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# Intro to Fluid Mechanics ME 6601

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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 6601 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/234254/me-6601-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.

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ME 6601 Introduction to Fluid Mechanics Module 6 Equations Table Of Contents Slide 2 o Diagam 1 from Slide 2 Coordinate De nition Slide 3 o Eguation 1 from Slide 3 Jacobian o Eguation 2 from Slide 3 Jacobian Slide 4 o Eguation 1 from Slide 4 Remolds Transport Theorem Slide 5 o Diagam 1 from Slide 5 Remolds Transport Theorem proof 0 Equation 1 from Slide 5 Re olds Transport Theorem proof 0 Modi cation to Enuation 1 from Slide 5 Revnolds Transport Theorem proof Slide 6 o Enuation 1 from Slide 6 Transport Theorem nrouf quot Slide 7 o Eguation 1 from Slide 7 Remolds Transport Theorem alternate form 0 Eguation 2 from Slide 7 Rem olds Transport Theorem alternate form Slide 8 o Eguation 1 from Slide 8 Rem olds Transport Theorem alternate form icontinuedl Slide 9 0 Equation 1 om Slide 9 Remolds Transport Theorem alternate form continued 0 Equation 2 from Slide 9 Remolds Transport Theorem alternate form continued Slide 10 o Eguation 1 from Slide 10 Remolds Transport Theorem physical interpretation Diagram 1 from Slide 2 Coordinate Definition This picture shows a three dimensional graph with x y and z axes respectively The origin is labeled zero where all three axes converge A vector labeled lowercase X subscript zero extends from the origin to a blue oval representing a uid particle A second vector extends from the origin to a second uid particle and this vector is labeled lowercase x A curved arrow points from the first particle to the second labeled phi superscript lowercase i Equation 1 from Slide 3 Jacobian Capital J equals the partial derivative of the coordinate triple lowercase X superscript one lowercase X superscript two lowercase X superscript three with respect to the coordinate triple lowercase X superscript one subscript zero lowercase X superscript two subscript zero lowercase X superscript three subscript zero which equals the determinate of the partial derivative of lowercase X subscript lowercase i with respect to lowercase X subscript lowercase j subscript zero Capital J the Jacobian is between zero and infinity Equation 2 from Slide 3 Jacobian Capital D capital J over capital D lowercase t equals capital J gradient dot lowercase v Equation 1 from Slide 4 Reynolds Transport Theorem Capital D over capital D lowercase t of the integral over capital V with a line through it as a function of time lowercase t of capital F as a function of lowercase X and lowercase t lowercase d capital V with a line through it equals the integral over capital V with a line through it as a function of lowercase t of open parentheses capital D capital F over capital D lowercase t plus F del dot lowercase v closed parentheses lowercase d capital V with a line through it Diagram 1 from Slide 5 Reynolds Transport Theorem proof This diagram shows a pair of ovals which contain smaller ovals inside of them The first oval labeled lowercase X subscript zero contains a smaller oval labeled capital V with a line through it subscript zero Underneath the bigger oval it eXpresses that lowercase t equals zero An arrow points from the first big oval to the second This arrow is labeled phi superscript lowercase i The second big oval is labeled lowercase X It contains a smaller oval which is labeled capital V with a line through it as a function of lowercase t Below the second big oval there is a lowercase t Equation 1 from Slide 5 Reynolds Transport Theorem proof Capital D over capital D lowercase t of the integral over capital V with a line through it as a function of lowercase t of capital F lowercase 1 capital V with a line through it equals capital D over capital D lowercase t of the integral over capital V subscript zero of capital F as function of capital J lowercase 1 capital V with a line through it subscript zero Modification to Equation 1 from Slide 5 Reynolds Transport Theorem proof The last equation is also equal to the integral over capital V with a line through it subscript zero of capital D over capital D lowercase t of capital F capital J lowercase 1 capital V with a line through it subscript zero Equation 1 from Slide 6 Reynolds Transport Theorem proof continued Capital D over capital D lowercase t of the integral over capital V with a line through it as a function of lowercase t of capital F lowercase 1 capital V with a line through it equals the integral over capital V with a line through it subscript zero of open parentheses capital D capital F over capital D lowercase t times capital J plus capital F times capital D capital J over capital D lowercase t close parentheses lowercase 1 capital V with a line through it subscript zero equals the integral over capital V with a line through it subscript zero of open parentheses capital D capital F over capital D lowercase t plus capital F del dot lowercase V close parentheses capital J lowercase 1 capital Vnot equals the integral over capital V with a line through it as a function of lowercase t of capital D capital F over capital D lowercase t plus capital F del dot lowercase V lowercase 1 capital V with a line through it equal to the integral over capital V with a line through it as a function of lowercase t of open parentheses the partial derivative of capital F with respect to t