  Flow over a surface can be stratified by imposing a fixed mean vertical
temperature (density) gradient profile throughout or via cooling at the
surface. These distinct mechanisms can act simultaneously to establish a stable
stratification in a flow. Here, we perform a series of direct numerical
simulations of open-channel flows to study adaptation of a neutrally stratified
turbulent flow under the combined or independent action of the aforementioned
mechanisms. We force the fully developed flow with a constant mass flow rate.
This flow forcing technique enables us to keep the bulk Reynolds number
constant throughout our investigation and avoid complications arising from the
acceleration of the bulk flow when a constant pressure gradient approach were
to be adopted to force the flow instead. When both stratification mechanisms
are active, the dimensionless stratification perturbation number emerges as an
external flow control parameter, in addition to the Reynolds, Froude, and
Prandtl numbers. We demonstrate that significant deviations from the
Monin-Obukhov similarity formulation are possible when both types of
stratification mechanisms are active within an otherwise weakly stable flow,
even when the flux Richardson number is well below 0.2. An extended version of
the similarity theory due to Zilitinkevich and Calanca shows promise in
predicting the dimensionless shear for cases where both types of stratification
mechanisms are active, but the extended theory is less accurate for gradients
of scalar. The degree of deviation from neutral dimensionless shear as a
function of the vertical coordinate emerges as a qualitative measure of the
strength of stable stratification for all the cases investigated in this study.
