SEM Inlet Reynolds-Stress Tensor and Spectrum

Why this case matters

The Synthetic Eddy Method (SEM) inlet is the workhorse turbulence generator for the solver’s atmospheric-boundary-layer and built-environment studies: it injects a fluctuating velocity field at the inlet whose statistics are set by a prescribed mean profile, Reynolds-stress tensor and length scales Jarrin et al.[1]. The existing ABL case (Atmospheric Boundary Layer) only verifies that the inlet reproduces the mean profile and the streamwise turbulence intensity - that is, the diagonal of the target tensor. It never checks that the full sampled covariance tensor \(\langle u'_i u'_j \rangle\) matches the prescribed \(R_{ij}\), and in particular it never checks the off-diagonal shear components \(R_{xz}\) and \(R_{yz}\), nor that the injected spectrum follows a target turbulence spectrum. A wrong sign or magnitude of the shear stress, or a wrong spectral distribution, can pass the mean-and-intensity check yet produce wrong gust loads downstream. This case closes that gap - it is the regression check that would have caught issue #901.

The setup is deliberately the simplest configuration in which the inlet statistics are exactly the thing measured. A short streamwise box (D3Q27, RRBGK collision, Smagorinsky LES) carries a SEM inlet on the West face and a regularized Neumann outlet on the East face. The spanwise and vertical directions are periodic, and the prescribed profile is constant with height, so the inlet plane is statistically homogeneous in both in-plane directions. A dense inlet-normal plane a few nodes downstream is sampled over time; the sampled second moments are pooled over the homogeneous \((y, z)\) and over time and compared, component by component, against the prescribed tensor.

Setup

• Fluid: SEM inlet on the West (\(x^-\)) face, regularized Neumann outlet (\(\rho = 1\)) on the East (\(x^+\)) face. Spanwise \(y\) and vertical \(z\) are periodic. Initialised from the inlet field (initialization.inlet_field).

• Mean flow: uniform with height, \(u_x = 0.05\) lattice units, so \(\mathrm{Ma} = u_x \sqrt{3} \approx 0.087 < 0.1\).

• Domain: \(64 \times 48 \times 48\) lattice units, single level, block size 8. No body.

• Turbulence model: Smagorinsky LES, \(C_s = 0.17\).

• SEM: isotropic eddy length scale \(\sigma = 8\) lattice units in all directions, eddy volume density 300, fixed random seed for reproducibility.

Prescribed target

The SEM profile (reference/target_profile.csv) prescribes, constant with height, the mean velocity and the full Reynolds-stress tensor:

(1)\[\begin{split}R_{ij} = \begin{bmatrix} R_{xx} & R_{xy} & R_{xz} \\ R_{xy} & R_{yy} & R_{yz} \\ R_{xz} & R_{yz} & R_{zz} \end{bmatrix} = \begin{bmatrix} 2.5\times 10^{-4} & 0 & -5\times 10^{-5} \\ 0 & 1.0\times 10^{-4} & 2\times 10^{-5} \\ -5\times 10^{-5} & 2\times 10^{-5} & 1.0\times 10^{-4} \end{bmatrix}\end{split}\]

This corresponds to turbulence intensities \(I_u = \sqrt{R_{xx}}/U = 31.6\%\) and \(I_v = I_w = 20\%\), and a streamwise-vertical shear correlation \(\rho_{uw} = R_{xz}/\sqrt{R_{xx} R_{zz}} = -0.316\) (the ABL-like negative sign). The tensor is symmetric positive-definite (all eigenvalues positive), so its Cholesky factor is real.

Cholesky relation

The SEM produces the correlated fluctuation by mapping an uncorrelated unit-variance field \(\tilde{u}_b\) through the lower-triangular Cholesky factor \(A_{ab}\) of the target tensor:

(2)\[u'_a = A_{ab}\, \tilde{u}_b, \qquad A A^{\mathsf{T}} = R, \qquad \langle u'_a u'_b \rangle = A_{ac} \langle \tilde{u}_c \tilde{u}_d \rangle A_{bd}^{\mathsf{T}} = A_{ac} \delta_{cd} A_{bd} = R_{ab}.\]

The solver builds \(A\) from the prescribed components as \(A_{xx} = \sqrt{R_{xx}}\), \(A_{xy} = R_{xy}/A_{xx}\), \(A_{yy} = \sqrt{R_{yy} - A_{xy}^2}\), \(A_{xz} = R_{xz}/A_{xx}\), \(A_{yz} = (R_{yz} - A_{xy} A_{xz})/A_{yy}\), \(A_{zz} = \sqrt{R_{zz} - A_{xz}^2 - A_{yz}^2}\) (Eq. 19 of Jarrin et al.[1]). The validation is therefore that the sampled \(\langle u'_i u'_j \rangle\) recovers the prescribed \(R_{ij}\) through this map and through the LBM transport from the inlet to the measurement plane.

