Lecture 16

[Music] [Music] hello everyone welcome back we have been discussing about the governing equations uh we discuss the equations for the incompressible flow compressor flows and then we also discuss the dimensional form as well as dial form okay so now it's time to basically start solving those equations numerically but before we solve these equations we need to basically identify the nature of this these equations Mass conservation equation momentum conservation equation and energy conservation equation what do you mean by I mean by Nature so as we have been discussing that for the incomers flow for example if you have any disturbance that propagates immediately why why because the speed of sound is infinity right but on the other hand when the flow is compressible it travels with a finite speed right so when you want to solve flows in different regimes your equations behave differently okay the character of the equation is different in different flow regimes we have got the the time dependent term we have got the adum term which is basically the transport term we have got the C term so the transport term thetion term has got the velocity component so it travels based on the magnitude and the direction of the Velocity on the other hand the c ststem or the any viscus stem uh would basically dissipate the heat in every direction right so the there's no directionality in your uh stress ter right so that has got a different character that term has got a different character right similarly you have the pressure gradient term it's very important to first understand how to basically numerically discretize each and every term in the N Stoke equations right only then we can finalize our numerical scheme that can be adopted to solve these kind of problems and uh then we'll be able to basically get a solution which is accurate as well as conversed okay so before we uh get into uh this we need to see some model equations right some of the equations you might have already studied in your uh uh basic mathematics course for example the heat equation and the wave equation and so on and so forth right some of the equations are the ordinary differential equations partial differential equations Etc right so if you compare uh certain terms of the ntic equations they may look similar to what you have already studied so what we're going to do we're going to look uh for some very basic model equations today okay and try to characterize them so that we can basically find the mapping into the nistic equations so that is the goal that we will classify uh the differential equation based on the uh basically what are the how many characteristics they have I will come to the characteristic what does it mean in in some some time but as you know that there are many ways of classifying a differential equation first of all it is PD or OD secondly whether it is linear or nonlinear or maybe the Quasi linear semi linear right and then you can have it is homogeneous or non-homogeneous and finally uh based on uh their physical character we can have whether the equation is elliptic or hyperbolic or uh parabolic it can be mixed of all three also okay or in certain regimes it may display one character more strongly than the other right so let's uh start with first just seeing the N equation right so uh for convenience I've used only the uh incompressible equ question but the discussion today does not depend on the what is incompressible or compressible what we are trying to see that what does each term indicate right uh uh whether what character they basically uh show right so if you look at all these equations the mass conservation momentum conservation and energy conservation also they can be basically written in a very uh simple uh form okay which is shown as below right so here what I've done I've taken this temporal term and the convection term and this is the diffusion term which can be discuss diffusion and there is a source term right now if you put 5 = to 1 right in this equation right and then let's put the diffusion term as equal to 0 this is your diffusion term for Z and Source term equal to Z right so what you'll get Del R by DT plus d by d x i row UI equal to0 or in the vector form d by DT plus d do r u = to Z what is this this is our Mass conservation equation and we can go to the incompossible Limit but let's not go there we have already discussed that point now you put FAL to U here right so now what we going to do we are going to put F equal to U okay and Gamma is equal to put mu Okay so putting pi = to U and Gamma is equal to Mu right so how will this equation look like del row U by DT plus Del let's write in the vector form only okay so that it is convenient for you row u u is equal to D do gamma row and plus s okay which can be the source term right so what is this what how does this equation look like this is like your momentum conservation equation I put U you can put V and W and you can get the equation for uh the Y momentum and G momentum okay so this is your X momentum equation right you can also in the sour ter you can also put the body forces also right if you replace 5 equal to e okay replace 5 by e now then what you'll get you'll get the energy conservation okay how will you do that Del by DT row e plus Dale do row u e and see I I've not put the pr gradient term but that can be absorbed here in five okay so here you can have uh uh like we have discussed that because you have uh d p by dxi which can be written as Delta i z Delta P by DX Z and on the left side you have d by DX Z row u y u j right so you can basically Club them together right so this will become d. r e is equal to D do uh gamma e okay and this is your uh the work done due to stress okay and plus the source term The Source term of course will vary between uh uh your energy conservation Mass conservation and momentum conservation so all the equations can be represented in in in the single form which has got some temporal term time dependent term which shows the uh the the change in the quantity uh the rate of change of quantity then the transport okay okay uh the transfer of the quantity from one point to another Point okay and then uh you have the diffusion uh due to the viscous forces okay the stress that gets developed and finally you can have some Sour right now uh let's uh look at very simple equations okay and then uh we will basically uh combine all these simple equations that you already know to see how this uh they can be represented in this form okay you might have already heard that the