Therefore if $\delta \vec{x}(t) \rightarrow 0$ , then $\delta \vec{x}(t+\delta t) \rightarrow 0$. In fact, for any time period T,

$$\delta \vec{x}(t+T) \propto \delta \vec{x}(t)+\int_{t}^{t+T}\delta\vec{x}\cdot\nabla \vec{v}dt \propto \delta\vec{x},$$

and $\delta \vec{x}(t+T) \rightarrow 0$ as $\delta \vec{x}(t) \rightarrow 0$ , therefore the particles will never be an infinite distance apart. Thus the flow is continuous and two particles that are neighbors will always be neighbors.

Material/Substantial Derivative: D/Dt

A material derivative is the time derivative - rate of change - of a property `following a fluid particle P'. The material derivative is a Lagrangian concept but we will work in an Eulerian reference frame.

Consider an Eulerian quantity $f(\vec{x},t)$. Taking the Lagrangian time derivative of an Eulerian quantity gives the material derivative. The Lagrangian time derivative is:

$$\begin{split} \frac{Df(\vec{x},t)}{Dt} & =\frac{df}{dt} =\le... ...vec{v}\delta t,t+\delta t)-f(\vec{x},t)}{\delta t} \end{split}$$

P is moving at an Eulerian velocity $\vec{v} = \frac{\delta\vec{x}}{\delta t}$. Performing a 3D Taylor series on f gives:

$$f( \vec{x},t+\delta t) = f(\vec{x},t)+\delta t\frac{\partial f(\vec{x},t)}{\partial t}+\delta \vec{x}\cdot\nabla f(\vec{x},t)+O(\delta^{2})$$

Therefore the substantial derivative is:

$$\frac{Df}{Dt} =\frac{\partial f}{\partial t} +\vec{v}\cdot\nabla f$$

With the generalized notation:

$$\underset{\mbox{Lagrangian}}{\underbrace{\frac{D}{Dt} }} \equiv ... ...mbox{Eulerian}}{\underbrace{\frac{\partial }{\partial t} +\vec{v}\cdot\nabla }}$$

Example: Lagrangian acceleration of a particle. Consider the Eulerian velocity $\vec{v}(\vec{x},t)$. Then the Lagrangian acceleration is:

$$\underset{\mbox{ \begin{tabular}{c} Lagragian acceleration \... ...lar}{c} Convective acceleration \end{tabular}}}{\vec{v}\cdot\nabla\vec{v}}$$

Difference between Lagrangian time derivative and Eulerian time derivative

Example: Consider an Eulerian quantity, temperature, in a room at points A and B where the temperature is different at each point.

At point C, the temperature rate of change is $\frac{\partial T}{\partial t}$ which is an Eulerian time derivative.

Example: Consider the same example as above: an Eulerian quantity, temperature, in a room at points A and B where the temperature varies with time.

Following a particle from point A to B, the Lagrangian time derivative would need to include the temperature gradient as time and position changed: $\frac{DT}{Dt} =\frac{\partial T}{\partial t} +\vec{v}\cdot\nabla T$

Concept of a Steady Flow

Assume a steady flow where the flow is observed from a fixed position. This is like watching from a river bank, i.e. $\frac{\partial }{\partial t} =0$ . Be careful not to confuse this with $\frac{D}{Dt}$ which is more like following a twig in the water. Note that $\frac{D}{Dt} =0$ does not mean steady since the flow could speed up at some points and slow down at others.