hello good afternoon everyone today we continue chapter one prince main definition or introductory thoughts on aerodynamics last time we were talking about flow similarity which are the criterias of law similarity we saw that to have similar flows we must have several conditions satisfied such as similar bodies but not just similar bodies right we need to have same reynolds number of the flow and the same mach number of the flow as well as same angle of attack so here we will have one example is it corresponds to example 1.5 from the book is a bit different because in example 1.5 anderson used imperial units and in this example i'm using c units for calculation because this are main units for us so the task is the following that we have airplane with known parameters of flight a mach number of flight 885 kilometers per hour altitude of light almost 12 kilometers we know density at this altitude we know temperature and pressure at this altitude of flight according to standard atmosphere table this data of raw infinity infinite and v infinite you can fly in appendix d of anderson textbook at the end of the book the question is if we would like to test such a body in a small wind tunnel we need to construct a small scale model of this airplane this small scale model is defined as 150 so size ratio is 50 as you can see from the screen and we need to create some specific conditions of the flow that would correspond to same reynolds and mach number that we have in usual flight one restriction for this particular wind tunnel is that we have fixed temperature of test inside wind tunnel which can be related to some uh structural limitations of the wind tunnel so it's 239.9 kelvin i think we can consider as 240 kelvin so let's have this example and calculate values of reynolds number mach number and after define same properties for the flow inside the wind tunnel i would like to remind you that we have some assumptions first assumption is that viscosity would be proportional to square root of temperature okay and and let's see if we need another we will we will use so uh reminding you definition of reynolds number and the mach number mach number is simply right is velocity divided by speed of sound and reynolds number is velocity size viscosity divided sorry density divided by this quantity i hope it's correct please let me know if you have some questions or you found some mistakes in my calculation in the comment or you can say during my presentation as well i will take a look on the comments and now i can i can see if some comment appears so let's um let's have this calculation so a mach number mach 1 mac infinite should be equal to mach 1 and reynolds infinite should be equal to reynolds 1. these are two conditions now we can have such such uh relation as a diameter or specific lengths for definition of reynolds number we use size size so 51 this is size ratio okay so let's see let's see what would be what would be simpler maybe we can start from mach number definition from infinite conditions i will write mach number definition as velocity infinite k mach number i will not use subscript infinite or 1 because it should be same mach number for both cases in infinite divided by a infinite and should be v1 divided by a1 we know that speed of sound is also proportional to square root of temperature from our previous discussion is known and in gas dynamic is elaborated such equation to prove it so i will write that velocity infinite divided by square root of temperature infinite is v1 divided by square root of temperature 1 temperature infinite and temperature 1 we know so from this relation we can see that v1 would be equal to v infinite okay and then multiplies by square root of t1 divided by t infinity i hope it's square now when we have all parameters known for this formula i will just simply calculate just a moment i will calculate here 885 okay now first should be probably the squared 140 i divide by 216.6 take square root and multiply by 885 931 at least from my calculation i received this value 930 okay to round 932 meters per second this is the velocity of test inside the wind tunnel you see that when we consider mach number it's not important how much you scale this body because scale of the body did not enter inside calculation of testing velocity so if you would like to keep same value of mach you need even to increase velocity inside the wind tunnel with respect to real flight because temperature is is bigger on the test than in the real flight 932 meters per second and now i would like to calculate the other parameters of flow to keep same reynolds number i will write my renals for peak body first okay okay we write proportional because since we do not know size of the body i will use it uh this scale scale of the body so i will write proportional to velocity 1 then c infinite instead of size so it will be 50. and uh it's not velocity one it should be velocity infinite density infinite and then we divide by square root of temperature infinite instead of using on viscosity we use temperature infinite and okay this should be equal to velocity one we already calculated so we will use it in our calculation then c one will be equal one because c infinity is 50 and density we do not know and we also divide by square root of temperature 1 as a proportionality coefficient for viscosity only one unknown here is density and we will calculate right now i will just do what i will multiply by square root of t1 and i will divide by v1 and c1 okay so i will write that density 1 is equal to the square root ratio of t 1 by t infinite multiplied by v sorry sometimes not so convenient to write here but it's it's very nice uh c infinite and the answer infinite divided by v1 and c1 this ratio just a moment this ratio c infinite by c one i will use from uh from this initial conditions here we'll just change color to be more understandable okay all other parameters we already have in numerical numerical values i will just need to substitute