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Exact filtered covariance and resolved-energy balance

Statement

Assume the Axiom of Countable Choice required by the normalized periodic torus convention. Let $d \in {1, 2, 3}$ , $L > 0$ , $T_{L}^{d} = (R / L Z)^{d}$ , and $d μ_{L} = L^{- d} d x$ . Fix $0 < ℓ < L / 2$ and let $K : R^{d} \to [0, \infty)$ be smooth, supported in a compact subset of $B_{1} (0)$ , with $\int K (s) d s = 1$ . Define $G_{ℓ} (r) = \sum_{n \in Z^{d}} ℓ^{- d} K ((r + L n) / ℓ)$ for $r \in T_{L}^{d}$ and, for every smooth periodic finite-dimensional field $q$ , ${\overline{q}}_{ℓ} (x) = \int_{T_{L}^{d}} G_{ℓ} (r) q (x - r) d x$ . Let $I \subseteq R$ be an open time interval and let $u \in C^{1} (I; C^{\infty} (T_{L}^{d}; R^{d}))$ be any smooth periodic vector field. Define

$τ_{ℓ} = {\overline{u \otimes u}}_{ℓ} - {\overline{u}}_{ℓ} \otimes {\overline{u}}_{ℓ}, Π_{ℓ} = - τ_{ℓ} : \nabla {\overline{u}}_{ℓ} .$

For $δ_{s} u (x, t) : = u (x - ℓ s, t) - u (x, t)$ and $m_{ℓ} (x, t) : = \int_{R^{d}} K (s) δ_{s} u (x, t) d s$ , the stress has the exact covariance form

$τ_{ℓ} (x, t) = \int_{R^{d}} K (s) δ_{s} u (x, t) \otimes δ_{s} u (x, t) d s - m_{ℓ} (x, t) \otimes m_{ℓ} (x, t) .$

It is symmetric and positive semidefinite at every $(x, t)$ .

For the additional conclusions below, suppose $ν > 0$ is a constant kinematic viscosity, $p$ and $f$ are smooth scalar and vector fields on $I \times T_{L}^{d}$ , respectively, and $u$ satisfies pointwise

$\partial_{t} u + \nabla \cdot (u \otimes u) + \nabla p = ν Δ u + f, \sum_{i} \partial_{i} u_{i} = 0,$

where $(\nabla \cdot T)_{i} : = \sum_{j} \partial_{j} T_{i j}$ , $\nabla \cdot v : = \sum_{i} \partial_{i} v_{i}$ , and $Δ : = \sum_{j} \partial_{j}^{2}$ . Here $p$ is pressure per unit mass and $f$ is force per unit mass. In SI dimensions, $x, L, ℓ$ are in m, $t$ in s, $u$ in m s $^{- 1}$ , $p$ in m $^{2}$ s $^{- 2}$ , $ν$ in m $^{2}$ s $^{- 1}$ , and $f$ in m s $^{- 2}$ .

The filtered field satisfies

$\partial_{t} {\overline{u}}_{ℓ} + \nabla \cdot ({\overline{u}}_{ℓ} \otimes {\overline{u}}_{ℓ}) + \nabla {\overline{p}}_{ℓ} = ν Δ {\overline{u}}_{ℓ} - \nabla \cdot τ_{ℓ} + {\overline{f}}_{ℓ}, \nabla \cdot {\overline{u}}_{ℓ} = 0.$

Writing $e_{ℓ} = ∣ {\overline{u}}_{ℓ} ∣^{2} / 2$ , the pointwise resolved-energy balance is

$\partial_{t} e_{ℓ} + \sum_{j} \partial_{j} ⁣ (e_{ℓ} {\overline{u}}_{ℓ, j} + {\overline{p}}_{ℓ} {\overline{u}}_{ℓ, j} - ν \sum_{i} {\overline{u}}_{ℓ, i} \partial_{j} {\overline{u}}_{ℓ, i} + \sum_{i} {\overline{u}}_{ℓ, i} τ_{ℓ, i j}) = - ν ∣ \nabla {\overline{u}}_{ℓ} ∣_{F}^{2} + τ_{ℓ} : \nabla {\overline{u}}_{ℓ} + {\overline{f}}_{ℓ} \cdot {\overline{u}}_{ℓ},$

where $∣ A ∣_{F}^{2} : = \sum_{i, j} ∣ A_{i j} ∣^{2}$ . Consequently, for every $t \in I$ ,

$\frac{d}{d t} \frac{1}{2} \int_{T_{L}^{d}} ∣ {\overline{u}}_{ℓ} ∣^{2} d μ_{L} + ν \int_{T_{L}^{d}} ∣ \nabla {\overline{u}}_{ℓ} ∣_{F}^{2} d μ_{L} = \int_{T_{L}^{d}} τ_{ℓ} : \nabla {\overline{u}}_{ℓ} d μ_{L} + \int_{T_{L}^{d}} {\overline{f}}_{ℓ} \cdot {\overline{u}}_{ℓ} d μ_{L} .$

The forward-transfer sign convention is $Π_{ℓ} = - τ_{ℓ} : \nabla {\overline{u}}_{ℓ}$ ; this theorem makes no sign assertion for $Π_{ℓ}$ .

