Interactive Lecture — Y2 Fluid Mechanics
Viscous Flow Near Surfaces, Transition, and Turbulent Boundary Layers
by Dr Arnaldo Delli Carri
When a real (viscous) fluid flows over a solid surface, the fluid in direct contact with the surface has zero velocity relative to it — this is the no-slip condition. Moving away from the wall, the velocity rises until it matches the undisturbed free-stream value \(U_\infty\). The thin region where this velocity change occurs is the boundary layer.
Prandtl's observation (1904): viscous effects are confined to a thin layer near the surface. Outside this layer the flow can be treated as inviscid. This single observation transformed fluid mechanics, splitting a hard problem into two tractable ones: an inviscid outer flow and a thin viscous inner layer.
The boundary layer thickness \(\delta\) is defined conventionally as the distance from the surface where the local velocity first reaches 99% of the free-stream value:
Boundary Layer Thickness Definition:
\[u\!\left(y = \delta\right) = 0.99\,U_\infty\]
For a smooth flat plate in zero pressure gradient, Blasius (1908) found an exact similarity solution. The boundary layer grows with distance \(x\) from the leading edge as:
Laminar Flat-Plate BL (Blasius):
\[\delta_\text{lam}(x) = \frac{5.0\,x}{\sqrt{Re_x}}, \qquad Re_x = \frac{U_\infty\,x}{\nu}\]
Because \(Re_x \propto x\), the thickness grows as \(\delta \propto \sqrt{x}\) — a parabolic envelope above the plate.
Local skin friction coefficient (laminar):
\[c_{f,\text{lam}} = \frac{0.664}{\sqrt{Re_x}}\]
Wall shear stress:
\[\tau_w = c_f \cdot \tfrac{1}{2}\rho\,U_\infty^2\]
As \(Re_x\) increases along the plate, small disturbances grow rather than decay. The laminar boundary layer eventually breaks down through a process called transition. For a smooth flat plate in low-turbulence flow, transition typically occurs near:
Critical Reynolds Number:
\[Re_{x,cr} \approx 5 \times 10^5\]
This value ranges from \(2\times10^5\) (rough surface, high free-stream turbulence) to \(3\times10^6\) (ultra-smooth, low turbulence). In engineering practice \(5\times10^5\) is the standard default.
A turbulent boundary layer has chaotic velocity fluctuations superimposed on the mean flow. It is characterised by a fuller mean velocity profile (more momentum close to the wall) and consequently higher wall shear stress. The 1/7th power-law approximation is widely used:
Turbulent BL Thickness (1/7th Power Law):
\[\delta_\text{turb}(x) \approx \frac{0.37\,x}{Re_x^{1/5}}\]
Turbulent Velocity Profile:
\[\frac{u}{U_\infty} = \left(\frac{y}{\delta}\right)^{1/7}\]
Local skin friction coefficient (turbulent):
\[c_{f,\text{turb}} = \frac{0.0592}{Re_x^{1/5}}\]
Note that the 1/7th power-law profile has an infinite slope at the wall (\(y=0\)), so the turbulent skin friction formula above is derived from experiments and integral methods rather than direct differentiation.
When the turbulent boundary layer starts growing at \(x = x_{cr}\), it must inherit the thickness the laminar layer had built up. Using the raw turbulent formula from \(x=0\) underestimates this. The standard correction is to find a virtual origin \(x_0\) such that:
\[\delta_\text{turb}(x_{cr} + x_0) = \delta_\text{lam}(x_{cr})\]
Then for \(x > x_{cr}\) the effective turbulent thickness uses \(x_\text{eff} = x + x_0\):
\[\delta_\text{turb}(x) = \frac{0.37\,(x + x_0)}{Re_{x+x_0}^{1/5}}\]
This ensures a smooth, physically correct BL thickness curve through transition.
