Boundary Layers

Concept

Boundary Layers and Flow Separation — Prandtl Theory and Viscous–Inviscid Interaction

Scope: derivation of the boundary-layer equations from first principles, analysis of laminar and turbulent regimes, integral methods, transition, and separation. Includes heat transfer analogies and physical interpretation of viscous–inviscid coupling.

1. Origin of the Boundary-Layer Concept

The boundary layer arises from the disparity between viscous and inertial forces in high-Reynolds-number flows.

At large Re=ρUL/μRe = \rho UL/\mu, viscous effects are confined to a thin region near solid surfaces, while the outer flow remains nearly inviscid.

Prandtl (1904) formulated the idea that viscous stresses matter only in a thin layer adjacent to the wall, simplifying the Navier–Stokes equations through scaling.

2. Scaling and Order-of-Magnitude Analysis

Consider steady, 2D incompressible flow over a flat plate: x: streamwise,y: normal to surface.x \text{: streamwise}, \quad y \text{: normal to surface}.

Let characteristic scales: U,  L,  δ≪L.U, \; L, \; \delta \ll L. Continuity: ∂u/∂x+∂v/∂y=0.\partial u/\partial x + \partial v/\partial y = 0.

Momentum in x: ρ(u∂u/∂x+v∂u/∂y)=−∂p/∂x+μ∂2u/∂y2.\rho(u \partial u/\partial x + v \partial u/\partial y) = -\partial p/\partial x + \mu \partial^2u/\partial y^2.

Compare terms: U2/LU2/L∼1,μU/δ2ρU2/L=LReδ2.\frac{U^2/L}{U^2/L} \sim 1, \quad \frac{\mu U/\delta^2}{\rho U^2/L} = \frac{L}{Re \delta^2}. To balance inertial and viscous effects: δ/L∼Re−1/2.\delta/L \sim Re^{-1/2}.

Thus the boundary-layer thickness grows with downstream distance as: δ(x)∼(x/Rex1/2).\delta(x) \sim (x/Re_x^{1/2}).

3. Prandtl’s Boundary-Layer Equations

Simplifying Navier–Stokes using δ/L≪1\delta/L \ll 1:

Continuity

∂u/∂x+∂v/∂y=0.\partial u/\partial x + \partial v/\partial y = 0.

Streamwise Momentum

u∂u/∂x+v∂u/∂y=−1ρ∂p/∂x+ν∂2u/∂y2.u \partial u/\partial x + v \partial u/\partial y = -\frac{1}{\rho}\partial p/\partial x + \nu \partial^2u/\partial y^2.

Normal Momentum

∂p/∂y=0.\partial p/\partial y = 0. Hence, pressure is constant across the boundary layer and equals the external inviscid pressure pe(x)p_e(x).

Boundary conditions: u=v=0 at y=0,u→Ue(x) as y→∞.u=v=0 \text{ at } y=0, \quad u \to U_e(x) \text{ as } y \to \infty.

4. Similarity and the Blasius Solution (Flat Plate, Zero Pressure Gradient)

For pe=constp_e = \text{const}, the governing equations reduce to: u∂u/∂x+v∂u/∂y=ν∂2u/∂y2.u \partial u/\partial x + v \partial u/\partial y = \nu \partial^2u/\partial y^2. Introduce similarity variable: η=yU2νx,ψ=2νUxf(η).\eta = y \sqrt{\frac{U}{2\nu x}}, \quad \psi = \sqrt{2\nu Ux} f(\eta).

Then: f′′′+ff′′=0,f(0)=f′(0)=0,  f′(∞)=1.f''' + f f'' = 0, \quad f(0)=f'(0)=0, \; f'(\infty)=1.

Numerical solution (Blasius, 1908): f′′(0)=0.332.f''(0) = 0.332.

Wall shear stress: τw=μ(∂u/∂y)y=0=0.332ρU2/Rex.\tau_w = \mu (\partial u/\partial y)_{y=0} = 0.332 \rho U^2 / \sqrt{Re_x}.

Boundary-layer thickness: δ(x)≈5.0xRex.\delta(x) \approx 5.0 \frac{x}{\sqrt{Re_x}}.

5. Displacement and Momentum Thickness

Define: δ∗=∫0∞(1−u/Ue)dy,θ=∫0∞(u/Ue)(1−u/Ue)dy.\delta^* = \int_0^\infty (1 - u/U_e) dy, \quad \theta = \int_0^\infty (u/U_e)(1 - u/U_e) dy.

The shape factor: H=δ∗/θ.H = \delta^*/\theta. For laminar Blasius flow: H=2.59.H = 2.59.

6. Integral Boundary-Layer Method (von Kármán Equation)

Integrate momentum equation across boundary layer: dθdx+(2+H)θUedUedx=Cf2,\frac{d\theta}{dx} + (2+H)\frac{\theta}{U_e} \frac{dU_e}{dx} = \frac{C_f}{2}, where Cf=2τw/(ρUe2)C_f = 2\tau_w/(\rho U_e^2) is the skin friction coefficient.

This integral approach allows approximate solutions for nonzero pressure gradients (Falkner–Skan family).

7. Adverse Pressure Gradient and Flow Separation

When dpe/dx>0dp_e/dx > 0, the outer flow decelerates, inducing a reversed pressure gradient. Within the boundary layer, velocity near the wall decreases and may reverse.

Separation condition: (∂u/∂y)y=0=0.(\partial u/\partial y)_{y=0} = 0.

In the separated region, shear stress changes sign and vorticity is shed into the wake.