Statement references

Positive periodic convolution filter and subfilter stress

Countable Choice axiom

Normalization, periodization, and derivative commutation for the positive fluid filter

Product rule for volume-form divergence

If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections

University of Washington, ME 544, Large-Eddy Simulation lecture notes, §3, printed pp. 10–12, filtered momentum, residual stress, resolved-energy equation (13)–(24)

Gregory L. Eyink, The ‘Multifractal Model’ of Turbulence and A Priori Estimates in Large-Eddy Simulation, §I–II, printed pp. 2–7, filtered equations, stress covariance, and transfer convention

Facts & Assumptions

Given: Countable Choice and $d \in {1, 2, 3}$ , $L > 0$ , $0 < ℓ < L / 2$ , $I$ open, a nonnegative unit-mass smooth kernel supported in a compact subset of $B_{1} (0)$ , and $u \in C^{1} (I; C^{\infty} (T_{L}^{d}; R^{d}))$ .

[A1]

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

[A2]

Additional adopted mathematical PDE premise, used only for the filtered momentum and energy conclusions: $ν > 0$ is constant, $p, f$ are smooth, $\partial_{t} u_{i} + \partial_{j} (u_{i} u_{j}) + \partial_{i} p = ν \partial_{j} \partial_{j} u_{i} + f_{i}$ and $\partial_{i} u_{i} = 0$ pointwise.

[F1]

Positive periodic convolution filter and subfilter stress: ${\overline{q}}_{ℓ} (x) = \int_{T_{L}^{d}} G_{ℓ} (r) q (x - r) d x$ , $τ_{ℓ} = {\overline{u \otimes u}}_{ℓ} - {\overline{u}}_{ℓ} \otimes {\overline{u}}_{ℓ}$ , and $Π_{ℓ} = - τ_{ℓ} : \nabla {\overline{u}}_{ℓ}$ .

[F2]

Normalization, periodization, and derivative commutation for the positive fluid filter: $G_{ℓ}$ has unit mass, the filter commutes with spatial and time derivatives, and its Euclidean representation is ${\overline{q}}_{ℓ} (x) = \int K (s) q (x - ℓ s) d s$ .

[F3]

Product rule for volume-form divergence: for a smooth scalar $g$ and smooth vector field $X$ on a volume-form manifold, $div (g X) = d g (X) + g div X$ .

[F4]

If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ : for differentiable functions with integrable derivatives on a closed interval, $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ .

[F5]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous real function on a compact interval is Riemann integrable.

[F6]

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections: an integrable function on a product rectangle with integrable sections has equal multiple and iterated Riemann integrals.

[F7]

Riemann's criterion on a nondegenerate rectangle in $R^{m}$ : integrability is equivalent to arbitrarily small Darboux gaps: bounded functions on a nondegenerate rectangle are Riemann integrable when every positive tolerance admits a grid with a smaller Darboux gap; closed rectangles are compact and continuous functions on them are uniformly continuous (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F8]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : the multidimensional Riemann integral is linear and order preserving.

[F9]

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral: a continuous integrand with a continuous parameter derivative on a compact rectangle can be differentiated under its one-dimensional Riemann integral.

[F10]

Lower and upper Darboux sums over a grid partition in $R^{m}$ : the Darboux gap is the sum of cell oscillations times cell volumes; Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh: grid cells cover the box and their volumes sum to its volume; Laws of finite sums and finite products: finite sums preserve inequalities.

[F11]

For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ : for each $η > 0$ there is $N \geq 1$ with $1 / N < η$ , so a period cell admits uniform grids of arbitrarily small mesh.

Proof

technique · direct

1.1A1F1F2F8algebra

Put $δ_{s} u (x, t) = u (x - ℓ s, t) - u (x, t)$ . Unit mass and the Euclidean filter representation give $m_{ℓ} = {\overline{u}}_{ℓ} - u$ . Expanding componentwise gives $\int K (s) δ_{s} u \otimes δ_{s} u d s = {\overline{u \otimes u}}_{ℓ} - {\overline{u}}_{ℓ} \otimes u - u \otimes {\overline{u}}_{ℓ} + u \otimes u$ , while $m_{ℓ} \otimes m_{ℓ} = ({\overline{u}}_{ℓ} - u) \otimes ({\overline{u}}_{ℓ} - u)$ . Subtraction gives the stated covariance formula and symmetry.

1.2A1A2F1F2algebra

Apply the filter to [A2]. By [F2], time and spatial derivatives pass through the filter, so $\partial_{t} {\overline{u}}_{ℓ} + \nabla \cdot {\overline{u \otimes u}}_{ℓ} + \nabla {\overline{p}}_{ℓ} = ν Δ {\overline{u}}_{ℓ} + {\overline{f}}_{ℓ}$ and $\nabla \cdot {\overline{u}}_{ℓ} = 0$ . Substituting ${\overline{u \otimes u}}_{ℓ} = {\overline{u}}_{ℓ} \otimes {\overline{u}}_{ℓ} + τ_{ℓ}$ gives the stated filtered system.

2.1F1F2step 1.1algebra

For $z \in R^{d}$ , contraction of the covariance formula gives $z^{T} τ_{ℓ} z = \int K (s) (z \cdot δ_{s} u)^{2} d s - (\int K (s) z \cdot δ_{s} u d s)^{2} = \int K (s) (z \cdot δ_{s} u - z \cdot m_{ℓ})^{2} d s \geq 0$ , where unit mass expands the square and $K \geq 0$ makes the last integrand nonnegative. Thus $τ_{ℓ}$ is positive semidefinite.

2.2step 1.2algebra

In components, $\partial_{j} ({\overline{u}}_{ℓ, i} {\overline{u}}_{ℓ, j}) = {\overline{u}}_{ℓ, j} \partial_{j} {\overline{u}}_{ℓ, i} + {\overline{u}}_{ℓ, i} \partial_{j} {\overline{u}}_{ℓ, j}$ ; the last term vanishes by $\nabla \cdot {\overline{u}}_{ℓ} = 0$ . Hence the filtered quadratic divergence is the incompressible transport term used in the energy balance.

3.1F2F3step 1.2step 2.2algebra

Multiply the filtered momentum equation by ${\overline{u}}_{ℓ, i}$ and sum over $i$ . The product rule [F3] and $\nabla \cdot {\overline{u}}_{ℓ} = 0$ give ${\overline{u}}_{ℓ, i} \partial_{i} {\overline{p}}_{ℓ} = \partial_{i} ({\overline{p}}_{ℓ} {\overline{u}}_{ℓ, i})$ and $- {\overline{u}}_{ℓ, i} \partial_{j} τ_{ℓ, i j} = - \partial_{j} ({\overline{u}}_{ℓ, i} τ_{ℓ, i j}) + τ_{ℓ, i j} \partial_{j} {\overline{u}}_{ℓ, i}$ . The scalar product rule gives $ν {\overline{u}}_{ℓ, i} \partial_{j} \partial_{j} {\overline{u}}_{ℓ, i} = ν \partial_{j} ({\overline{u}}_{ℓ, i} \partial_{j} {\overline{u}}_{ℓ, i}) - ν (\partial_{j} {\overline{u}}_{ℓ, i})^{2}$ , and [F2] supplies $\partial_{t} (∣ {\overline{u}}_{ℓ} ∣^{2} / 2) = {\overline{u}}_{ℓ, i} \partial_{t} {\overline{u}}_{ℓ, i}$ . Collecting the divergence terms yields the stated pointwise local energy equation.

4.1F7F10F11step 3.1

On the closed period cell $Q = [0, L]^{d}$ , each component of the local energy flux and its divergence is continuous. By [F7], each is bounded and uniformly continuous. Given $ε > 0$ , let $δ > 0$ be a uniform-continuity radius for tolerance $ε / (2 L^{d})$ . Choose $N \geq 1$ with $1 / N < δ / (L \sqrt{d})$ by [F11] and divide each coordinate interval into $N$ equal parts. Each cell's Euclidean diameter is at most $L \sqrt{d} / N < δ$ , so its oscillation is below that tolerance; by [F10] the Darboux gap is at most $(ε / (2 L^{d})) \sum_{i} vol (Q_{i}) = ε / 2 < ε$ . The criterion [F7] makes the flux components and their divergence Riemann integrable on $Q$ .

5.1F4F5F6step 4.1

Fix all coordinates but $x_{j}$ . Apply [F4] to $u = 1$ and $v$ equal to the $j$ -th local flux component on $[0, L]$ ; its derivative is continuous and integrable by [F5], and periodicity makes the endpoint term zero. By [F6], integrating these zero one-dimensional integrals over the other coordinates gives zero integral for the total divergence on $T_{L}^{d}$ .

6.1F6F7F9step 3.1step 5.1algebra∎

Fix $t_{0} \in I$ and choose a compact interval $J \subset I$ with $t_{0}$ in its interior. On $J \times Q$ , $e_{ℓ}$ and $\partial_{t} e_{ℓ}$ are continuous, so [F7] gives boundedness and uniform continuity and its grid criterion gives Riemann integrability of each spatial section. By [F6] the spatial integral is an iterated integral; applying [F9] successively to its $d$ one-dimensional integrations passes $\partial_{t}$ through each integral at $t_{0}$ . Integrating the local identity from step 3.1 over $Q$ , using step 5.1 to make the total-divergence integral zero, and dividing by $L^{d}$ for normalized Haar measure gives the stated global resolved-energy identity at $t_{0}$ .

Remarks

The covariance identity is exact and the stress is positive semidefinite because the kernel is nonnegative. The transfer $Π_{ℓ}$ is a separate contraction with the resolved gradient; this theorem makes no universal sign assertion. The filtered stress is an exact residual of the stated smooth equation, not an adopted closure model.

Exact filtered covariance and resolved-energy balance

Statement

Statement references

Facts & Assumptions

Proof

Remarks

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