Two integral measures characterise the boundary layer's effect on the outer flow:
Displacement Thickness — how far streamlines are pushed outward:
\[\delta^* = \int_0^\infty \!\left(1 - \frac{u}{U_\infty}\right) dy\]
Momentum Thickness — proportional to the drag force per unit width:
\[\theta = \int_0^\infty \frac{u}{U_\infty}\!\left(1 - \frac{u}{U_\infty}\right) dy\]
Shape Factor:
\[H = \frac{\delta^*}{\theta} \approx \begin{cases} 2.59 & \text{laminar (Blasius)} \\ 1.3\text{--}1.4 & \text{turbulent} \end{cases}\]
In an adverse pressure gradient (pressure rising in the flow direction), the near-wall fluid decelerates. The wall shear stress \(\tau_w\) falls progressively until it reaches zero — the separation point — beyond which the flow reverses and a recirculation zone forms. The large, low-pressure wake behind a separated body is the dominant source of pressure drag on bluff bodies.
Turbulent boundary layers resist separation far better than laminar ones. Their 1/7th-power velocity profile carries significantly more momentum close to the wall than the laminar quartic, so the flow can push further into an adverse gradient before reversing.
Flow past a circular cylinder is the classic demonstration of how a boundary layer transition can reduce drag. At low Reynolds numbers (\(Re_D \lesssim 10^5\)) the boundary layer is laminar and separates early, at roughly 80° from the front stagnation point, leaving a wide, turbulent wake and a drag coefficient \(C_D \approx 1.2\).
As \(Re_D\) approaches a critical value (typically \(\sim 5 \times 10^5\) for a smooth cylinder), the laminar boundary layer undergoes transition to turbulent before it separates. The turbulent BL, with its fuller profile, remains attached much longer — separation is pushed back to around 120–140° — and the wake narrows dramatically. The result is a sudden drop in \(C_D\) to about 0.3: a fourfold reduction in pressure drag. This abrupt drop is called the drag crisis.
Making the boundary layer more turbulent — and therefore raising skin friction — cuts total drag by eliminating most of the pressure wake. Golf ball dimples, cricket ball seams, and surface roughness on submarines all exploit this effect deliberately. The critical Reynolds number depends strongly on surface roughness — a rough surface trips transition earlier, shifting the drag crisis to lower speeds.
The simulation shows the flat plate from leading to trailing edge. Coloured particles flow from left to right, their speed proportional to the local velocity (blue = slow, warm = fast). Velocity profiles at six stations show the shape of the boundary layer. The δ(x) envelope grows parabolically in the laminar region and more steeply after transition. Switch to the Skin Friction tab to see \(c_f(x)\).
| Concept | Equation | Notes |
|---|---|---|
| Local Reynolds number | \(Re_x = U_\infty x / \nu\) | Inertia / viscous forces |
| Laminar BL thickness | \(\delta = 5.0\,x / \sqrt{Re_x}\) | Blasius, flat plate, \(\delta \propto \sqrt{x}\) |
| Turbulent BL thickness | \(\delta = 0.37\,x / Re_x^{1/5}\) | 1/7th power law, \(\delta \propto x^{4/5}\) |
| Critical Re (transition) | \(Re_{x,cr} \approx 5 \times 10^5\) | Smooth plate, low turbulence |
| Laminar skin friction | \(c_f = 0.664 / \sqrt{Re_x}\) | Decreases along plate |
| Turbulent skin friction | \(c_f = 0.0592 / Re_x^{1/5}\) | Slower decrease; higher than laminar |
| Laminar velocity profile | \(u/U_\infty = 2\eta - 2\eta^3 + \eta^4\) | Pohlhausen quartic, \(\eta = y/\delta\) |
| Turbulent velocity profile | \(u/U_\infty = \eta^{1/7}\) | Fuller near-wall shape |
| Displacement thickness | \(\delta^* = \int_0^\infty (1 - u/U_\infty)\,dy\) | Streamline displacement |
| Momentum thickness | \(\theta = \int_0^\infty (u/U_\infty)(1-u/U_\infty)\,dy\) | Related to drag |
| Shape factor | \(H = \delta^*/\theta\) | Laminar ≈ 2.59; turbulent ≈ 1.3–1.4 |
| Wall shear stress | \(\tau_w = c_f \cdot \tfrac{1}{2}\rho U_\infty^2\) | Viscous drag per unit area |
| Mixed drag coefficient | \(C_D = 0.074/Re_L^{1/5} - 1742/Re_L\) | Both sides; transition at \(Re_{cr}=5\times10^5\) |
Air at 20°C (\(\nu = 1.5 \times 10^{-5}\) m²/s, \(\rho = 1.204\) kg/m³) flows over a flat plate at \(U_\infty = 5\) m/s. At \(x = 0.5\) m from the leading edge, calculate:
Show Solution
Given: \(\nu = 1.5\times10^{-5}\) m²/s, \(\rho = 1.204\) kg/m³, \(U_\infty = 5\) m/s, \(x = 0.5\) m.