8. Turbulent Boundary Layers

At Rex>5×105Re_x > 5\times10^5, laminar flow transitions to turbulent.

8.1 Reynolds Decomposition

u=uˉ+u′,v=vˉ+v′.u = \bar{u} + u', \quad v = \bar{v} + v'.

Averaging Navier–Stokes yields Reynolds-Averaged Navier–Stokes (RANS) equations: ρ(uˉ∂uˉ/∂x+vˉ∂uˉ/∂y)=−∂pˉ/∂x+μ∂2uˉ/∂y2−ρ∂(u′v′‾)/∂y.\rho(\bar{u} \partial\bar{u}/\partial x + \bar{v} \partial\bar{u}/\partial y) = -\partial\bar{p}/\partial x + \mu \partial^2\bar{u}/\partial y^2 - \rho\partial(\overline{u'v'})/\partial y.

The Reynolds stress term −ρu′v′‾- \rho\overline{u'v'} acts as an additional shear stress.

8.2 Eddy Viscosity Concept

Assume turbulent stress proportional to mean velocity gradient: −ρu′v′‾=μt∂uˉ/∂y,-\rho\overline{u'v'} = \mu_t \partial\bar{u}/\partial y, where μt\mu_t is eddy viscosity (not a property of the fluid, but of the flow field). Empirical models (Prandtl mixing-length): μt=ρ(lm)2∣∂uˉ/∂y∣,lm=κy.\mu_t = \rho(l_m)^2 |\partial\bar{u}/\partial y|, \quad l_m = \kappa y.

8.3 Log-Law of the Wall

Integrating momentum balance gives: u+y+=1κln⁡y++B,\frac{u^+}{y^+} = \frac{1}{\kappa} \ln y^+ + B, where: u+=u/uτ,y+=yuτ/ν,uτ=τw/ρ.u^+ = u/u_\tau, \quad y^+ = y u_\tau/\nu, \quad u_\tau = \sqrt{\tau_w/\rho}. Constants κ≈0.41,  B≈5.0.\kappa \approx 0.41,\; B \approx 5.0.

Skin friction coefficient (smooth wall): Cf=0.0576Rex−1/5.C_f = 0.0576 Re_x^{-1/5}.

9. Thermal Boundary Layer

Coupled energy equation: ρcp(u∂T/∂x+v∂T/∂y)=k∂2T/∂y2.\rho c_p(u \partial T/\partial x + v \partial T/\partial y) = k \partial^2 T/\partial y^2.

Define thermal boundary-layer thickness δT\delta_T: δT/δ=(Pr)−1/3  (laminar),(Pr)−1/5  (turbulent).\delta_T/\delta = (Pr)^{-1/3} \; (\text{laminar}), \quad (Pr)^{-1/5} \; (\text{turbulent}).

Analogies Between Momentum and Heat Transfer

AnalogyExpressionAssumption
Reynoldsf/2=Stf/2 = StSimilar velocity–temperature profiles
PrandtlSt=(f/2)Pr−2/3St = (f/2)Pr^{-2/3}Laminar flat plate
Chilton–ColburnjH=StPr2/3=f/2j_H = St Pr^{2/3} = f/2Turbulent flow

10. Flow Separation and Reattachment

In adverse pressure gradient, wall shear decreases and may reach zero: τw=0→separation.\tau_w = 0 \to \text{separation}.

Separated shear layer rolls up into vortices; pressure recovery is poor →\to form drag increases.

Reattachment occurs if momentum is reintroduced (e.g., via turbulent mixing or suction).

Control methods:

  • Boundary-layer suction
  • Vortex generators
  • Surface roughness or riblets

11. Transition to Turbulence

Instability mechanism:

  • Tollmien–Schlichting waves: viscous instability in laminar boundary layer.
  • Amplified by adverse pressure gradients and surface roughness.

Transition location often predicted by empirical correlation: Reθ≈400.Re_\theta \approx 400.

12. Energetics and Entropy Production

Local entropy generation within boundary layer: σs=μT(∂u/∂y)2+kT2(∂T/∂y)2.\sigma_s = \frac{\mu}{T} (\partial u/\partial y)^2 + \frac{k}{T^2} (\partial T/\partial y)^2. Integrating over the wall-normal direction quantifies viscous and thermal irreversibility, directly linked to skin-friction and heat-transfer losses.

13. Summary Equations

ConceptExpression
Boundary-layer thicknessδ/x∼Rex−1/2\delta/x \sim Re_x^{-1/2}
Blasius solutionτw=0.332ρU2/Rex\tau_w = 0.332 \rho U^2/\sqrt{Re_x}
Displacement thicknessδ∗=∫0∞(1−u/Ue)dy\delta^* = \int_0^\infty (1 - u/U_e) dy
Momentum thicknessθ=∫0∞(u/Ue)(1−u/Ue)dy\theta = \int_0^\infty (u/U_e)(1 - u/U_e) dy
Shape factorH=δ∗/θH = \delta^*/\theta
von Kármán integral eq.dθ/dx+(2+H)(θ/Ue)(dUe/dx)=Cf/2d\theta/dx + (2+H)(\theta/U_e)(dU_e/dx) = C_f/2
Turbulent law of wallu+=(1/κ)ln⁡y++Bu^+ = (1/\kappa)\ln y^+ + B
Skin frictionCf=0.0576Rex−1/5C_f = 0.0576 Re_x^{-1/5}
  • 01_Potential_and_Viscous_Flows.md — laminar flow and viscous drag foundations.
  • transport-equations.md — PDE and numerical approaches to boundary-layer modeling.
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy generation in shear-driven systems.

See Also