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Exact viscous diffusion layer and material-circulation decay — formal worked fields

Example

For the wall layer take $ν = U = 1$ . At time $t = 1$ and distance $y = 2$ from the wall, the exact solution obeys $∣ u (1, 2) ∣ = E (1) \leq e^{- 1}$ . In the separate periodic example, take $ν = 1$ , $L_{x} = 4$ , $L_{y} = 2 π$ , $L_{z} = 1$ , $m = k = 1$ , $A = 2$ , $c = 1$ , and $b = π / 2$ . Then $α = k b = π / 2$ and the material loop initialized as an ellipse in the Statement has circulation $Γ (t) = 2 C_{π / 2} e^{- t}$ , where $C_{π / 2} = \int_{0}^{2 π} \cos θ \sin ((π / 2) \cos θ) d θ > 0$ . Its circulation strictly decreases between times $0$ and $1$ .

Example references

Exact viscous diffusion layer and material-circulation decay

Facts & Assumptions

Given: The wall layer is considered only for $t > 0, y > 0$ ; the periodic shear and the loop initialized as an ellipse are on the indicated flat three-torus.

[A1]

Wall layer: $ν = U = 1$ , $t = 1$ , $y = 2$ , $ε = e^{- 1}$ . Periodic field: $ν = 1$ , $L_{x} = 4$ , $L_{y} = 2 π$ , $L_{z} = 1$ , $m = k = 1$ , $A = 2$ , $c = 1$ , $b = π / 2$ , and $α = π / 2$ .

[F1]

Exact viscous diffusion layer and material-circulation decay: $E (s) \leq e^{- s^{2}}$ ; for the smooth material loop initialized as an ellipse with $0 < α = k b < π$ , circulation is $Γ (t) = A c C_{α} e^{- ν k^{2} t}$ with $C_{α} = \int_{0}^{2 π} \cos θ \sin (α \cos θ) d θ > 0$ .

Verification

1.1givenA1F1algebra

The wall-layer parameters give $y = 2 \sqrt{ν t} \sqrt{\log (1 / ε)} = 2$ , so the theorem's tail bound yields $∣ u (1, 2) ∣ \leq e^{- 1} ∣ U ∣ = e^{- 1}$ . The corner is not involved because $t = 1$ and $y = 2$ are both positive.

2.1givenA1F1algebra∎

For the periodic field, $k L_{y} = 2 π$ , $0 < c < L_{x} / 2$ , $0 < b < L_{y} / 2$ , and $0 < α = k b = π / 2 < π$ , so [F1] applies. Substitution gives $Γ (t) = 2 \cdot 1 \cdot C_{π / 2} e^{- t} = 2 C_{π / 2} e^{- t}$ . Since $C_{π / 2} > 0$ , $Γ (0) = 2 C_{π / 2}$ and $Γ (1) = 2 C_{π / 2} / e$ are distinct. This is a formal viscous circulation change for this loop only.

Depends on

Exact viscous diffusion layer and material-circulation decay