welcome you all again to discourse on foundations of computational fluid dynamics we are on to module 2 of the first week last class we basically did will put out the syllabus for the course and then outcome what you obtain from this course some of the mathematical operations then definition of velocity field vorticity viscosity definition of fluid based on viscosity and we still have to do review of equations and there are other topics like non dimensionalization vorticity stream function and so on whenever a fluid is moving we have to characterize them visualize them so this is given by flow visualization and there are four such characteristic lines define for flow description they are time line path line streak line and stream line so we will see the definition of them and later when we take example we will see that how these are used to define a flow so time line or the lines or curves formed by number of fluid particle part at particular instant of time this is very important so it's unsteady flow it changes with the time and you are looking at one particular instant how different fluid particles are that in stand behave and if you are able to connect all the particle at that instant then similarly you proceed with the time you are able to get timeline for that flow so they get displaced other particles proceed further in the flow path line is another definition this is a contour apart followed by fluid moving fluid particles they are traced by individual fluid particles and in experiment also in CFD you can inject a dye then it traverses then if you call connect all the contour then you get path line so bot path lines or characteristics Lagrangian description of the fluid motion such that the coordinates of the fluid particles can be expressed in terms of time as well as initial coordinates there are other two definitions streak line and stream line so streak lines are the locus of all fluid particles which passes a particular point and after some instant whatever particles that is crossed at that particular point and you connect all of them then you get a straight line so in experiment if a dye is injected and it would continuously move only along the straight line stream lines for this is a standard definition which all of you know a family of curves which are tangent to the direction of the flow at every point at a given instant so if we draw a tangent to a stream line then you get velocity component of the vector at that particular instant your streamlines show the direction in which the fluid Edmond would tend to move at any point of the flow field we know our flow can be steady or unsteady so in steady problem all this for prescription path line time line streamline streak line they all will be same they will be different for unsteady flow and this is to be noted very carefully now next important topic giving of basic equations we will not do detailed derivation will only touch few steps and explain importance of terms detail derivation is available any in any standard undergraduate fluid mechanics textbook in general there are three basic equations one is the conservation of mass which is also known as continuity equation consideration of momentum and then conservation of energy primarily consideration of mass and conservation of momentum or necessary for any fluid mechanics problem and consideration of energy he is used whenever you you are interested in compressible flows or to do with the heat transfer problem these are only primary equations you also need to have additional equations depending on the problem for example ideal gas equation is used in compressible flows and if you are interested to find out a concentration distribution for example pollutant dispersal problem then you want interested to find out particular concentration then you saw additional species transport equation governing patience can be expressed in two different forms one is a differential form and integral form similarly we have already seen last class different coordinate system Cartesian coordinate system cylindrical coordinate system and spherical coordinate system so one is able to convert or derive known equation in any coordinate system and convert from one coordinate system to other coordinate system so we before going to the derivation one important concept that we have to learn in fluid mechanics is what is known as a system and control volume system is basically related to a fixed identifiable mass so with the time the boundary may change but the mass remains then the boundary which contains that mass may change but the quantity of mass that remains same so across the boundary you can have a work or energy transfer the usual example given is piston and cylinder assembly certain quantity of gas is inside the cylinder and piston moves front and back the mass that is remained the same and you have a energy transfer happening across the boundary and piston moves back and forth now though it looks simple for a problem where there is a continuous change of the boundary it is difficult to apply idea of system or concept of system and get a solution so there is alternate what is known as a control volume where you don't really focus on a fixed quantity of mass only the boundary focus on fixed window and then observe what is happening to that window so we can have a mass transfer also in addition to momentum as well as energy transfer crossing the boundary control volume can change shape or move it need not be fixed in a flow he can also move for example you want to find out flow passing aircraft or flow moving over a four wheeler and you define control volume her own the aircraft or four-wheeler as the aircraft moves or as the four-wheeler moves the control volume also moves similarly if you take a balloon then the balloon as you're deflating it changes its shape so the boundary which is defined here as a control volume it can change shape this is helpful for example you have a elastic material and you want to find out fluid structure interaction problem later then there is a boundary it's not fixed it undergoes changes as a function of time control volume his study is very helpful the boundary which is enclosing the control volume is called control surface so there is a schematic available so here this red boundary is a control surface and the fluid crosses through the control volume and you find out what is happening to the control volume because of the flow that is happening through this control volume so for example you want to find out in the case of aircraft what are the lift generated what is the drag force exerted on the curve then you find out net and then get a drag estimate so as I mentioned before one can get all the governing questions even in integral form or in differential form we will see some of them in integral form then later we will move to differential form so to get an integral form we can relate the system and control volume derivatives for any general extensive property n then one can get all basic laws by instantaneously identifing system and that coincides with the control