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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 14 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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Date Created: 11/02/15

ME 6601 Introduction to Fluid Mechanics Module 6 Table of Contents Slide 1 7 Reynolds Transport Theorem Slide 2 7 Coordinate De nition Slide 3 Jacobian Slide 4 7 Reynolds Transport Theorem Slide 5 7 Reynolds Transport Theorem proof Slide 6 Reynolds Transport Theorem 7 proof quot A Slide 7 Reynolds Transport Theorem 7 alternate form Slide 8 Reynolds Transport Theorem 7 alternate form continued Slide 9 Reynolds Transport Theorem 7 alternate form continued Slide 10 7 Reynolds Transport Theorem 7 physical integpretation Slide 11 7 Reynolds Transport Theorem summary Slide 1 Reynolds Transport Theorem Welcome back Now that we have finished our study of kinematics we have to begin the process of deriving the governing equations which we are going to be needing in order to proceed further with the subject of uid mechanics Two main principles we will be employing this semester are conservation of mass and balance of linear momentum N ewton s Second Law of Motion In order to derive these principles we are going to require the use of one other thing called Reynolds Transport Theorem That is what we are going to be spending this module deriving So first before we do the Reynolds Transport Theorem which by the way in this second bullet tells us that it allows us to take the time derivative of the integral over a material volume of the uid Before we actually derive the Reynolds Transport Theorem we are going to need a couple of very preliminary results so lets start with those Slide 2 Coordinate Definition Lets start by assuming that we have a parcel of uid which is colored blue and some coordinate system and that we allow capital Xnot to represent the continuum of vectors pointing to every parcel of this blue uid at some initial time t equal to zero And remember we have talked in the past about the fact that this initial time is totally arbitrary At any later time the parcel of uid may be located over here and we will use the vector lowercase bold X to describe the same material uid points at some later time There is some transformation going from the initial time to the later time that we are labeling phi superscript lowercase i Slide 3 Jacobian Now a quantity you have probably have not seen since freshman calculus but I assure you that you did see it there is known as the Jacobian The Jacobian of our transformation X superscript i equals phi superscript i of the vector xnot represents what is called the dilatation which is the growing or shrinking of an infinitesimally small volume as that volume follows the motion And that Jacobian is given by this expression This is the notational expression We call the Jacobian J This is just notation and what this means is the determinate of the tensor which is the derivative of lowercase X superscript lowercase i with respect to lowercase xnot superscript j The Jacobian is a quantity that has a value that lies between zero and infinity So it is always bounded from above by infinity And again if you haven t seen this in a while I can refer you back to your elementary calculus text We need to prove something about the Jacobian because the Jacobian is usually useful for transforming our integrals over moving volumes The thing that we need to have about the Jacobian is a simple lemma I say that we need to prove it but we are not going to prove it We are just going to state it in this particular course So we state it here without proof You can find the proof in a variety of locations but the proof is nothing more than a brute force differentiation The lemma says that material derivative notice our capital D notation the material derivative of the Jacobian is given by the Jacobian times the divergence of the spatial velocity field We will be using the same notation that we have used in the earlier modules So we need this in order to derive in order to prove the Reynolds Transport Theorem that we will be looking at next So again the material derivative of the Jacobian is equal to the Jacobian itself times the divergence of the velocity field Slide 4 Reynolds Transport Theorem Lets now start by letting V with a bar through it which is a function of time be an arbitrary uid material volume arbitrary is important here Lets let this function capital F which is going to depend upon both spatial position the vector lowercase bold X and time t lets let that function be either scalar function or vector function of both position and time It does not matter which it is The proof of the theorem does not change and the applicability of the theorem is the same to either scalar or vector functions If we look at the integral of this function F over our volume V of t and if that integral is welldefined which means if it eXists if it is not infinite then the Reynolds Transport Theorem tells us the following So this is a statement of the Reynolds Transport Theorem We are going to be proving it after we state it The time rate of change following the motion that s the material derivative of the integral over the volume of F is equal to the integral over the volume of capital D capital F time rate of change of F following the motion plus capital F times the divergence of the velocity field We are going to find this very useful in deriving a principle of conservation of mass and later the balance of linear momentum So again the Reynolds Transport Theorem due to Osborne Reynolds English hydrodynamicist tells us that the material derivative of the volume integral of any vector or scalar function capital F is given by the volume integral over the same moving material volume of the material derivative of capital F plus capital F times the divergence of the velocity field Slide 5 Reynolds Transport Theorem proof All right lets look at the proof In order to start the proof we need to talk about two different volumes We have our volume capital V of lowercase t an arbitrary time but lets define also a volume capital Vnot this lavender region highlighted here which is associated with capital V of lowercase t but back at the time of t equals zero Okay so this is the initial position of the uid material volume that will later transform itself to this position and be capital V of lowercase t This is what the Jacobian allows us to do The Jacobian allows us to write our integral over the moving volume capital V of lowercase t of capital F as the integral over the fixed volume and fixed is the important word here capital Vnot of capital F times the transformed lowercase d capital V which is just the Jacobian times lowercase d capital Vnot Okay so here is the