Hello, and welcome to our ninth lecture in aerodynamics. Last time, we started to explore specifically incompressible and inviscid flows, with a hint of irrotationality sprinkled throughout. This led us to two gigantic equations in the field. The Bernoulli equation, which relates the pressure and velocity along a streamline for rotational flow, and if the flow is irrotational, Bernoulli works everywhere.
Also, If the flow is irrotational, we showed that both the velocity potential and the stream function satisfy the Laplace equation. Today, we are going to continue our dive into incompressible and inviscid flows and start to look at elementary flows. In a way, these are small packages of unique flow features that represent building blocks for aerodynamics. The main four we will cover today are the uniform flow, the source and sink, the doublet, and the vortex.
Before we jump in, let's recall some things we've learned in the past. If flow is incompressible and irrotational, we know that we can use the Laplace equation on the stream function and the velocity potential. These are second order linear partial differential equations.
There are probably many important features of these types of equations, But what's most important to us is that it means we can add solutions to these equations together. For example, if we want to recreate the flow over an oval, which we will learn shortly, we can simply superimpose three separate elemental flows by adding them together. We can add their stream functions, and we can add their velocity potentials. And in the end, we have equations that describe the oval flow.
So? Let's remember that we can add the solutions together, and we'll put this in the back of our heads and call it A. Now, a useful feature of inviscid flow is that we do not have boundary layers on our surfaces. This means that the fluid can move freely without viscosity slowing it down, and there's no longer a no-slip condition to worry about.
If we have an object we would like to learn about aerodynamically, we can recreate that object shape in streamlines. We can fully describe the flow field over that object assuming flow is irrotational. And that's really the point of today, is trying to make quote unquote objects in a flow with streamlines using building blocks. We're going to essentially build aerodynamic flows from scratch.
So let's jump right into the elementary flows. Elementary flows continue to fall under our umbrella of inviscid and incompressible flow, meaning viscosity is zero and density is constant. These are the building blocks of aerodynamic flows, allowing us to create complex flow fields completely analytically.
Note, today is mostly in cylindrical coordinates. It's just unavoidable. So many of our elementary flows are round. so it makes more sense to use a polar coordinate system than a Cartesian one. There are four main basic flows that we'll cover today.
The uniform flow, the source and sink, the doublet, and the vortex. We're going to consider each individually, but build them in a way that they're easy to compare to one another. We'll look at the velocity field, the stream function, and the velocity potential for each flow.
The uniform flow is just a constant velocity field where the streamwise velocity is a constant u infinity and the vertical velocity is zero. We can build the stream function from the velocity field by definition. u is d psi dy which is u infinity, v is d psi dx which is zero. Using these two equations to solve for psi we can find the stream function in Cartesian coordinates.
However, the rest of our fields will be in cylindrical coordinates, so let's transform it. Next up we have the velocity potential. Similarly, we can build it from the velocity field using the definition of the velocity potential.
This leads us to the Cartesian definition of the velocity potential for a uniform flow, which will turn into cylindrical coordinates again. This elemental flow represents translational movement of an object. and will be the main ingredient for almost all of our flows. Next, we have the source and the sink, which are identical flows but opposite in direction.
They are characterized by having streamlines that emit outwards or inwards along straight streamlines coming from a single point. Sources flow out and sinks suck in. The velocity field is characterized by the volumetric flow rate per unit span, or depth, represented by lambda. If we are working with a source, the volumetric flow rate is positive, and for a sink it's negative.
Note, we're starting now with the cylindrical coordinates outright. The radial velocity, u sub r, is the volumetric flow rate divided by the local circumference, 2 pi r. The azimuthal velocity, u theta, is by definition 0. Again, we build the streamlines by starting with the velocity field and using the definition of the stream function in cylindrical coordinates. This gives us a simple expression for the source and sink stream function. If it's a source, this is positive, and if it's a sink, this is negative.
And finally, we build the velocity potential from the velocity field using its defined equations. Next, we move on to the doublet. The doublet is not a unique building block because it's really just a source and a sink combined and shoved really close together.
Consider a source and sink both with equal and opposite strengths, separated by the distance L. What we do to make a doublet is shrink L to zero while simultaneously keeping the parameter L times lambda a constant. This means that as L goes to zero, lambda approaches infinity.
This constant lambda L represents the strength of the doublet. Instead of building the stream function from the velocity as we've done before, we can build it by putting a source and sink stream function together and taking it to these limits and constraints. Start by adding the stream function of a source and sink with equal and opposite strength.
Note here that each get their own theta coordinate. Let's define the difference between theta1 and theta2 to be delta theta and rewrite the equation. From trigonometry and the small angle approximation, we can define delta theta into L sine theta divided by R.
