# Spille-Kaltenbach Control Planes

Synthetic turbulent methods provide a reasonable first estimate of the fluctuating turbulent velocity field at the LES inlet. Downstream of the inlet, however, many of the relevant turbulent scales may have been dissipated retarding the transition to fully turbulent flow. Local control planes which introduce a volumetric forcing term to the Navier-Stokes equations regulate the turbulent production in the shear stress budget [24]. As discussed, for example, in the work of Keating et al. [14] or Zhang et al. [29], local flow events such as bursts and sweeps are enhanced or damped by the local forcing thus contributing to the Reynolds shear stress (u’v’)

e (y, t) = (u’v’)*’ (x0,y) – (u’V)z, t (x0,y, t) (12)

where (u’v’)* is the target Reynolds shear stress at the control plane which is provided by the RANS solution and (u’v’)z, t is the current Reynolds shear stress in the LES domain which is averaged over the spanwise direction and time. For the time average a window function with a time constant equal to « 10050/u,5 is used. The force magnitude is given by

f (xq, y, z, t) = r (y, t) [u (xq, y, z, t) – (u)z’‘]

with

r (y, t) = ae (y, t) + в e(y, t’) dt’ . (14)

The proportional part is the main contributor to the force when the error e in Eq. 12 is high at the beginning of the simulation. Proceeding in time, the integral part gives the force the necessary response to enhance or damp the local flow events. The constants a and в were set to 10 and 25 respectively, to ensure on the one hand, a rapidly decreasing error e and on the other hand, a stable simulation process. In subsequent sections the STGM of Jarrin et al. combined with the control plane approach is referred to as ’zonal SEM’ and the STGM of Batten et al. combined with the control plane approach is referred to as ’zonal Batten’.

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