(a) Local Reynolds number:
\[Re_x = \frac{U_\infty\,x}{\nu} = \frac{5 \times 0.5}{1.5\times10^{-5}} = \frac{2.5}{1.5\times10^{-5}} = 1.667\times10^5\]
Since \(Re_x = 1.667\times10^5 < 5\times10^5\), the boundary layer is laminar.
Answer: \(Re_x = 1.667\times10^5\) (laminar).
(b) Boundary layer thickness:
\[\delta = \frac{5.0\,x}{\sqrt{Re_x}} = \frac{5.0 \times 0.5}{\sqrt{1.667\times10^5}} = \frac{2.5}{408.3} = 6.12\times10^{-3}\ \text{m}\]
Answer: \(\delta = 6.12\) mm.
(c) Skin friction coefficient and wall shear stress:
\[c_f = \frac{0.664}{\sqrt{Re_x}} = \frac{0.664}{408.3} = 1.626\times10^{-3}\]
\[\tau_w = c_f\cdot\tfrac{1}{2}\rho U_\infty^2 = 1.626\times10^{-3}\times\tfrac{1}{2}\times1.204\times5^2\]
\[\tau_w = 1.626\times10^{-3}\times15.05 = 0.0245\ \text{Pa}\]
Answer: \(c_f = 1.63\times10^{-3}\), \(\tau_w = 0.0245\) Pa.
Water at 20°C (\(\nu = 1.004\times10^{-6}\) m²/s) flows over a smooth flat plate at \(U_\infty = 2\) m/s. Use \(Re_{x,cr} = 5\times10^5\). Calculate:
Show Solution
Given: \(\nu = 1.004\times10^{-6}\) m²/s, \(U_\infty = 2\) m/s, \(Re_{x,cr} = 5\times10^5\).
(a) Transition location:
\[x_{cr} = \frac{Re_{x,cr}\cdot\nu}{U_\infty} = \frac{5\times10^5\times1.004\times10^{-6}}{2} = \frac{0.502}{2} = 0.251\ \text{m}\]
Answer: \(x_{cr} = 0.251\) m.
(b) Laminar BL thickness at transition:
\[\delta_{lam} = \frac{5.0\times x_{cr}}{\sqrt{Re_{x,cr}}} = \frac{5.0\times0.251}{\sqrt{5\times10^5}} = \frac{1.255}{707.1} = 1.775\times10^{-3}\ \text{m}\]
Answer: \(\delta_{lam}(x_{cr}) = 1.78\) mm.
