2. Problem Solving on Turbulent Flow

So, if we use equation 14 in equation 15, what was equation 14? l m was kappa into y, this was what we said in equation number 14. So, we can simply write and this is equation number 16, very simple. So, for small values of y it can be assumed. So, if the y is very small we can assume that tau is equal to tau not, where tau not is the shear stress at the pipe wall and can be assumed to be a constant. So, at the wall the shear stress is assumed to be constant and equal to tau not.

In this chunk, we see an application of substitution in equations used for fluid dynamics. Equation 14 involves defining a variable 'l m' as the product of 'kappa' and 'y'. This relationship helps simplify calculations for turbulent flow conditions. The text then states that for small values of 'y', the shear stress ('tau') can be approximated as equal to a constant value known as 'tau not', which is the shear stress at the wall of the pipe.

Imagine you have a garden hose. When you place your thumb partially over the opening, you allow a small amount of water to flow out smoothly (small 'y'). Here, the pressure at the opening (analogous to 'tau not') remains constant, even as you adjust your thumb. Similarly, in turbulent flow, there is a predictable relationship between the shear stress at the wall and the flow conditions.

And therefore, what we can say, if we substitute tau is equal to tau not in equation 16, we can obtain or du / dy, this quantity actually tau not can be written as rho. So, but the catch here is, what is the catch? We have considered small value of y. So, du / dy can be written as, 1 / kappa y and under root tau / rho is rho u *. So, it becomes and this u * under root tau not / rho is the sheer velocity and this has the dimension of velocity.

In this part, we learn how substituting shear stress ('tau') as 'tau not' into equation 16 allows us to express the derivative of the velocity ('du/dy') in simpler terms. The equation shows that 'du/dy' is inversely proportional to 'y' and directly related to other factors, such as 'kappa' and density ('rho'). The derived expressions help define 'u *', which represents the shear velocity. This concept is key in characterizing turbulent flow characteristics.

Think of it like adjusting the speed of a conveyor belt carrying boxes. If you know the belt's speed (shear velocity), and you can measure how tightly the boxes are packed ('y'), you can estimate how quickly they should move past a certain point ('du/dy'). This relationship is essential in engineering applications involving fluid transport.

Then using the boundary conditions, what are the boundary conditions? So, u at y is equal to R, where R is the radius of the pipe. We will get, u is equal to u max. That is what we have seen at the center line of the pipe the velocity is going to be the maximum. So, if we use this boundary condition u at y is equal to R is u max, we can get u is equal to, you know, we put u max here, y will be R and therefore, we can obtain C.

This chunk discusses how boundary conditions affect the solutions to the equations governing turbulent flow. By establishing that at the wall (where 'y' equals the radius 'R' of the pipe), the velocity 'u' equals 'u max' (the maximum velocity), we can later determine coefficients in our equations. This fundamental principle allows for accurate modeling of velocities within the flow.

Consider a racetrack where cars have maximum speed at the center due to less friction. If we define the racetrack's border as having a certain radius (the wall), knowing the maximum speed in the center helps us figure out speeds throughout the track, similar to how we use boundary conditions in fluid flow equations.

So, this is ln. So, we can put it in form of log. This is simple manipulation, we can get u max minus u / u * equals to 2.5 ln R / y. And, so, this is ln. This is just simple manipulation, we can get u max minus u / u * is equal to 5.75 log to the base 10 R / y. u max minus u is called the velocity defect or velocity defect law, this is velocity defect law. This is just simple, you know, manipulation of these terms here.

In this section, we learn how to express the relationship between maximum flow velocity and actual flow velocity in logarithmic terms. The equation identifies the difference between the two velocities as the 'velocity defect'. Understanding this law is important for predicting how velocity changes with distance from the wall of the pipe in turbulent flow scenarios.

Imagine a car racing down a straight road. The car's speed (u max) varies as it nears the edges due to increased air resistance and friction. The difference between the maximum speed and the speed at any point on the road (u) gives a measure of lost potential speed, or the 'velocity defect'. This analogy helps understand why maintaining speed in turbulent conditions is crucial.

So, now we are going to solve one of the problems, problem number 7. And what it says is, the velocity of water. So, what we have learned in this particular lecture is about the turbulent flow and this problem 7 will help you in solving any problem that is based on this particular concept. So, it says the velocities of water through a pipe of diameter 10 centimeter are 4 meters per second and 3.5 meters per second at the center of the pipe and 2 centimeters from the pipe center, respectively. Considering turbulent flow in pipe, determine the sheer stress at the wall. So, we need to determine tau not.

