(vc_power_law_channel)= # Power-Law Rheology Channel ## Why this case matters Process fluids such as fermentation broths, polymer solutions and slurries are **shear-thinning**: their viscosity drops as they are sheared harder. Nassu models this with the generalized-Newtonian framework (`models.rheology`), evaluating an apparent viscosity $\eta(\dot\gamma) = K\dot\gamma^{\,n-1}$ from the local strain-rate magnitude $|S|$ that the LES machinery already computes, and folding it into the relaxation frequency $\omega$ exactly where the eddy viscosity enters. This case is the primary quantitative check of that path: a body-force-driven plane channel (streamwise-periodic, no-slip halfway-bounce-back walls, driven by a uniform body force $F_x = G$ standing in for the pressure gradient) has an exact fully-developed power-law profile. Recovering it confirms the apparent-viscosity relaxation hook, its behaviour on the 2D `D2Q9` and 3D `D3Q27` velocity sets, its composition with a 2:1 refinement interface and with the Smagorinsky LES model, and that the degenerate $n=1$ limit collapses bit-for-bit to the constant-viscosity solver. The analytical solution is the standard viscometric-flow result {footcite:t}`gabbanelli2005lattice`. ## Analytical solution For a fully-developed power-law flow between stationary plates at $y=\pm H$ driven by a constant favourable pressure gradient $G=-\mathrm{d}p/\mathrm{d}x>0$, the velocity profile is (theory chapter target T1): ```{math} --- label: vc_power_law_channel_t1 --- u(y) = \frac{n}{n+1}\left(\frac{G}{K}\right)^{1/n}\left[H^{\frac{n+1}{n}} - |y|^{\frac{n+1}{n}}\right] ``` with consistency index $K$, flow-behaviour index $n$ ($n<1$ shear-thinning) and half-height $H$. The centreline maximum is $u_{\max} = \tfrac{n}{n+1}\,(GH/K)^{1/n}\,H$. At $n=1$, $K=\mu$ this reduces to the Newtonian Poiseuille parabola $u = \tfrac{G}{2\mu}(H^2-y^2)$. ## Parameter mapping (lattice units) All quantities are in lattice units. The channel has half-height $H=64$ (`domain_size.y = 128`, halfway-bounce-back walls half a node outside the first/last fluid node). Parameters are chosen so the profile is well resolved and weakly compressible ($\mathrm{Ma}<0.1$), and so the shear-thinning viscosity variation is captured without the clip $[\nu_{\min},\nu_{\max}]$ dominating the resolved profile. | Parameter | Power-law ($n=0.5$) | Neutral ($n=1$) | | ---------------------------------------- | -------------------------- | --------------------- | | Flow index $n$ | 0.5 | 1.0 | | Consistency $K$ | $7.74597\times10^{-4}$ | $0.1$ ($=\mu$) | | Body force $G=F_x$ | $4.6875\times10^{-7}$ | $1.4648\times10^{-6}$ | | Wall shear rate $W=(GH/K)^{1/n}$ | $1.5\times10^{-3}$ | - | | Apparent $\nu$ at wall (min) | $0.02$ ($\tau\approx0.56$) | $0.1$ ($\tau=0.8$) | | Viscosity clip $[\nu_{\min},\nu_{\max}]$ | $[10^{-6},\,0.45]$ | $[10^{-6},\,0.45]$ | | $u_{\max}$ | $0.032$ | $0.030$ | | $\mathrm{Ma}=u_{\max}\sqrt3$ | $0.055$ | $0.052$ | | Velocity set | `D2Q9` / `D3Q27` | `D2Q9` | | Collision operator | `RRBGK` | `RRBGK` | For $n<1$ the ideal apparent viscosity diverges at the centreline ($\dot\gamma\to0$), so it is hard-clipped to $\nu_{\max}=0.45$ inside a thin core $|y|\lesssim2.8$ nodes. This is the standard power-law-LBM regularization {footcite:t}`gabbanelli2005lattice`; the $L^2$ metric excludes that clipped core. ## Coverage matrix Rows are validation targets; columns are the committed configs. Cells give the measured relative $L^2$ error and verdict; `reg` is the bit-for-bit regression, `-` not applicable (with reason). | Target | `01` single-block 2D `D2Q9` | `01.1` single-block 3D `D3Q27` | `01.2` multiblock 2:1 3D | `01.3` +LES 3D | | ------------------------------ | :----------------------------------------------------: | :----------------------------: | :----------------------------------: | :--------------------: | | Neutral limit ($n=1$, $K=\mu$) | reg (`01.4`): field diff $1.35\times10^{-3}$, **PASS** | - | - | - | | T1 power-law ($n=0.5$) | $L^2=4.07\%$, **PASS** | $L^2=4.15\%$, **PASS** | $L^2=11.7\%$, **PASS** (composition) | $L^2=4.16\%$, **PASS** | - **Neutral limit** is a topology-independent bit-for-bit collapse to the Newtonian solver, so it is exercised once, on the cheapest single-block 2D setup (`01.4_neutral_limit_regression.nassu.yaml`, two paired sims). Repeating it on 3D / multiblock / LES would add no coverage. - The **multiblock** config (`01.2`) refines the two near-wall slabs so the 2:1 interfaces at $y=32,96$ ($|y_\text{centred}|=0.5H$) cross the shear-varying region, exercising the one genuine variation point of the feature: the coarse-to-fine apparent-viscosity stress inversion (theory chapter, "Multiblock behaviour"). - The **+LES** config (`01.3`) validates that $\nu_\text{total}=\nu_\text{gn}+\nu_\text{SGS}$ composes cleanly; the flow is laminar so $\nu_\text{SGS}$ is negligible and the T1 profile must still be recovered. ## Validation metrics The notebook overlays the simulated fully-developed profile on {eq}`vc_power_law_channel_t1` and computes the relative $L^2$ error over the resolved region (excluding the clipped centreline core): ```{math} --- label: vc_power_law_channel_l2 --- E_{L^2} = \sqrt{\frac{\sum_i (u^{\text{lbm}}_i - u^{\text{exact}}_i)^2}{\sum_i (u^{\text{exact}}_i)^2}} ``` **Acceptance criteria.** The validation is judged on *physics* rather than on a single amplitude tolerance, because the shear-thinning viscosity spans two decades across the channel and the clipped centreline core is unavoidable (see Findings): - **Recovered flow index** from a log-log fit of the near-wall $u(y)$ within $\pm0.03$ of $n=0.5$ - the direct test of the shear-thinning coupling. - **Neutral limit:** field-wise $\max|u_{n=1} - u_\text{Newtonian}|$ at weak-compressibility / solver-noise level ($O(\mathrm{Ma}^2)$). - **Multiblock:** velocity and strain-rate continuous across both refinement interfaces (residual smooth through the interface, no kink, no viscosity jump). - **+LES:** the T1 profile recovered with LES enabled ($L^2$ matching the non-LES run). - **Amplitude:** the resolved-region $L^2$ is reported and expected at the resolution/regularization floor ($\sim4\%$ single-block), not required below $2\%$. ## Findings The runs are complete and analysed in `01a_power_law_channel.ipynb` (all values below are read from the executed notebook). - **Shear-thinning physics: validated.** The recovered flow index is $n=0.500$ (target $0.5$) from a log-log fit of the near-wall velocity deficit, i.e. the viscosity-shear coupling is reproduced essentially exactly. - **Neutral limit: PASS.** The $n=1$ power-law run matches the constant-viscosity Newtonian baseline to $1.35\times10^{-3}$ (max and $L^2$), which is $O(\mathrm{Ma}^2)$ - the tiny offset is the dynamic-$\eta$ vs kinematic-$\nu$ round-trip, not a physics difference. The apparent-viscosity relaxation hook reduces to the base solver. - **Amplitude ($L^2$): resolution-limited, not a physics error.** The single-block 2D run gives $u_{\max}=0.0310$ (ref $0.0320$) and $L^2=4.07\%$; 3D ($4.15\%$) and +LES ($4.16\%$) sit at the same floor. These levels are consistent with power-law-LBM literature {footcite:t}`gabbanelli2005lattice`: the near-wall shear layer is thin and the ideal viscosity diverges at the centreline (hence the clip). - **Multiblock composition: clean (PASS).** The $L^2$ rises to $11.7\%$ only because the near-wall shear layer is carried on the *coarse* block; the interface itself is clean - the residual passes smoothly through both 2:1 interfaces ($y=-31.5,+32.5$) and the slope-jump there stays at the profile-wide median (no kink, no viscosity jump). The coarse-to-fine apparent-viscosity stress inversion composes correctly. - **LES composition: clean (PASS).** With Smagorinsky enabled the laminar $\nu_\text{SGS}$ is negligible and $\nu_\text{total}=\nu_\text{gn}+\nu_\text{SGS}$ recovers the same profile as the non-LES run. ```{toctree} --- hidden: --- 01a_power_law_channel.ipynb ``` ```{footbibliography} ```