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For the following velocity profiles, determine whether the flow has separated or on the verge of separation or will be attached with the surface.

For the following velocity profiles, determine whether the flow has separated or on the verge of separation or will be attached with the surface.

  1. uU=32(yδ)12(yδ)2

  2. uU=2(yδ)2(yδ)3

  3. uU=2(yδ)+(yδ)2

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Solution :

Given :

1st Velocity Profile

uU=32(yδ)12(yδ)3u=3U2(yδ)U2(yδ)3

Differentiating w.r.t. y, uy=3U2×1δU2×3(yδ)2×1δ

At y=0,

(uy)y=0=3U2δ3U2(0δ)2×1δ=3U2δ

As (uy)y=0 is positive.

Hence flow will not separate or flow will remain attached with the surface.

2nd Velocity Profile

uU=2(yδ)2(yδ)3u=2U(yδ)2U(yδ)3

Differentiating w.r.t. y, uy=4U(yδ)×1δ3U(yδ)2×1δ

At y=0,

(uy)y=0=4U×(0δ)1δ3U×(0δ)21δ=0

As (uy)N=0=0, the flow is on the verge of separation.

3rd Velocity Proflle

uU=2(yδ)+(yδ)2u=2U(yδ)+U(yδ)2

Differentiating w.r.t. y, uy=2U(1δ)+2U(yδ)×1δ

at y=0,

(uy)y=0=2Uδ+2U(δ)×1δ=2Uδ

As (uy)y=0 is negative the flow has separated.

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