Chaos, Turbulence and the Navier-Stokes equations

big rills of the worlds that feed on their velocity and little wheels have lesser wheels and so on so viscosity we was fry Richardson greetings flame mechanics enthusiasts engineer Leo here on this video we're going to talk about turbulence and its relation to the thing is navier-stokes equations as a member for an advanced species one may wonder how is it possible that we managed to go to the moon when at the same time we cannot accurately predict something singly trivial as the weather we can barely predict weather conditions within seven days or so everyone why does that happen well the answer is it is apparent in consistence in our technology and knowledge lies the theory of chaos which is closely related to the theory of turbulent flows there is a philosophical debate about whether we live in a random or in a deterministic universe but what does that even mean well to put it simple in a random universe that would be no consistent laws or equations that govern what happens there only upper-end laws that we as intelligent beings could infer to approximate and establish some sense to it we would only be able to constantly describe it using statistics as the opposite a deterministic universe will have consistent laws and equations governing it following this knowing the initial state of a physical system and its governing set of equations it will be possible to predict its future at any given time it happens that in a system that has many variables and is highly unstable and sensible to perturbations even if we do know the initial state and its governing laws we would only be able to predict a near future state of the system furthermore as a system evolves from its initial state it would appear that there are no deterministic laws whatsoever governing it appearing to be increasingly chaotic and random that is what chaos theory is about and there in the middle of a random and deterministic nature is where we find turbulent Falls or simply turbulence there was a recurring debate over whether turbulence was a random or deterministic phenomenon this resulted in two different approaches to study turbulence the statistical and deterministic description of turbulence even though Noah days these two different views are not systematically distinct from each other they provided different tools to study turbulence let's talk about the origin of turbulence we have discussed in previous videos the famous navier-stokes equations but I will summarize again its main characteristics here our description of turbulence will be limited to Newtonian in compressive and Isel thermal flow that is Newtonian is a fluid which has linear relationship between velocity gradients and shear stress an incompressible fluid is a fluid that has no variation no considerable variation of volume due to pressure and isothermal flow is a thought that occurs without heat exchanges well then a Stokes equations are the equations that govern these types of fluid flow being answers the Newton's second law which is the conservation of momentum and here on the left hand side we have the fluid acceleration which is two terms here we have two types of acceleration the convective acceleration and the temporal acceleration and on the right hand side we have the sum of a forces acting on this volume online infinitesimal volume of fluid which are the pressure and the viscous force of course in other types of flows you can have other forces acting such as gravity but here we will simplify things and consider only the pressure and viscosity so this from here of the navier-stokes equation is presented on its dimensionless form so all the flow parameters like density viscosity the mean flow velocity the mean scale of velocity and the length scale like for example diameter of a pipe or the length of an obstacle all these parameters are confined the only one which is their famous Reynolds number well the Reynolds number is given by this expression here we have the velocity scale which is given by u D which is which is a length scale and [Music] this ladder here is a Greek letter nu which is the kinematic viscosity the kinematic viscosity is the dynamic viscosity divided by the density of the fluid physically speaking the Reynolds number represents the ratio between inertial and viscous forces okay on the numerator we have the inertial forces and on the denominator we have the viscous forces we'll talk more about that later so following our discussion on turbulence that the flows that occur in nature can be divided basically in three types namely laminar transitional and turbulent the laminar flows occur smoothly and without lateral mixing being the minority of flows in nature they are purely deterministic meaning that knowing the initial and boundary conditions of our four where study we can determine it states at any future time in addition the velocity and pressure of these flows are well-defined in every point in space they occur typically for low Reynolds number meaning that the viscous force prevail over the forests so typically their anus number is much less than one so this means that the viscous forces are much larger than the inertial forces so the product of the velocity scale and the line scale is much smaller than the kinematic viscosity if we increase the Reynolds number that means we are increasing the inertial term the inertial forces office flow the fluid flow becomes unstable so this means that the transversal flow starts developing from in waves which distorts the main flow this is seen in the famous Raynaud's experiment where he has this pipe and he places a die in the center line of the flow and as he increases the Reynolds number of this pipe flow the is died this line of tie which occurs on the centerline begins to develop some waves and then becomes totally fully unstable and occurring strong mixing of this time so as the inertial forces start to prevail these fluctuations gain more strength the initial stability is create secondary instabilities and the secondary creates a tertiary instabilities and so on resulting in a wide range of wavelengths so this intermediate state between