Introduction to classical large eddy simulation (LES) of turbulent flows Andrés E. Tejada-Martínez Center for Coastal Physical Oceanography Old Dominion University

Outline - introduction to spatial filters Part I: Theory Part II: Computations - introduction to spatial filters - equations governing the large eddies (filtered (LES) equations) - importance of numerical discretization (i.e. the numerical solver) - subgrid-scale (SGS) models/approximations in the filtered equations LES of isotropic turbulence and unstratified/stratified channel flows LES of Langmuir turbulence - animations (flow over an airfoil, flow over a cavity)

Steps in large-eddy simulation (LES) The Navier-Stokes equations are filtered with a low-pass filter Filtering presents a closure problem as an unknown residual (subgrid-scale) stress which may be modeled or approximated The modeled filtered Navier-Stokes equations governing the largest scales (or the large eddies) are numerically solved. - filtering attenuates smallest (residual) scales, while preserving largest - stress represents effect of attenuated smallest scales on largest scales

Large-eddy simulation large eddies resolved in LES

Large-eddy simulation Residual scales not resolved in LES, and must be modeled Residual scales are modeled under assumption of isotropy

Filtering in real space A filtered function is defined as: Examples of homogeneous filter kernels, , are Filtering attenuates scales less than and splits the function as 1/2h 1/h box hat y y x-h x x+h x-h x x+h residual (small) component large component

Filtering in real space Scales < Note that in general except for the sharp cutoff filter Filtering attenuates or removes (depending on the shape of the filter kernel) scales on the order of the filter width:

Filter kernels in real space The width of G(r) may be defined with respect to the box filter as See Turbulent Flows by S.B. Pope for functional forms of kernels

Filters in Fourier space: transfer function The Fourier transform (F.T.) of a filtered function is the transfer function of the filter multiplied by the F.T. of the un-filtered function: For the sharp cutoff filter but in general this is not true

Sketch of filtered energy spectra Both filters leave low wavenumber content untouched ln(E(k)) un-filtered spectrum In the low wavenumber range filtered spectrum using cutoff filter filtered spectrum using box filter un-filtered spectrum In the high wave- number range ln(k)) Filtering with the box filter leads to an attenuation of scales around Filtering with the sharp cutoff filter preserves scales at less than and completely erases scales at higher wavenumbers

LES and other approaches For a turbulent flow, the Navier-Stokes (N-S) equations contain a large number of scales. While solving these equations numerically, the grid must contain a great number of points in order to represent (resolve) all of the scales present. In LES we filter the equations, thereby suppressing the smaller scales. With fewer scales, the filtered equations need less grid points. In direct numerical simulation (DNS) no filtering is performed, as the simulation attempts to represent all scales down to the dissipative ones In Reynolds-averaged N-S simulation (RANSS), we solve the ensemble-averaged N-S. Averaging suppresses all of the scales except for the largest, thus RANSS requires much fewer grid points than LES and DNS. LES resolves many more scales than RANSS, but not as many as DNS. No. of grid points

Unfiltered equations

Unfiltered equations

Filtered momentum equation Filter the momentum eqn. with an arbitrary homogenous filter of width . Homogeneity of filter allows commutation with differentiation:

Filtered momentum equation Filter the momentum eqn. with an arbitrary homogenous filter of width . Homogeneity of filter allows commutation with differentiation:

Filtered momentum equation Filter the momentum eqn. with an arbitrary homogenous filter of width . Homogeneity of filter allows commutation with differentiation: leads to is an unknown stress accounting for the effect of the filtered-out small scales on the large scales governed by the filtered equation

Residual (subgrid-scale (SGS)) stress Note that in general: Decompose the SGS stress as

Residual (subgrid-scale (SGS)) stress Note that in general: Decompose the SGS stress as

Residual (subgrid-scale (SGS)) stress Note that in general: Decompose the SGS stress as deviatoric (trace-free) component isotropic component

Residual (subgrid-scale (SGS)) stress Note that in general: Decompose the SGS stress as deviatoric (trace-free) component isotropic component This decomposition leads to The modified filtered pressure contains the isotropic part of the SGS stress

Filtered equations

Filtered equations SGS stress SGS stress:

Filtered equations SGS stress: SGS density flux: SGS stress (obtained in same way as the SGS stress)

Comments on the filtered equations The filtered equations are numerically solved for the filtered variables describing the large scales The SGS stress and SGS density flux present closure problems and must be modeled or approximated in terms of filtered variables only In theory, the filter used to obtain the filtered equations is arbitrary In practice, the filter is inherently assumed by the discretization (i.e. the numerical method used to solve the filtered equations and the SGS models) The discretization can only represent (resolve) down to scales on the order of 1,2, or 3 times the grid cell size, h, thereby “filtering-out” smaller scales.

Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales ln(k))

Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales spectrum based on scales in inertial range ln(k))

Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales spectrum based on scales in inertial range resolved (large) scales sub-grid (residual) scales ln(k)) We choose the grid size, h, to fall within the inertial range to facilitate SGS modeling

Role of discretization in LES Both A and B do a good job representing low wavenumber spectrum of ln(E(k)) Spectrum of Spectrum of obtained with discretization A Spectrum of obtained with discretization B ln(k)) Discretization A behaves more like a sharp cutoff filter, while B behaves more like a box filter Ideally we would aim for a discretization like A

Smagorinsky SGS model Recall that the SGS stress and density buoyancy flux must be modeled or approximated

Smagorinsky SGS model Recall that the SGS stress and density buoyancy flux must be modeled or approximated Both are trace-free Smagorinsky (1967) model:

Smagorinsky SGS model Recall that the SGS stress and density buoyancy flux must be modeled or approximated Both are trace-free Smagorinsky (1967) model: Smagorinsky coefficient

Smagorinsky SGS model Recall that the SGS stress and density buoyancy flux must be modeled or approximated Both are trace-free Smagorinsky (1967) model: Smagorinsky coefficient Analogously:

The eddy (turbulent) viscosity The turbulent viscosity has units . Because we are working with the smallest resolved scales, we can set In LES the SGS range starts at the inertial range, thus we may invoke Kolmogorov’s 2nd hypothesis: Statistics of scales of size, say, within the inertial range have a universal form uniquely determined by the rate of energy transfer, And we may have In a global sense, the rate of energy transfer within the inertial range is roughly equal to the SGS dissipation. Here we assume it locally:

Difficulties with the Smagorinsky model For isotropic turbulence, Lilly (1967) showed that Major difficulty: The constant coefficient allows for a non-vanishing turbulent viscosity at boundaries and in the presence of relaminarization The Smagorinsky coefficient should be a function of space and time In 1991, Germano and collaborators derived a dynamic expression for the Smagorinsky coefficient

Dynamic Smagorinsky model Recall filtering the N-S equations with an homogeneous filter of width

Dynamic Smagorinsky model Recall filtering the N-S equations with an homogeneous filter of width Consider a new filter made up from successive applications of the 1st filter (above) and a new “test” filter. This “double” filter has width Application of this “double” filter is denoted by a “bar-hat” in the form of

Dynamic Smagorinsky model Recall filtering the N-S equations with an homogeneous filter of width Consider a new filter made up from successive applications of the 1st filter (above) and a new “test” filter. This “double” filter has width Application of this “double” filter is denoted by a “bar-hat” in the form of With this new filter, the filtered momentum equation becomes:

Dynamic Smagorinsky model Recall filtering the N-S equations with an homogeneous filter of width Consider a new filter made up from successive applications of the 1st filter (above) and a new “test” filter. This “double” filter has width Application of this “double” filter is denoted by a “bar-hat” in the form of With this new filter, the filtered momentum equation becomes:

Dynamic Smagorinsky model Recall filtering the N-S equations with an homogeneous filter of width Consider a new filter made up from successive applications of the 1st filter (above) and a new “test” filter. This “double” filter has width Application of this “double” filter is denoted by a “bar-hat” in the form of With this new filter, the filtered momentum equation becomes: Scale invariance: Both and are in the inertial range, thus

Dynamic Smagorinsky model Consider the following tensor proposed by Germano :

Dynamic Smagorinsky model Consider the following tensor proposed by Germano : (resolved)

Dynamic Smagorinsky model Consider the following tensor proposed by Germano : (resolved) (modeled)

Dynamic Smagorinsky model Consider the following tensor proposed by Germano : (resolved) (modeled) Minimization of the difference between these two with respect to Cs leads to: - Averaging in statistically homogenous direction(s)

Dynamic Smagorinsky model Consider the following tensor proposed by Germano : (resolved) (modeled) Minimization of the difference between these two with respect to Cs leads to: - Averaging in statistically homogenous direction(s) Explicit application of test filter (denoted by a “hat”) is required, unlike 1st filter

Dynamic Smagorinsky model Sketch of spectra ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on ln(k))

Dynamic Smagorinsky model Sketch of spectra ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on Spectrum based on ln(k))

Dynamic Smagorinsky model Sketch of spectra Subtest scales ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on Spectrum based on ln(k)) By applying the test filter, the Germano formulation samples the field between the subgrid scales and the subtest scales in order to obtain the model coeff.

Dynamic mixed model Recall and true SGS stress subgrid component Inserting the former into the latter leads to + subgrid-scale terms The subgrid-scale terms can be approximated via the Smagorinsky model A dynamic coefficient in the Smagorinky model can be derived here as well This mixed approach leads to a modeled SGS stress better correlated with the true SGS stress. Both approaches lead to good correlation with true SGS energy dissipation

LES methodology used in computations SGS stress SGS density flux SGS stress model: SGS density flux model: Model coefficients in SGS models are computed dynamically as described

Numerical scheme used in computations Horizontal derivatives (in x and y) are treated spectrally Vertical (z-) derivatives are treated with 6th or 5th order implicit stencils To prevent spurious high wavenumber content not resolvable by the grid, advection terms are: The high order accuracy of this discretization allows for it to behave like the sharp cutoff filter - restriction to periodic boundaries in x and y - allows Dirichlet and Neumann boundaries in z 1. de-aliased in x and y 2. filtered in z with a high order implicit filter

LES of decaying isotropic turbulence Isotropic turbulent fluctuations decay in time due to viscous dissipation in the absence of energy source Comte-Bellot & Corrsin (1971) studied this flow by passing air at U=10 m/s through a bar grid with cells of size M = 2 in. Data in the form of energy spectra is available at tU/M=42,98 Each side of computational box is 55cm. Grid is 33x33x33 periodic. Size of smallest resolved scales about 35 x (size of dissipative scales (0.07cm)) Energy spectrum of initial condition matches that recorded in experiment at tU/M=42. Numerical solution will be compared to data at tU/M=98.