This chunk introduces a practical example related to the theoretical concepts discussed. The problem involves a turbulent flow of water through a pipe with specified dimensions and velocities. The objective is to calculate the wall shear stress ('tau not'), linking theoretical principles with real-world application. This problem reinforces the concepts of velocity profiles and shear stress previously discussed.

Imagine a plumbing system in a house. When water flows through a pipe, understanding how fast the water moves at different points helps in calculating the pressure and potential damage from leaks. This problem puts you in a plumber's shoes, making calculations relevant for ensuring safe and efficient plumbing.

As always what we do we solve, we write given, diameter is given as 10 centimeter, try to always write down in SI units. So, we write 0.1 meter. So, diameter is 10. So, radius is going to be 0.05 meter. u max is given, is given as 4 meters per second, that is, at y is equal to R. And this is also given, u at r is equal to 2 centimeter is given 3.5 meters per second, that is, y is equal to R - r. So, y is going to be 5 - 2 is equal to 3 centimeter.

In this chunk, we detail the approach to solving the previously stated problem. The steps involve converting given measurements (such as diameter) into SI units, identifying parameters like radius and velocities required for the problem-solving process. Proper unit conversion is critical for maintaining consistency and accuracy in engineering calculations.

Just like a chef prepping ingredients, you start by measuring everything precisely to ensure the recipe comes out right. Similarly, converting measurements in engineering to a standard format (SI units) is just as crucial for the 'recipe' of successful fluid dynamics solutions.

So, now u max we are using the minus u / u * was 5.75 log R / y. So, substituting the values here, this from here, this equation, 4 - 3.5 divided by u * is equal to 5.75 log base 10 5 / 3. This will give us, u * as 0.392 meters per second. We also know, u* is under root tau not / rho or tau not is rho u* whole square. Therefore, tau not rho is 1000 and u * we already got, 0.392 whole square.

This segment covers the calculations leading to the wall shear stress ('tau not'). The provided equation assists in solving for the frictional velocity ('u *') using the previously established velocities and logarithmic relationships. Once determined, the shear stress at the wall can be calculated using the derived values, showcasing the application of theory to arrive at tangible results.

Based on our earlier plumbing example, once you've measured the flow rates, you can calculate the pressure exerted at the walls of the pipes. Just like a doctor diagnosing a condition based on observed symptoms, calculating shear stress reveals the pressures at play in fluid movement.

So, now, the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. Actually this is the flow, this is the true picture, this is a laminar flow that we have seen before. But below is, this is the V average and the velocity fluctuates or deviates from these depending upon the flow condition.

This chunk highlights the differences in velocity profiles between turbulent and laminar flow. While laminar flow shows a parabolic profile, turbulent flow exhibits a fuller velocity profile with fluctuations due to various flow conditions. These visual differences are key in understanding how fluids behave under different circumstances.

Think of water flowing quietly down a narrow stream (laminar) versus rushing through a wider river with waves and eddies (turbulent). The way water moves visually indicates how smoothly or chaotically it flows, impacting things like erosion and sediment transport.

So, this is the V average line. There are several other layers, viscous sublayer, buffer layer, overlap layer and turbulent layer. So, as I told you in the last slide, there are different layers, different layers in turbulent flow.

This part introduces the concept of various layers present in turbulent flow. These layers include the viscous sublayer (where viscosity has a significant effect), buffer layer, overlap layer, and turbulent layer. Each of these layers has distinct characteristics and behaviors that influence the overall fluid dynamics.

Imagine a multi-layer cake. Each layer contributes to the overall flavor and texture. In a similar manner, different layers in turbulent flow contribute to its unique properties and behaviors depending on factors like turbulence and viscosity.

Now, when it comes to these beds and these regimes, some of the important terms that are there is hydro dynamically rough and smooth boundaries. So, this is the, if you see, there is a term called k. Here, if in here, so, k here is the mean height of the surface irregularities. We talked in the beginning that the turbulence could occur due to the presence of irregularities on the surface.