name equations are nonlinear equations okay and then uh they are basically uh nonlinear equations okay hom non-homogeneous equations it may be non homogeneous equation if you have the source term present okay uh it is multi-dimensional so it it is quite complicated uh expression it is the uh it is a PD because it has the time as well as space dependence so all those things are there right so let's look at some simple equations in the beginning so I have uh put together uh two equations that you have I'm sure you must have studied uh in your math course so one is your heat equation which is nothing but DT by DT and if your thermal cond is constant you are not considering the variation of the conductivity you can take this in the one dimensional uh uh flow you can take this G Dy DX squ right so let's say if you have a rod okay and if you heat it up on one side and see how the uh the temperature is changing with time so you can use it this is the unsteady heat conduction equation right and uh I'll be using the short form uh at many times so this Delta by DT can be WR as T subscript T okay okay that means subscript indicates that this is the partial derivative with respect to time then this is K okay and T XX that means it is uh basically differentiate twice with respect to X right so this is your unsteady heat conduction okay and if it is a stady straight heat conduction then this TTT term goes away and what you have is txx okay equal to 0 right which is nothing but del2 T Del x² equal to 0 right and this is uh a very well-known form uh wellknown equation that you know is the laas in equation okay this is the form of a laas equation right the second equation that uh we have studied is the wave equation okay suppose the there is a string here okay and then you just perb it okay at an some initial condition the string will vibrate and the waves will travel so you'll be solving uh this problem using the wave equation and the wave equation is given as Delta U and use some initial disturbance some some function okay can you can write five also c² del2 U by x² but here the C plays a very very important role okay unlike in the heat equation where the information travels at an infinite speed so here information travel at a infinite speed okay here it travels at a finite speed okay which is given by C here right so for example let's say uh you put a Delta function or a point source let's say in in this room I put a point source somewhere let let me first consider the heat equation put a heat Source okay immediately uh uh it will work in a way that it it will be distributed everywhere at an infinite speed and it will reach to an equilibrium toally solution right which is which does not have any discontinuity right on the other hand if you have uh you have if I'm speaking here I'm speaking to you the sound is traveling and it is traveling at a constant speed which is the speed of sound okay which is can be determined based on the room temperature and we have talked about this Cal to ro of G for the isentropic process right which is fairly valid because we are we are not adding heat when we are speaking right so asent Tropic process can be uh assumed right so the sound basically travels so if I put a Delta function let's say I speak and then I stop speaking the it will take some time for the sound to reach to you and if further you are it will basically take more time to reach there okay because it is traveling at a finite speed okay now for uh the Delta function let's say if I start with a Delta function so my U is equal to Delta X okay uh for the uh wave equation and similarly uh that is a point source I'm giving point source uh in the web equation similarly I can give the point source in the heat equation also and if you solve those equations analytically which is presented in the last slide Okay so so the solution will look very very different okay so this is the solution that you will find out uh for the heat equation uh without going into the derivations I'm sure you must have derived it in your math course you know how to solve the uh uh the partial differential equations okay uh finding the the general solution and particular integral and you can basically solve it okay so you can see that there is this sum which is exponential minus 4T okay so even if at a very very small T okay it will Decay very fast okay and basically it will dissipate everywhere okay it will distribute everywhere in a very very small time and that's why we say that it is it happens at an infinite speed on the other hand uh if you try to solve for the wave equation for the same Delta function disturbance okay the initial disturbance at t equal to Z Your solution will look something like like this half x- CT and half X Delta X Plus CT and half Delta x- CT which means that this C which is the speed of sound okay and which has got both plus C and minus C okay the some on the front and some on the back okay and uh you can see that it depends both x and t okay and it will be only be valid when xus uh because it's a Delta function right so it will be only point which is x- CT or X Plus CT right so you will have so we'll have wave traveling you'll feel the presence of the wave at x minus CT then there is no wave then again there's a wave presence of the wave is fa and so on and so forth right so that's how your sound wave equation I mean presents the solution for a Delta function right the other equation that you know is the bond layer equation so if there is a flat plate okay I'm just looking at some of the equations that you already know in your heat transfer and fluid mechanics scores right so if the flow is happening from the left and this is the edge of the plate so the bond will start developing and if you have read about the bond lay Theory right so the bond Theory describes the development of the bond layer provided the condition that your Delta is much much smaller than l so if this is distance is L and this distance is Delta okay so the bond Theory tells you how this Delta will evolve but in the space right you can see that as the L is changing your Delta will change right and for the flat plate you have all kind of assumptions one is that DP by d y equal to Z right so there is no variation of pressure in the bond layer so if I fix my coordinate system this is my X and this is my y okay so Delp equal to Z and similarly Del to U by Del x² is smaller than Del to U by D y² so you and we have already derived the N equation so there are some conditions it it is a stady state okay and the condition