This gives us our final form of the stream function for a doublet in cylindrical coordinates. Now, we'll define the velocity potential from the stream function. Essentially, with one step, we're going to convert the stream function to the velocity field, and then the velocity field to the velocity potential. This gives us two equations for the velocity potential with unknown functions of integration. These can be solved for the final form of the velocity potential for a doublet in cylindrical coordinates.
You'll see the strength of the doublet. L times lambda, which is a constant, called k or kappa in a lot of places. And now we move on to our final elemental flow, the vortex. This is characterized by having a fluid orbiting around some central point. The flow field is defined by having no radial velocity and a constant angular velocity, thus constant c over the radial coordinate.
An interesting feature of this vortex, sometimes referred to as a point vortex, is that it is irrotational everywhere except at the center point, much like a ferris wheel. The constant that defines the azimuthal velocity, uθ, is the circulation divided by 2π. The strength of the vortex is tuned via this circulation.
You might notice that each elemental flow has a parameter to tune the strength. The uniform flow has u∞, The sources and sinks have lambda, the doublet has k or kappa, which is lambda times L, and the vortex has circulation. From the velocity field, as we've done before, let's build the stream function. And finally, let's build the velocity potential from the velocity field.
You might be wondering, Why do we bother with all of this? Well, now we can use these stream functions and velocity potentials to build complex flows by adding them together. Let's get started by building some relatively simple flows from our building blocks. Today we'll cover the semi-infinite body, the Rankine oval, the stationary cylinder, and the rotating cylinder. First up, we have the semi-infinite body.
which is what you get when you add together a uniform flow and a source. You'll notice that the final streamlines of our flow take the shape of what looks like a body. What we'll do here for each flow that we build is first define the stream function, and then velocity field if we can.
Then, talk about the stagnation points and the streamlines that conveniently represent the body. From above, we know the stream function for a uniform flow, and a source, separately. Adding them together gives us the stream function for this complex flow.
Let's mark this down as the final form of the stream function for a semi-infinite body. Once we have the stream function, we can take spatial derivatives of it to get the velocity field. Keep in mind, we're still using the cylindrical coordinate system, so the cylindrical version of the stream function definition is used. In these flows, the stagnation points are important because they often lie on the streamline that best represents a body's boundary for that particular flow. Looking above, we can identify the stagnation point.
The streamline with the stagnation point, after splitting, looks a lot like the boundary that represents our semi-infinite body. So, we want to find our stagnation point or points in the flows when we build them. By definition, stagnation has a zero radial and azimuthal velocity. Let's write down our velocity field again, and determine from the two equations where in the field both equations are zero. The coordinate for stagnation in this flow is at a theta angle of pi, or 180 degrees, and a distance from the origin of lambda over 2 pi u infinity.
Note here that the origin is defined by the center of the source. Now, the streamline that has stagnation point is found by taking the stream function and evaluating it at this coordinate. From above, we know the stream function. Sine of pi is zero, so the entire first term goes away.
The second term simplifies, and we get that psi equals lambda over two. and that's the constant that represents the streamline with the boundary on it. Using the streamline, we can define the flow outside of it to be equal to the flow around a semi-infinite body if it existed in our flow.
Next, we consider our first closed body, the Rankine Oval. This is defined as a uniform flow plus a source and sink of equal and opposite strength. The resulting streamlined field looks a lot like flow around an oval. Here, let's note the center of the source and sink, and define the origin to be at the center of our oval. The source and sink are both a distance b away from the origin.
Now, the fact that the source and sink are off of the origin will present some coordinate difficulties for us pretty soon. Consider some point in our flow. The origin, the source, and the sink all have different angles and radial locations relative to this point. It's important to keep this in mind moving forward.
First up, we define the stream function by adding the stream function of each component individually. In our stream function, theta1 and theta2 are the angles from the source and sink's reference frames. Since the source and sink are not at the origin, these theta1 and theta2 angles are functions of the azimuthal and radial coordinate from the true origin, theta and r, and the distance of the source and sink from the origin, b. Let's quickly sketch the system and see if we can figure out theta1 and theta2 with trigonometry.
What we find is that these angles are inverse tangent functions of the origins, coordinates, and b. Now, if we truly define the velocity field, we need to incorporate these angle functions into the stream function in order to take accurate derivatives. And, that would be pretty nasty. Lucky for us, we rarely need the velocity field itself, and we can skip that here. If you ever did need it, you could just do it out with the derivatives as defined.