Decaying isotropic turbulence behind a bar grid isotropic far field

Effect of sub-grid scale (SGS, residual) model 3-D energy spectra

Geometry and grid for LES of channel Channel geometry: Reynolds No. based on friction velocity, Periodicity in x and y. No-slip velocity and fixed density at walls Grid is 33x33x65 (# of points in x, y, and z), stretched in z. DNS of Kim, Moin & Moser (192x160x129) is compared with our results z x y

Body force Flow is driven by a body force. There are 2 ways to determine this force: 1. Force control to achieve desired z x

Body force Flow is driven by a body force. There are 2 ways to determine this force: 1. Force control to achieve desired z x 2. Mass control to achieve desired - Body force is dynamically adjusted towards desired bulk velocity

Basic relationships Recall the classical Reynolds decomposition and let

Basic relationships Recall the classical Reynolds decomposition and let In LES:

Basic relationships Recall the classical Reynolds decomposition and let turbulence intensities (not subgrid components) In LES:

Basic relationships Recall the classical Reynolds decomposition and let turbulence intensities (not subgrid components) In LES: We want to study: resolved Reynolds shear stress

Wall forces in unstratified channel flow Theoretical mean force = 0.435 DNS mean force = 0.426

Mean velocity in unstratified channel

Variances in unstratified channel flow

Shear stresses in unstratified channel

1-D spectra in unstratified channel Spectra obtained from autocorrelation in x in the LES with force control Our high order discretization behaves like a sharp cutoff filter

1-D spectra in unstratified channel Results from Najjar and Tafti, Physics of Fluids, 1996 Ref. 34 is DNS data used earlier from Kim, Moin & Moser, J. Fluid Mech., 1987 Cases 9, 10, and 11 use the same # of grid points as our LES. Low order discretization used by Najjar and Tafti behaves like a box filter as spectra deviates from DNS data at around kx=4.

Stratification effects

Stratification effects strength of stratification Ri based on bulk velocity Ri based on friction velocity

Stratification effects strength of stratification Ri based on bulk velocity Ri based on friction velocity Ri based on friction velocity is more convenient to track We will look at with fixed density at top and bottom

Wall Forces Theoretical mean force = 0.435 DNS mean force = 0.426

High/low speed streaks near wall y x Fully turbulent Laminarized

Mean velocity

Instantaneous streamwise velocity contours z Ri = 0 x Ri = 60

Root mean squares of fluctuations

Shear stress and turbulence structure

Nusselt numbers Nusselt No. = measure of wall mass transport due to turbulence

Density statistics

Instantaneous density contours z Ri = 0 x Ri = 60

Mixing Efficiency

Observations Stratification suppresses turbulence intensities. Stratification leads to a density interface separating density into two layers. Within the two layers density is well mixed Stratification leads to a quasi-periodic flow structure in the core co-existing with turbulent structures near the boundaries

Langmuir cells in wind-driven channel z x no-slip wall

Langmuir cells in wind-driven channel z x no-slip wall Surface stress is applied such that Craik-Leibovich (C-L) vortex forcing is added to the filtered momentum equation to account for Langmuir cells

Langmuir cells in wind-driven channel z x no-slip wall Surface stress is applied such that Craik-Leibovich (C-L) vortex forcing is added to the filtered momentum equation to account for Langmuir cells Simulations are distinguished by the turbulent Langmuir number - Stokes drift vel. in vortex forcing

Langmuir cells in wind-driven channel z x no-slip wall Surface stress is applied such that Craik-Leibovich (C-L) vortex forcing is added to the filtered momentum equation to account for Langmuir cells Simulations are distinguished by the turbulent Langmuir number - Stokes drift vel. in vortex forcing - Simulations with and use (33x33x65)-grid with z-stretching - Simulation with uses (33x33x97)-grid with z-stretching

Mean velocity Langmuir cells tend to homogenize the upper water column

Variances Langmuir cells increase horizontal fluctuations in the upper water column

Time-averaged x-component of vorticity z/h y/h Counter rotating cells are seen in the case with C-L vortex forcing,

Engineering applications of LES LES of flow around an airfoil Simulation by Kenneth Jansen, at Rensselaer Polytechnic Institute Velocity contours on 4 planes parallel to wing and 1 plane normal to wing High/low speed streaks appear on parallel planes closer to wing