plus del dot lowercase v capital F close parentheses lowercase 1 capital V with a line through it Equation 1 from Slide 7 Reynolds Transport Theorem alternate form Capital D capital F over capital D lowercase t equals the partial derivative of capital F with respect to lowercase t plus lowercase bold V dot del capital F Equation 2 from Slide 7 Reynolds Transport Theorem alternate form Capital D over capital D lowercase t of the integral over capital V with a line through it as a function of t of capital F as a function of lowercase bold X and t lowercase 1 capital V with a line through it equals the integral over capital V with a line through it as a function of t of the partial derivative of capital F with respect to lowercase t lowercase 1 capital V with a line through it plus the integral over capital S as a function of lowercase t of capital F times lowercase v dot lowercase n lowercase 1 capital S Equation 1 from Slide 8 Reynolds Transport Theorem alternate form continued The integral over capital V with a line through it as a function of lowercase t of del dot lowercase v capital F lowercase 1 capital V with a line through it equals the integral over capital S as a function of lowercase t of capital F lowercase v dot lowercase n lowercase 1 capital S The integral over capital V with a line through it as a function of lowercase t of del dot lowercase v capital F lowercase 1 capital V with a line through it equals the integral over capital S as a function of lowercase t of capital F lowercase v dot lowercase n lowercase 1 capital S Equation 1 from Slide 9 Reynolds Transport Theorem alternate form continued ME 6601 Introduction to Fluid Mechanics Module 2 Table of Contents Slide 1 7 Kinematics I Slide 2 7 Manv Particles Make Up a C quot Slide 3 7 Material Coordinates Slide 4 7Material C J39 quot J Slide 5 7 Spatial Coordinates 7 why Slide 6 7 Material I 39 I and Spatial Fularian 139 Slide 7 7 Material and Spatial C J39 quot AI Slide 8 Material and Spatial Coordinates quot Slide 9 7 The Material Derivative Slide 10 The Material Derivative l quot 39 Slide 11 7 Kinematics I Summa Slide 1 Kinematics 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 uids 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 masspoint mechanics it is ver easy to talk about the motions of a single particle Lets call capital X and that is a BOLDfaced 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 l 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 masspoint 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 uid 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 uid 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 uid 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 itt 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 Xnot 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 navpoint 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 Xnot but now at time t Again capital X is the position at time t of the particle that was initially at Xnot 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 navpoint mechanics In uid 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 uid around Alright so this is a more convenient way to work for most situations in uid 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 uid is owing by the measurement device We are not attaching a measurement device to a particle of uid that is owing through the system Rather we are sitting at a fixed location watching the ow go by and making measurements of the uid particles that happen to pass our sensor When we do analysis of ow 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 controlvolume If you have had undergraduate uid mechanics you know that when you do a controlvolume momentum balance to determine say the force that a owing jet has on this particular obstacle that is de ecting 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 uid comes into the control volume and exits rather than trying to follow an individual parcel of uid 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 in ated 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 uid 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 uid 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 Xnot 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 Xnot 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 xnot 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 uid particle Slide 7 Material and Spatial Coordinates continued If we use the chain rule for differentiation we know that theta is a function of xnot 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 xnot 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 subscriptl with respect to t holding xnot 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 Xsubscript 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 Xnot and time That is the velocity of that particular uid particle that at time t was located at Xnot 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 righthand 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 1 times the partial of theta with respect to x subscript 1 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 uid 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 uid 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 xnot 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 uid from one location in the ow to another location in the ow That is often referred to as the convective rate of change Slide 11 Kinematics 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

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