The Falkner–Skan boundary layer comprises a family of steady, two‑dimensional laminar boundary‑layer flows that develop along an inclined surface (a wedge), with similarity formulations introduced by Victor M. Falkner and Sylvia W. Skan. In such wedge geometries the plate’s inclination relative to the oncoming flow produces a longitudinal pressure gradient in the outer inviscid flow; this spatial variation in pressure drives either acceleration or deceleration of the near‑wall flow and thereby alters the boundary‑layer velocity profile and thickness. Falkner–Skan solutions capture these effects while retaining the steady, two‑dimensional laminar assumptions of classical boundary‑layer theory and thus describe how the boundary layer responds to imposed favorable or adverse pressure gradients. The family generalizes the Blasius flat‑plate solution, which appears as the special case with zero pressure gradient, and provides a convenient idealization for modeling flows in wind‑tunnel experiments and practical configurations where a flat plate experiences a prescribed longitudinal pressure variation.
Prandtl’s boundary‑layer framework partitions flow over a solid surface into a thin, near‑wall viscous layer and an outer inviscid region, permitting the omission of many Navier–Stokes terms inside the layer except in a small neighbourhood of the leading edge where the full equations remain required. Under the classical assumptions adopted here — steady, incompressible flow with constant density ρ and kinematic viscosity ν — the governing boundary‑layer equations reduce to a two‑dimensional continuity relation and a simplified momentum balance in the streamwise direction. With x oriented along the plate in the flow direction and y normal to the wall toward the free stream, and with velocity components u(x,y) and v(x,y), the mass conservation reads ∂u/∂x + ∂v/∂y = 0. The streamwise momentum equation expresses a balance among convective inertia, the externally imposed pressure gradient and wall‑normal viscous diffusion:
u ∂u/∂x + v ∂u/∂y = −(1/ρ) ∂p/∂x + ν ∂^2u/∂y^2.
The transverse momentum reduces to 0 = −∂p/∂y, implying that pressure is uniform across the thin boundary layer (p = p(x)) and that the streamwise pressure gradient appearing above is determined by the outer inviscid flow.
A useful class of solutions arises when the velocity profile at different streamwise positions can be collapsed by appropriate boundary‑layer thickness and velocity scales; this similarity transformation converts the boundary‑layer partial differential equations into ordinary differential equations. Falkner and Skan (1930) exploited this approach to derive a parametric family of self‑similar laminar boundary‑layer profiles for flow over wedges, thereby capturing how different external pressure‑gradient conditions associated with wedge geometries alter the boundary‑layer structure.
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Falkner–Skan equation — first‑order boundary layer
Consider high‑Reynolds‑number flow past a wedge of semi‑angle πβ/2 immersed in a uniform reference velocity U0. Outside the thin viscous layer the inviscid (Euler) solution prescribes an edge velocity u_e(x) whose streamwise pressure gradient in the Prandtl x‑momentum balance is approximated by Bernoulli’s relation,
−(1/ρ) ∂p/∂x = u_e du_e/dx. To obtain similarity solutions one assumes a power‑law edge velocity u_e(x) = U0 (x/L)^m, where L is a reference length and the exponent m characterizes the imposed pressure gradient (m < 0 adverse, m > 0 favorable).
Compatibility of this outer law with boundary‑layer scaling leads to a local thickness
δ(x) = [2 ν L/(U0 (m+1))]^{1/2} (x/L)^{(1−m)/2},
so that the viscous region grows algebraically with x with exponent (1−m)/2. Introducing the similarity coordinate η = y/δ(x) and a stream function ψ that enforces mass conservation, one writes
ψ(x,y) = u_e(x) δ(x) f(η),
or, in explicit combined form,
ψ(x,y) = [2 ν U0 L/(m+1)]^{1/2} (x/L)^{(m+1)/2} f(η),
where f(η) is the dimensionless shape function to be determined. Velocities follow from ψ by u = ∂ψ/∂y = u_e(x) f′(η) and v = −∂ψ/∂x, giving the transverse component in similarity form
v(x,y) = −[(m+1) ν U0/(2 L)]^{1/2} (x/L)^{(m−1)/2} [ f + ((m−1)/(m+1)) η f′ ].
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Substituting the similarity ansatz into Prandtl’s streamwise momentum equation reduces the PDE to the Falkner–Skan ordinary differential equation
f‴ + f f″ + β [1 − (f′)^2] = 0,
with boundary conditions f(0)=0, f′(0)=0 (no slip and impermeability at the wall) and f′(∞)=1 (matching the outer flow). The similarity parameter β is related to the outer exponent by
β = 2 m/(m+1),
so that physically important limits are recovered: m = β = 0 yields the Blasius flat‑plate problem, while β = 1 corresponds to Hiemenz flow. The sign of m (equivalently β) indicates the pressure‑gradient character: negative m (adverse pressure gradient) promotes separation tendencies, whereas positive m (favorable gradient) tends to stabilize the boundary layer.
Not all m produce physically admissible similarity profiles. Douglas Hartree showed that solutions satisfying the wall conditions remain physically acceptable only for −0.090429 ≤ m ≤ 2 (equivalently −0.198838 ≤ β ≤ 4/3); for m < −0.090429 the similarity solution can predict f′(η) > 1 somewhere in the flow, i.e. local velocities exceeding the imposed outer edge velocity, indicating breakdown of the similarity formulation.
