1.2. Department of Civil Engineering

Hydraulic Engineering Prof. Mohammad Saud Afzal Department of Civil Engineering Indian Institute of Technology – Kharagpur Lecture – 20 Boundary Layer Theory (Contd..)

This chunk serves as the introduction to the lecture on Boundary Layer Theory within the Hydraulic Engineering course under the Civil Engineering department. It establishes context by identifying the professor and the educational institution, which helps in setting the stage for the technical discussion that follows.

Think of this introduction as the title of a chapter in a book. Just like a title gives a glimpse into what to expect, this introduction sets the academic context and prepares students for a detailed discussion on boundary layers.

The question says that, the velocity profile for laminar boundary layer is given as u / U equal to 2 y / delta - y / delta whole square. Now, we have to find an expression for boundary layer thickness delta and the wall shear stress.

This chunk presents a specific problem related to the velocity profile of a laminar boundary layer. With the equation given as u/U = (2y/δ) - (y/δ)², it sets the stage for calculating two important fluid dynamics concepts: the boundary layer thickness (δ) and the wall shear stress (τ). Understanding this mathematical description is essential for analyzing fluid flow conditions in civil engineering applications.

Imagine water flowing smoothly along the surface of a flowing river. The water close to the riverbed moves slower (due to friction), while water above it moves faster. This difference in velocities creates a 'boundary layer', and this problem focuses on quantifying that layer mathematically.

Therefore, the momentum thickness can be written as, 0 to delta u / U 1 - u / U dy. This is the definition of momentum thickness.

Momentum thickness is a key concept in boundary layer theory, defined mathematically as the integral from 0 to δ of the velocity profile modified by the term (1 - u/U). This definition allows engineers to quantify the momentum deficit due to the viscous effects near the boundary, which is crucial for assessing shear stress and drag forces.

Consider padding on the wall of a water slide. Just as the slide's smoothness affects how fast you can go, the momentum thickness represents how sticky the fluid feels at the boundary, affecting the speed of the flow above it.

Also tau 0, we know, can be written as mu du / dy at y is = 0. And how do we get this? From Newton's law of viscosity.

Wall shear stress (τ₀) is expressed using Newton's law of viscosity as τ₀ = μ (du/dy) at y=0. This relationship indicates how the viscosity of the fluid (μ) influences the rate of change of velocity (du/dy) right at the wall. Studying this helps in understanding how easily a fluid slides over a surface, which has practical implications in civil engineering.

Think of the wall of a pipe and the water flowing through it. If you imagine the water as a series of layers, the closest layer to the wall moves the slowest due to friction. The shear stress is a measure of that friction, like how hard you have to push against syrup to make it flow across a surface.

So, now to find delta what we must do? We must simply integrate it.

To solve for the boundary layer thickness (δ), one needs to perform an integration based on the relationships established earlier in the discussion. This involves rearranging the terms, expressing them appropriately, and applying limits to connect the theoretical expression of δ with practical parameter values. Integration helps derive a formula that can be used to calculate δ based on other known quantities.

Just like a baker needs to mix ingredients thoroughly to get the right dough's thickness, engineers need to integrate the velocity profile over the boundary layer to accurately determine its thickness.

At end of the plate x = 1 meter, end of plate. Reynolds number at this position will be U x / nu or rho U x / mu.

The Reynolds number (Re) is a dimensionless quantity crucial in fluid mechanics that helps predict flow patterns in different fluid flow situations. The calculation at this stage is given by the relationship Re = U * x / ν or Re = ρ * U * x / μ, where 'U' is the flow velocity, 'x' is the characteristic length, 'ν' is the kinematic viscosity, and 'μ' is the dynamic viscosity. This analysis helps determine whether the flow is laminar or turbulent.

Imagine driving different vehicles on a road. Some maintain smooth speeds (laminar flow), while others jostle around with turbulence depending on their speed—this is similar to how Reynolds number helps to predict fluid behavior in flow dynamics.

As always, we will write given. So, length is given as 1 meter, b is given as 0.8 meter, U is 150 millimeters per second or we can write 150 into 10 to the power -3 meters per second, mu is given as 0.01 poise.

In this step, actual values are assigned to the parameters and plug into the earlier derived formula for boundary layer thickness. By defining specific cases, students learn to apply theoretical knowledge to practical scenarios. It emphasizes the idea of transforming abstract concepts into concrete computations.

Think of this like a recipe where you add specific amounts of ingredients to get a dish just right. Similarly, using real values will help determine the exact boundary layer thickness in this engineering problem.

So, delta by formula is 5.48 x divided by under root R e x.

This concludes the calculation by expressing the final boundary layer thickness (δ) entirely in terms of known parameters, specifically emphasizing how δ can be quantified as proportional to 'x' and inversely related to the square root of Reynolds number. This highlights the interconnected nature of the variables involved in boundary layer theory.

Imagine using a formula to create a perfect batch of cookies. The relationship between dough thickness and ingredients is akin to how boundary layer thickness relates to flow conditions and physical parameters.

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