(theory_LES)= # Large-Eddy Simulation Almost every flow of engineering interest is turbulent. Air and water have low viscosity, so even modest velocities and length scales push the Reynolds number high enough that the flow carries eddies spanning a wide range of sizes, from the largest structures set by the geometry down to millimetre scales, and all of them matter: the large eddies set the bulk transport and unsteady loads, the small ones set the mixing and dissipation. The central problem is that you cannot afford to compute all of these scales at once. Large-eddy simulation (LES) is the compromise that makes the problem tractable. It computes the large, energy-carrying eddies directly and replaces the small, universal eddies with a model. This page explains why that split is the natural one, where to place the cut, and why LES is the right level of fidelity for high-Reynolds-number flows. The discussion is illustrated throughout with computational wind engineering (CWE), the application Nassu is built for, where the approaching atmospheric boundary layer makes the scale range especially wide. ## The energy cascade Turbulence is not a single phenomenon at a single scale. It is a population of eddies spanning a wide range of sizes that constantly exchange energy. Energy enters the flow at the large scales, where the mean flow is unstable and sheds the biggest eddies. Those eddies are themselves unstable and break into smaller ones, which break into smaller ones still. This one-way transfer of kinetic energy from large to small scales is the **energy cascade** {footcite:p}`davidson2015turbulence`. The cascade ends only when the eddies are small enough that viscosity can act on them. At that point, viscous friction converts their kinetic energy into heat. The size at which this happens is the **Kolmogorov scale** $\eta$, the smallest eddy that survives in the flow {footcite:p}`kolmogorov1941local`. The decisive fact for simulation is how far apart the largest and smallest scales are. Kolmogorov's 1941 theory predicts that the ratio of the integral length scale $L$ (the size of the energy-carrying eddies) to the dissipation scale $\eta$ grows with the Reynolds number as $$ \frac{L}{\eta} \sim Re^{3/4} $$ (les_scale_separation) At high Reynolds number, $Re \sim 10^{6}$ or more (the regime of full-scale wind engineering, but also of vehicles, turbomachinery and most industrial aerodynamics), the scale separation is enormous. Resolving every scale in three dimensions and in time would require on the order of $Re^{3}$ grid-time operations, which is far beyond reach at that scale. Equation {math:numref}`les_scale_separation` is the reason a direct attack on the problem is hopeless and a modelling strategy is unavoidable. ```{admonition} Key idea --- class: note --- The cost of resolving every eddy scales like $Re^{3}$. Each tenfold rise in Reynolds number multiplies the work a thousandfold, so at engineering Reynolds numbers fully resolved simulation is far out of reach for the foreseeable future. This single fact, not a lack of accurate equations, is what forces a modelling strategy. ``` ## The energy spectrum The cascade is most clearly read from the **energy spectrum** $E(\kappa)$, which reports how much turbulent kinetic energy resides at each wavenumber $\kappa$ (small $\kappa$ = large eddies, large $\kappa$ = small eddies). Consider the spectrum below. ```{figure} /_static/img/theory/les_vs_dns_spectrum.svg --- align: center width: 70% --- The energy spectrum $E(k)$: a low-wavenumber energy-containing hump, a straight $-5/3$ inertial subrange, and a steep high-wavenumber dissipation roll-off. A DNS resolves all of it; an LES resolves only the scales below the grid cutoff $k_c$ and models the rest. ``` Three regions appear. At the lowest wavenumbers sit the **energy-containing eddies**: large, anisotropic, and shaped by the specific geometry of the flow (the body, the terrain, the approaching boundary layer). These carry most of the kinetic energy and are not universal, so they must be computed for each case. At the highest wavenumbers lies the **dissipation range**, where viscosity drains the cascade. Between the two is the **inertial subrange**, where eddies are too large to feel viscosity directly yet too small to remember the geometry that created them. Here the only active process is the cascade itself: energy is handed from one scale to the next at a constant rate $\varepsilon$. Kolmogorov argued that this transfer is statistically