laminar and turbulent flow is what's called a transitional flow and it will be a subject for another series of videos it is important to note here that the transitional flow has many stages this stage would correspond to development of the initial stability secondary tertiary instabilities and so on so there is in fact a critical value of the Reynolds number for which a flow becomes turbulent and this varies from flow to flow for the type of flow and it's not the same for example I flow around and like oh like a cylinder circular cylinder and flow within a pipe and for more complex fools like involving density variations and compressive flows and heat exchanges there are other parameters other dimensionless numbers that governs the transition to turbulence so the reynolds number is not the only parameter that makes the flow unstable so if we increase further the reynolds number then stabilities become fully developed and they finally reaches States which is called turbulence in this case Arenas number is much larger much greater than 1 which means that the product of the velocity scale of the flow which is usually the free stream velocity and the length scale of the flow is much larger than the kinematic viscosity so the inertial forces prevail over the viscous forces and on this flow we have velocity fluctuations in all directions of space and thus a wide range of wavelengths in a manner that is almost impossible to visually recognize then separately from one another as we did in transitional flows one can argue that a flow has become chaotic so in order to find order within the chaos we have to rely on sophisticated mathematical tools we can visualize this dependence of the foraging on the Reynolds number in one example the flow around the circular cylinder so for very low Reynolds numbers the flow occurs very smoothly with no vortex formation down stirring up the cylinder if we increase the Reynolds number for rayna's number around of order 10 to the power of 1 there occurs the formation of a pair of vortices down string of the cylinder increasing the Reynolds number further these vortices become unstable and form a vortex treat downstream of the cylinder but still laminar if we continue to increase the rayna's number even beyond the transition to the turbulence of course so what is turbulence is it possible to define turbulence so turbulent flows are the majority of flows in nature with countless examples we have human breeze airplane flight river flows oceanic currents earth atmospheric phenomena solar flares and even the flow of the Great Red Spot on Jupiter and much more even galaxies look to us like gargantuan ages meaning that the universe itself is turbulent turbulent flows play a fundamental role in the nature surrounding us and on our life itself the study of turbulence has applications to the fields of aeronautics Daleks nuclear and chemical engineering which geography meteorology astrophysics and much more but how do we define turbulence a turbulent flow is a flow which is disorder in Thailand space but that is not a precise definition Lizzie on his book turbulence in fluid says the following quote the flows one called turbulent may possess fairly different dynamics may be three-dimensional or sometimes quasi two-dimensional make exhibit where organized structures are otherwise a common property which is required of them is that they should be able to mix transported quantities much more rapidly than if only molecular diffusion processes were involved and quote considering this the same outlet proposed the following definition of turbulence one a turbulent flow must be unpredictable in the sense that is Mao uncertainty as to its knowledge at a given initial time will amplify so as to render impossible or precise a terminus t'k prediction of its evolution - it has to satisfy the increase in mixing property and three it most involve a wide range of spatial wavelengths so regarding this third characteristic of the wide range of spatial wavelengths we can understand how turbulence generates this range by analyzing the navier-stokes equations themselves so here we have again an have a Stokes equation and we will only look at one dimensional flow so we just consider all the other two dimensions and we look at one dimensional flow so we will soon 0 pressure gradients and consider a very large Reynolds number so the pressure term and the viscous term vanishes and we arrive at this equation here which is known as the inviscid burgers equation we will talk more about this equation on another feature another future video we will consider an initial condition for the flow given by this cosine function here which is a simple wave here ki is the wave number and capital a is the amplitude of this wave we can expand the solution of this differential equation in the Taylor series Center at T 0 so here it is the Taylor series up to the first order term involving the first derivative and here are confined on this term here of quarter t minus t0 square the high order terms with higher order derivatives so given that the time derivative on the left hand side can be isolated in this manner and that we can obtain our derivatives from the initial condition we obtain the following okay so using the identity from the sign the relationship between sign of two K which is two times the wave number we arrive finally at this expression here so we can see that due to the nonlinear term in the navier-stokes equation the perturbation will amplify the velocity field over time and generate new wavelengths hence the nonlinear term or the navier-stokes equation is the primary source of turbulence thanks for watching don't forget to like subscribe to this channel and leave your comment please see you next time and for those who do not know yet I recently launched patreon page where you can contribute to this channel by as low as one dollar a month so your help will be very highly appreciated thank you bye [Music]

Channel: Engineer Leo

Views: 13,771

Rating: 4.9542336 out of 5

Keywords: navier-stokes, chaos, turbulence, butterfly effect

Id: DK2GvzWie4E

Length: 15min 59sec (959 seconds)

Published: Sun Dec 08 2019