Math 2.0

Fluid mechanics Definition

Large eddy simulation

Builds on Navier–Stokes equationsTurbulenceBoussinesq eddy viscosity hypothesisStrain rate tensorReynolds decomposition

Definition

Consider an incompressible fluid with constant density ρ\rho and kinematic viscosity ν\nu in ℝ3\R^3, whose velocity uiu_i and pressure pp satisfy the Navier–Stokes equations. Let GΔG_\Delta be a filter kernel of width Δ>0\Delta > 0, an integrable function with ∫GΔ=1\int G_\Delta = 1, and write f~=GΔ∗f\tilde f = G_\Delta * f for the filtered field. Filtering the equations gives ∂u~i∂t+∂(u~iu~j)∂xj=−1ρ∂p~∂xi+νΔu~i−∂τij∂xj,∂u~i∂xi=0,\frac{\partial \tilde u_i}{\partial t} + \frac{\partial (\tilde u_i \tilde u_j)}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \tilde p}{\partial x_i} + \nu \Delta \tilde u_i - \frac{\partial \tau_{ij}}{\partial x_j}, \qquad \frac{\partial \tilde u_i}{\partial x_i} = 0, with the subgrid-scale stress τij=uiuj~−u~iu~j\tau_{ij} = \widetilde{u_i u_j} - \tilde u_i \tilde u_j. Large eddy simulation is the numerical solution of these filtered equations for u~i\tilde u_i and p~\tilde p on a grid with spacing comparable to Δ\Delta. In it, τij\tau_{ij} is replaced by a subgrid-scale model expressed in terms of the filtered fields.

Remarks

Let S~ij\tilde S_{ij} be the strain rate tensor of u~\tilde u and CsC_s a constant, typically between about 0.10.1 and 0.20.2. The most common model is Smagorinsky's eddy viscosity model, in the spirit of the Boussinesq hypothesis: τij−13τkkδij=−2νtS~ij,νt=(CsΔ)2∣S~∣,∣S~∣=2S~ijS~ij.\tau_{ij} - \tfrac13 \tau_{kk} \delta_{ij} = -2 \nu_t \tilde S_{ij}, \qquad \nu_t = (C_s \Delta)^2 \abs{\tilde S}, \qquad \abs{\tilde S} = \sqrt{2 \tilde S_{ij} \tilde S_{ij}}. The filter commutes with derivatives on ℝ3\R^3, which is why the filtered equations take this form.

Example: use a Gaussian kernel, or a sharp cutoff in Fourier space at wavenumber π/Δ\pi / \Delta, with Δ\Delta in the inertial range of the flow. Then the resolved field contains the energetic large eddies, and the model mainly has to drain energy at the right rate. Non-example: solving the mean equations of the Reynolds decomposition with an ensemble average is not large eddy simulation. That average removes all fluctuations, while the filter removes only scales smaller than Δ\Delta. Also, the filter in general does not satisfy f~~=f~\tilde{\tilde f} = \tilde f.

Reading path

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Boussinesq eddy viscosity hypothesis, concept, no evidence levelBoussinesq eddy visco…conceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptReynolds decomposition, concept, no evidence levelReynolds decompositionconceptStrain rate tensor, concept, no evidence levelStrain rate tensorconceptTurbulence, concept, no evidence levelTurbulenceconceptLarge eddy simulation, definition, no evidence levelLarge eddy simulationdefinition
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  1. Boussinesq eddy viscosity hypothesis concept
  2. Navier–Stokes equations concept
  3. Reynolds decomposition concept
  4. Strain rate tensor concept
  5. Turbulence concept
  6. Large eddy simulation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Large_eddy_simulation
About this page
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MSC76F65
DefinesLarge eddy simulation
UsesNavier–Stokes equations, Turbulence, Boussinesq eddy viscosity hypothesis, Strain rate tensor, Reynolds decomposition
Changed3 November 2026

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