5.4 Blasius solution

The Blasius solution is a cornerstone of boundary layer theory in fluid dynamics. It describes laminar flow over a flat plate, providing insights into velocity profiles, boundary layer thickness, and drag forces. This solution forms the foundation for understanding more complex flow scenarios.

The Blasius approach uses similarity variables to simplify the boundary layer equations into a single ordinary differential equation. By solving this equation numerically, we can determine key flow characteristics and their dependence on factors like Reynolds number and distance from the plate's leading edge.

Blasius boundary layer

• Fundamental concept in fluid dynamics describing the behavior of fluid flow near a solid surface
• Provides insights into the development of boundary layers and the associated flow characteristics
• Essential for understanding drag force, heat transfer, and flow separation in various engineering applications

Laminar flow over flat plate

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• Considers a steady, incompressible, and laminar flow over a flat plate with zero pressure gradient
• Flow is initially uniform and parallel to the plate surface
• Viscous effects are confined to a thin region near the plate called the boundary layer
• Boundary layer thickness increases with distance from the leading edge of the plate

Prandtl boundary layer equations

• Simplified form of the Navier-Stokes equations for flow within the boundary layer
• Assumes that the boundary layer thickness is much smaller than the plate length
• Neglects the streamwise diffusion and pressure gradient terms
• Consists of the continuity equation and the streamwise momentum equation

Similarity solution approach

• Seeks a solution to the boundary layer equations that is independent of the streamwise coordinate
• Introduces a similarity variable that combines the streamwise and normal coordinates
• Reduces the partial differential equations to an ordinary differential equation (ODE)
• Enables the determination of the velocity profile and other flow properties

Blasius similarity variable

• Defined as $\eta = y \sqrt{\frac{U_\infty}{\nu x}}$, where $y$ is the normal coordinate, $U_\infty$ is the freestream velocity, $\nu$ is the kinematic viscosity, and $x$ is the streamwise coordinate
• Represents a dimensionless distance from the plate surface
• Captures the growth of the boundary layer with increasing $x$

Blasius differential equation

• Obtained by transforming the boundary layer equations using the similarity variable
• Third-order nonlinear ODE: $f''' + \frac{1}{2}ff'' = 0$
• Boundary conditions: $f(0) = 0$, $f'(0) = 0$, and $f'(\infty) = 1$
• Describes the self-similar velocity profile within the boundary layer

Numerical solution methods

• Blasius equation cannot be solved analytically due to its nonlinearity
• Numerical methods, such as the shooting method or finite difference methods, are employed
• Shooting method involves guessing the missing initial condition $f''(0)$ and iteratively solving the ODE
• Finite difference methods discretize the domain and solve the resulting system of algebraic equations

Blasius velocity profile

• Obtained by solving the Blasius equation numerically
• Represents the dimensionless streamwise velocity $u/U_\infty$ as a function of the similarity variable $\eta$
• Exhibits a smooth transition from zero velocity at the plate surface to the freestream velocity far from the plate
• Provides insights into the shape and development of the boundary layer

Boundary layer thickness

• Defined as the distance from the plate surface where the velocity reaches 99% of the freestream velocity
• Increases with the square root of the streamwise coordinate: $\delta \propto \sqrt{x}$
• Represents the extent of the viscous effects near the plate surface
• Depends on the Reynolds number, which characterizes the ratio of inertial to viscous forces

Displacement thickness

• Quantifies the distance by which the external flow is displaced due to the presence of the boundary layer
• Defined as $\delta^* = \int_0^\infty (1 - \frac{u}{U_\infty}) dy$
• Represents the mass deficit in the boundary layer compared to the inviscid flow
• Plays a crucial role in calculating the effective shape of the body in the presence of a boundary layer

Momentum thickness

• Measures the loss of momentum in the boundary layer compared to the inviscid flow
• Defined as $\theta = \int_0^\infty \frac{u}{U_\infty}(1 - \frac{u}{U_\infty}) dy$
• Relates to the drag force experienced by the flat plate
• Used in the calculation of the skin friction coefficient

Wall shear stress

• Represents the viscous stress exerted by the fluid on the plate surface
• Defined as $\tau_w = \mu (\frac{\partial u}{\partial y})_{y=0}$, where $\mu$ is the dynamic viscosity
• Determined by the slope of the velocity profile at the plate surface
• Contributes to the drag force acting on the plate

Skin friction coefficient

• Dimensionless parameter that characterizes the frictional drag on the plate surface
• Defined as $C_f = \frac{\tau_w}{\frac{1}{2}\rho U_\infty^2}$, where $\rho$ is the fluid density
• Depends on the Reynolds number and decreases with increasing distance from the leading edge
• Blasius solution provides an analytical expression for the skin friction coefficient

Drag force on flat plate

• Resultant force acting on the plate due to the combined effects of pressure and shear stress
• For a laminar boundary layer, the drag force is primarily due to skin friction
• Calculated by integrating the wall shear stress over the plate surface
• Depends on the plate length, fluid properties, and freestream velocity

Blasius solution assumptions

• Steady, incompressible, and laminar flow
• Flat plate with zero pressure gradient
• Negligible streamwise diffusion and pressure gradient terms in the boundary layer equations
• No-slip condition at the plate surface
• Freestream velocity remains constant

Validity of Blasius solution

• Applicable to laminar boundary layers with zero pressure gradient
• Accurate for moderate Reynolds numbers (typically up to $Re_x \approx 5 \times 10^5$)
• Breaks down when the flow transitions to turbulence or in the presence of adverse pressure gradients
• Provides a good approximation for the initial development of the boundary layer

Transition to turbulence

• Laminar boundary layer becomes unstable and transitions to turbulence at high Reynolds numbers
• Transition occurs when the critical Reynolds number is exceeded (typically around $Re_x \approx 5 \times 10^5$)
• Influenced by factors such as surface roughness, freestream turbulence, and pressure gradients
• Characterized by the appearance of turbulent fluctuations and increased mixing within the boundary layer

Turbulent boundary layers

• Exhibit irregular and chaotic flow behavior with enhanced mixing and momentum transfer
• Velocity profile is fuller and has a higher velocity gradient near the wall compared to laminar boundary layers
• Characterized by increased skin friction and heat transfer rates
• Require different mathematical models and empirical correlations to describe their behavior

Blasius solution vs turbulent flow

• Blasius solution is valid for laminar boundary layers, while turbulent boundary layers require different treatment
• Laminar boundary layers have a parabolic velocity profile, while turbulent boundary layers have a logarithmic profile
• Turbulent boundary layers exhibit higher skin friction and heat transfer rates compared to laminar boundary layers
• Transition from laminar to turbulent flow occurs at a critical Reynolds number and affects the overall flow behavior