Boundary Layer, Drag and Dimensional Analysis
Boundary layer thickness, laminar and turbulent boundary layers, separation, drag and lift, Buckingham pi theorem, important dimensionless numbers and model similarity.
When fluid flows past a surface, friction acts only in a thin layer near it: the boundary layer. Dimensional analysis then lets small models predict the behaviour of full-size machines and ships.
Boundary layer on a flat plate
- At the plate, the fluid velocity is zero (no-slip). The boundary layer thickness δ is where u reaches 99% of the free stream U.
- Local Reynolds number Re_x = Ux/ν. The layer turns turbulent at about Re_x ≈ 5 × 10⁵.
- Laminar (Blasius): δ/x = 5/√Re_x. So δ ∝ √x.
- Turbulent: δ/x ≈ 0.37/Re_x^(1/5).
- Displacement thickness δ* = ∫(1 − u/U) dy; momentum thickness θ = ∫(u/U)(1 − u/U) dy. Shape factor H = δ*/θ (2.59 for Blasius).
Separation
In an adverse pressure gradient (pressure rising along the flow, dp/dx > 0), the slow fluid near the wall can stop and reverse; the flow separates at the point where (∂u/∂y) at the wall = 0. Separation causes a large wake and pressure drag. Streamlining delays it.
Drag and lift
- Drag F_D = C_D (½ρU²) A; lift F_L = C_L (½ρU²) A.
- Drag = skin friction + pressure (form) drag.
- Stokes' law for a sphere at very low Re: F_D = 3πμVd, giving C_D = 24/Re.
Dimensional analysis
Buckingham π theorem: a relation among n variables with m fundamental dimensions (M, L, T) can be written with n − m dimensionless groups.
| Number | Ratio | Formula | Important for |
|---|---|---|---|
| Reynolds | inertia/viscous | ρVL/μ | Pipe flow, drag |
| Froude | inertia/gravity | V/√(gL) | Ships, open channels, spillways |
| Euler | pressure/inertia | Δp/(ρV²) | Pressure-driven flow |
| Weber | inertia/surface tension | ρV²L/σ | Droplets, capillary waves |
| Mach | inertia/elastic | V/c | Compressible flow |
Model similarity
A model is dynamically similar to the prototype when the relevant dimensionless numbers match:
- Reynolds similarity (pipes, submarines): V_m L_m/ν_m = V_p L_p/ν_p.
- Froude similarity (ships, spillways): V_m/√L_m = V_p/√L_p, so velocity scales as √(length scale) and discharge as L^2.5.
Practice
A relation involves 7 variables with 3 fundamental dimensions. How many dimensionless π groups are there?
Show answer and explanation
Answer: 4Buckingham π: n − m = 7 − 3 = 4.
For a laminar boundary layer, at Re_x = 10⁴ and x = 0.5 m, what is δ in mm?
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Answer: 25δ = 5x/√Re_x = 5 × 0.5/100 = 0.025 m = 25 mm.
A spillway model is built at 1 : 25 scale under Froude similarity. The prototype velocity is 10 m/s. What is the model velocity in m/s?
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Answer: 2V_m = V_p/√25 = 10/5 = 2 m/s.
Flow separation from a surface occurs when:
Show answer and explanation
Answer: AAn adverse pressure gradient slows the near-wall fluid until (∂u/∂y) at the wall = 0, where it separates.