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Today, we're exploring laminar boundary layers and how to derive expressions for boundary layer thickness and shear stress from a given velocity profile. Let’s start with the velocity profile equation.
What does the velocity profile tell us about the flow characteristics?
Great question! The velocity profile indicates how the fluid velocity varies with respect to the distance from the surface. In our case, it's u/U = 2y/δ - (y/δ)². Here, 'U' is the free stream velocity and 'y' is the distance from the surface.
So, what does 'δ' represent?
Good point! 'δ' is the boundary layer thickness, which represents the region where the fluid velocity transitions from the free stream velocity to zero at the wall.
Can we use this profile to find wall shear stress?
Exactly! By applying the momentum integral equation, we can also derive the wall shear stress from this velocity profile.
What’s the momentum thickness we can calculate?
Great follow-up! The momentum thickness can be computed as θ = 2δ/15. Let's move forward and further analyze the wall shear stress.
Now that we have θ, let's express shear stress using the von Karman momentum integral equation: τ₀ = ρU²(dθ/dx).
How do we derive dθ/dx from θ?
We calculate dθ/dx from θ = 2δ/15 using the derivative rules. This leads us to incorporate velocity and fluid properties into the equation.
What are the key properties needed to solve this?
We need viscosity μ and density ρ of the fluid. For example, if we're looking at water, we can substitute these values directly.
So, how do we express τ₀ in terms of δ?
By setting τ₀ to μ(du/dy) at y=0 and equating with our earlier equation, we can find a relationship for τ₀ based on δ and other parameters.
And then we can derive the boundary layer thickness for practical applications?
Absolutely! We can use the derived formulas to make calculations for specific scenarios.
Now, let’s apply all this knowledge to solve a real-world problem involving boundary layer thickness in water flowing over a flat plate.
What is our starting point for this problem?
We have a plate that is 1 meter long and a width of 0.8 meters, with water flowing at a velocity of 150 mm/s.
What units should we use for consistency?
Convert velocities to meters per second and viscosity in SI units for calculations. Remember to check the Reynolds number to confirm it’s laminar.
Calculating δ must involve using the Reynolds number in the equation we derived?
Exactly! You would substitute the values into δ = 5.48 x /√Re_x and calculate accordingly.
Once we have δ, how do we find the shear stress?
You’ll apply τ₀ = 0.365μU/√Re_x. Simply plug in your values after calculating δ for final results!
This application reinforces our understanding of fluid dynamics!
The height at which the flow transitions to free stream velocity.
A parameter to quantify the thickness of the boundary layer.
Affects drag forces experienced by the surface in contact with the fluid.
Crucial for determining if the flow is laminar or turbulent.
In the flow, the δ does grow, where velocity slows, that’s how it goes.
Imagine a train on tracks, moving smoothly at a high speed. The air around it has layers, some rushing, some not. The layer closest moves slow, and that’s the boundary - where speed drops and shear starts to occur.
Vegan Shear: V as in velocity, S as in shear - remember shear starts at zero when y is near.
BLS (Boundary Layer Shear): To fine-tune your understanding of boundary layer shear and its dynamics.
Term
What role does the Reynolds number play in fluid mechanics?
Click to flip
A thin layer of fluid in immediate contact with a surface where velocity transitions from zero at the surface to the free stream velocity.
A measure of the thickness of the boundary layer defined as the integral of the velocity profile.
The tangential force per unit area exerted on a surface by the fluid.
A dimensionless quantity that measures the ratio of inertial forces to viscous forces in fluid flow.
This section discusses the calculation of the boundary layer thickness and wall shear stress for a laminar boundary layer using the momentum integral equation.
The section presents a problem solving process to find the boundary layer thickness (B4) and wall shear stress (C40) from a given velocity profile for a laminar boundary layer. Key equations such as the von Karman momentum integral equation are applied in the derivation of results.
In this section, we examine the process of deriving expressions for boundary layer thickness and wall shear stress based on a given velocity profile for a laminar boundary layer. Starting with the laminar boundary layer velocity profile represented by u/U = (2y/B4) - (y/B4)², we calculate the momentum thickness by integrating the definition of momentum thickness from 0 to B4. The result gives us the momentum thickness, B8 = (2/15)B4. Using the von Karman momentum integral equation, we establish a relation for C40 and subsequently integrate to find B4 in terms of the flow parameters. The eventual formula for B4 is derived as B4²/2 = (15BCx)/( C1U), leading to the formula B4 = 5.48 x /√Re_x. The section concludes with a real-world problem applying these concepts to find boundary layer thickness and wall shear stress for specific conditions.