MCG 4345 - Aerodynamics – p. 1 / 3 – Professor S. Tavoularis 4.6 Laminar Boundary Layer Separation – Stratford Criterion Separation may only occur if dp > 0 (or, equivalently, dx due dx < 0). Criteria for separation: • These conditions correspond to τw = 0 • Falkner-Skan solutions: m = −0.091 • Pohlhausen’s solution: λP = −7 to − 8 (the solution gives λ = −12) • Thwaites’method: λ = −0.0842 • Stratford criterion: see below • Beyond the separation point, the boundary layer approximation is no longer valid and none of these methods can be used for calculating τw . The main type of drag in the separated region is form (pressure) drag. MCG 4345 - Aerodynamics – p. 2 / 3 – Professor S. Tavoularis Stratford criterion: Separation occurs when C p dC p x dx 2 = 0.0104 where x is an effective origin and the pressure recovery coefficient C p is defined in terms of the minimum pressure pm along the wall and the corresponding free stream velocity um as p − pm Cp = 1 2 = 1 − ρum 2 ue um 2 The definition of x depends on the variation of dp/dx > 0 along the b.l.. Three different cases may be considered, as follows. (a) dp/dx > 0 from the start of the boundary layer (x = 0): x = x ; um , pm are the values at x = 0. 2 dC p as it increases with increasing x. At x = 0, C p = 0. Compute C p x dx When this becomes 0.0104, separation occurs. ( dp/dx = 0 , 0 ≤ x ≤ xm (b) , then use um , pm at xm but x = x (from start). dp/dx > 0 xm < x MCG 4345 - Aerodynamics – p. 3 / 3 – Professor S. Tavoularis (c) The pressure gradient is initially favourable and then adverse, as on the top surface of ( dp/dx < 0 , 0 ≤ x ≤ xm an airfoil, i.e. dp/dx > 0 , xm < x To define x, consider an equivalent problem with dp/dx = 0 upstream of xm and such that it has the same θ (momentum thickness) at xm as the actual problem. Then, from Thwaites’ method: Z xm 5 ue xm = dx = ... (can be computed for a given ue (x)) um 0 and x = x − (xm − xm ) 2 dC p Notice that C p = 0 at x = xm . Compute C p x dx till it becomes 0.0104.
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