This video from The Efficient Engineer is sponsored by Brilliant. One of the very first things you learn in fluid mechanics is the difference between laminar and turbulent flow. And for good reason – these two flow regimes behave in very different ways and, as we’ll see in this video, this has huge implications for fluid flow in the world around us Here we have an example of the laminar flow regime. It’s characterised by smooth, even flow. The fluid is moving horizontally in layers, and there is a minimal amount of mixing between layers. As we increase the flow velocity we begin to see some bursts of random motion. This is the start of the transition between the laminar and turbulent regimes.
If we continue increasing the velocity we end up with fully turbulent flow. Turbulent flow is characterised by chaotic movement and contains swirling regions called eddies. The chaotic motion and eddies result in significant mixing of the fluid. If we record the velocity at a single point in steady laminar flow, we’ll get data that looks like this. There are no random velocity fluctuations, and so in general laminar flow is fairly easy to analyse. For turbulent flow we’ll get data that looks like this. This flow is much more complicated. We can think of the velocity as being made up of a time-averaged component, and a fluctuating component. The larger the fluctuating component, the more turbulent the flow. Because of its chaotic nature, analysis of turbulent flow is very complex. Since laminar and turbulent flow are so different and need to be analysed in different ways, we need to be able to predict which flow regime is likely to be produced by a particular set of flow condition We can do this using a parameter which was defined by Osborne Reynolds in 1883.
Reynolds performed extensive testing to identify the parameters which affect the flow regime, and came up with this non-dimensional parameter, which we call Reynolds number. It’s used to predict if flow will be laminar or turbulent. Rho is the fluid density, U is the velocity, L is a characteristic length dimension, and Mu is the fluid dynamic viscosity. The equation is sometimes written as a function of the kinematic viscosity instead, which is just the dynamic viscosity divided by the fluid density.
The characteristic length L will depend on the type of flow we are analysing. For flow past a cylinder it will be the cylinder diameter. For flow past an airfoil it will be the chord length. And for flow through a pipe it will be the pipe diameter. Reynolds number is useful because it tells us the relative importance of the inertial forces and the viscous forces. Inertial forces are related to the momentum of the fluid, and so are essentially the forces which cause the fluid to move. Viscous forces are the frictional shear forces which develop between layers of the fluid due to its viscosity. If viscous forces dominate flow is more likely to be laminar, because the frictional forces within the fluid will dampen out any initial turbulent disturbances and random motion.
This is why Reynolds number can be used to predict if flow will be laminar or turbulent. If inertial forces dominate, flow is more likely to be turbulent. But if viscous forces dominate, it’s more likely to be laminar. And so smaller values of Reynolds number indicate that flow will be laminar. The Reynolds number at which the transition to the turbulent regime occurs will vary depending on the type of flow we are dealing with. These are the ranges usually quoted for flow through a pipe, for example. Under very controlled conditions in a lab the onset of turbulence can be delayed until much larger Reynolds numbers. Most flows in the world around us are turbulent.
The flow of smoke out of a chimney is usually turbulent. And so is the flow of air behind a car travelling at high speed. The flow of blood through vessels on the other hand is mostly laminar, because the characteristic length and velocity are small. This is fortunate because if it were turbulent the heart would have to work much harder to pump blood around the body. To understand why this is, let’s look at how the flow regime affects flow through a circular pipe. The flow velocity right at the pipe wall is always zero. This is called the no-slip condition. For fully developed laminar flow, the velocity then increases to reach the maximum velocity at the centre of the pipe. The velocity profile is parabolic. For turbulent flow the profile is quite different.
We still have the no-slip condition, but the average velocity profile is much flatter away from the wall. This is because turbulence introduces a lot of mixing between the different layers of flow, and this momentum transfer tends to homogenise the flow velocity across the pipe diameter. Note that I have shown the time-averaged velocity here. The instantaneous velocity profile will look something like this. In pipe flow one thing we are particularly interested in is pressure drop. Across any length of pipe there will be a drop in pressure due to the frictional shear forces acting within the fluid. The pressure drop in turbulent flow is much larger than in laminar flow, which explains why the heart would have to work harder if blood flow was mostly turbulent! We can calculate Delta-P along the pipe using the Darcy-Weisbach equation. It depends on the average flow velocity, the fluid density and a friction factor f.
For laminar flow the friction factor can be calculated easily. It is just a function of the Reynolds number. If we combine these two equations we can see that the pressure drop is proportional to the flow velocity. But for turbulent flow calculating f is more complicated. It is defined by the Colebrook equation. f appears on both sides of the equation, so it needs to be solved iteratively. Unlike laminar flow, for which the pressure drop is proportional to the flow velocity, it turns out that for turbulent flow it is proportional to the flow velocity squared. And it also depends on the roughness of the pipe surface. Epsilon is the height of the pipe surface roughness, and the term Epsilon/D is called the relative roughness. Surface roughness is important for turbulent flow because it introduces disturbances into the flow, which can be amplified and result in additional turbulence. For laminar flow it doesn’t have a significant effect because these disturbances are dampened out more easily by the viscous forces.
