Summary of "Fluid Mechanics | Module 5 | Fluid Flow I Boundary Layer Theory | Part 3 (Lecture 49)"
Summary of "fluid mechanics | Module 5 | Fluid Flow I boundary layer Theory | Part 3 (Lecture 49)"
This lecture by Gopal Sharma focuses on the derivation and application of the von Kármán momentum integral equation in fluid mechanics, specifically related to drag forces on a flat plate in fluid flow. The lecture builds upon previous concepts such as energy thickness, displacement thickness, and momentum thickness in boundary layers.
Main Ideas and Concepts
- Recap of Previous Lecture:
- Definitions and concepts of energy thickness, displacement thickness, and momentum thickness in boundary layers.
- Objective of Current Lecture:
- To derive and understand the von Kármán momentum integral equation.
- To analyze drag forces on a flat plate due to fluid flow.
- To distinguish between different types of drag and their calculation.
- drag force on a flat plate:
- drag force arises due to the fluid resisting the motion of the plate.
- It is composed of two main components:
- Pressure Drag (Form Drag): Due to pressure differences around the body.
- Skin Friction Drag (Viscous Drag): Due to shear stress from fluid viscosity acting along the surface.
- The total drag is the sum of pressure drag and skin friction drag.
- Shear Stress and Drag:
- At the fluid-plate interface, the fluid velocity is zero (no-slip condition).
- Shear stress develops due to velocity gradients near the surface.
- Shear stress multiplied by the surface area gives the skin friction drag force.
- Direction and Sign Convention of Drag:
- Drag acts opposite to the direction of fluid flow relative to the plate.
- The lecture explains the importance of correctly defining the direction of forces and momentum changes.
- Velocity Profile and boundary layer Thickness:
- Velocity varies from zero at the plate surface to free stream velocity outside the boundary layer.
- boundary layer thickness grows along the plate length.
- Velocity profiles at various sections along the plate are considered to understand momentum changes.
- Mass Flow Rate and Momentum Flux in boundary layer:
- Mass flow rate varies along the plate due to fluid entering the boundary layer.
- Momentum flux changes due to velocity gradients and mass inflow.
- Taylor series expansion is used to approximate changes in mass flow rate along the plate.
- Momentum Balance on a Differential Element:
- A control volume element within the boundary layer is analyzed.
- Momentum influx and outflux are calculated on all sides.
- Net rate of change of momentum in the control volume is related to the drag force on that element.
- von Kármán momentum integral equation:
- Derived by applying momentum conservation to the boundary layer control volume.
- Expresses the relationship between boundary layer parameters and drag force.
- Valid for laminar, turbulent, and transitional flows over flat plates.
- Application of the Equation:
- The equation helps calculate local and total skin friction drag.
- Requires knowledge of velocity distribution within the boundary layer.
- Can be used to estimate boundary layer thickness and drag coefficients.
- Practical Notes:
- Velocity profiles need to be known or assumed for solving problems.
- Newtonian fluid behavior and constant properties are assumed.
- The lecture hints at upcoming discussions on velocity profiles for laminar and turbulent flows.
Methodology / Step-by-Step Instructions (Derivation and Application)
- Define the system: Consider a flat plate aligned with fluid flow, with length \( L \), and fluid velocity \( U_\infty \).
- Identify control volume: Select a small differential element of length \( dx \) along the plate surface within the boundary layer.
- Establish velocity profile: Define velocity \( u(y) \) from the plate surface (where \( u=0 \)) to the edge of the boundary layer (where \( u=U_\infty \)).
- Calculate shear stress:
- Use Newton’s law of viscosity: \( \tau_w = \mu \left(\frac{\partial u}{\partial y}\right)_{y=0} \).
- Calculate drag force on element:
- \( dF_D = \tau_w \times \text{area} = \tau_w \times b \times dx \), where \( b \) is plate width.
- Calculate mass flow rate and momentum flux:
- Mass flow rate into element: \( \dot{m} = \rho b \int_0^\delta u dy \).
- Use Taylor series expansion to express changes in mass flow rate along \( x \).
- Apply momentum conservation:
- Momentum influx and outflux are calculated on control volume faces.
- Account for momentum entering from the free stream side (outside boundary layer).
- Derive von
Category
Educational