This chunk discusses the concept of boundary conditions in turbulent flows, identifying 'hydrodynamically rough' and 'smooth boundaries'. The variable 'k' represents the average height of the surface irregularities, illustrating how these irregularities influence flow characteristics. Understanding whether boundaries are smooth or rough is key in predicting flow behavior.

Consider a river with a smooth bed versus one with rocks and boulders. The smooth river allows water to flow calmly, while the rough bed causes turbulence. Thus, the state of the boundary impacts flow patterns significantly.

But when the surface irregularities are much smaller than delta dash, the height of the viscous sublayer, the eddies are unable to reach the surface irregularities when the roughness height is much less. Therefore, we define that boundary as smooth boundary. So, smooth boundary are the one, where the thickness of the viscous sublayer is much larger than the surface irregularities.

In this segment, we delve deeper into the classification of boundaries based on surface irregularities. If the irregularities (k) are significantly smaller than the thickness of the viscous sublayer (delta dash), we categorize the boundary as smooth. This distinction helps in assessing how effectively the turbulent flow interacts with the surface during movement.

Think of a freshly painted wall versus a textured wall. The smooth wall allows paint to glide smoothly (analogous to turbulent flow over a smooth boundary), while the textured wall disrupts the flow, causing uneven patterns (similar to turbulent flow over a rough boundary).

When k is much larger than the delta dash, that is, the thickness of viscous sub layer, the irregularities are above the laminar sublayer leading to the interaction of eddies with the surface irregularities and therefore, these are called rough boundaries. From Nikuradse's roughness, k / delta dash if it is less than 0.25. So, these values which we are going to talk about, has been derived from experiments by Nikuradse.

Here, we elaborate on the transition to rough boundaries. If the height of irregularities (k) is much larger than the thickness of the viscous sublayer (delta dash), we classify those as rough boundaries. The work of Nikuradse helps define critical thresholds for classifying boundaries based on the ratio of k to delta dash.

Picture a well-maintained road versus a pothole-ridden one. The smooth road allows cars to glide easily, while the rough one causes bumps and jolts, analogous to the distinction between smooth and rough boundaries in fluid dynamics.

In terms of roughness Reynolds number, so actually, there is something called roughness Reynolds number that is dependent upon k the height of the irregularities. So, in terms of roughness Reynolds number, if this Reynolds number is less than the 4, the boundaries is smooth, if it is more than the 100 then the boundary is rough and if it lies between 4 and 100 the boundaries is transitional.

This chunk presents the concept of the roughness Reynolds number, which helps identify boundary conditions based on flow characteristics. Values below 4 indicate smooth boundaries, while those exceeding 100 signify rough boundaries. Values in between classify boundaries as transitional. This helps engineers evaluate flow behavior based on known parameters.

Consider river water under different conditions: a stream with minimal rocks may flow smoothly (less than 4), while a rapid, rocky section creates turbulence (more than 100). This classification aids in predicting flow patterns and outcomes.

Now, we will solve one problem about this particular concept. So, the question is, a pipeline carrying water has average height of irregularities projecting from the surface of the boundary of the pipe as 0.15 millimeter. What type of boundary it is? We have to estimate the rough or smooth or transitional boundary. The shear stress at the pipe wall is 4.9 Newton per meter square and the kinematic viscosity is 0.01 Stokes.

This problem presents a practical application of the roughness concept. Students must determine if the boundary identified by irregularities is rough, smooth, or transitional based on provided measurements, including shear stress and kinematic viscosity, linking theory to real-world scenarios.

Think of examining the surface of different pipes in a household. By noticing the roughness, you can predict how efficiently water will flow through. This real-life observation helps in associating theoretical knowledge with practical implications.

So, best is to calculate the roughness Reynolds number Re* and that is given as, u* k / nu. So, Re* is, u* is 0.07, k is 0.15 into 10 to the power - 3 and nu is 0.01 into 10 to the power - 4 and that comes to be 10.5. So, as Re* lies between 4 and 400, this implies that the boundary is transitional.

This final chunk involves calculating the roughness Reynolds number (Re*), incorporating previously calculated values and given parameters. By inserting these values into the formula (u* k / nu), students can classify the boundary based on the results of their calculations, demonstrating the application of theoretical material in practical scenarios.

Imagine measuring the depth and flow rate of a stream to understand its behavior and safety for crossing. By applying this method, you predict outcomes reliably, just as using the roughness Reynolds number helps predict fluid behavior in different boundary conditions.

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