volume this is theory what is known as the Reynolds transport theorem right now we are not going to the details of the derivation in mathematical form it is expressed and given here the extensive property n is related to intensive property heater and it is related here as n equal to integral or mass which is for actually system ETA into DM is related to volume which is for the control volume and ETA Rho into DV now you can see by substituting different values for ETA you get different governing equations for example if you say ETA equal to 1 so if we substitute ETA equal to 1 here and then the resulting quantity corresponding extensive property is actually the mass so if you put 1 ETA equal to 1 it is integral Rho into DV you know Rho is a density kg per meter cube and DV is meter cube product of them will give you mass so that n is actually mass and that will give you consideration of mass equation in integral form similarly if you define kita its intensive property as V then if you substitute here V so we into Rho into DV you know mass into velocity will give you the linear momentum and similarly on the right hand left hand side extensive property and will give you the linear momentum so we can substitute different value for ETA and they are given here one is next one is for angular momentum then total energy and total entropy so one can obtain from this general relationship between extensive property and intensive property to this integral relationship one can obtain basic laws in integral form as I mentioned before this is what is known as the Reynolds transport theorem now we will see how to get expression in detail for consideration of mass we know this equation is stated as rate of increase of mass within the fixed volume must be equal to net flux crossing the boundary so if you consider a control volume mass that is inside the control volume is Rho into DV V is a volume it is given here we and then if you define at any point da is a normal vector and V is a velocity vector and the area is a area of the boundary volume so if you express conservation of mass in integral form so dou by dou T of Rho into DV over volume V is equal to Rho V dot Da because the area is also vector velocity is also vector you do the dot product you get a flux crossing and here it is da is continuously changing for this illustration figure that is given so you find out net flux crossing the boundary it may be coming in some place it may be entering the volume in some place it may be going outside that volume and you sum them up you get net that's why we do integral the integral will correspond to some e and you get that sum and that is related to rate of increase of mass within that control volume now the differential form of the equation is d Rho dou Rho by dou T plus del dot Rho V equal to 0 we have already seen this different operators so V is a velocity vector and in Cartesian coordinate system you have UV W as a component define of the velocity so we already seen we will now see in detail so let us take a volume three-dimensional representation here fluid element and with elemental length define in each direction for example DX is a length of the element in X direction dy is a length of element in y direction similarly DZ is a length of the element in s a direct direction so Reynolds transport theorem you have already seen so if you substitute ETA equal to 1 then you get n on the left hand side to be mass so that equation is put here again and volume is taken inside so dou by dou T of control volume Rho into DV will give you the mass this will give you the flux crossing the control surface and you find out flux crossing at each control surface and you sum them up so in this case if you consider this is a rectangular volume element and you define for example this is one phase and this is another phase front phase back phase and so on so there are six phases here top and bottom so one has to get explanation of this term on all the control surface so the first term is dou Rho by dou T into DX into dy into DZ DX dy DZ corresponds to volume and d'Oro by dou T is multiplying the DX new ideas will give you a rate of change of the mass now if you consider the next term second term in this equation so as I mentioned if you consider Rho if you consider center of this volume XYZ then from the center if you go to the left in X direction it becomes DX by 2 similarly from the center in the X direction if you go to the right that elemental length is DX by 2 positive right similarly you can go from the center come front and come so go to the back there will be DS at half similarly top and bottom that will be dy by 2 half in either direction so this term which is accounting control which is accounting mass flux through control surface and you find out the term for each of this phase so x minus DX by 2 will be the left phase X plus DX by 2 will be the right phase similarly Y minus dy by 2 will be the bottom phase and Y dy by 2 you'll be the top face similarly from the front and back so there are six terms for this cuboidal element that you're considered and if you write for example only for the left phase mass flux through the left phase row X Y Z T that is defined here assume that is a density available at Center of element and you go left hand side and this will give you that flux that is crossing and this will be the flux that is leaving so you can say mass flux through the left phase of CV and mass flux through the right phase of CV so you get term defined like this for each phase and put them in this equation and finally in all you have dy DZ and DX term which is actually the volume so if you do that arithmetic substitute for all phases and sum them up if you do that arithmetic and divide by volume DX dy DZ the term appearing in all the three terms of the spatial as well as on temporal if you divide by the volume then you get dou Rho by dou T he plus dou by dou X of Rho u plus dou by dou Y of Rho V Plus dou by dou R is it of Co W and you also know how to convert or write in other form is dou Rho by dou T plus del dot Rho into be equal to zero okay now what we have done is a generic sense it is not particular to any situation generic equation gets reduced or simplified for different situation for example in steady flow we know the parties don't change with the time so any term with the time derivative is their equal to zero in this case it is dou Rho by dou T so that first term dou Rho by dou T goes to zero and mass conservation equation gets reduced with only a special derivative term that is given here and in terms of vector del dot Rho V equal to zero another simplification if it is incompressible flow you know density is a constant so any derivative any special derivative of density will not be there and that term also is removed and you get only the velocity component so dou u by dou X plus dou V by dou y plus dou W by dou is at equal to zero and in terms of vector it is Del dot be equal to zero so the next equation we said we will do three equation conservation of mass consideration of momentum and conservation