utility of the Jacobian here Notice that the outside is the time rate of change following the motion We haven t done anything with that yet However we started with the time rate of change over a volume which was itself moving in time this is the time rate of change following the motion and now we have this time rate of change of an integral over a fixed volume That means we can now take this derivative inside the integral sign and operate with it there Slide 6 Reynolds Transport Theorem proof continued And so that is what we have done now We have now said that since we have fixed volume we have the time rate of change of this product capital F times capital J integrated over this fixed volume lowercase d capital Vnot We can apply to that the usual product rule of differentiation The product rule works even with the material derivative So keeping the same lefthand side that we had before inside we are now over the fixed volume and using the product rule we have capital D capital F over capital D lowercase t times J plus capital F times capital D capital J over capital D lowercase t lowercase d capital Vnot So that is nothing more than the simple product rule Now we have to remind ourselves that we have a lemma that we just stated without proof that tells us what the material derivative of the J acobian does So from that lemma we can substitute for capital D capital J over capital d lowercase t capital J which has been factored out of these parentheses times the divergence of the spatial velocity field So I have factored a J from here and I have factored a J that has come from the capital D capital J over capital D lowercase t and that appears on the outside In the inside of the parentheses I just have a material derivative of F plus F times the divergence of the velocity field We still have the integral over the fixed volume Vnot but now that we have it in this form we can transform back to the moving volume and if we transform back to the moving volume we get this We now have the integral over capital V of lowercase t of capital D capital F over capital D lowercase t plus capital F divergence of lowercase v and that completes the proof of the Reynolds Transport Theorem So now that we have that available a very powerful theorem we can use this in deriving our principles of the conservation of mass and the balance of linear momentum Slide 7 Reynolds Transport Theorem alternate form Lets first look at an alternate form of the Reynolds Transport Theorem Remember that the definition of the material derivative of any quantity capital F vector or scalar is the partial of capital F with respect to lowercase t plus lowercase v dot gradient operating on capital F And if we use that use the expression for capital D capital F over capital D lowercase t in the right hand side of the Reynolds Transport Theorem then we replace capital D capital F over capital D lowercase tby the right hand side that we have on the line above So we can rewrite the Reynolds Transport Theorem in this form Slide 8 Reynolds Transport Theorem alternate form continued And now we may regroup some terms If we note that v dot the gradient of capital F plus capital F divergence of lowercase v is nothing more than the divergence of v times F then we may rewrite this line in this form and now we have the integral over a volume of the divergence of a quantity and now we can apply one of Green s theorems the one we call the divergence theorem to rewrite that piece of the right hand side as a surface integral So we have the divergence of some quantity lowercase v capital F integrated over a volume That can be rewritten using the divergence theorem as the integral over the surface which bounds the volume capital V of lowercase t of the quantity capital F times lowercase v dot n lowercase d capital S a normal component of that quantity Slide 9 Reynolds Transport Theorem alternate form continued Therefore if we substitute that into the Reynolds Transport Theorem our alternate form becomes the following We have the time rate of change following the motion the integral over some volume of a function capital F which depends on space and on time is equal to the time rate of change capital F integrated over the moving material volume plus capital F times the moving component of the velocity lowercase v dot lowercase n integrated over the bounding surface of the moving volume The bounding surface is called capital S of lowercase t If we used the notation of the Cartesian tensors we can write this expression in the following way The time rate of change of capital F of lowercase X subscript lowercase i and lowercase t we now indicate the vector lowercase bold X by lowercase X subscript lowercase i is equal to the partial of capital F with respect to t integrated over the volume that term does not change going to tensor notation because we are assuming that capital F is just an arbitrary function plus capital F times lowercase v subscript lowercase j lowercase n subscript lowercase j remembering that in this term since lowercase j is a repeated indeX it is a summation indeX so this is going to be capital F lowercase v subscript one times lowercase n subscript one plus lowercase v subscript two times lowercase n subscript two plus lowercase v subscript three times lowercase n subscript three integrated over the surface Slide 10 Reynolds Transport Theorem physical interpretation One thing that is nice about the alternate form involving the surface integral is that we get a nice physical interpretation of the Reynolds Transport Theorem from this form So let us look at this term by term lets look first at the lefthand side where we have the terms colored red So on the lefthand side we have the material derivative of the volume integral of capital F which is nothing but the time rate of change of capital F and in all of these cases I mean integrated over the volume following the motion The second term is the time rate of change of capital F integrated over the volume I have left out these words just because they take up too much space within the volume So I have here the rate of change of capital F with respect to lowercase t integrated over the volume and then finally the last term represents the net ef uX or flow out of capital F through the surface of the volume capital V of lowercase t So thenI take capital F and multiply by the inner product of the velocity with the unit outward normal vector Integrate that over the surface and that tells me the rate at which capital F is leaving our material volume capital V of lowercase t through the surface capital S of lowercase t Slide 11 Reynolds Transport Theorem summary All right so that is our Reynolds Transport Theorem an alternate form of it

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