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Questions and Answers
What is the effect of a dimpled surface on a golf ball?
What is the effect of a dimpled surface on a golf ball?
What happens to the velocity profile when the pressure gradient is negative?
What happens to the velocity profile when the pressure gradient is negative?
Which condition is likely to cause flow separation?
Which condition is likely to cause flow separation?
What characterizes the stability of flow at a point of separation?
What characterizes the stability of flow at a point of separation?
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What is indicated by the presence of an inflection point in a velocity profile?
What is indicated by the presence of an inflection point in a velocity profile?
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How does a scour hole in a river bed affect the flow velocity?
How does a scour hole in a river bed affect the flow velocity?
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What can happen when there is a slight pressure difference in a steady flow?
What can happen when there is a slight pressure difference in a steady flow?
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What does the Strouhal number measure in fluid dynamics?
What does the Strouhal number measure in fluid dynamics?
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In what scenarios is the position of separation easier to determine?
In what scenarios is the position of separation easier to determine?
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What primary concept does boundary layer theory allow to be separated in fluid flows?
What primary concept does boundary layer theory allow to be separated in fluid flows?
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Which of the following statements is true regarding boundary layers at the Earth's surface?
Which of the following statements is true regarding boundary layers at the Earth's surface?
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What parameter is indicated to be small in the boundary layer approximation?
What parameter is indicated to be small in the boundary layer approximation?
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What type of flow is NOT included in the description of flows treated in the chapter?
What type of flow is NOT included in the description of flows treated in the chapter?
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What is assumed about the flow velocity in the boundary layer as it approaches a solid boundary?
What is assumed about the flow velocity in the boundary layer as it approaches a solid boundary?
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How does the introduction of boundary-fitted coordinates affect the solution of fluid flow problems?
How does the introduction of boundary-fitted coordinates affect the solution of fluid flow problems?
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What does a low Reynolds number indicate about the flow?
What does a low Reynolds number indicate about the flow?
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What is the effect of increasing the Reynolds number on the boundary layer thickness?
What is the effect of increasing the Reynolds number on the boundary layer thickness?
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What phenomenon is referred to as 'stall' in fluid flow?
What phenomenon is referred to as 'stall' in fluid flow?
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What does the boundary layer thickness ($ ext{y} = ext{$ ext{δ}$}$) represent in fluid dynamics?
What does the boundary layer thickness ($ ext{y} = ext{$ ext{δ}$}$) represent in fluid dynamics?
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In the context of boundary layer behavior, what is significant about the no-slip condition?
In the context of boundary layer behavior, what is significant about the no-slip condition?
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Which of the following statements is true about the terms in the equation of continuity?
Which of the following statements is true about the terms in the equation of continuity?
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What is the relationship between pressure gradient and velocity in the boundary layer?
What is the relationship between pressure gradient and velocity in the boundary layer?
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Why is the pressure variation across the boundary layer considered small?
Why is the pressure variation across the boundary layer considered small?
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How does the surface roughness of an object like a golf ball affect its boundary layer?
How does the surface roughness of an object like a golf ball affect its boundary layer?
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What does a smaller boundary layer thickness imply about shear forces on an object?
What does a smaller boundary layer thickness imply about shear forces on an object?
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What does the last term of the equations of motion generally correspond to?
What does the last term of the equations of motion generally correspond to?
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In the context of boundary layer theory, what does the term flow separation refer to?
In the context of boundary layer theory, what does the term flow separation refer to?
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Which condition best describes the behavior of fluid velocity near the surface of an object?
Which condition best describes the behavior of fluid velocity near the surface of an object?
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What is the relation of the Reynolds number to boundary layer thickness?
What is the relation of the Reynolds number to boundary layer thickness?
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What is the significance of Prandtl's work on boundary layers?
What is the significance of Prandtl's work on boundary layers?
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In which circumstance might the boundary layer approximation break down?
In which circumstance might the boundary layer approximation break down?
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What happens to the velocity in the boundary layer as pressure decreases in the downstream direction?
What happens to the velocity in the boundary layer as pressure decreases in the downstream direction?
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What are the implications of the boundary layer equations being of parabolic type?
