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ME 6601: Introduction to Fluid Mechanics Module 2 Table of Contents Slide 1 – Kinematics I Slide 2 – Many Particles Make Up a Continuum Slide 3 – Material Coordinates Slide 4 – Material Coordinates (continued) Slide 5 – Spatial Coordinates – why? Slide 6 – Material (Langrangian) and Spatial (Eularian) coordinates Slide 7 – Material and Spatial Coordinates (continued) Slide 8 - Material and Spatial Coordinates (continued) Slide 9 – The Material Derivative Slide 10 - The Material Derivative (continued) Slide 11 – Kinematics I Summary Slide 1 – Kinematics I Welcome to Module Two. We have now concluded our introduction into Cartesian tensor notation. We now need to start talking about kinematics because we need to describe the motions of fluids, and we are all used to describing the motion of particles and matter at this point in mechanics. The term “Kinematic” itself refers to relationships between spatial position, orientation, and time. In mass-point mechanics, it is very easy to talk about the motions of a single particle. Lets call capital X – and that is a BOLD-faced capital X, indicating it is a vector – a function of time. We are going to use that notation, capital X, to refer to the position history of the particle. So the particle is indicated in red. This is an arbitrary coordinate system that we have chosen. And as that particle moves along a curved trajectory, capital X, that vector will be changing, being drawn from the origin up to the particle. To get the particle velocity, which we have now indicated as a vector, we have a very simple matter to differentiate the position of that particle, X, which is only a function of t. That means that this is an ordinary derivative, d divided by d t. So if we differentiate that position vector X, we obtain the velocity vector, V, that is indicated on the figure. It is drawn tangent to the trajectory because, at any instance, the particle is moving along the trajectory, so the velocity vector has to be tangent to it. So that is what we have been doing since high school, for those of you who have had vector physics, and some calculus physics, in high school. But now we need to do things a little differently. If we have a countable number of particles -- and by countable, it can be an infinite number. It just means that we need some way to assign each particle to an integer. So we start counting 1, 2, 3, 4, 5 … Slide 2 – Many Particles Make Up a Continuum If we take a countable number of particles, we have an easy extension from mass-point mechanics. We just assign an index, n, to every one of those particles, and n may go from one to infinity, or it may go from one to fifteen, if we only have fifteen particles in our system that we are trying to keep track of. Associated with each one of those trajectories, and each one of those is a separate trajectory which depends upon t, we have a velocity vector V, superscript lowercase n. By the way, I don’t mean to imply here by putting the superscript n in parentheses that there is differentiation involved of the capital V. Capital V is a differentiated version of capital X, but that notation is just a way to write individual particles and individual velocities. So lowercase n is an integer, and we use that to count the particles. Instead of writing it with superscript notation, we could write it a different way. We could say the trajectory X is equal to the trajectory, which is a function of t and n, depending upon which particle we happened to be following. And then, of course, with each of those particles, we can say that we have a velocity vector, which is a function of time, and of the particle as well. So the extension from one particle to a countable number of particles, even if it is infinite, is no problem. But in fluid mechanics, we don’t do that – we can’t do that. How can we count every particle? We have an uncountable number of particles. Between any two particles, there is yet another one. Alright, so integers are insufficient in order to deal with a continuum, especially with fluid mechanics. So for a continuum we will require continuous identification variables. So we have indicated those here. So now, the trajectory is a function of not only time, but also on lowercase a, b, and c, where lowercase a, b, and c are some triple of scalars that span the space. By spanning the space, we mean that any particle in the space is representable in terms of lowercase a, b, and c, written in that order. Slide 3 – Material Coordinates Now how do we choose lowercase a, b, and c? How do we identify every particle? We could paint each one a different color, but that is not very practical. We want to deal with mathematics here, so instead, we are going to have to come up with some continuous way of identifying the particles so that they will be known for all subsequent time. By the way, those variable a, b, and c are known as material coordinates. We use that phrase material coordinates a lot to refer to an actual piece of the fluid continuum -- a piece of material. So how should they be chosen? That is what we have just started talking about. It turns out that a convenient choice for this triple, lowercase a, b, c, is to let those values designate the positions of the material particles at some