Since we don't have the velocity field to truly calculate our stagnation locations, we have to do it by inspection. If our flow represents the flow around an oval, we can expect that the stagnation will happen at the upstream and downstream tips of our body. This means that theta equals theta1 equals theta2, which is at the angular coordinates of 0 and pi, which represent the back and front of the oval respectively. Knowing this, we can plug it into our stream function equation and simplify. The first term is zero because the sine function for these theta coordinates is zero.
And the last two terms are always equal and opposite when they have shared theta values, or when theta1 equals theta2. So, we find that psi equals zero for the stagnation points. This constant represents the streamline that best represents the body because it has the stagnation points on it.
So we set the original stream function equation equal to zero to define the streamline. Now we move to a special flow where essentially the source and sink are merged. This is a flow over a stationary cylinder and is the sum of a uniform flow and a doublet. Here we see that when we add the streamlines of a doublet and uniform flow, we get a stream pattern that represents a two-dimensional circle. which is a cylinder when considered extension in the third dimension.
As we've done in the past, add the stream functions of the individual flows together to get the stream function of the cylinder flow. Let's do some convenient rearranging and bundle some constants that we'll define as being the square of the circle radius. Then, we can rewrite our simplified expression for the stream function. We get the velocity field by taking the spatial derivatives of the stream function.
By now, we're experts at estimating the stagnation point locations, which we can see happens at the radial and azimuthal coordinates r and 0 and r and pi. Notice that, if we were to plug these values into the stream function, for both cases the stream function would equal zero. This means that the constant zero represents the streamline of the cylindrical body when the stream function is set equal to that constant.
Everything outside of this boundary streamline represents flow over a stationary cylinder. And last, we have our first opportunity to use the vortex flow. If we take stationary cylinder flow and add a vortex, we get the case of a rotating cylinder.
Adding rotation adds a lot of interesting features to the flow, which we'll explore more later in our studies. But here we'll explore what it does to the streamline field. With rotation, we can see that it pulls more of the streamlines up above the cylinder and less passes underneath.
To define the stream function officially, we add the stationary cylinder case to the vortex stream function. Note here the vortex cylinder stream function has this extra capital R representing normalization by the cylinder radius. We can do this because there is an arbitrary constant in the way the stream function is defined. And if we add a special form of this constant to the vortex stream function, it puts an r conveniently inside this natural log. We define the velocity field by taking the spatial derivatives of the stream function yet again.
And lastly, we come to the stagnation points. Interestingly, We have two cases to consider. If the circulation is low enough, the stagnation points remain on the cylinder, and there are two of them. These two stagnation points happen when the radial coordinate is equal to the cylinder radius, which gives us theta coordinates defined by an inverse sine function.
We get this by plugging big R into the velocity fields, setting them to zero, and solving for theta. However, if the circulation is above some limit, 4 pi u infinity r, then the stagnation points are pushed off of the cylinder surface and into the flow. This gives us one single stagnation point outside of our cylinder. Since we assume that it happens at the center of the cylinder location, by inspection we could say that the stagnation points happen at negative pi over 2. Plugging this theta value into the velocity field and setting it equal to zero, lets us solve for the radio location of this stagnation point, which is a function of the free stream, the circulation strength, and the radius of the cylinder.
So, defining the boundary streamline is not as useful here because our stagnation points can hop off of the boundary. And that finishes our exploration of building relatively simple aerodynamic flow fields from our elemental flows. In practice, you will come across the elementary flows most commonly in Computational Fluid Dynamics, or CFD, because many flow-solving techniques have elementary flows built into them. For example, if we had an arbitrary-shaped body in a flow that we assumed to be non-lifting, we could employ the source-panel method. This essentially is defining a surface of your body as a series of many little sources put together, and then you can recreate the boundary using a stream function instead of defining the boundary in the code itself.
Similarly, if you think your object is a lifting object, you would instead apply the vortex panel method. Here, you represent the surface of your lifting object as a series of point vortices of varying circulation. These are These are both common flow-solving techniques you will likely come across as an aerodynamicist.
Okay, let's review. We started today by realizing that we can add potentials and stream functions together because the equations and their solutions are linear, and that streamlines are representative of boundaries in the inviscid flow. Four elemental flows are introduced.
where we defined their velocity field, stream function, and velocity potential. These were the uniform flow, the source and sink, the doublet, and the vortex. Using these elementary flows, we built more complex aerodynamic flows representative of flow around bodies.
We constructed the semi-infinite body from uniform flow at a source, the Rankine oval from a uniform flow plus a source and sink, the stationary cylinder from a uniform flow plus a doublet, and the rotating cylinder by adding a vortex flow to our stationary cylinder. As always, we end with a practical note about how you might come across these techniques in modern flow solvers. I hope you enjoyed the video, and thanks for watching!