What are the implications of the boundary layer equations being of parabolic type?
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Which statement best describes the relationship between Reynolds number and laminar flow?
Which statement best describes the relationship between Reynolds number and laminar flow?
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What boundary condition applies at the boundary of the object?
What boundary condition applies at the boundary of the object?
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What is the consequence of assuming steady state in the analysis of boundary layer flow?
What is the consequence of assuming steady state in the analysis of boundary layer flow?
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What can result from a strong enough pressure gradient in the boundary layer?
What can result from a strong enough pressure gradient in the boundary layer?
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How does the characteristic boundary layer thickness affect other flow dynamics?
How does the characteristic boundary layer thickness affect other flow dynamics?
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In the boundary layer equations, which term is influenced by changes in fluid viscosity?
In the boundary layer equations, which term is influenced by changes in fluid viscosity?
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How is the dimensionless height $y^+$ related to the Reynolds number?
How is the dimensionless height $y^+$ related to the Reynolds number?
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What does a small characteristic boundary layer thickness imply about the fluid flow?
What does a small characteristic boundary layer thickness imply about the fluid flow?
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What did Lanchester predict about fluid dynamics in relation to Prandtl's findings?
What did Lanchester predict about fluid dynamics in relation to Prandtl's findings?
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What is the primary outcome of applying the boundary conditions on the fluid flow at the object?
What is the primary outcome of applying the boundary conditions on the fluid flow at the object?
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Why might the boundary layer grow thicker as it moves downstream?
Why might the boundary layer grow thicker as it moves downstream?
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Study Notes
Boundary Layer Theory in Fluid Mechanics
- Scope: This chapter focuses on viscous flows with inertia, excluding turbulent and inertialess flows. Geophysical flows at the Earth's surface are generally not covered as they often involve turbulence.
- Reynolds Number: The Reynolds number is assumed low enough to maintain laminar flow.
- Prandtl's Boundary Layer Theory: This theory allows separating many flows into potential and viscous components. The potential component is relatively simple to solve, plus some approximations can be used for solving the viscous component. Matching the two components yields an overall solution, but this isn't a universal solution method. This approach, while not completely applicable to geophysical flows due to high Reynolds numbers, offers a valuable theoretical framework for analyzing field data and modelling results.
- Boundary Layer: This is a thin area near a solid boundary where viscous effects are dominant, reducing the velocity to zero at the boundary. The viscous layer always exists and often determines the flow pattern, albeit extremely thin at high Reynolds numbers. Boundary layer thickness is defined by where the velocity reaches a certain percentage (e.g., 99%) of the potential flow velocity.
- Coordinate System: Boundary-fitted coordinates (such as the 𝑥 and 𝑦 axes) are used to simplify calculations, assuming minimal boundary curvature relative to boundary layer thickness. This avoids the need for complex curvilinear coordinates.
Scaling and Simplification of Equations
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Scaling Variables: Velocities are scaled using potential flow velocity or free stream velocity (𝑈). Lengths are scaled by a characteristic dimension (𝐿) or radius of curvature (R).
- 𝑥∗ = 𝑥/𝐿
- 𝑦∗ = 𝑦/Δ
- 𝑢∗𝑥 = 𝑢𝑥/𝑈
- 𝑢∗𝑦 = 𝑢𝑦/𝑈
- 𝑡∗ = t𝑈/𝐿
- ℎ∗ = ℎ/𝐿
- 𝑝∗ = 𝑝/(𝜌𝑈^2)
Continuity and Motion Equations
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Continuity Equation: The equation of continuity, 𝜕𝑢∗𝑥/𝜕𝑥∗ + 𝜕𝑢∗𝑦/𝜕𝑦∗ = 0, is a fundamental equation describing mass conservation.
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Motion Equations (x-direction): 𝜕𝑢∗𝑥/𝜕𝑡∗ + 𝑢∗𝑥𝜕𝑢∗𝑥/𝜕𝑥∗ + 𝑢∗𝑦𝜕𝑢∗𝑥/𝜕𝑦∗ = − 𝜕𝑝∗/𝜕𝑥∗ − (𝑔𝐿/𝑈^2)𝜕ℎ∗/𝜕𝑥∗ + (𝜈𝑈/𝐿)(𝜕^2𝑢∗𝑥/𝜕𝑥∗^2 + (𝐿^2/Δ^2) 𝜕^2𝑢∗𝑥/𝜕𝑦∗^2).