Simulation parameters

A single simulation runs for 30,000 steps; the dense plane is sampled every 5 steps over the last 25,000 steps (after a 5,000-step development window), giving roughly 5,000 snapshots. Because the field is homogeneous in \((y, z)\), the second moments are pooled over the whole plane and over time, which converges the statistics at a modest sample count.

Parameter	Value
Fluid velocity set	D3Q27
Fluid collision operator	RRBGK
Fluid relaxation time \(\tau\)	0.51 (\(\nu = 1/300\), \(\omega \approx 1.961\))
LES model	Smagorinsky, \(C_s = 0.17\)
Mean inlet velocity \(U\)	0.05 (uniform with height)
Domain (lattice)	\(64 \times 48 \times 48\), block size 8
Boundary conditions	SEM inlet (W), regularized Neumann outlet (E); \(y, z\) periodic
SEM length scale \(\sigma\)	8 (isotropic), eddy volume density 300, seed 1
Measurement plane	\(x = 8\), full \((y, z)\) span, spacing 1
Sampling	every 5 steps from step 5,000 to 30,000 (\(\approx 5{,}000\) snapshots)

Validation metrics

Reynolds-stress tensor. The notebook removes the homogeneous mean from each velocity component and forms the six independent sampled covariances over the pooled \((y, z, t)\) samples:

(3)\[\langle u'_i u'_j \rangle = \frac{1}{N_{\text{samples}}} \sum_{y, z, t} \left(u_i - \overline{u_i}\right)\left(u_j - \overline{u_j}\right).\]

The pass criterion is agreement within \(\sim 10\%\) on every component \(R_{xx}, R_{yy}, R_{zz}, R_{xy}, R_{xz}, R_{yz}\), with special attention to the shear terms \(R_{xz}\) and \(R_{yz}\) (correct sign and magnitude).

Energy spectrum. The 1D longitudinal energy spectrum \(E_{11}(k)\) of the streamwise fluctuation, computed along the homogeneous in-plane directions and averaged over the plane and time, should follow the von Karman spectral form in the inertial range von Kármán[2]:

(4)\[\frac{k\, S_u(k)}{\sigma_u^2} = \frac{4\, (k L_u / U)}{\left[1 + 70.8\, (k L_u / U)^2\right]^{5/6}},\]

where \(S_u\) is the one-sided streamwise velocity spectrum, \(\sigma_u^2 = R_{xx}\), \(L_u\) the streamwise integral length scale (set by the SEM eddy size \(\sigma\)), and \(k\) the wavenumber. The same comparison may be made against the Kaimal surface-layer spectrum Kaimal et al.[3] for an ABL context. The expected behaviour is a \(-5/3\) inertial-range slope rolling off into the energetic large scales near \(k L_u / U \sim 1\).

Note

Results pending GPU execution. This case has not yet been run; the analysis notebook is wired against the prescribed target and the export schema, but its figures and the quantitative pass/fail verdict will be filled in once the case is run on GPU. Run it with:

uv run nassu run validation/inlet/01_sem_reynolds_stress/01_sem_reynolds_stress.nassu.yaml

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[1] (1,2)

Nicolas Jarrin, Sofiane Benhamadouche, Dominique Laurence, and Robert Prosser. A synthetic-eddy-method for generating inflow conditions for large-eddy simulations. International Journal of Heat and Fluid Flow, 27(4):585–593, 2006.

[2]

Theodore von Kármán. Progress in the statistical theory of turbulence. Proceedings of the National Academy of Sciences, 34(11):530–539, 1948.

[3]

Jagadish C. Kaimal, J. C. Wyngaard, Yasushi Izumi, and O. R. Coté. Spectral characteristics of surface-layer turbulence. Quarterly Journal of the Royal Meteorological Society, 98(417):563–589, 1972.