Since the Colebrook equation is so difficult to use, engineers usually use its graphical representation, the Moody diagram, to look up friction factors for different flow conditions. Where flow is laminar the friction factor is only a function of Reynolds number, so we get a straight line on the Moody diagram. For turbulent flow you select the curve corresponding to the relative roughness of your pipe, and you can look up the friction factor for the Reynolds number of interest.
So we know that if Reynolds number is large, inertial forces dominate, and the flow is turbulent. But even for turbulent flow viscous forces can be significant in the boundary layers that develop at solid walls. Because of the no-slip condition, shear stresses are large close to a wall. This means that in a turbulent boundary layer there remains a very thin area close to the wall where viscous forces dominate and flow is essentially laminar. We call this the laminar, or viscous, sublayer. Its thickness decreases as Reynolds number increases. Above the laminar sublayer there is the buffer layer, where both viscous and turbulent effects are significant. And above the buffer layer turbulent effects are dominant. If the roughness of a surface is contained entirely within the thickness of the laminar sublayer, the surface is said to be hydraulically smooth, because the roughness has no effect on the turbulent flow above the sublayer.
This is important in pipe flow because, as can be seen from the Moody diagram, flow in smooth pipe has a lower friction factor and so smaller pressure drop than flow in rough pipe. We can see that for a given roughness the friction factors converge to a constant value to the right of this dashed line, meaning that at high Reynolds number the friction depends only on the relative roughness. At these high Reynolds numbers the thickness of the laminar sublayer is extremely thin, and so the effect of the surface roughness is governing. Modelling turbulent flow through a pipe is fairly simple, but most scenarios are far more complex. It’s worth talking more about why analysis of turbulent flow is so complicated, and a lot of it has to do with the turbulent eddies we saw at the start of the video. Large eddies contain a lot of kinetic energy. Xem them chuyen phat nhanh di singapore tai Viexpress
Over time the energy in these large eddies feeds the creation of progressively smaller eddies, until at the smallest scale the turbulent energy in minuscule eddies dissipates as heat, due to frictional forces caused by the fluid viscosity. We can think of the energy in the flow as cascading from the largest to the smallest eddies, and so this concept is called the energy cascade. The energy cascade was summarised in a very elegant way by the physicist Lewis Fry Richardson, who wrote that “Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls, and so on to viscosity”. Because of this behaviour, turbulence involves a huge range of length and time scales. This makes analysis of turbulent flow very complex, to the point that it is probably the most significant challenge facing the field of Fluid Mechanics.
For complex scenarios like flow past an airfoil, we can’t accurately describe the fluid behaviour using simple equations. So to analyse the flow we have to use either experimentation or numerical methods, or a combination of the two. Modelling flow using numerical methods is the field of Computational Fluid Dynamics. It essentially involves using computational power to solve the Navier-Stokes equations, which is a system of partial differential equations that describes the behaviour of fluids, but is difficult to solve. To do this we model the fluid domain around the airfoil as a mesh of discrete elements, define boundary conditions and fluid properties, and apply an appropriate assessment technique to find a solution.
I mentioned earlier that one of the main challenges when dealing with turbulence is capturing the wide range of length scales associated with the turbulent eddies. There are three main techniques which are used to simulate flow in CFD, and they differ mainly in how they treat turbulence on these different scales. First we have Direct Numerical Simulation. This involves solving the Navier-Stokes equations down to even the smallest scales, and so all turbulent eddies are fully resolved, meaning that they are simulated explicitly. This is very computationally expensive, and isn’t a practical solution for the vast majority of fluid flow problems. Next we have Large Eddy Simulation. This technique resolves the large scale eddies explicitly, but small scale eddies are filtered out and are modelled, using what is known as a subgrid-scale model.
LES is much less computationally expensive than DNS. Finally we have the Reynolds-Averaged Navier-Stokes technique, which is the least computationally expensive of the three techniques. This is a time-averaged method which doesn’t resolve eddies explicitly at all. Instead it models the effect of eddies using the concept of turbulent viscosity. Several different turbulence models exist, like the K-Epsilon or K-Omega models, with different models being better suited to different problem types. As is so often the case in engineering, experience and intuition will need to be used to determine which techniques and models are best suited to a particular problem. When it comes to troubleshooting problems in the real world, the importance of engineering intuition can’t be overstated. And that’s why I’d like to introduce you to Brilliant. Brilliant is a math and science learning website and app that has courses covering a wide range of topics, including differential equations, energy, momentum, and even dimensional analysis, to name just a few which are relevant to Fluid Mechanics.
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