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Constant velocity has zero subfilter stress

Example

Assume the Axiom of Countable Choice required by the normalized periodic torus convention. Let $d \in {1, 2, 3}$ , $L > 0$ m, $I \subseteq R$ an open time interval measured in s, $w \in R^{d}$ a constant velocity measured in m s $^{- 1}$ , and constant kinematic viscosity $ν > 0$ measured in m $^{2}$ s $^{- 1}$ . Set $u (x, t) = w$ , $p (x, t) = 0$ , and $f (x, t) = 0$ on $T_{L}^{d} \times I$ . Use the dimensional incompressible equation $\partial_{t} u + \nabla \cdot (u \otimes u) + \nabla p = ν Δ u + f$ , with pressure per unit mass $p$ and force per unit mass $f$ , where $\nabla \cdot (u \otimes v)_{i} = \sum_{j} \partial_{j} (u_{i} v_{j})$ and $Δ = \sum_{j} \partial_{j}^{2}$ . This is a smooth divergence-free solution. For every $0 < ℓ < L / 2$ , the positive periodic convolution filter gives ${\overline{u}}_{ℓ} = w$ , $τ_{ℓ} = 0$ , and $Π_{ℓ} = 0$ . Every velocity increment vanishes, so the Hölder transfer estimate applies with $A (t) = 0$ and has zero right-hand side. This is an exact mathematical example, not a reported observation.

Example references

Positive periodic convolution filter and subfilter stress

Countable Choice axiom

Hölder bound for filtered subfilter energy transfer

Facts & Assumptions

Given: The constant fields and parameters in the example.

[A1]

The adopted mathematical system is $\partial_{t} u + \nabla \cdot (u \otimes u) + \nabla p = ν Δ u + f$ with $\nabla \cdot u = 0$ .

[A2]

Adopt Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the Fourier-Sobolev torus convention is stated and proved under this axiom.

[F1]

Positive periodic convolution filter and subfilter stress: the filter has a nonnegative unit-mass kernel, $τ_{ℓ} = {\overline{u \otimes u}}_{ℓ} - {\overline{u}}_{ℓ} \otimes {\overline{u}}_{ℓ}$ , and $Π_{ℓ} = - τ_{ℓ} : \nabla {\overline{u}}_{ℓ}$ .

[F2]

Hölder bound for filtered subfilter energy transfer: under the stated periodic Hölder increment hypothesis, $∣ Π_{ℓ} ∣ \leq C_{d, K, α} A^{3} ℓ^{3 α - 1}$ .

Verification

technique · direct

Given: $u (x, t) = w$ , $p = 0$ , $f = 0$ , and the filter width satisfies $0 < ℓ < L / 2$ .

1.1givenA1algebra

Every spatial and time derivative of $u$ vanishes, so $\nabla \cdot u = 0$ , $\partial_{t} u = 0$ , $\nabla \cdot (u \otimes u) = 0$ , and $Δ u = 0$ . Thus [A1] holds with $p = f = 0$ for every $ν > 0$ .

2.1A2F1step 1.1algebra

Since the kernel has unit mass, filtering the constant field gives ${\overline{u}}_{ℓ} = w$ and filtering $u \otimes u = w \otimes w$ gives ${\overline{u \otimes u}}_{ℓ} = w \otimes w$ . Hence $τ_{ℓ} = 0$ , $\nabla {\overline{u}}_{ℓ} = 0$ , and $Π_{ℓ} = 0$ .

3.1A2F2step 2.1algebra∎

For all $s, x, t$ , $u (x - ℓ s, t) - u (x, t) = 0$ , so the Hölder condition holds with $A (t) = 0$ for any $0 < α < 1$ . Applying [F2] yields $∣ Π_{ℓ} ∣ \leq 0$ , consistent with the exact value in step 2.1.

Constant velocity has zero subfilter stress

Example

Example references

Facts & Assumptions

Verification

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