of energy the second important equation is conservation of momentum which is based on Newton's second law of motion F is equal to Ma like we did for consideration of mass here also we need to consider elemental volume find out net force acting on the control volume and then net momentum flux crossing the control volume you are own for both of them then you get finally consideration of momentum equation so it is stated as net force on the control volume equals rate of change of momentum within the control volume and net flux crossing the control volume so in vector form here again we are not doing the detailed derivation that is there in any standard fluid mechanic textbook or any other open source material or other impotent course so in vector form the consideration of momentum is given here Rho into dou V by dou T plus V dot del into B which is on the left hand side equals pressure gradient viscous force and any other force now V is a velocity field and this del squared B which accounts for shear force is a laplacian operator and given here as del square by Del x squared del square by Del Y square plus del square by Del X square and mu you know is the dynamic viscosity of the fluid so if you divide equation by Rho then you get mu by Rho nu 1 upon Rho into del P and only the external force so if a problem you are applying the problem on say involving with a magnetic field then this external force Fe represent additional magnetic force similarly a problem involving in gravity then you have a gravity force appearing as additional source term so this is given as a source term and for different problem will have a different definition of the source term or external force so if you look at this equation on the left hand side you have one component for time derivative and this del 3 component as a special derivative and this is a derivative of time you know that gives acceleration similarly this term also supposed to be acceleration we will get the definition clarity next slide so the left-hand side is acceleration and the right-hand side pressure is a force viscosity of the shear force or any other force so that's why I said Newton's second law F is equal to M into a he's actually is a first principle based on which the conservation of momentum is direct and you are able to see here the same acceleration terms and force term on other side we have already seen in the first class different review you have seen total derivative which is d by DT capital D by DT equal to time derivative dou by dou T plus V dot del so if you substitute the definition of material derivative or substantial delay to told derivative in the momentum equation then it is really done as given here so d by DT of V on the left hand side on the right hand side is the same pressure for viscous force and any other force it is convenient to write in vector form sometime it is easy to write equation with scalar form so we will also get to see how great momentum equation in scalar form so for example we know three dimensional representation X momentum equation a detail expression of the same momentum equation for x momentum equation we know the velocity component along the X is U so dou u by dou T Plus u into dou u by dou X plus V into dou u by dou by a plus W into dou a by dou Z and this corresponds to expansion of this material derivative D by DT of viii on the left hand side and pressure term dou P by dou X is appearing here for pressure viscous force is appearing here and then the external force is considered as gravity force when we take a component you get Rho into GX so where GX is acceleration due to gravity which is considered here as external force one can also obtain similarly by substituting for V vector in terms of velocity component for Y as V and W for Z you get momentum equation the corresponding direction Y momentum equation and Z momentum equation so if you could do that you are able to write a complete momentum equation and you are able to see here corresponding V momentum equation or Y momentum equation and third Direction W momentum equation or Z momentum equation okay now you see advantage of writing in vector form writing in scalar form and in vector form these three equations with the so many terms or they turn an elegantly simple one equation with only two terms there is a scalar form you are able to write in detail and one can actually locate what component responsible afford so while writing a code also you write in detail form then write the code it's easier to take decode later and this set of equation that is all the three momentum equations with the full components written all the terms written he generally called navier-stokes equations Navy and Stokes or two scientists who independently developed these equations and as a credit to them these equations are named as navier-stokes equations and you can again understand this equation if you take left hand side all the time derivatives which is given a dou u by dou T dou V by dou T and dou W by dou T they are all called local acceleration and u into dou u by dou X V into dou u by dou Y as W into dou u by dou Z so those three terms in X momentum equation similarly three terms in momentum equation three terms in W momentum equation are called convective acceleration and you put them together all the four terms will give you a total acceleration so that is what is explained left hand side first term is a local acceleration that is this term and remaining three terms are called convective acceleration put together it is called total acceleration now if we look at this particularly convective acceleration indeed closely we see here U is a velocity field which is a function main quantity in the X momentum equation and dou u by dou X is the derivative of the same velocity right in other words also we into dou u by dou V is a velocity field in y component and this is a velocity derivative so function multiplying the derivative of the same function is resultant what is known as a non linear in nature so such non linear nature of this convective acceleration actually attracts a special attention because the behavior of how to solve how to represent you multiplying dou by dou X in discretization is important and we are going to focus special attention on this convective term because of its non-linearity the behavior and the treatment of this convective term can change the solution results in some unstable also accuracy to some extent so we are going to be we are going to see in detail treatment of this nonlinear convection term later so in this class we did important topic flow description consideration of mass equation consideration of momentum equation detail in differential form and writing the momentum equation in vector form and scalar form and importance of non linear convection term and why one should focus on convection term discretization how to represent different external force for different problem so next class we are going to see little more detail on this momentum equation and go on to the next equation conservation of energy equation thank you you