Laminar Boundary-Layer Thickness on a Flat Plate: 22-Agric-A3 Heat Engineering, May 2018, Q3

Problem 3: Laminar Boundary-Layer Thickness on a Flat Plate (25 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Free-stream air velocity U∞ = 1.5 m/s at 27°C; distance from the sharp leading edge x = 0.5 m; kinematic viscosity ν = 1.566×10^-5 m²/s (μ = 1.853×10^-5 kg/m·s is extraneous to this part, needed only if wall shear stress were also asked for); Table 6.1 gives the Blasius similarity solution f′(η) = u/U∞ against the similarity variable η=y√(U∞/νx).

Find. The boundary-layer thickness δ at x = 0.5 m, taken (by convention) as the wall-normal distance y at which u/U∞ = 0.99.

Approach. Read the edge of the boundary layer off Table 6.1 as the similarity coordinate η where f′(η)=0.99, then invert the similarity variable's definition to convert that η back to a physical distance y = δ at the given x and U∞.

Blasius similarity profile f′(η)=u/U∞ against η; the boundary-layer edge is conventionally taken at f′(η)=0.99, i.e. η=4.918 from Table 6.1.

Locate the 99%-velocity point in similarity coordinates. Table 6.1 lists f′(η)=0.99000 exactly at $$\eta_{99} = 4.918$$
Convert η back to a physical distance. The Blasius similarity variable is defined as $\eta = y\sqrt{U_\infty/(\nu x)}$, so $$\delta = \eta_{99}\sqrt{\frac{\nu x}{U_\infty}} = 4.918\sqrt{\frac{(1.566\times10^{-5})(0.5)}{1.5}}$$ $$\delta = 4.918\times(2.285\times10^{-3}\ \text{m}) = \boxed{0.01124\ \text{m} = 11.2\ \text{mm}}$$ (For reference, $Re_x = U_\infty x/\nu = 4.79\times10^4$, comfortably laminar, so the Blasius solution applies.)

Quantity	Result
Reynolds number at x = 0.5 m, Re_x	4.79×10⁴ (laminar)
Boundary-layer thickness, δ	11.2 mm (0.01124 m)

Concept & Theory

Topic: The Blasius similarity solution for laminar boundary-layer flow over a flat plate with zero pressure gradient.

Key relations: Similarity variable $\eta = y\sqrt{U_\infty/(\nu x)}$; velocity profile $u/U_\infty = f'(\eta)$ tabulated numerically (no closed form) by solving the Blasius ODE $2f'''+ff''=0$; boundary-layer thickness conventionally taken where $f'(\eta)=0.99$.

Why this works: Prandtl's boundary-layer equations for this flow admit a similarity solution because the velocity profile at every streamwise station x collapses onto a single curve when plotted against the combined variable η — physically, viscous diffusion of momentum away from the wall grows as √(νx/U∞), so scaling y by that same length collapses all stations onto one profile. Since δ is defined by convention (there is no sharp physical edge to the boundary layer), the 99% criterion is simply the point where Table 6.1 is entered and inverted.

Common pitfall: Using the dynamic viscosity μ in place of the kinematic viscosity ν in the similarity variable (they differ by the fluid density, which is not needed here since ν is given directly), or reading η from the y column instead of solving for it from the f′(η) column.

Source: F.P. Incropera & D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Ch. 6 (laminar boundary layers, Blasius solution, Table 6.1).

Question 3 of 4: Laminar Boundary-Layer Thickness on a Flat Plate

Problem 3: Laminar Boundary-Layer Thickness on a Flat Plate (25 points)