arbitrary reference time. We can call that reference time anything we like. We can call it three point four, but lets call it the initial time. Lets start our stopwatch running at the time we identify all the particles. So we will call it t equal to zero. That is then, that lowercase a, b, c, represents the position, capital X subscript zero, is the notation we are using. This is the position at time, t, equal to zero, of the particle that is at capital X subscript zero. Now, this looks like it is circular reasoning, but it is not. This says the initial position is the position at time equals zero of the particle that was initially at capital X subscript zero. Okay, so you can read these equations in that way if you like. Or we can say X-not is the position of the particle that was initially at X equal to zero at time, t, equal to zero. Okay, so it is a little hard to get used to. Your textbook has some additional material discussing this, in a little bit different form than the way that I present it. Alright, what we are essentially doing is we are taking a snapshot of the flow at time, t, equal to zero, and because of taking that snapshot at time, t, equal to zero, we can identify every particle in the field. We can point to this particle, that particle, another one… wherever. Every particle is identified, and we can later track each one of those particles as the flow proceeds. So, in schematic form, here is our vector capital X as a function of lowercase t -- the position at time, t, of this particle. That particle started out at time, t, equal to zero. Before time, t, equal to zero was back up here. If we like, we could have set this as time, t, equal to zero. It doesn’t matter because, remember, we said that this is an arbitrary reference time. Alright, so now, because at time, t, equal to zero we can draw a position vector to any point in the flow field, we have identified, we have a way now, of identifying every particle in the flow field, and consequently, of identifying every particle at some later time. At this point it may be a good point to just pause for a second, and ask, “Why are we so worried about particles? We are dealing with a continuum. Who cares about all these particles?” Well, it turns out that our laws of motion, one particularly that we are going to be using most often is Newton’s Second Law of Motion, is written from the standpoint of nav-point or particle mechanics. So we need a means of converting our statements about particle mechanics into statements associated with continuum. In addition, we are going to be trying to work in a more convenient coordinate system, then identifying all of these particles and following them all around as they progress through the flow field. Slide 4 – Material Coordinates (continued) So that is our motivation for doing this. So for our traditional material coordinates, then, of mass point mechanics, we have the position field X is the position the particle that was initially at capital X-not, but now at time, t. Again, capital X is the position at time, t, of the particle that was initially at X-not. We can read them either way. Okay, because we can differentiate these X’s to get velocity and to get acceleration, we have got a pretty easy situation working with material coordinates in terms of nav-point mechanics. In fluid mechanics, however, we are going to find that it is much more convenient, USUALLY -- now put that caveat in there – USUALLY, much more convenient to work with what we call spatial coordinates that are fixed in some point in space, rather than coordinates that are fixed at particles, and are following individual parcels of fluid around. Alright, so this is a more convenient way to work for most situations in fluid mechanics. Slide 5 – Spatial Coordinates – why? We will talk about a couple that may be different in a moment. Why would we want to use spatial coordinates? We usually make our measurements in a spatial frame of reference. Here is a flow through a pipe, and we have some sort of a device, maybe a pito tube, sticking into the pipe. The pito tube is fixed at some point in space, it could be moving as well, it does not matter, but the fluid is flowing by the measurement device. We are not attaching a measurement device to a particle of fluid that is flowing through the system. Rather, we are sitting at a fixed location watching the flow go by, and making measurements of the fluid particles that happen to pass our sensor. When we do analysis of flow past an object, it is easier, in some respects, to use a spatial frame of reference. Here is an example. This would be a global spatial frame, that we would call a control-volume. If you have had undergraduate fluid mechanics, you know that when you do a control-volume momentum balance to determine, say, the force that a flowing jet has on this particular obstacle that is deflecting it, it is much more convenient to draw a control volume, which we have indicated by a dotted line, in contact with the device, and watch what happens when fluid comes into the control volume, and exits, rather than trying to follow an individual parcel of fluid that for only an instant of time is in contact with our turning pane. Okay, so those are the reasons why it is usually more convenient to use spatial frame of reference. Do we