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Boundary Layer Theory

Field: Aerodynamics

Sequence of Expressions

Let u(x,y)=(u,v)\mathbf{u}(x, y) = (u, v) be the velocity vector field in the domain Ω\Omega. On the solid surface Γ\Gamma defined by y=0y=0, the no-slip boundary condition mandates that the fluid velocity matches the wall velocity uwall\mathbf{u}_{wall}. Assuming a stationary wall, this is expressed as:\nu(x,0)=(u(x,0) v(x,0))=(0 0 0) \mathbf{u}(x, 0) = \begin{pmatrix} u(x, 0) \ v(x, 0) \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}
For an incompressible, Newtonian fluid (∇⋅u=0\nabla \cdot \mathbf{u} = 0), the governing equations are:\n\n1. **Continuity Equation:**\n∇⋅u=0 \nabla \cdot \mathbf{u} = 0 \n2. **Momentum Equation:**\nρ(u⋅∇)u=−∇p+μ∇2u+f \rho (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f} \nwhere ρ\rho is the density, μ\mu is the dynamic viscosity, u\mathbf{u} is the velocity vector, pp is the pressure, and f\mathbf{f} represents external body forces.
Consider the flow domain Ω\Omega with coordinates (x,y)(x, y), where xx is the streamwise direction and yy is the normal direction. The boundary layer approximation assumes that the characteristic length scale normal to the surface (δ\delta) is much smaller than the characteristic length scale parallel to the surface (LL), i.e., δ/L≪1\delta/L \ll 1. This allows the simplification of the full Navier-Stokes equations by neglecting terms involving second derivatives with respect to xx (or higher-order derivatives in xx) relative to those involving yy. Specifically, the assumption is that the flow gradients are dominated by the normal direction: ∣∂∂y∣≫∣∂∂x∣\left| \frac{\partial}{\partial y} \right| \gg \left| \frac{\partial}{\partial x} \right|. This leads to the reduction of the governing PDEs to a simplified set valid near the wall.
Assuming steady, incompressible flow (ρ=constant\rho = constant) and neglecting the yy-momentum equation, Prandtl's boundary layer equations are:\n\n1. **Continuity Equation (2D):**\n∂u∂x+∂v∂y=0 \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \n2. **xx-Momentum Equation:**\nρ(u∂u∂x+v∂u∂y)=−∂p∂x+μ∂2u∂y2 \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu \frac{\partial^2 u}{\partial y^2} \nThese equations govern the velocity components (u,v)(u, v) within the boundary layer, subject to the no-slip condition at y=0y=0.
The Reynolds number (ReRe) is a dimensionless ratio quantifying the relative importance of inertial forces to viscous forces. It is defined based on the fluid properties and characteristic scales:\nRe=ρULμ Re = \frac{\rho U L}{\mu} \nwhere ρ\rho is the fluid density (kg/m3\text{kg/m}^3), UU is the characteristic free-stream velocity (m/s\text{m/s}), LL is the characteristic length scale (m\text{m}), and μ\mu is the dynamic viscosity (Pa⋅s\text{Pa} \cdot \text{s}). ReRe is a dimensionless quantity.