Boundary Layer Theory

Boundary layer integral thicknesses

The velocity approaches the free-stream value U_\infty only asymptotically at the edge of a boundary layer, so the thickness \delta is defined by a conventional cut-off, such as u = 0.99\,U_\infty. Because the velocity deficit inside the layer reduces the mass flow and momentum flux carried by the fluid, two integral thicknesses are used to describe these losses. The displacement thickness is \delta^* = \int_0^\infty \left(1 - \dfrac{u}{U_\infty}\right)dy, the distance by which the outer streamlines are pushed away from the wall to conserve mass flow. The momentum thickness is \theta = \int_0^\infty \dfrac{u}{U_\infty}\left(1 - \dfrac{u}{U_\infty}\right)dy, which expresses the streamwise momentum deficit as an equivalent thickness of free-stream flow. The integrand of \theta is the product of the two factors: u/U_\infty alone is not a momentum deficit, and 1 - u/U_\infty alone gives \delta^*.

For a laminar flat-plate layer the three thicknesses are ordered as \delta > \delta^* > \theta. The Blasius similarity solution gives \delta \approx 5.0\,x/\sqrt{Re_x}, \delta^* \approx 1.72\,x/\sqrt{Re_x} and \theta \approx 0.664\,x/\sqrt{Re_x}, so the shape factor H = \delta^*/\theta is about 2.6.

The ratio of these thicknesses depends on the assumed profile. For a linear profile u/U_\infty = y/\delta, the momentum thickness becomes \theta = \delta\int_0^1 \eta(1-\eta)\,d\eta = \delta/6, so \theta/\delta = 1/6.

When reasoning about how quantities vary across the layer, note that along a normal line from the wall to the edge u rises from zero to U_\infty, v goes from zero at the wall to a nonzero value at the edge, and \partial u/\partial x is nonzero inside the layer while vanishing at the wall and at the edge. The pressure, however, stays constant across the layer.

Laminar flat-plate boundary layer growth

For steady, laminar, incompressible flow over a flat plate, the boundary layer thickness grows as the square root of the distance from the leading edge, \delta \propto x^{1/2}. Equivalently \delta \propto \sqrt{\nu x/U_\infty}, so \delta/x \propto Re_x^{-1/2} with Re_x = U_\infty x/\nu.

Because the kinematic viscosity appears only under the square root, the thickness at a fixed station and fixed free-stream speed scales as \sqrt{\nu}. Increasing \nu by a factor of four therefore doubles \delta.

Approximate profiles fix the numerical coefficient. The quadratic profile u/U_\infty = \dfrac{3y}{2\delta} - \dfrac{y^2}{2\delta^2} satisfies the wall and edge velocity conditions and gives \delta \approx 4.64\,x/\sqrt{Re_x}. A linear profile, combined with the momentum integral relation and the wall shear \tau_w = \mu U_\infty/\delta, gives \delta \approx 3.5\,x/\sqrt{Re_x}.

The \sqrt{x} law applies only to laminar flow. A turbulent layer grows faster, with \delta \propto x^{4/5}, so the laminar exponent should not be used for turbulent flow.