ever use material frames of reference? Sure. One example that you could think of is a weather balloon. A weather balloon is inflated, it has an instrument package attached to it, it is put aloft, it is allowed to flow with the prevailing winds, and the assumption is the weather balloon is carried along with the local fluid velocities. So that is making a material measurement, or a material reference frame measurement. Are we ever interested in doing an analysis in terms of material coordinates? The answer is “Yes” again. There are situations. For instance, maybe we have a smokestack that has pollutants coming out of the smokestack. What do we want to know? We want to know where those pollutants end up, if they happen to end up in our neighborhood, or if they end up some place else. So we would like to track the fluid that comes out of the smokestack in doing a material frame analysis in that kind of situation. Slide 6 – Material (Langrangian) and Spatial (Eularian) coordinates Now this is what we have already said. Since our fundamental principles are expressed in terms of material coordinates, and we would like to work in more of a convenient spatial coordinate reference frame, we have got to be able to handle the transition between the two frames of reference. We have got to be able to transfer from a material frame of reference over to a spatial frame of reference, and back if necessary. We are going to be examining that in this module, and we are going to be doing some examples in the subsequent module. Lets now start out by assuming that we have some variable, we are going to call it theta, maybe to represent the temperature. It could be a scalar variable, like the temperature, or it could be a vector variable like velocity or acceleration. But for right now, lets just assume we have this variable, theta. Here in red I have indicated a capital theta, which is a function of initial position and time. You will notice now I have reversed position and time because we usually put position and time at the tail end of a list of independent variables. We have also got lowercase theta, which is the same thing, (temperature, if you like) designated at various spatial locations, which I am representing by lowercase variables. The lowercase x (that is a vector) is the spatial position at which we happen to be making a measurement. So again, the capital theta would refer to theta – lets call it the temperature – the temperature, of this particle that was initially at X-not, at any later time, t. So if we have a sensor riding along with that particular particle, that is the temperature we would be measuring. The lowercase theta represents the temperature at some location, specified location, lowercase x, at the same time, t. If this particle happens to be at that location, those temperatures are exactly the same. What do we measure when we measure with temperature? We are measuring the temperature of the particle that happens to be at the location at the instance of the measurement. So again, we are using material, we are designating the material coordinates with capital letters, and we are designating spatial coordinates with lowercase letters. The capital here, the X-not, and the lowercase letters being over here, the x. Lowercase t doesn’t matter. Alright, so in all the things we do in this module, we are going to use upper and lower case letters, respectively, to designate material and spatial coordinates. What are we really after? The bottom line in this analysis, is that we would like to determine the rate of change of our quantity theta, temperature, with respect to time, following a material particle. That is, with x-not, the particle identity, held fixed. Remember, this is the initial position of a particular particle. So if we say that is fixed, it means the particle is fixed. So again, it is the derivative of capital theta, holding the material identity fixed. That is what we want because that is how Newton’s Second Law is written, in terms of force equals mass times acceleration. It is the mass times the acceleration of a given fluid particle. Slide 7 – Material and Spatial Coordinates (continued) If we use the chain rule for differentiation, we know that theta is a function of x-not and lowercase t. It is a capital theta because it is a material variable. What we want is a partial derivative of theta with respect to t, holding the particle identity fixed. Using the chain rule now, we know that that is the partial derivative of lowercase theta with respect to time, holding the position fixed plus the derivative of time with respect to time, holding x-not, the particle identity, fixed, plus the partial of theta with respect to x subscript I – this is a lowercase x, which means a spatial position – holding time fixed, times the partial of x subscript I with respect to t, holding x-not, the particle identity, fixed. And now again, you will see that this is the first place we have employed our Cartesian tensor notation because the summation rule is in effect here. I have a lowercase i subscript here, and a lowercase i subscript there. So this would be a partial derivative of theta with respect to x subscript one, partial derivative of x-subscript one with respect to t, partial derivative of theta with respect to x subscript two, partial