so just t1 would be equal to square root of 240 divided by 216 by n6 and here the infinite 200 eight eight five fifty and the density is zero three three three kilogram per meter meter cubic for this altitude and we divide by 932 meters per second c is one and probably that's it yeah that's it we need to to see the following uh the following formula that units should be units of density since we have density on the left side of this formula we need to have also density on the right side of this formula this is how we can check our calculation let's see we divide temperature by temperature kelvin by kelvin here then we divide meters per second per meters per second here this proportionality coefficient is dimensionless 52 1 and we have just density in the numerator of this formula i will calculate just a moment we will receive some value proportional to our density and proportional to scale coefficient so it will be bigger we can see that we have a numerator 50 and the denominator just one so it will be something proportional to density and all approximately 50 times bigger but also we consider change of temperature and we consider change of velocity a moment i will calculate here okay i think we have a good answer 16.6 kilogram per meter cubic this is the answer for density we must have in our system so from practical point of view what does it mean for us it means that we need to have not just high velocity inside the wind tunnel but also high density inside this wind tunnel and with the low temperature if we need a high density it means that also high temperature oh so just sorry no no not temperature uh high pressure temperature we have high pressure that would correspond to this high density uh as a result we must have a wind tunnel that allows us to control not just the velocity in its test section but also to control density in instance in this test section in a big range in big range 16.6 is approximately 16 times or more than let's say more than 10 times bigger than our normal ambient temperature of air so we need significantly to increase pressure in the test section this can be done with for example such kind of wind tunnel which is called a variable density tunnel it allows to change several parameters as temperature pressure and as a consequence change of of density this equipment is extremely complex extremely expensive so it's quite valuable and difficult to operate inside part of this wind tunnel variable density tunnel is shown on this picture there is a vent turbine that operates that creates a flow velocity and there is a passage around the test section that allows to to bypass to circulate here of course since we need to create some specific value of pressure inside this is hermetic structure such a wind tunnel is not something simple nowadays there are not so many wind tunnels of this type and in order to reduce complexity of aerodynamic tests not to have same mach and reynolds kept in the in the test section it is possible to test separately with the different values of mach number and to test separately with different values of reynolds number keeping just one parameter uh constant for example for big and small model we keep mach number of test and for another test we keep reynolds number for this test it requires less much expensive wind tunnel and can reduce total cost of development of aerospace structure as a disadvantage this technique should consider some value of error associated with not constant macro renault's number and have some kind of correction methodology of similarity similarity theory so as we see as we see as a result of application of uh buckingham pa theorem we can find any aerodynamic coefficient such as lift coefficient drag coefficient or moment coefficient as a value of three independent parameters reynolds mach and angle of attack typically typically uh aerodynamic coefficients uh as a function of angle of attack are represented on figure 132 we can see that uh there are two two parameters there is no momentum coefficients is another discussion uh that lift coefficient is usually proportional to angle of attack has some kind of linear proportionality right starting from some negative value until positive high value a relatively high value and it has some maximum at some specific value of angle of attack very soon we will talk about why we can have maximum of lift coefficient this is related to physics of flow and flow separation when we see drug coefficient it's usually much smaller values than lift coefficient and it can have its minimum value usually in some small positive values of angle of attack some small values you can find it for from zero to five degrees for example is not always until five degrees mass but for most of the airflows we can find that minimum value of that coefficient is in these limits between 0 and 5 degrees where the lift coefficient maximum angle of attack is much higher it can be 10 can be 15 can be 20 for some airfoils or some wings for better windsor and earth wells from point of view of maximum values of angle of attack this uh maximum angle of attack can be high okay can be high and definitely much higher than for minimum for drug coefficient why it's important maximum and minimum for each function it will be very obvious very soon when we will talk about efficiency from point of view of aerodynamics behavior is this if you remember we have already talked about definition of aerodynamic coefficient i will write you here for example lift coefficient is cl is lift force divided by q infinite see for example which where c is s or area of aerodynamic body or could be diameter in each case is different for in general aerodynamic body requires some reference reference area for example q infinite is a dynamic pressure which is calculated as one half of density multiplied by velocity in square and l is the lift force um this lift coefficient is not just a function of lift force and but can be represented as a function of angle of attack for example