4.2 Boundary layer equations

Boundary layer equations are crucial in aerodynamics, describing fluid behavior near solid surfaces. They simplify the Navier-Stokes equations, focusing on the thin region where viscous effects dominate and velocity transitions from zero to freestream.

These equations help predict drag, heat transfer, and flow separation. By understanding boundary layer dynamics, engineers can optimize aerodynamic designs, improve efficiency, and control flow characteristics in various applications.

Boundary layer concept

• The boundary layer is a thin region near a solid surface where viscous effects are significant and the velocity transitions from zero at the surface to the freestream value
• Understanding boundary layers is crucial in aerodynamics as they influence drag, heat transfer, and flow separation
• The boundary layer concept was first introduced by Ludwig Prandtl in 1904 and revolutionized the field of fluid mechanics

Viscous effects near surfaces

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• Near solid surfaces, fluid particles adhere to the surface due to the no-slip condition, resulting in a velocity gradient normal to the surface
• Viscous forces dominate within the boundary layer, while outside the boundary layer, the flow can be treated as inviscid
• The viscous effects in the boundary layer lead to skin friction drag and heat transfer between the fluid and the surface

Boundary layer thickness

• The boundary layer thickness $(\delta)$ is defined as the distance from the surface where the velocity reaches 99% of the freestream velocity
• The boundary layer thickness increases along the surface in the flow direction, starting from zero at the leading edge
• Factors influencing the boundary layer thickness include the Reynolds number, surface roughness, and pressure gradient

Displacement thickness

• The displacement thickness $(\delta^*)$ is the distance by which the external inviscid flow is displaced outwards due to the presence of the boundary layer
• It represents the mass deficit in the boundary layer compared to the inviscid flow
• The displacement thickness is defined as: $\delta^* = \int_0^{\delta} \left(1 - \frac{u}{U_{\infty}}\right) dy$

Momentum thickness

• The momentum thickness $(\theta)$ is a measure of the momentum deficit in the boundary layer compared to the inviscid flow
• It is defined as the distance by which the boundary layer should be displaced to compensate for the reduction in momentum due to viscous effects
• The momentum thickness is given by: $\theta = \int_0^{\delta} \frac{u}{U_{\infty}} \left(1 - \frac{u}{U_{\infty}}\right) dy$

Boundary layer equations

• The governing equations for boundary layer flows are derived from the conservation of mass and momentum principles
• These equations are simplified versions of the Navier-Stokes equations, taking into account the characteristics of boundary layer flows
• Boundary layer equations are used to analyze the velocity and pressure fields within the boundary layer

Continuity equation

• The continuity equation represents the conservation of mass in a fluid flow
• For a two-dimensional incompressible boundary layer flow, the continuity equation is: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$
• The equation states that the rate of change of velocity in the x-direction is balanced by the rate of change of velocity in the y-direction

Navier-Stokes equations

• The Navier-Stokes equations are the fundamental governing equations for viscous fluid flows
• They represent the conservation of momentum and are derived from Newton's second law of motion
• For a two-dimensional incompressible flow, the Navier-Stokes equations in the x and y directions are:
  • x-momentum: $\rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}\right) = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)$
  • y-momentum: $\rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}\right) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right)$

Boundary layer approximations

• In boundary layer flows, the following approximations are made to simplify the Navier-Stokes equations:
  • The flow is assumed to be steady and two-dimensional
  • The boundary layer thickness is much smaller than the characteristic length of the surface
  • The velocity component normal to the surface (v) is much smaller than the velocity component parallel to the surface (u)
  • The pressure gradient across the boundary layer is negligible
• These approximations are valid for high Reynolds number flows and thin boundary layers

Simplified boundary layer equations

• Applying the boundary layer approximations to the Navier-Stokes equations leads to the simplified boundary layer equations:
  • x-momentum: $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{dP}{dx} + \nu \frac{\partial^2 u}{\partial y^2}$
  • Continuity: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$
• These equations, along with appropriate boundary conditions, can be solved to determine the velocity and pressure fields within the boundary layer

Laminar boundary layers

• Laminar boundary layers are characterized by smooth, orderly flow with no mixing between fluid layers
• They occur at low Reynolds numbers and are governed by the laminar boundary layer equations
• Understanding laminar boundary layers is essential for predicting drag and heat transfer in low-speed flows

Blasius solution

• The Blasius solution is an exact solution to the laminar boundary layer equations for flow over a flat plate with zero pressure gradient
• It assumes a similarity variable $(\eta)$ and a stream function $(\psi)$ to transform the partial differential equations into an ordinary differential equation
• The resulting non-linear ODE is solved numerically to obtain the velocity profile and boundary layer characteristics (thickness, displacement thickness, and momentum thickness)

Falkner-Skan solution

• The Falkner-Skan solution generalizes the Blasius solution to include the effect of a non-zero pressure gradient on the laminar boundary layer
• It introduces a pressure gradient parameter $(\beta)$ that can be positive (favorable pressure gradient), negative (adverse pressure gradient), or zero (zero pressure gradient, Blasius solution)
• The Falkner-Skan equation is solved numerically to obtain the velocity profiles and boundary layer characteristics for different pressure gradient conditions

