The numerical simulation of a flow through a duct requires an externally
specified forcing that makes the fluid flow against viscous friction. To this
aim, it is customary to enforce a constant value for either the flow rate (CFR)
or the pressure gradient (CPG). When comparing a laminar duct flow before and
after a geometrical modification that induces a change of the viscous drag,
both approaches (CFR and CPG) lead to a change of the power input across the
comparison. Similarly, when carrying out the (DNS and LES) numerical simulation
of unsteady turbulent flows, the power input is not constant over time.
Carrying out a simulation at constant power input (CPI) is thus a further
physically sound option, that becomes particularly appealing in the context of
flow control, where a comparison between control-on and control-off conditions
has to be made.
We describe how to carry out a CPI simulation, and start with defining a new
power-related Reynolds number, whose velocity scale is the bulk flow that can
be attained with a given pumping power in the laminar regime. Under the CPI
condition, we derive a relation that is equivalent to the
Fukagata--Iwamoto--Kasagi relation valid for CFR (and to its extension valid
for CPG), that presents the additional advantage of natively including the
required control power. The implementation of the CPI approach is then
exemplified in the standard case of a plane turbulent channel flow, and then
further applied to a flow control case, where the spanwise-oscillating wall is
used for skin friction drag reduction. For this low-Reynolds number flow, using
90% of the available power for the pumping system and the remaining 10% for the
control system is found to be the optimum share that yields the largest
increase of the flow rate above the reference case, where 100% of the power
goes to the pump.Comment: Accepted for publication in J. Fluid Mec