looking back to this figure 132 we see that is changing the lift coefficient okay linearly smaller angle of attack smaller lift coefficient bigger angle of attack bigger is the lift coefficient when we apply it here and calculate velocity for example from this equation we will see the following let me write you we will use definition of velocity from dynamic pressure this was equal to lift divided by here 2 goes here density velocity in square okay multiplied by heart or area word or area so what we see when we combine this equation with the cl function as function of angle of attack we say that smaller cl smaller angle of attack this part becomes smaller right and it means that velocity becomes bigger right and vice versa when we increase angle of attack our cl becomes bigger okay goes in this direction when cl becomes bigger velocity which is in the denominator of the right side of the equation becomes smaller so we if we would like to fly with higher angle of attack automatically our velocity is decreasing more and more and more minimal uh limit of this velocity is called stall velocity it corresponds to maximum possible angle of attack which is called alpha stall stall angle of attack okay and vice versa if we have zero angle of attack somewhere here we have minimal value of lift coefficient and we have maximum value of flight velocities or maximum value of light velocity corresponds to close to zero or zero value of the angle of attack okay just uh when we combine equi function of dependence of uh lift coefficient to angle of attack and the formula to calculate lift coefficient we may find such dependences such dependencies higher angle of attacks smaller is the velocity not possible to keep the velocity uh as high as it as we had it in zero angle of attack for example just looking to to this equation but not just because we don't want just because we can't our energy of engine is limited so we cannot produce uh very different very different very high thrust to keep high velocity on high angle over time looking to another picture figure 134 we may find also dependence on the drug coefficient same formula as we had for for the lift coefficient i can rewrite for the drug coefficient as well okay let me write here it will be let's use this color here that cd is drag divided by q infinite s as well so we write that q infinite is one half of density infinite velocity infinity n squared and we multiply by our area okay that's that's it and we remember function of drug coefficient as we had in one of previous slides this is alpha and this is cd axis so somewhere close to zero we have minimal value of drug coefficient and it means that at this point we will have maximum value of our velocity okay which corresponds to minimal drag coefficient when we increase to stall we have maximum value of drug coefficient cd maximum and it will correspond to the velocity of stall which is minimum velocity according to this this equation increasing coefficient aerodynamic coefficient we automatically decrease the velocity of light of velocity of light in this case is more visible i think because drug d is a value that is acting against uh against trust so we need to have trust of the engine power of the engine corresponding to required drug if we have engine with constant trust or maximum trust that we cannot change that value of d becomes independent on the angle of attack and just enters inside this equation as constant so nothing depends on drug value here only velocity and uh value of drug coefficient are important to determine its dependence one from the other there are different types of aerodynamics body aerodynamic bodies related to drag there are so called high aerodynamic shape or high drug aerodynamic shapes aerodynamic shapes that have a higher drag which can be optimized for some specific regime can work well or very nice for example supersonic airplanes but when you try to fly supersonic jet with the subsonic velocity is not so well optimized for subsonic velocity and can be considered as high drag aerodynamic shape and there are low drug aerodynamic shapes for example which are nicely profiled airplanes for example created for laminar or for long range airplanes very efficient planners that are considered as low drag aerodynamic shapes that have much smaller values of drug low drag aerodynamic shapes with the same value of drug or where the same value of trust can reach higher maximum velocity so you can see it from from this picture here of course in all these conditions i did not involve uh trust yet so because force definitely for supersonic why supersonic airplanes fly faster because they have much more powerful energy units but if we compare airplanes with same level of trust we will see that airplanes which are more perfect from a point of view of efficiency they will fly differently faster than just regular or high drug aerodynamic shapes main criteria of aerodynamic performance is shown on this figure 135 is a ratio between lift and drag uh usually we ask i ask question okay which aerodynamic body you consider as the best how we can compare two different airplanes if we can can we say that one airplane from aerodynamic point of view is better than the other and the answer is these criteria here lift to drug criteria is not important if you have just high lift but you have very high drug as well and vice versa it's not important to reduce just drug to reduce just losses you also need to have this airplane to carry some weight from some distance so the correct criteria from aerodynamic point of view is ratio between lift and drag it's a ratio also between weight and thrust of your airplane which is same lift to drug is same as weight to trust for constant velocity and constant altitude flights we have for each airplane such graphics not just for the best just for any type of airplane or airflow we can find such graphics where we can see ratio between lift and drag for various flight velocities we can find that there