is it is incompressible and there are few more okay and finally you come to this equation which is the bondar equation right we'll look at the character of this Bond layer equation also so the bonday equation describes the flow it basically uh it's the ratio of the viscous force uh uh and the inner Cal force is it's the balance between these two forces so if the runos number is very very high then the the role of the viscous force is limited to very small area so if the bond runos is quite High uh then you will have the bonday which is very close to this region although you will have the gradient you have to have the bond layer you cannot ignore the bond layer okay even no no matter how how high your initial force is okay you will have the gradient changing you will have the Velocity going to zero here and velocity going to U Infinity here okay but this distance may become very very small the Delta may become very very small but Delta has to be there okay uh so we'll look at the nature of this equation also when we go to the classification of the PDS okay right now I'm just describing what are the equations that you already know okay is the linear convection equation uh and it is actually another form of the wave equation that we already saw okay so linear convection equation uh there's a big role of linear convection equation in the development of different schemes which are currently used in cfd so basically two schemes one is the convection scheme okay and another is the diffusion scheme okay so these two basically this relates to the wave and this relates to the heat okay so these two equations have played a big role in devising schemes which can be used for the N equations because uh your the viscus ter basically uh is similar uh I mean it ulates the the diffusion process and your Adent term because you have the uh du by DX partial U by partial X is there so that is the adduction term is detected by the convection equation although it is not linear conviction but the nonlinear conviction right so how it is related to the wave equation if you just take a derivative of with respect to let's say time here so you can write this D by DT equal to minus c d by DX right so you can take a derivative with respect to time again okay okay that will become minus C Del to U by DX DT right and then what you do you can uh basically interchange the d uh derivatives become C Del by D X and this is d by DT right and this D by DT nothing was minus c d by DX right so minus c d by d x minus C d u by DX right which is nothing but c² Del to U by D x² so Del to U by Del t² = to C c² x² okay so uh this is the first order and this is the second order but both represent the the wave equation okay or basically the propagation of wave in a medium okay so now that we have some uh we just reviewed some the equation that we have already studied or gone through in our math course or even in the flu Dynamics course let's look at how to classify these speeds okay so first of all we need to find out I mean there are some very basic whether it's a PD or OD right so uh and then we'll see what is linear or nonlinear then it is homogeneous or nonhomogeneous and finally we will go to a little bit more advanced topic which is uh uh whether the equation can be called elliptic parabolic or hyperbolic and it has got a physical significance that's why we will spend some time on uh this characterization okay uh so before going into the obpd or even other things let's have some Basics first is the order of the differential equation so the order of differential equation is basically defined to be that of the highest order derivative it contains so let's say if you have a function Del to Q X Q Plus uh U Del to U by D x² and plus u = to Z something like that okay so the order of this is your uh three yeah this is the third order differential equation right and now the degree of the differential equation is defined as the power to which the highest order derivative is R so remember this highest derivative is very very important when you're characterizing a BD right and this will this will come uh to again and again right so for example in this if you see the degree is the one right so let's see if I raise this TOS Delta ² sare right still the degree remains one why because uh the the power to the highest order derivative is one right it degree will change only when we have this delt Q by Q if I change the uh uh degree of this right uh the power of this otherwise the degree will not change okay so this now it's order is three and degree is 2 right so we learned about the order and the degree okay and now about PD and OD so uh let's define this independent and dependent variables okay so for example if you look at the N St equations okay to Del U by DT plus d by d x z r u y u z = to minus d p by d x z plus d by d x z y z okay or you can write in terms of velocity gradient there okay can be written in terms of velocity gradient right so here you can see what are the independent variables you have t so the independent variables you have are basically T and XJ which are basically X Y and G if it's a three dimensional flow right and what are the dependent variables of course you know okay row u v p and all those things okay these are basically your dependent variables these are the dependent variables Okay so if a equation has just one independent variable that means for example it's a 1D study State flow okay let's say it's a study State 1 D flow right then what will happen the T will disappear okay the t y z all disappears okay and then what is the dependent independent variable dep that we have is only X okay in that case it will become an OD right so if it becomes an OD you have many advantages as you already know it becomes very easy to solve but unfortunately most of the cases most of the realistic cases that we come across in our flid and mix are basically uh the cases where it depends on more than one special direction as well as on time right so the ntic equations are are basically the the partial differential equations right and it's called spao temporal PD because it depends on space as well as it depends on uh time so PD when the the independent variables are more than one okay here it is uh x y z and T but it can be anything anything which comes at the denominator in derivative they are basically the independent variables right and in the numerator they are the the dependent variable which are depending on those independent variables right so for example here row can be expressed as row uh x y z