universal, which fixes the slope of the spectrum at the celebrated $-5/3$ power law {footcite:p}`kolmogorov1941local,pope2000turbulent`: $$ E(\kappa) = C_K\, \varepsilon^{2/3}\, \kappa^{-5/3} $$ (les_kolmogorov_spectrum) with $C_K \approx 1.5$ the Kolmogorov constant. The universality of the inertial subrange is what makes LES work: because the small scales behave the same way in every high-Reynolds-number flow, they can be captured by a single, geometry-independent model rather than computed explicitly. ```{admonition} Inertial subrange vs dissipation range --- class: note --- The inertial subrange is the scale range over which energy is only transferred from large to small eddies (the cascade), with negligible direct viscous dissipation. The actual viscous dissipation happens at smaller scales still, in the dissipation range at the right of the spectrum. ``` ## DNS, RANS and LES: choosing the altitude There are three established ways to confront turbulence numerically, distinguished entirely by how much of the spectrum they resolve. **Direct numerical simulation (DNS)** resolves the whole spectrum, from $L$ down past $\eta$, with no turbulence model at all. It is the most accurate option and the reference against which models are judged, but by Equation {math:numref}`les_scale_separation` its cost scales like $Re^{3}$. At the Reynolds numbers of full-scale engineering this is decades out of reach, so DNS is a tool for canonical low-Reynolds-number flows, not for engineering design {footcite:p}`pope2000turbulent`. **Reynolds-averaged Navier-Stokes (RANS)** sits at the opposite extreme. It resolves none of the turbulent fluctuations: it solves for the time-averaged flow and models the effect of the *entire* turbulence spectrum through a closure such as $k$-$\varepsilon$. RANS is cheap and adequate for attached, mildly separated flows, but it is consistently weak where it matters most, namely the massively separated, unsteady wakes behind bluff bodies, where it cannot represent the large-scale vortex shedding that drives peak and fluctuating loads {footcite:p}`pope2000turbulent,sagaut2006large`. **Large-eddy simulation (LES)** is the middle ground, and it is the level Nassu targets. The reasoning is a direct consequence of the spectrum: - The large, energy-containing eddies are case-specific, carry the transport and the loads, and are comparatively few. **Resolve them.** - The small, inertial- and dissipation-range eddies are universal and only act to drain energy. **Model them**, with a single closure that needs to reproduce the right dissipation rate rather than the exact small-scale motion. This division places the computational effort exactly where the physics is case-dependent and removes it from where the physics is universal. The cut between resolved and modelled scales should fall in the **lower part of the inertial subrange**: far enough up the spectrum that the resolved eddies include all the energetic, geometry-dependent motion, and low enough that the modelled scales lie in the universal range where a simple closure is valid {footcite:p}`sagaut2006large`. Placing the cut too low leaves energetic eddies unresolved and corrupts the loads; placing it too high wastes grid points resolving scales a model could have handled. ```{admonition} DNS, RANS or LES --- class: important --- The three methods differ only in how much of the spectrum they resolve. DNS resolves everything and models nothing, at $Re^{3}$ cost. RANS resolves nothing and models the whole spectrum, missing the large-scale unsteadiness behind bluff bodies. LES resolves the energetic, case-specific large eddies and models only the universal small ones. For the separated, unsteady wakes that set wind-engineering loads, LES is the least expensive method that still captures the physics that matters. ``` This trade-off is decisive for bluff-body aerodynamics, and CWE in particular. The phenomena that govern the loads, namely separation, reattachment, vortex shedding and gust response, live in the large resolved scales, so LES captures them directly. The remaining small scales are handled by a subgrid model at a fraction of the DNS cost. The pages that follow build this idea up in order: first what the filtering operation that defines the resolved/subgrid split actually is and the closure problem it creates, and then the subgrid models that close it. ```{toctree} --- hidden: true maxdepth: 1 --- Filtering Subgrid Models ``` ```{eval-rst} .. footbibliography:: ```