Thermal boundary layer

• The thermal boundary layer develops when there is heat transfer between the fluid and the surface
• It is the region where the temperature gradient is significant, and the fluid temperature transitions from the surface temperature to the freestream temperature
• The thermal boundary layer thickness $(\delta_T)$ is defined as the distance from the surface where the temperature reaches 99% of the freestream temperature

Heat transfer in laminar flow

• In laminar boundary layers, heat transfer occurs through conduction and convection
• The local heat transfer coefficient $(h_x)$ and the Nusselt number $(Nu_x)$ are used to characterize the heat transfer performance
• For flow over a flat plate with constant surface temperature, the local Nusselt number is given by: $Nu_x = \frac{h_x x}{k} = 0.332 Re_x^{1/2} Pr^{1/3}$
• The average Nusselt number over the entire plate can be obtained by integrating the local Nusselt number

Turbulent boundary layers

• Turbulent boundary layers are characterized by chaotic, fluctuating flow with intense mixing between fluid layers
• They occur at high Reynolds numbers and are more common in practical engineering applications than laminar boundary layers
• Turbulent boundary layers have higher skin friction drag and heat transfer rates compared to laminar boundary layers

Transition from laminar to turbulent

• The transition from a laminar to a turbulent boundary layer occurs when the Reynolds number based on the distance from the leading edge exceeds a critical value (typically around $5 \times 10^5$)
• The transition process is influenced by factors such as surface roughness, pressure gradient, and freestream turbulence
• During the transition, the flow becomes unstable, and small disturbances grow, leading to the formation of turbulent spots that eventually merge into a fully turbulent boundary layer

Turbulent velocity profile

• The turbulent velocity profile consists of three distinct regions: the viscous sublayer, the buffer layer, and the outer layer
• In the viscous sublayer, the velocity profile is nearly linear, and viscous effects dominate
• The buffer layer is a transition region where both viscous and turbulent effects are important
• In the outer layer, the velocity profile follows a logarithmic law, and turbulent mixing dominates

Logarithmic law of the wall

• The logarithmic law of the wall describes the velocity profile in the outer layer of a turbulent boundary layer
• It is given by: $\frac{u}{u_*} = \frac{1}{\kappa} \ln \left(\frac{y u_*}{\nu}\right) + B$
• Here, $u_*$ is the friction velocity, $\kappa$ is the von Kármán constant (≈ 0.41), and $B$ is a constant that depends on the surface roughness
• The logarithmic law is valid for $30 < y^+ < 500$, where $y^+$ is the non-dimensional wall distance $(y^+ = \frac{y u_*}{\nu})$

Turbulent heat transfer

• In turbulent boundary layers, heat transfer is enhanced by the turbulent mixing, resulting in higher heat transfer coefficients compared to laminar flow
• The Stanton number $(St)$ is used to characterize the heat transfer performance in turbulent flow
• For flow over a flat plate with constant surface temperature, the local Stanton number is given by: $St_x = \frac{h_x}{\rho c_p U_{\infty}} = 0.0296 Re_x^{-1/5} Pr^{-2/3}$
• The average Stanton number over the entire plate can be obtained by integrating the local Stanton number

Boundary layer separation

• Boundary layer separation occurs when the fluid particles in the boundary layer are unable to overcome the adverse pressure gradient and detach from the surface
• Separation leads to the formation of a recirculating flow region behind the separation point, which can significantly increase drag and affect the overall flow field
• Understanding and predicting boundary layer separation is crucial for the design of aerodynamic surfaces and flow control strategies

Adverse pressure gradient

• An adverse pressure gradient is a situation where the pressure increases in the flow direction $(dP/dx > 0)$
• In an adverse pressure gradient, the fluid particles in the boundary layer experience a force that opposes their motion, causing them to decelerate
• If the adverse pressure gradient is strong enough, the velocity near the surface can become zero or even reverse, leading to boundary layer separation

Separation point

• The separation point is the location on the surface where the boundary layer separates from the surface
• At the separation point, the wall shear stress becomes zero $(\tau_w = 0)$, and the velocity gradient at the wall is zero $(\partial u/\partial y|_{y=0} = 0)$
• The separation point can be determined by solving the boundary layer equations with the appropriate boundary conditions

Separated flow regions

• Downstream of the separation point, a recirculating flow region forms, known as the separated flow region
• In the separated flow region, the flow is highly unsteady and characterized by vortex shedding and turbulent mixing
• The size and shape of the separated flow region depend on factors such as the Reynolds number, the pressure gradient, and the surface geometry

Effects of separation on drag

• Boundary layer separation can significantly increase the pressure drag (form drag) acting on a body
• The separated flow region creates a low-pressure zone behind the body, resulting in a pressure difference between the front and rear surfaces
• The increased pressure drag can lead to reduced aerodynamic efficiency and performance, especially in applications such as airfoils, diffusers, and ground vehicles