derivative of x subscript two with respect to t, plus partial derivative of theta with respect to x subscript three, partial derivative of x subscript three with respect to t, respectively. We go to the next part. Now, we use color down in the new equation and color up in the original equation. The partial of lowercase t with respect to itself is always one. So this quantity is one, respective to what is being held fixed here. And the partial of x subscript lowercase i with respect to lowercase t, holding particle identity fixed, which appears there, is the same as the partial of the position, capital x subscript I with respect to t, holding particle identity fixed. That is just the material velocity, capital V subscript I of initial position capital X-not and time. That is the velocity of that particular fluid particle that at time t was located at x-not. That is the material velocity, which also happens to be the velocity that we measure at a certain location where that particle happens to be. Slide 8 - Material and Spatial Coordinates (continued) Now we want to express the right-hand side of our equation using only spatial variables, so we are going to use lowercase notation to right that the partial of capital theta with respect to t holding particle identity fixed, is given by the partial of lowercase theta with respect to t at a fixed position, plus v subscript I, times the partial of theta with respect to x subscript I. Again, the summation convention can hold. This is with t fixed. We can drop the subscripts because there is no confusion as to what we are holding fixed. Slide 9 – The Material Derivative So finally, if we take this, and we write it in tensor or vector form, if we just drop all of the subscripts, then we just get this expression, with the lowercase v subscript i (the material velocity), times the partial derivative of theta with respect to lowercase x subscript lowercase i. Or in vector notation, we can identify this term as “v dot the gradient operating on theta”. It is easier to think here in terms of the scalar, but v dot the gradient is a scalar operator, so we don’t have to worry about whether theta is a scalar or a vector because a scalar operator can operate easily on either scalar field or vector field. So this is the quantity we were after. That spatial quantity, the partial of theta with respect to t plus v dot the gradient of theta, is known in fluid mechanics as the material derivative. It is also called the substantial derivative, because it is the derivative following the substance, and it is also called the Stokes derivative, because it is named in honor of Sir George Stokes, a prominent English mathematician and fluid analyst from the late Nineteenth century. We designate this summation usually by the notation either total derivative d with respect to time, but that is usually confusing to a lot of people, so it is more convenient, most of the time, to use the notation capital D, theta, with respect to lowercase t. So, when we write capital D, theta, over capital D capital T, we mean this expression. Slide 10 - The Material Derivative (continued) All right, so let us look at what physically these terms mean. Our overall expression – you remember we had it originally indicated as the partial of capital theta with respect to t, holding x-not fixed. That is the derivative, holding the material point fixed. Or, time rate of change of lowercase theta following the material. That is what we call the material derivative. It consists of two parts: now highlighted in magenta is the partial derivative of lowercase theta with respect to t. That is the time rate of change of theta at a fixed point in space. So if we were measuring with our pito tube stuck in the pipe, which would be the time rate of change that we would be measuring at that location. Finally, v dot the gradient of lowercase theta is the time rate of change of theta due to the movement of fluid from one location in the flow to another location in the flow. That is often referred to as the convective rate of change. Slide 11 – Kinematics I Summary So we have now a more complicated situation than we had before with material coordinates because this is not simply a straight kind of derivative. We have a time derivative at a fixed location plus the velocity dot the gradient operating on theta. So we local derivative and we have the convective rate of change, thus forming the two components of the material derivative. So to summarize what we have gotten so far in this module, we have discussed the notions of material and spatial frames of reference, and certainly there value to the study of fluid mechanics. We have indicated that there are times in which we would prefer to use a material frame of reference following fluid around, with the example of following pollution coming out of a smokestack. But most of the time in fluid mechanics, we will find it more convenient to work with spatial frames of reference. We have also now determined how we can represent time derivatives following the material, which we need for Newton’s Law of Motion, and the usually more convenient spatial frame of reference. What we are going to do in the next module is to pick up at this point and to carry on with an example.

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