is always a maximum of such flight velocity that produces maximum possible lift to drag ratio in case of work one i propose you to calculate for your chosen airfoil uh this this curve in terms of flight velocity or maybe better in terms of angle of attack because this this quantities also can be calculated for various various angles of attack also we can see that exists a maximum of lift to drug ratio for any kind of airfield or wind or air complete airplane that characterizes aerodynamic efficiency of the body aerodynamic efficiency of the body usually this maximum angle of attack of maximum lift to drag ratio is small is between zero and seven zero five zero to ten maximum for each specific airfoil or specific wing this value is different but it always can be obtained numerically when you have data for lift coefficient and drug coefficient i can also extend this here and i can tell you that lift to drag is same as weight to trust ratio for constant velocity and constant altitude flight and as we may see from definition of aerodynamic coefficients it will be also same as lift coefficient to draw coefficient ratio so it's not important which or you take just force or you take aerodynamic coefficient ratio will be same between them because we divide by same dynamic pressure and we divide by same area we cancel pressure and dynamic area for both of these definitions and we have this relation okay another example here is also from anderson book is in pdl units here i translate you to c units asking to calculate aerodynamic coefficient of lift for such airplane and the lift to drag ratio this is very simple let's do quite fast just to show you how can we calculate these parameters so we have velocity of flight altitude of flight we have weight we have density of flight on this altitude we have drug coefficient and we have area this is a planar area of airplane which is used to calculate aerodynamic coefficient so to get the lift coefficient we need to practically have everything right let's calculate let's try at least if i make some error please write me or tell me in a comment okay cl cl is lift divided by q infinite s and the same as weight divided by q infinite s okay and why i make equal lift to weight because i have already constant altitude so is not accelerating in vertical direction and it means that force of weight is equal to force of lift okay i have 67 200 and below i have here i have two density is zero four one four and velocity is nine hundred or seven hundred ninety two kilometers per hour we need to calculate in meters per second to be all in c units how we calculate in meters per second we have to [Music] multiply right wait how we can do it we divide by 3.6 probably we divide by 3.6 7 9 2 and my 3.6 will be here i hope it's correct and we forgot to write area it's 31 87 i will use 32 meter square multiply by 32 let's see i hope it will be correct ah no no no no it's um square it's in square so velocity and square right and 3.6 also should be in square because of this so okay i hope now it's better please check if some error appeared let me know and meanwhile i will calculate i will have 67 200 multiplied by two multiply 3.6 and three point six and divide it by zero four one four divided by seven nine 2 times and divided by 32 do we have it yeah probably yeah we have i received 021 this is a good value for lift coefficient i mean good because it's not so small not negative it cannot be negative right but it can be very very small which usually is not possible to have for uh sustained flight it must be something of realistic and cannot be high about exactly how big or how small could be values we will speak very very soon 021 lift coefficient and lift to drag ratio lift to drug ratio is c l by c d and 0 21 divided by 0 0 15. let me calculate 0 21 0 0 15 8 14 also nice value okay this is lift to drug ratio this is lift coefficient what is asked to calculate in example 1.6 okay okay okay what we can say about these values um when we calculate that lift or drug ratio we can compare with the some typical values for airplanes this is uh okay can be between 10 and 20 for typical airplanes is something between 10 and 20 so we are having 15 is good and it means that physical meaning of this 14 it means that a trust of our engine can be 14 times smaller than weight of our engine so to flight in these conditions we need to divide our weight by 14 and we obtain trust of the engine which we need to have this flight at constant velocity and constant altitude and that's quite quite acceptable another example that anderson shows to calculate such aerodynamic coefficients and such a dynamic properties is conditions on takeoff before we calculated maximum velocity and here we calculate take off all tall conditions we see that in such case weight is higher it means that our airplane is still full of fuel okay stall velocity is much smaller than uh cruise velocity uh the altitude is zero meters so we are just taking off our airplane from ground and here is the density of the pair of air on zero altitude this density you can also find in standard atmosphere table in appendix of the book area planner area of the airplane is same is the same structure so not important if we calculate a structure for high altitude flight or on the ground when it just takes off of course reference area is same because this airplane has constant geometry okay as constant geometry is asked to calculate maximum value of lift coefficient for this airplane we can do it in the same way that we did the previous calculation we just write that cl maximum would be equal to to weight okay maximum weight that we have on takeoff we divide by okay we multiply by 2 and we multiply by density again velocity of stall in square and the area i already will try to do transformation of of velocity from kilometers per hour to meters per second please in your calculations do not forget to do such kind of transformation if you use some reference velocity and i will use