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Motion Equations (y-direction): 𝜕𝑢∗𝑦/𝜕𝑡∗ + 𝑢∗𝑥𝜕𝑢∗𝑦/𝜕𝑥∗ + 𝑢∗𝑦𝜕𝑢∗𝑦/𝜕𝑦∗ = − (𝐿^2/Δ^2)𝜕𝑝∗/𝜕𝑦∗ − (𝑔𝐿/𝑈^2)(𝐿^2/Δ^2)𝜕ℎ∗/𝜕𝑦∗ + (𝜈𝑈/𝐿)(𝜕^2𝑢∗𝑦/𝜕𝑥∗^2 + (𝐿^2/Δ^2)𝜕^2𝑢∗𝑦/𝜕𝑦∗^2).
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Pressure Across Boundary Layer: Pressure variation across the thin boundary layer is negligible (𝜕𝑝∗/𝜕𝑦∗ + 𝑔𝐿/(𝑈^2)𝜕ℎ∗/𝜕𝑦∗ = O(Δ^2/𝐿^2). The pressure is governed by the potential flow solution.
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Simplified Equations (final boundary layer equations):
- 𝜕𝑢𝑥/𝜕𝑥 + 𝜕𝑢𝑦/𝜕𝑦 = 0
- 𝜕𝑢𝑥/𝜕𝑡 + 𝑢𝑥𝜕𝑢𝑥/𝜕𝑥 + 𝑢𝑦𝜕𝑢𝑥/𝜕𝑦 = − (1/𝜌)𝜕𝑝/𝜕𝑥 − 𝑔 𝜕ℎ/𝜕𝑥 + 𝜈 𝜕^2𝑢𝑥/𝜕𝑦^2
- (1/𝜌)𝜕𝑝/𝜕𝑦 + 𝑔 𝜕ℎ/𝜕𝑦 = 0
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Boundary Conditions: 𝑢𝑥 = 0, 𝑢𝑦 = 0 at 𝑦 = 0; 𝑢𝑥 = ̃𝑢𝑥 at 𝑦 = ∞
Boundary Layer Equations Type
- Parabolic Type: Small changes on the body (e.g., a bump) affect the entire boundary layer downstream. Unsteady changes on the body propagate instantaneously across the boundary layer but are advected to downstream positions.
Limitations of Boundary Layer Theory
- High Reynolds Number: The theory becomes less accurate at higher Reynolds numbers, particularly in the wake of an object and in low Reynolds number flows. Boundary layer approximations may be inaccurate in flows with thick boundary layers (e.g., pipe flow, where the boundary layer occupies the whole pipe).
Flow Separation
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Adverse Pressure Gradient: Separation occurs when the downstream pressure increases. This leads to reverse flow near the boundary.
- Separation Point: The point where shear stress and velocity gradient are zero (𝜏𝑦𝑥 = 0)
- Mixing High Momentum: Increased momentum mixing can reduce the separated region (e.g., dimples on a golf ball).
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Infllection Point: The velocity profile might display an inflection point, which frequently indicates flow instability and possible transition to turbulence.
- Pressure Gradient and Velocity Profile: A negative pressure gradient (pressure decreasing downstream) means 𝜕^2𝑢𝑥/𝜕𝑦^2 is negative, and no inflection point exists. A positive pressure gradient (pressure increasing downstream), on the other hand, can create a velocity profile with an inflection point.
- Sharp Corners: Separation occurs predictably at sharp corners of objects.
- Unsteady and 3D Separation: Flow separation can be unsteady, with oscillations in the separation point, causing vortex shedding in a Kármán vortex street pattern.
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Description
Explore the fascinating realm of Boundary Layer Theory as applied to fluid mechanics. This quiz delves into viscous flows, the significance of the Reynolds number, and Prandtl's theory of separating potential and viscous flow components. Test your understanding of the principles that govern laminar flow near solid boundaries.
