Fluid Mechanics

Boundary Layer Theory

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Contents:

  1. Concept of the Boundary Layer
  2. Boundary Layer Thicknesses
  3. Boundary Layer Integral Equations
  4. Boundary Layer Differential Equations
  5. Law of the Wall

Concept of the Boundary Layer

When fluid flows over a solid object a thin viscous layer is established. This thin layer grows in size the longer the fluid has been flowing over the solid. This thin layer is termed the boundary layer. Within the boundary layer the flow is dominated by viscous forces. Above the boundary layer (free-stream) the flow is dominated by inertial forces and velocity is uniform.

Boundary Layer Thicknesses

We may define several length scales relevant to the boundary layer.

Boundary Layer Thickness

This parameter is the easiest to understand. The boundary layer thickness is defined as the distance from the wall where velocity stops changing. To simplify analysis we say that the boundary layer thickness is the distance away from the wall where:

u=0.99U∞u=0.99U_\infty

Displacement Thickness

The displacement thickness, δ∗\delta^*, is the distance perpendicular to the wall that a free-stream pathline must be displaced to uphold the conservation of mass.

δ∗=∫0δ(1−uU)dy\displaystyle\delta^*=\int_0^\delta \left(1-\dfrac{u}{U}\right)dy

Momentum Thickness

The momentum thickness, θ\theta, is the distance perpendicular to the wall that momentum must be displaced to uphold the conservation of momentum due to the momentum loss within the boundary layer.

θ=∫0δuU(1−uU)dy\displaystyle\theta=\int_0^\delta \dfrac{u}{U}\left(1-\dfrac{u}{U}\right)dy

Boundary Layer Integral Equations

The boundary layer integral equations can be determined by taking the boundary layer as a control volume and applying a mass and momentum balance. Through this process the following equation is achieved:

−dpdxδ−τw=ddx∫0δρu2dy−U∞ddx∫0δρudy\displaystyle-\dfrac{dp}{dx}\delta-\tau_w=\dfrac{d}{dx}\int_0^\delta \rho u^2dy-U_\infty\dfrac{d}{dx}\int_0^\delta\rho udy

Recognizing that the pressure gradient can be expressed as −dpdx=ρU∞dU∞dx-\frac{dp}{dx}=\rho U_\infty\frac{dU_\infty}{dx} and rearranging terms we arrive at the boundary layer integral equation:

τwρ=d(U∞2θ)dx+U∞δ∗dU∞dx\dfrac{\tau_w}{\rho}=\dfrac{d(U_\infty^2\theta)}{dx}+U_\infty\delta^*\dfrac{dU_\infty}{dx}

Using the definitions of shape factor, H=δ∗θH=\frac{\delta^*}{\theta}, and skin friction coefficient, Cf=2τwρU∞2C_f=\frac{2\tau_w}{\rho U_\infty^2}, we obtain:

Cf2=dθdx+(H+2)θU∞dU∞dx\dfrac{C_f}{2}=\dfrac{d\theta}{dx}+(H+2)\dfrac{\theta}{U_\infty}\dfrac{dU_\infty}{dx}

Boundary Layer Differential Equations

By evaluating the two-dimensional continuity and Navier-Stokes equation and recognizing that the following velocity and velocity gradients are dominant in a laminar boundary layer we can arrive at the differential boundary layer equations.

u>>vu>>v

∂u∂y>>∂u∂x\dfrac{\partial u}{\partial y}>>\dfrac{\partial u}{\partial x}

∂v∂y>>∂v∂x\dfrac{\partial v}{\partial y}>>\dfrac{\partial v}{\partial x}

The laminar boundary layer differential equations are thus:

∂u∂x+∂v∂y=0\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}=0

u∂u∂x+v∂u∂y=U∞∂U∞∂x+ν∂2u∂y2u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}=U_\infty\dfrac{\partial U_\infty}{\partial x}+\nu\dfrac{\partial^2 u}{\partial y^2}

∂p∂y=0\dfrac{\partial p}{\partial y}=0

Law of the Wall

By non-dimensionalizing the boundary layer differential equations we can attempt to find a semi-analytical solution for describing a zero pressure gradient (ZPG) boundary layer. We first introduce a non-dimensional length scale and a non-dimensional velocity scale:

uτ=τwρu_\tau=\sqrt{\dfrac{\tau_w}{\rho}}

y+=uτyνy^+=\dfrac{u_\tau y}{\nu}

u+=uuτu^+=\dfrac{u}{u_\tau}

It is found that the following relationship exists between u+u^+ and y+y^+:

u+=y+u^+=y^+

u+=1κln⁡y++5.0u^+=\dfrac{1}{\kappa}\ln y^++5.0

for y+<10.8\text{for }y^+< 10.8

for y+>10.8\text{for }y^+> 10.8