these numbers here seven one two hundred thirty seven one two hundred twenty and multiplied by 2 i will divide by 1 23 here and 31.87 and i will transform velocity to meters per second you know this coefficient of transformation we just needed to multiply by one thousand divided by three three thousand six hundred hours two seconds and kilometers to meters transformation and it goes in square just a moment i will calculate again this value 7 1 2 3 okay mm-hmm now something is wrong okay i need to check what i calculated wrongly here and if you see some error in my calculation please let me know some mistake a few times strange just a moment i will try to find this error and we can continue but if you found first please write in the chat yeah i found already it was in my calculation i divide two times by area receive two small values smaller than previous lift coefficient but now it's correct and is equal 1.82 approximately i i make rounding 182 you see it's almost two okay this is the value of of maximum lift coefficient for such case unfortunately we do not have some data to calculate drug coefficient than to compare lift to drug ratio in this case but it will be smaller much smaller than then for the first case for stall flight because if you remember lift to drag coefficient has such a such a curve so it goes down when you increase angle of attack it decreases so maybe we have much smaller maybe four or five ten something like this let's see it depends if we would have a trust of the engine at this moment we would be capable to calculate this value lift drug ratio but since we do not have trust or we do not have drug coefficient we cannot extract anything about it in this in this example but okay in future i think we can have more examples then under some proposes to talk about different types of laws i will pass quite uh fast that you'll know this information and i think it's not the first time that you will hear different types of flows free molecular flow and continuous flow then inviscid and viscous flows then incompressible and compressible flows definition and the mach number regimes this is preliminary introduction uh in uh first chapter of anderson because after he speaks about uh all these types of flaws in details in further chapters and not just about this but also about some specific aspects uh concerning these chapters but not about one just one type of flow is not covered in one type of flow is not covered in aerodynamics is free molecular flow which is outside of continuous medium which we do not consider as an appropriate type of flow for uh solution by navier-stokes equation or any other continuous flow method so this is another another type of flow which is not covered in this book aerodynamics classical that is working with the continuum flow where we cannot see separate molecules and we see just a medium with average properties this flow this medium can be viscous considered as viscose and inviscid and for both of these types of flow it can be considered as incompressible flow or compressible flow in its in this division compressible flow also can be separated to subsonic flows transonic flow supersonic and hypersonic flow depending on a value of dimensions criteria mach number okay continuous versus molecular flow i will not tell you probably you already know this is not so much interesting at least we do not have any theory about it in aerodynamics so i see no reason to talk about you can study this type of flow in other subject for example propulsion we have a propulsion course which talks about engines working the vacuum plasma plasma engines right and there is a theory of verified gas there uh and investment invested versus viscous flow is extremely important especially in application for aerodynamic high lift to drug ratio structures like airfoils where we can see two regions with flow outside inviscid and flow near the body which is very thin and considered as viscose where we can take into account skin friction and forces and energies related to interaction of body and flow such processes are extremely important because in many cases especially for airfoils most part of drug force is formed by viscos viscosity and this is a quantity which is which exists only inside the boundary layer okay i had here one video but i think now this video is not working because it's just slides you can find lots of such videos or visualization of or flow flow separation is a process that happens in almost all aerodynamic structures depending on angle of attack for low small angle of attack this is smaller effect but for a higher angle of attack for high lift values also increase flow separation flow separation is not desirable not a desirable process but it naturally appears right and we need to understand how to predict how to manage with such kind of flow we see that almost there are no models of analytical calculation so this is why very important become numerical simulations of separated flows and experiments experimental methods of study of aerodynamic bodies four decades let's say almost for 100 years flow separation is widely studied there are many works on this area and in this course we will also try to do such a simulation inside work three you will face a problem of flow separation you will be capable to study by yourself and to present your research about flow separation in airfoil of your choices extremely interesting physical phenomena extremely important i also would like to tell you that this semester i am working with some student to to build aerodynamic tunnel small scale aerodynamic tunnel especially for visualization of flow separation so i hope that maybe middle of this year or end of this year we will have a structure to visualize and to study experimentally flow separation in airfoils and if it works one day we could have some kind of practical work to to see this phenomena by our eyes this work is not simple not fast so what is revised that maximum in the middle of the year this structure