Velocity and Acceleration Fields; Coordinate Systems; Stagnation Points

In this lesson, we will be discussing the following topics:

1. Coordinate Systems

2. Velocity Field and Types of Flow

3. Kinematic Properties of Fluids

4. Acceleration Field of a Fluid

5. Example – Finding acceleration of a particle using the above equations

6. Stagnation Points

7. Example – Stagnation Points

Video coming soon

Coordinate Systems

In this course, we use two different coordinate systems:

(1) Cartesian/Rectangular Coordinate System(x, y, z). Velocity components of V are now u, v, w (respectively it is velocity in the x, y, z direction).

We would write out the velocity components like so:

(2) Cylindrical Polar Coordinate System (r, theta, z) – This refers to radius (Radial distance from the z-axis), angle (measured from a line parallel to the x-axis, counterclockwise is positive) and coordinates along the z-axis. The velocity components are now (Vr, V theta, Vz) – respectively, it is the radial velocity, tangential velocity and axial velocity. We use this system when we have circular geometries, such as a pipe.

We would write the velocity components like so, where the “e” terms are the unit vectors in each direction (analogous to i, j, k from the cartesian system).

Here is a visual of this coordinate system labelled on a pipe:

Note that this coordinate system is more difficult to use because while the r- and z-coordinates are for spatial/position changes, theta is for an angular change as you can see in this picture above. Because of this angular change, the unit vectors for r and theta also change their spatial orientation with changes in the angular coordinates. Therefore, when we do derivation of these equations, the derivative of the unit vector is also taken into account.

For example, taking the partial derivative with respect to the coordinate systems in both cases would look like the following. You’ll notice these changes have been implemented in the equations that we use for polar coordinates in this course:

Converting Between Cartesian and Cylindrical Coordinates

You can convert from one coordinate system to another if you wish. The following equations relate these two coordinate systems:

Velocity Field and Types of Flow

Velocity fields can be defined by different types of flow. There are three types of flows that can occur in any situation:

(1) Steady Flow: Velocity does not change with respect to time (is not a function of time).

(2) Unsteady Flow: Velocity changes with respect to time (velocity components are functions of time as well as the geometries x, y, z).

(3) Uniform Flow: No change in velocity over a certain area.

Kinematic Properties of Fluids

Let’s consider any fluid property alpha (this can refer to velocity, temperature, pressure and more). Equations that can be used in certain situations throughout this course are summed up here to understand the difference between all of the equations:

(1) Total Differential Change in alpha

(2) Spatial changes

(3) Time Derivative (change with respect to time) of alpha on a Particle

(4) Substantial Derivative, Partial Derivative or Material Derivative

(5) Particle Acceleration Vector

Acceleration Field of a Fluid

When we say acceleration field of a fluid, it is referring to a concept learned in Fluids I – the velocity field of a fluid! Look at the picture of fluid flowing in a pipe below – there are many, many tiny water molecules flowing in here, but what if we were able to know the velocity and acceleration of each individual water molecule, or of the overall system?

Note: Remember that both velocity and acceleration are vector variables, so they have a magnitude (eg. the velocity is 5 meters/second or the acceleration is 10 meters/second^2), but they also have a direction.

In Fluids I, we learned the equation for the cartesian vector form of a velocity field, that mathematically shows how velocity changes over time and 3D space. This means that it is a function of space and time, and the equation is as follows:

Here’s a breakdown of what each part of the equation means:

When we look at the acceleration (including the direction) of the fluid is flowing, and then draw those vectors, it is the acceleration field of the fluid. There is a complicated equation to figure this out though! Equation 2 is what we can use to obtain the acceleration at any point in a fluid mathematically. Remember from physics that acceleration is velocity divided by time. But remember from calculus that if you have an equation for velocity (eg. if the velocity is 5x+ 2 and not just 5) then you need to start taking derivatives. This means acceleration is instead equal to the derivative of the velocity with respect to time (dV/dt term on the left):

Here’s a breakdown of what each part of the equation means:

We can break this equation down further, since from Equation 1, since velocity is made up of components (u, v, w), but each are dependant on position coordinates (x,y,z) and time (t). This means the equation will then be broken down into the following, using the chain rule for derivatives. Here I have shown the equation specifically for the u-component, du/dt (you can see it’s du/dt on the left side of the equation) but the same will apply to the dv/dt and dw/dt as well.

Here is the general form of the chain rule for derivatives, which is what we use to go from Equation 2 to Equation 3. I won’t cover how the chain rule came about, because that’s a topic in Differential Calculus. But we use this rule when we want to find the derivative of an equation that is dependent on other variables. Specifically, from Equation 2 we simplified the du/dt term in Equation 3. Because u itself was dependent on (x,y,z,t), we used the chain rule here.

Now that we have recalled the chain rule, here are the steps to how we obtained Equation 3 from Equation 2. Note that we are using the curvy letter d in the derivatives because we are taking partial derivatives of each variable, since the equation is dependent on multiple variables x, y, z, and t.

Now writing Equation 3 for the x, y, and z components of acceleration (which are the derivatives of u, v, and w with respect to time – du/dt, dv/dt, and dw/dt):

Here are the steps on how Equation 3 became Equation 4:

Note that the left side of the equation is the Lagranian frame of reference, and the right side is the Eulerian frame.

This overall equation for the acceleration field of a fluid can be separated into different parts, as shown below in Equation 5. This is just taking the same equations from Equation 4, and rewriting them as one overall acceleration, rather than components ax, ay, and az as above.

These different accelerations are called the local and the convective accelerations. The local acceleration (aka unsteady acceleration) is there if the flow is unsteady, because it is due to change of the velocity over the flow field. It disappears when there is steady flow – this then means that the variables are constant over time. Usually, we do assume steady flow in many problems.

The convective acceleration results when the velocity varies with position (eg. in a nozzle). Both of these terms combined give the total acceleration on the left side of the equation, a.

This equation can be applied to any other variable, such as pressure too.

Example: Finding acceleration of a particle using the above equations

We are asked to find the acceleration, and we are given velocity. Knowing that acceleration is the change in velocity over the change in time, and the velocity components with i, j, and k are not numbers – they are equations that depend on x, y, z, and t.

Stagnation Points

Stagnation points are the points in the flow field of a fluid where there is no movement (velocity is zero, so acceleration is also zero). For example, take a look at the following flow:

In this example above, the fluid is moving towards a solid wall, but then gets deflected because of the wall and moves around it instead. Right in the center is where there would be a stagnation point – meaning, if you placed an object there, it wouldn’t move with the fluid because there is no fluid moving at that point.

Example – Stagnation Points

Let’s find the stagnation points in this velocity field. To begin, we need to understand what a stagnation point is. As described above, it is the point where the velocity equals zero. Thus, we can sent each individual velocity component to zero (since the velocity is given to us in i and j components).

After setting each component equal to zero, isolate each equation for y (or x, but I just did it for y):

Set both equations equal to one another (this is using substitution since both equations are equal to y, they must be equal to one another):

Solve for x, then substitute that value back into any one of the two equations to get y.

Therefore, there is a stagnation point, and these are the coordinates: