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Covariance versus effective Reynolds stress

Physical scope

A two-member constant-velocity example in a formal constant-density incompressible model, used only to distinguish covariance from the signed effective stress contribution and to check SI units. It reports no measurement and proposes no closure law.

Example

Let $τ > 0$ s, $I = (- τ, τ)$ , and use initial and current Cartesian coordinates measured in metres. Let $B \subseteq R^{3}$ be a nonempty bounded open reference region with $C^{1}$ boundary and let $U \subseteq R^{3}$ be an open neighborhood of $\overline{B}$ . Choose $e \in R^{3}$ with Euclidean norm $∣ e ∣ = 1$ and a speed $c > 0$ measured in m s $^{- 1}$ . Define $χ_{1}, χ_{2} : I \times U \to R^{3}$ by $χ_{1} (t, a) = a + c t e$ and $χ_{2} (t, a) = a - c t e$ . On the open spatial region $Ω = R^{3}$ , define $u_{1}, u_{2} : I \times Ω \to R^{3}$ by $u_{1} (t, x) = c e$ and $u_{2} (t, x) = - c e$ . Take equal dimensionless weights $a_{1} = a_{2} = 1 / 2$ , zero pressure-per-unit-mass fields $p_{j} : I \times Ω \to R$ (units m $^{2}$ s $^{- 2}$ ), zero body-force-per-unit-mass fields $f_{j} : I \times Ω \to R^{3}$ (units m s $^{- 2}$ ), and common kinematic viscosity $ν = 0$ m $^{2}$ s $^{- 1}$ in the per-unit-mass finite Reynolds identity. For the formal constant-density interpretation, adopt a constant density parameter $ρ_{0} > 0$ in kg m $^{- 3}$ for momentum scaling. Define the covariance $R = \sum_{j = 1}^{2} a_{j} (u_{j} - \overset{ˉ}{u}) \otimes (u_{j} - \overset{ˉ}{u})$ and define the effective Reynolds-stress correction $σ_{R} = - ρ_{0} R$ when the momentum equation is written with a $+ div σ$ contribution on the right. Then $\overset{ˉ}{u} = 0$ , $R = c^{2} e \otimes e$ , and $ρ_{0} R$ is a nonzero positive-semidefinite tensor with units Pa, while $σ_{R}$ has the opposite sign. This is an exact finite-field illustration, not an empirical observation or a statistical turbulence model.

Example references

finite weighted Reynolds identity and its hypotheses

Newtonian velocity, density, and tension-positive stress conventions

Flierl and Ferrari, MIT OpenCourseWare 12.820, Chapter 3, §§3.6–3.7, complete lecture notes

Tennekes and Lumley, A First Course in Turbulence (1972), Chapter 2, §2.1, full-text digital copy

Facts & Assumptions

[F1]

Exact finite Reynolds averaged equation: for an open time interval and spatial domain, a finite family of divergence-free velocity fields that are $C^{1}$ in time and space with continuous second spatial derivatives, $C^{1}$ pressure fields, continuous forcing fields, fixed nonnegative weights summing to one, and one common $ν \geq 0$ , the pointwise averaged equation contains $- div R$ , where $R = \sum_{j} a_{j} (u_{j} - \overset{ˉ}{u}) \otimes (u_{j} - \overset{ˉ}{u})$ . Its current Step-1 readiness record is ready; the proof was inspected at this exact interface.

[F2]

Classical frames, primitive quantities and SI units: a material motion is a $C^{2}$ map whose time slices are bijections from $U$ onto open images with $C^{2}$ inverses and positive Jacobian determinant, with $χ (0, a) = a$ in the initial chart; the reference label $a$ and current position $x = χ (t, a)$ are measured in metres.

[F3]

Classical frames, primitive quantities and SI units: the Eulerian velocity is defined by $u (t, χ (t, a)) = \partial_{t} χ (t, a)$ and has units m s $^{- 1}$ .