should be constructed it's just for the future for people who would like to work in flow separation i'm telling that there is a possibility not just to simulate in our university but also to see by eyes and analyze experiments related to flow separation okay now such a process of fall separation or viscous and non-viscous flow can happen also in other types of aerodynamic bodies not just aerofoils but also blunt bodies like cylinder we will see definition of blunt body very soon is very interesting that for example drug can be a source of pressure or can be a source of viscosity or can be a source of both depending on how much is influence of each of these components aerodynamic body can be called or streamlined body or blunt body this will be defined very soon just for you to to understand that such structures exist and um theoretical calculation can provide also uh experimental effects so we can see we can see it from theoretical point of view and from experimental point of view this kind of flaws okay incompressible versus compressible flow compressible flow is much more complex we will study it in gas dynamics and i think i think it's stuck yo the screen is stuck okay let's wait a moment okay it returned we will study compressible flowing gas dynamic difference between incompressible and compressible flow is one more equation at least one more equation that uh describes effect of compressibility okay usually it's compressibility related to derivative of density by pressure change or it can be a function of mach number white function of mach number we also will see next semester and often it's also included in the energy equation compressibility changes energy significantly or redistributes energy balance significantly so one more equation is required to calculate compressible system with respect to incompressible system also more complexity more difficult to calculate sometimes when we can we consider our flow incompressible because in such way uh flow model is much more simple and easier to manage but in many in several cases not possible mach number regime subsonic flow you already know right some flow which has values of mach number everywhere smaller than one mach number is uh as we saw today is the ratio between velocity and speed of sound uh why point eight here because if we work around airfoil we have local increase of velocity on upper part especially of the airfoil and because of this local increase of velocity we need to work initially with lower values of mach number not to reach mach 1 or some supersonic mach number on this upper edge of the earth so this is why subsonic is more or less until mach 0.8 starting from mach eight we receive uh so-called transonic region gransonic regime it means that can be several cases of of several regions of supersonic flow around airflow or around any other aerodynamic structure not just could be airflow could be other structures right we have transonic regime in two different configurations a case upper case is when we have subsonic infinite flow and the local supersonic regions for example you can see here supersonic region uh also supported by shock wave transition from the supersonic region back to subsonic region on the upper side of the earth and also shockwave on the lower side of the airfoil and transition from subsonic to supersonic regime and then transition back to subsonic regime is the first type of transsonic flow second type of transonic flow is when we have initially supersonic have supersonic flow from mach 1 to mach point 1.2 approximately and near airfoil is created such a bubble of subsonic flow this bubble is will be discussed widely in gas dynamics why it appears uh just anticipating i tell you that it appears because there is a reaction of of the body on the flow and when flow impacts body body generates pressure waves and these pressure weights accumulate when they accumulate they create pressure gradient and flow starts to stop because it's a pressure gradient that x acts against the flow so flow starts to stop and appear such subsonic region then all pressure waves they are stopped in some specific position which it forms this uh shock wave bow shock wave so bow shock wave is accumulated big number of pressure waves that were generated by body and they go against the flow uh position of both shock wave is extremely uh complex and shape of power shockwave we will discuss in with the many details in uh next uh subject in gas dynamics it's too early now before we started to talk about compressible flow equations and at the end of such flow we usually can have trailing edge shock this trailing edge shock is the shock wave which is connected to the trailing edge of aerodynamic body and is a transition is it the sort of transition when flow leaves the body to equilibrate the pressure distribution such kind of natural physical equilibrium happens only through a shock wave it always exists it's not possible to avoid it can make only harder or softer this kind of shock wave but this process will exist always because we have body in the flow and the flow reacts on the body or body reacts on the flow which is equivalent let's say equivalent similar things that i've discussed you case one is shown on the picture you can see visualization by schlieren photos of shock waves or lambda shock waves or which are connected to airfoil and also related to flow separation shock wave big one is generated by several smaller weaker shock waves here they are called lambda shock waves because they look like letter lambda on the upper side on the lower side like they have two legs at least at least two legs and here we see visualized flow separation high turbulence zone which appears because of existence of the shock wave shock wave distorts a lot it creates huge gradients of flow parameters of thermodynamic parameters and intensifies a lot uh turbulence in the flow so it's not possible to to have a very nice flow in transonic regime i mean nice from point of view of