(c) Virtual-origin correction:
Find \(x_0\) such that the turbulent formula matches the laminar thickness at \(x_{cr}\):
\[\frac{0.37\,(x_{cr}+x_0)}{Re_{x_{cr}+x_0}^{1/5}} = \delta_{lam}\]
Substituting \(Re_{x_{cr}+x_0} = U_\infty(x_{cr}+x_0)/\nu\) and rearranging:
\[0.37\,(x_{cr}+x_0)\cdot\left(\frac{\nu}{U_\infty(x_{cr}+x_0)}\right)^{1/5} = 0.001775\]
\[0.37\left(\frac{\nu}{U_\infty}\right)^{1/5}(x_{cr}+x_0)^{4/5} = 0.001775\]
Let \(A = 0.37\left(\frac{1.004\times10^{-6}}{2}\right)^{1/5} = 0.37\times(5.02\times10^{-7})^{0.2}\)
\[(5.02\times10^{-7})^{0.2} = e^{0.2\ln(5.02\times10^{-7})} = e^{0.2\times(-14.504)} = e^{-2.901} = 0.05498\]
\[A = 0.37\times0.05498 = 0.02034\]
\[(x_{cr}+x_0)^{4/5} = \frac{0.001775}{0.02034} = 0.08726\]
\[x_{cr}+x_0 = 0.08726^{5/4} = 0.08726^{1.25}\]
\[= e^{1.25\ln(0.08726)} = e^{1.25\times(-2.439)} = e^{-3.048} = 0.04739\ \text{m}\]
\[x_0 = 0.04739 - 0.251 = -0.2036\ \text{m}\]
(Negative \(x_0\) is typical — the virtual origin is upstream of the leading edge.)
Turbulent thickness at \(x = x_{cr} + 0.5 = 0.751\) m, using \(x_{eff} = 0.751 + x_0 = 0.751 - 0.2036 = 0.547\) m:
\[Re_{x_{eff}} = \frac{2\times0.547}{1.004\times10^{-6}} = 1.090\times10^6\]
\[\delta_{turb} = \frac{0.37\times0.547}{(1.090\times10^6)^{0.2}}\]
\[(1.090\times10^6)^{0.2} = e^{0.2\ln(1.090\times10^6)} = e^{0.2\times13.90} = e^{2.780} = 16.12\]
\[\delta_{turb} = \frac{0.2024}{16.12} = 1.255\times10^{-2}\ \text{m}\]
Answer: \(\delta_{turb} = 12.6\) mm at \(x = 0.751\) m.
A thin flat plate, 3 m long and 1.2 m wide, is aligned with an air stream at \(U_\infty = 12\) m/s. Air: \(\rho = 1.204\) kg/m³, \(\nu = 1.5\times10^{-5}\) m²/s. Use \(Re_{x,cr} = 5\times10^5\). Calculate:
Show Solution
Given: \(L = 3\) m, \(W = 1.2\) m, \(U_\infty = 12\) m/s, \(\rho = 1.204\) kg/m³, \(\nu = 1.5\times10^{-5}\) m²/s.
(a) Reynolds number at trailing edge:
\[Re_L = \frac{U_\infty L}{\nu} = \frac{12\times3}{1.5\times10^{-5}} = \frac{36}{1.5\times10^{-5}} = 2.4\times10^6\]
Since \(Re_L > 5\times10^5\), the flow is mixed laminar–turbulent. Transition at:
\[x_{cr} = \frac{5\times10^5\times1.5\times10^{-5}}{12} = \frac{7.5}{12} = 0.625\ \text{m}\ (20.8\%\ \text{of plate})\]
Answer: \(Re_L = 2.4\times10^6\), \(x_{cr} = 0.625\) m (20.8% of plate).
(b) Mixed drag coefficient:
\[Re_L^{1/5} = (2.4\times10^6)^{0.2}\]
\[= e^{0.2\ln(2.4\times10^6)} = e^{0.2\times14.691} = e^{2.938} = 18.88\]
\[C_D = \frac{0.074}{18.88} - \frac{1742}{2.4\times10^6} = 3.921\times10^{-3} - 7.258\times10^{-4}\]
Answer: \(C_D = 3.195\times10^{-3}\).
(c) Total drag force (one side):
\[A = L\times W = 3\times1.2 = 3.6\ \text{m}^2\]
\[F_D = C_D\cdot\tfrac{1}{2}\rho U_\infty^2\cdot A = 3.195\times10^{-3}\times\tfrac{1}{2}\times1.204\times12^2\times3.6\]
\[= 3.195\times10^{-3}\times86.69\times3.6 = 3.195\times10^{-3}\times312.1\]
Answer: \(F_D = 0.997\) N \(\approx 1.00\) N.