[F4]

Classical frames, primitive quantities and SI units: stress uses the tension-positive traction convention $σ n$ and stress/traction have units Pa.

[A1]

For this example only, identify the positive constant density used in the momentum scaling with the constant reference value $ρ_{0} (a) \equiv ρ_{0}$ , with $ρ_{0}$ measured in kg m $^{- 3}$ ; this is an adopted constant-density interpretation, not a deduction about density transport or an empirical premise.

[A2]

The phrase “effective Reynolds-stress correction” denotes the signed tensor contribution whose divergence appears in a momentum equation written with $+ div σ$ on the right. This sign convention does not identify $σ_{R}$ with a measured total Cauchy stress or assert its uniqueness from its divergence.

Given: $I = (- τ, τ)$ , $B$ , $U$ , $e$ , $c$ , the two translation maps, the constant velocity fields on $Ω = R^{3}$ , $a_{1} = a_{2} = 1 / 2$ , $p_{j} = f_{j} = 0$ , $ν = 0$ , the constant-density interpretation [A1], and the sign convention [A2]. Apply [F1] to the specified fields using the physical conventions [F2]–[F4].

Verification

1.1givenF1F2F3algebra

For each fixed $t$ , $χ_{1} (t, \cdot)$ and $χ_{2} (t, \cdot)$ are translations with inverses $x \mapsto x - c t e$ and $x \mapsto x + c t e$ , respectively; $D_{a} χ_{j} = I_{3}$ and $J_{j} = 1$ . They are $C^{2}$ on $I \times U$ , orientation preserving, and equal the identity at $t = 0$ . Since $c$ has units m s $^{- 1}$ and $t$ has units s, the translations preserve the stated metre coordinates. By [F2]–[F3], their Eulerian velocities on the images are $c e$ and $- c e$ , as extended by the same constant fields to $Ω$ . Each field is $C^{\infty}$ , divergence free, time independent, and has constant $u_{j} \otimes u_{j}$ ; hence every derivative in the per-unit-mass member equation is zero, so the hypotheses of [F1] hold with the given zero pressure, forcing, and viscosity.

2.1givenF1step 1.1algebra

The equal-weight mean is $\overset{ˉ}{u} = (c e - c e) / 2 = 0$ , so $v_{1} = c e$ and $v_{2} = - c e$ . Therefore $R = \frac{1}{2} (c e \otimes c e) + \frac{1}{2} ((- c e) \otimes (- c e)) = c^{2} e \otimes e$ .

3.1A1F3F4step 2.1algebra

For every $z \in R^{3}$ , $z^{T} R z = c^{2} (z \cdot e)^{2} \geq 0$ , and $R$ is symmetric and nonzero. Since each velocity has units m s $^{- 1}$ by [F3], $R$ has units m $^{2}$ s $^{- 2}$ ; multiplying by $ρ_{0}$ in kg m $^{- 3}$ gives kg m $^{- 1}$ s $^{- 2}$ =Pa. Thus $σ_{R} = - ρ_{0} R$ is a negative-semidefinite effective correction with units Pa under [F4].

4.1F1A1A2step 2.1algebra∎

The supplier equation [F1] contains $- div R$ on the right. Multiplying by the constant $ρ_{0}$ gives $- ρ_{0} div R = - div (ρ_{0} R) = div (- ρ_{0} R)$ , because $ρ_{0}$ is spatially constant. Under [A2] the corresponding effective stress correction is therefore $σ_{R} = - ρ_{0} R$ ; the covariance $R$ itself remains positive semidefinite.

Authoring status

The current Step-1 readiness record for Classical frames, primitive quantities and SI units marks it ready for authoring. This records authoring readiness only; independent review and the native gates remain outstanding, and the readiness disposition is not a premise of this calculation.

Covariance versus effective Reynolds stress

Example

Example references

Facts & Assumptions

Verification

Authoring status

Depends on

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Sources