lift to drug ratio it's not good absolutely it's not good so we would like to avoid such kind of process and we cannot use conventional airfoils for such kind of flows because they have very bad performance of lift to drug coefficient in this case and there are some special types of of airfoils which we will study also in gas dynamics in transonic flow section and especially how to predict and how to design such airfoils for performance and efficient operation in transonic region this is for another another course also and supersonic flows probably you know everywhere should be a mach number higher than one we can start from stomach number uh could be 1.2 1.5 all depending a lot on on shape of the body because not always 1.2 is enough to have everywhere supersonic flow and then flow passes through series of shock waves which change significantly the flow structure this theory of shock waves we study in deep in gas dynamics and then flow also must pass through expansion waves flow shock waves and expansion waves they always go together it's not possible that we have just uh for example shock wave and we will not have expansion waves it's physics makes uh or nature equilibrates everything so shock wave increases pressure inside inside flow and you know probably that far distance from this body will also have same pressure as it had before right for example airplane flies before airplane we had one atmosphere and very far from this airplane we also have one atmosphere how it's done if we have waves it just increases increases pressure now we have another opposite process not opposite from point of view of physics because expansion wave has very different physics but in sense of relaxation of of the flow shock wave creates tension on the flow and the expansion wave relaxes flow to its initial state and we will study both of this process that analytically you will be capable to calculate and i hope you will like because models are very nice and not so difficult to use in figure you also can see picture schlieren photo that shows shock wave and expansion waves after the body one comment here that of course if we have uh supersonic flow there is it could be some region where flow is still subsonic but these regions are small small very small with respect to global volume of of calculation same as here we have also subsonic regions at the end of the body some circulation vertex is high turbulence cannot be calculated analytically can be just simulated with very big difficulties if you want a high quality simulation and can be tested experimentally can be also only studied experimentally if you want uh extremely high quality and the hypersonic hypersonic flow something that we will also study there is one chapter of gas dynamic devoted to this not in a big and long discussion hypersonic flows are extremely complex there is a theory of hypersonic flow and we have several books in the library you will see if when you open this book you will be surprised with the quantity of mathematical and physical models that describe hypersonic flow and we will not pay too much attention on this huge amount of models because it's a question of probably more advanced subject but we will discuss basics of hypersonic flow so you will know if you like this subject you will know from where you could start studying and what to expect approximately in this section these models of hypersonic flow they involve not just theory of aerodynamics and gas dynamics they also involve theory of chemistry dissociation and ionization plasma physics so this is why so much complex concept cannot be seen just in aerodynamics and should not be seen in aerodynamics you can see very short chapter of hypersonic flow in anderson book so these are types of flaws i wanted to present you today guide map of studying aerodynamics under some gears and the last thing very important also introduction to boundary layers you will probably already know what is boundary layer in portuguese commander limit right and you studied in fluid mechanics i'm sure that you know just to refresh your memory not to explain from zero right boundary layer is some structure that deals with change of velocity change of velocity happens near the body when flow interacts with the body it stacks on molecules of gas stuck on the body of airfoil for example on the wall and velocity on the wall va for example here we can see that velocity is equal to zero on the wall far from this body velocity is equal to infinite velocity so how this transition happens this question is answered in theory of boundary layers there are different types of boundary layers at least you know that there are laminar and turbulent flows right which have different structures and can be described by different models theory of boundary layer we discuss also in gas dynamics in details there is a chapter about laminar flow there is chapter about turbulent boundary layer some models some analytical solutions uh boundary layer is also characterized by its thickness delta as we may see here this thickness uh says to us when at this velocity stopped to change due to influence of viscosity forces if we draw profile of velocity for example like this we will see that this thickness here is a point where velocity does not change anymore this is profile of velocity and drop okay so usually we can calculate thickness of bond very layer just by seeing gradient of velocity like dv by dn this great normal normal direction to the wall right at inside the boundary layer this gradient will be always positive but when we are outside of boundary layer this dv by dn will be close to zero almost here okay velocity still can change in the normal direction it happens due to uh general flow and body interaction that is not so huge with respect to velocity change inside the boundary layer usually we have a reasonable estimation of thickness of boundary layer we say that when gradient is smaller than five percent of maximum of its maximum or initial value we