and T So row depends on all x y z and T So row is the dependent variable and XY ZT becomes the independent variable okay now uh let's look at the the different nature of the the PDS okay so we'll start with so typically in the lit we have linear and nonlinear okay but then you have some subclassification of the nonlinear process which are basically the the Quasi linear or the semi linear okay so we'll briefly go into the discussion of these quasi linear and semi linear uh equations okay although what I've seen is that in the literature they have basically mixed these two terminologies and they have different they may have different definitions uh on how you Define the qu linear process or the semi linear process right uh but the the difference between a nonlinear and linear is quite clear okay so uh the qu linear is example for example the derivatives of the highest order linear okay so for example if we take an example so which is basically Del to U by D x² or just write this yeah okay let me write here Delta U by D y² plus Delta U by squ is equal to 0 right so if you look at this expression okay so what are the derivatives of the highest order okay so these are I mean here they are all of the same order but you can have other terms also so if they are linear then it is a qu linear thing so okay this is a qu linear process right uh you can have other Expressions also let me write some other Expressions okay let's say I have expression which is u xx x + x² u y y y + uh u² okay uh UT TT equal to zero right so here uh you can as you can see here the order was two okay here the order is three okay and this is also equation of the uh uh the qu linear okay but let's uh take an example which is basically u x y² and plus u x + u y = z let's take this so here the derivative of the where is the highest order the order is two and thetive of the highest order is uh basically not linear this is uxy s so this is not Cy linear so this is not Cy linear okay these two are C linear but this is not the qu linear okay within the quas linear you can have uh more uh uh basically uh classifications so one is the semi semi linear which is that it should be equal linear and second is that coefficient of the highest derivative okay should not contain either dependent variable or its derivative right so let me give you one example which is a uh basically uh F linear but not semi linear okay uh so this is by by default your semi linear right it it is quas linear and the coefficient of high dity does not contain any either dependent variable or its derivative okay but if you look at this example okay this is neither quas linear this is neither semi linear because uh the precondition for semi linearity is that it should be uh C linear so if it is not C linear it cannot be semilinear but let me take you for one example which is uh basically qu linear but not semilinear okay so that example could be let's say U utt plus uxt okay is equal to something okay let's say x² okay so this you can see U that uh derivatives of the highest order are linear okay so this is c a linear okay but this is not semilinear okay so uh this is very important to know that this is not semilinear because it should not contain the either dependent variable or its derivative here it contains the dependent variable which is U okay so this is not semilinear and if it is not qu linear okay then it will be it has to be the nonlinear okay so I'll give you just example so this is for example this is a nonlinear equation okay this equation that we see a nonlinear equation okay and for other classification that I can talk about for example this is a non-homogeneous equation because there is an x² coming on the right side so what is the difference between homogeneous and non homogeneous equation so in the nonhomogenous equation so all the terms contain dependent variable or its derivative right it should not contain an independent variable here in this particular case you have an independent variable coming x² Okay so uh you can write an expression I mean think of any expression uh let's say if you are talking about the poison equation okay if let's say WR Del s u is equal to X okay or Y whatever it is so this is your non non homogeneous equation because it has a an uh uh independent variable coming uh in the equation uh the linear equations are very simple okay uh it should not multiply I mean the the dependent variable and all its partial derivative must occur linearly right so all I mean it the precondition it it has to be qu linear and it has to be sem linear then only it can become linear right and it should not uh the derivative should not multiply uh any of this uh dependent variable right and then only it can become the linear equation and I'll give you some example I mean we already saw many examples for example the heat equation or the linear convection equation the all the equation that we saw so this is let's say uh U = to cux okay minus ux okay this is your uh linear equation but if I just replace let me write in the form that we already wrote plus cux = 0 so this is your linear equation and if I just write it U + uux = to Z this is a linear equation but here um uh the the dependent variable and it's partial so this is the partial derivative okay it is not occurring L linearly okay so then this is becomes a nonlinear equation okay because the derivative U so in this particular case your W is traveling the speed of sound C right here if you want want to solve this equation it will it is traveling at a speed of sound is the speed which is U and that U has to be determined from the equation right so it is non linear depending in other way you can see if you write this expression U + uux equal to 0 okay so this is d u by DT plus u d by DX = to 0 right so this is d u by DT and you can write d by D X U ² by 2 equal to Z right so this is occurring nonlinearly right you can clearly see here so that's why this is a nonlinear equation this is nonlinear pde and this is homogeneous okay so uh uh in this particular class we basically reviewed some of our basic ideas of uh linear equation nonlinear equation homogeneous equation non- homogeneous equation uh then what are the PDS and ODS uh in the next class uh what we'll do we'll go to further classification and we'll see uh the equations how we can classify these equations as whether they're elliptic parabolic or hyperbolic and what does it mean okay so with that we'll stop today and we'll continue our discussion in the next class thank you very much [Music] [Music] [Music]