Boundary layer control

• Boundary layer control refers to techniques used to manipulate the boundary layer to achieve desired flow characteristics, such as delaying separation, reducing drag, or enhancing heat transfer
• These techniques can be classified into active and passive methods, depending on whether external energy is required
• Effective boundary layer control can significantly improve the performance of aerodynamic surfaces and flow systems

Suction vs blowing

• Suction and blowing are active boundary layer control methods that involve removing or injecting fluid through the surface
• Suction removes the low-momentum fluid near the surface, making the boundary layer thinner and more resistant to separation
• Blowing injects high-momentum fluid near the surface, energizing the boundary layer and delaying separation
• The effectiveness of suction and blowing depends on factors such as the location, intensity, and distribution of the suction/blowing slots

Vortex generators

• Vortex generators are passive devices that create streamwise vortices in the boundary layer
• These vortices promote mixing between the high-momentum fluid in the outer layer and the low-momentum fluid near the surface
• The enhanced mixing energizes the boundary layer, making it more resistant to separation
• Vortex generators are commonly used on aircraft wings, turbine blades, and diffusers to delay separation and improve performance

Riblets and surface roughness

• Riblets are small, streamwise grooves on the surface that can reduce turbulent skin friction drag
• They work by limiting the spanwise motion of the streamwise vortices in the turbulent boundary layer, reducing the turbulent mixing and momentum transfer
• Surface roughness, on the other hand, can be used to promote turbulent transition and enhance heat transfer
• Roughness elements, such as sand grains or dimples, create local disturbances that trigger the transition from laminar to turbulent flow

Laminar flow control techniques

• Laminar flow control techniques aim to maintain a laminar boundary layer over a larger portion of the surface to reduce skin friction drag
• These techniques include shaping the surface geometry (e.g., using a favorable pressure gradient), suction, and active wave cancellation
• Laminar flow control is particularly useful for high-speed applications, such as aircraft wings and turbine blades, where the skin friction drag is a significant component of the total drag

Computational methods

• Computational methods are essential tools for analyzing and predicting boundary layer flows in complex geometries and flow conditions
• These methods solve the boundary layer equations or the full Navier-Stokes equations numerically, providing detailed information about the velocity, pressure, and temperature fields
• The choice of the computational method depends on factors such as the flow regime, the geometry, and the desired accuracy and computational cost

Integral boundary layer methods

• Integral boundary layer methods are based on solving the integral form of the boundary layer equations
• These methods assume a parametric velocity profile (e.g., polynomial or power law) and solve for the boundary layer parameters, such as the displacement thickness and the momentum thickness
• Integral methods are computationally efficient and can provide quick estimates of the boundary layer characteristics, but they are less accurate than differential methods

Finite difference schemes

• Finite difference schemes are used to discretize the boundary layer equations on a structured grid
• The derivatives in the equations are approximated using finite difference formulas, such as central differences or upwind schemes
• The resulting system of algebraic equations is solved iteratively or directly to obtain the velocity and pressure fields
• Finite difference schemes are relatively simple to implement and can handle complex geometries with body-fitted grids

Turbulence modeling for boundary layers

• Turbulence modeling is necessary for simulating turbulent boundary layers, as directly resolving all the turbulent scales is computationally prohibitive
• Reynolds-Averaged Navier-Stokes (RANS) models, such as the k-ε and k-ω models, are commonly used for turbulent boundary layer simulations
• These models solve for the mean flow quantities and model the effect of turbulence using additional transport equations for turbulence kinetic energy and dissipation/specific dissipation rate
• More advanced turbulence models, such as the Reynolds Stress Models (RSM) and Large Eddy Simulation (LES), can provide more accurate results but at a higher computational cost

Direct numerical simulation (DNS)

• Direct Numerical Simulation (DNS) is a computational method that solves the full Navier-Stokes equations without any turbulence modeling
• DNS resolves all the spatial and temporal scales of turbulence, from the largest eddies to the smallest Kolmogorov scales
• Due to the extremely high resolution required, DNS is computationally expensive and limited to low Reynolds number flows and simple geometries
• DNS is mainly used for fundamental research and validation of turbulence models, rather than for practical engineering applications

Experimental techniques

• Experimental techniques are essential for measuring and visualizing boundary layer flows in real-world conditions
• These techniques provide valuable data for validating computational models and gaining insights into the physics of boundary layer flows
• The choice of the experimental technique depends on factors such as the flow regime, the quantity of interest (e.g., velocity, pressure, temperature), and the spatial and temporal resolution required

Hot-wire anemometry

• Hot-wire anemometry is a technique for measuring the instantaneous velocity in a flow
• It consists of a thin wire (typically tungsten or platinum) heated by an electric current and exposed to the flow
• As the flow passes over the wire, it cools the wire, and the change in wire resistance is related to the