already are outside of uh boundary layer so we see it was initial gradient at point a and then it at any other point we compare with this point a gradient when it's smaller than five percent we cut we cut boundary layer model why i'm saying boundary layer so important because for example for simulation we not we but ansys is using two different two different models to simulate flow inside the boundary layer it uses boundary layer model with viscosity and outside of boundary layer it used just in this inviscid fluid program automatically decides if uh some calculation uh point is inside or outside so we choose between two different models so and i quite sure that since we would like to use ansys as advanced user we need to understand how this choice is made at least what are the models how this choice is made this is why i tell you that we must pay attention to some aspects work tree you will calculate with inclusion of boundary layer model okay not just velocity changes inside the boundary layer but also other thermodynamic properties for example for inviscid for incompressible incompressible flow is just the velocity change is enough because that's the only parameter we need parameter we need to to calculate but for compressible flow we also have change of temperature is so called thermal boundary layer which does not uh totally correspond to velocity boundary layer thermal boundary layer uh is important because we start to have heat exchange with the wall which changes not just temperature distribution but also changes velocity distribution seems viscosity proportional to gradient of velocity okay i will probably add here that um yeah this quantity is a function of temperature since we have heat transfer between body and flow we have redistribution of energy uh it becomes very important because impacts not just thermal processes by itself but also impacts on on viscosity uh viscosity and shear stress right shear stress i remember now this tau is mu multiplied by this velocity gradient you remember this definition of hair stress and mu is a function of temperature so the thermal boundary layer is related to the velocity boundary layer we will discuss about uh these quantities in gas dynamics here just preliminary preliminary but i hope enough information for you to start understanding at least something this is what i already told you thermal boundary layer and velocity boundary layer dependence on viscosity and temperature you can see that for example for air with increase of temperature we increase we have increase of viscosity typical values uh these which we use in the normal conditions because it corresponds to model as 15 degrees we use 1.8 10 minus 5 in our calculations of reynolds number for example as you may see also in my other videos where i explain how to calculate reynolds number and size of the body for simulation i used 1.8 as viscosity okay growth of thickness of boundary layer quite obviously i hope you you know if something is not clear about thickness of boundary layer please let me know i think now is i for my opinion information is enough and definitely this thickness increases we can see from experiments two different types of boundary layers you also should know that depending on reynolds number which is calculated as a function of distance of this x coordinate is used to calculate reynolds number okay it's called reynolds x is equal to uh density velocity uh infinite right then see infinite and here is this x and divided by this quasi-c okay like this so we see that reynolds is directly related to coordinate of flow and depending on this reynolds number we have first small values on small coordinates corresponding to laminar flow and then increasing reynolds number it makes transition to turbulent flows okay laminar flow thickness smaller than turbulent boundary layer thickness okay about uh values of this thickness we will speak very soon so don't worry just important to to understand it okay laminar flow is more say predictable and maybe friendly for aerodynamic applications and turbulent is more chaotic and consumes much more energy but not in all cases okay i would like to see if you have some questions usually i finish at this moment lecture and next time we will talk about magnitudes and variations of aerodynamic coefficients which for my opinion must be started from fresh right because it's a lot of interesting data and on tuesday we can start talking about such important concept how big should be could be aerodynamic coefficients of momentum lift and drug please let me know if you have some questions in the chat or say by voice i can answer your questions because we have some time and ready to answer please delay me no questions okay thank you but no need to tell that you have no you need to say that you don't have questions okay thank you people for at least saying that you have no questions but i honestly expect that some questions will appear because some information maybe not so much new for you but from aerodynamic point of view you will see that uh things are different when we compare just fluid mechanics and aerodynamics because aerodynamics is a lot more applied science that you can use your everyday study or your everyday research than just fluid mechanics if there are no questions we could finish for today and we meet next week tomorrow we will not have class as because we had last week and have a nice end of the week see you next time i expect also questions about work one you still have something when we finish chapter one i will uh explain you better how to do work one okay i do not forget about work one i would like to talk much more about it because it's important to understand the best way of presentation of aerodynamic quantities and analysis of aerodynamic quantities which is not so much clear from a theoretical book we need to calculate something in practice to show this so no questions thank you very much and see you next time goodbye goodbye thank you professor goodbye