Integral and local boundary‑layer quantities follow directly from f. The displacement thickness is
δ1(x) = [2/(m+1)]^{1/2} (ν x/U)^{1/2} ∫_0^∞ (1 − f′) dη,
expressing the mass‑deficit in terms of the universal profile f′(η). The wall shear stress is
τ_w(x) = μ [(m+1)/2]^{1/2} (U^3/(ν x))^{1/2} f″(0),
giving the skin friction in terms of the similarity curvature at the wall. The transverse (y) pressure gradient also admits a non‑dimensional representation in the similarity variables and is expressible through f and its derivatives, consistent with the two‑dimensional similarity reduction.
Compressible Falkner–Skan boundary layer
Under the low‑Mach approximation the boundary‑layer problem is formulated allowing spatial variation of density ρ, dynamic viscosity μ and thermal conductivity κ, with the wall specific enthalpy h_w prescribed. The two‑dimensional continuity equation retains density within the advective terms,
∂(ρ u)/∂x + ∂(ρ v)/∂y = 0,
while the streamwise momentum balance assumes the boundary‑layer scaling that keeps the imposed pressure gradient dp/dx as a streamwise forcing and discards streamwise viscous diffusion, yielding
u ∂u/∂x + v ∂u/∂y = −(1/ρ) (dp/dx) + (1/ρ) ∂/∂y ( μ ∂u/∂y ).
Thermal effects are expressed in the enthalpy form of the energy equation,
ρ ( u ∂h/∂x + v ∂h/∂y ) = ∂/∂y ( (μ/Pr) ∂h/∂y ),
so that heat diffusion appears coupled to momentum diffusion through the factor μ/Pr. The Prandtl number is defined with freestream reference values as Pr = c_{p_∞} μ_∞ / κ_∞, where the subscript ∞ denotes evaluation in the freestream state.
Boundary conditions are the usual no‑slip and no‑penetration at the wall y = 0 (u = 0, v = 0) together with the prescribed wall enthalpy h = h_w(x). In the far field (or at the inlet) the solution matches freestream values: u → U and h → h_∞ as y → ∞ (or at x = 0).
A self‑similar reduction of the governing PDEs is possible only if the equations and boundary conditions are invariant under the scaling transformation x → c^2 x, y → c y, u → u, v → v/c, h → h, ρ → ρ, μ → μ. This invariance enforces that the wall enthalpy cannot vary in x; h_w must be constant for similarity to hold. Physically, the low‑Mach formulation preserves an incompressible‑like momentum structure while coupling it to a variable‑property enthalpy equation: density variations enter advective and continuity terms, and thermal diffusion is governed by μ/Pr. Only when the stated scaling symmetry and a streamwise‑uniform wall enthalpy obtain can the partial differential system be reduced to the ordinary (similarity) form characteristic of Falkner–Skan solutions.
Howarth–Dorodnitsyn transformation and self-similar formulation
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The Howarth–Dorodnitsyn transformation introduces self-similar coordinates for compressible boundary layers by defining
η = sqrt[ Uo(m+1) / (2 ν∞ L^m) ] x^{(m-1)/2} ∫_0^y (ρ/ρ∞) dy,
ψ = sqrt[ 2 Uo ν∞ / ((m+1) L^m) ] x^{(m+1)/2} f(η),
together with dimensionless fields h̃(η) = h/h∞, h̃_w = h_w/h∞, ρ̃ = ρ/ρ∞ and μ̃ = μ/μ∞. The coordinate η explicitly contains the vertical integral ∫_0^y (ρ/ρ∞) dy, so variations in density enter the similarity coordinate and thereby alter the boundary-layer structure relative to constant-density cases. The parameters Uo, ν∞, L and the exponent m set the streamwise similarity scaling through the factors x^{(m-1)/2} and x^{(m+1)/2}.
Reduced ODE system and closure
Application of this transformation reduces the boundary-layer PDEs to two coupled ordinary differential equations for f(η) and h̃(η):
(ρ̃ μ̃ f”)’ + f f” + β [ h̃ − (f’)^2 ] = 0,
(ρ̃ μ̃ h̃’)’ + Pr f h̃’ = 0,
where primes denote d/dη and β and Pr are dimensionless coupling parameters. Closure requires constitutive relations ρ̃ = ρ̃(h̃) and μ̃ = μ̃(h̃): specifying these algebraic dependencies is essential before the ODEs can be integrated because ρ̃ and μ̃ appear inside the derivative operators and thus modify effective momentum and thermal diffusion.
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Boundary conditions and common property models
The self-similar problem is posed with no-slip and prescribed wall enthalpy at η = 0, f(0) = 0, f'(0) = 0, h̃(0) = h̃_w, and with free-stream conditions f'(∞) = 1, h̃(∞) = 1. For air it is common to take γ = 1.4 and Pr = 0.7 and to adopt simple algebraic models such as ρ̃ = h̃^{-1} and μ̃ = h̃^{2/3}; these supply the functional forms ρ̃(h̃) and μ̃(h̃) needed for numerical integration. If c_p is constant then h̃ equals the dimensionless temperature, h̃ = θ̃ = T/T∞, allowing temperature and enthalpy to be used interchangeably in property relations.
Physical interpretation
In the self-similar variables f'(η) is the nondimensional streamwise velocity profile with f'(∞)=1 matching the outer flow. The β-term in the momentum equation couples momentum and thermal fields through the combination [h̃ − (f’)^2], while the appearance of ρ̃ and μ̃ inside the derivative operators, (ρ̃ μ̃ f”)’ and (ρ̃ μ̃ h̃’)’, indicates that variable density and viscosity directly alter both the diffusion terms and the similarity structure compared with constant-property Falkner–Skan flows.