Initial and boundary conditions for LES In LES computations the unsteady Navier–Stokes equations are solved, so

3.8.5 Initial and boundary conditions for LES In LES computations the unsteady Navier–Stokes equations are solved, so

suitable initial and boundary conditions must be supplied to generate a well- posed problem.

Initial conditions For steady flows the initial state of the flow only determines the length of time

required to reach the steady state, and it is usually adequate to specify an initial field that conserves mass with superimposed Gaussian random fluctuations with the correct turbulence level or spectral content. If the development of a time-dependent flow depends on its initial state it is necessary to specify it more accurately using data from other sources (DNS or experiments).

Solid walls The no-slip condition is used if the LES filtered Navier–Stokes equations

are integrated to the wall, which requires fine grids with near-wall grid points y + ≤ 1. For high Reynolds number flows with thin boundary layers it is necessary to economise on computing resources by means of graded non- uniform meshes. As an alternative it is possible to use wall functions. Schumann (1975) proposed a model that takes the fluctuating shear stress to

be in phase with the fluctuating velocity parallel to the wall and links the shear stress to the instantaneous velocity through logarithmic wall functions of the same type as equation (3.49) used in the RANS k– ε model and RSM.

Moin (2002) reviewed an advanced method of dynamically computing von Karman’s proportionality constant κ in his near-wall RANS mixing length model by matching the values of the RANS and LES eddy viscosities at matching points. This avoids excessive shear stress predictions associated with the standard value κ = 0.41.

Inflow boundaries Inflow boundary conditions are very challenging since the inlet flow propert-

ies are convected downstream, and inaccurate specification of the inflow

3.8 LARGE EDDY SIMULATION 107

boundary condition can strongly affect simulation quality. The simplest method is to specify measured mean velocity distributions and to super- impose Gaussian random perturbations with the correct turbulence intensity, but this ignores the cross-correlations between velocity components (Reynolds stresses) and two-point correlations (i.e. spatial coherence) in real turbulent flows. Distortions of turbulence properties can take considerable settling distance before the mean flow reaches equilibrium with the turbu- lence properties, and the settling distance is problem dependent. Several alternatives are available:

1 Represent the inlet flow with the correct geometry using a RANS

turbulence model. The commonest method is to perform an unsteady flow calculation with the RSM to obtain estimates of all the Reynolds stresses at the inlet plane and impose these by maintaining correct values of the relevant autocorrelations and cross-correlations during the generation of the Gaussian random perturbations.

2 Extend the computational domain further upstream and use a

turbulence-free inflow (by developing the flow from a large reservoir). This requires a long upstream distance, typically of the order of

50 hydraulic diameters, until a fully developed flow is reached, but is feasible if an inlet flow with thin boundary layers is required.

3 Specify a fully developed inlet profile as the starting point for internal

flows in complex geometry. Such profiles can be economically computed from an auxiliary LES computation with streamwise periodic boundaries (see below).

4 Specify a precise inlet profile with prescribed shear stress, momentum thickness and boundary layer thickness. Lund et al. (1998) proposed a technique to extract inlet profiles for developing boundary layers from auxiliary LES computations. Other methods with this objective have been developed by Klein et al. (2003) and Ferrante and Elgobashi (2004). The former is based on digital filtering of random data and the specification of length scales in each co-ordinate direction to generate two-point correlations. The latter proposes a refinement of the Lund et al. procedure to ensure that the correct spectral energy distribution is reproduced across the wavenumbers. Both algorithms are reported to reduce the settling length between the inflow boundary and the location in the computational domain of the actual LES calculations where the turbulence reaches equilibrium with the mean flow.

Outflow boundaries Outflow boundary conditions are less troublesome. The familiar zero gradient

boundary condition is used for the mean flow, and the fluctuating properties are extrapolated by means of a so-called convective boundary condition:

∂φ ∂φ +R n =0 ∂t

∂n Periodic boundary conditions

All LES and DNS calculations are three-dimensional because turbulence is three-dimensional. Periodic boundary conditions are particularly useful in directions where the mean flow is homogeneous (e.g. the z-direction in

CHAPTER 3 TURBULENCE AND ITS MODELLING

two-dimensional planar flow). All properties are set to be equal at equivalent points on pairs of periodic boundaries. The distance between the two peri- odic boundaries must be such that two-point correlations are zero for all points on a pair of periodic boundaries. This means that the distance should

be chosen to be at least twice the size of the largest eddies so that the effect of one boundary on the other is minimal.