Transition to stably stratified states in open channel flow with radiative surface heating

Abstract

Direct numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer-Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height δ to L , a bulk Obukhov length scale for the flow. We cover the range λ = δ/L = 0-2.0, from neutral conditions to the onset of the stable regime, with the Reynolds number range Re τ = 200-800, at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At λ 1 the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance l ν from the surface, with l ν /δ ∼ Re −1/2 . In this equilibrium region the flow can be characterised by the flux Richardson number R f and the local Obukhov length scale Λ. At higher λ local mixing limit conditions are observed over an extended region. At λ = 2 the flux Richardson number approaches critical limit values of R f ,c 0.18 and gradient Richardson number Ri c 0.2. At high λ, we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter Re L ,c = L u τ /ν 200-400 (where u τ is the friction velocity and ν the kinematic viscosity), which is related to the L * parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241-259) and buoyancy Reynolds number R. With energetic equilibrium attained, the local buoyancy Reynolds number, Re Λ = Λ u w 1/2 /ν, is directly related to the separation of the Ozmidov (l O ) and Kolmogorov (η) length scales in the outer boundary layer by Re Λ R ≡ (l O /η) 4/3 . The inner wall region has the behaviour R ∼ Re L Re τ , in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and R ∼ Re L .

Evolution of thermally stratified turbulent open channel flow after removal of the heat source

“…We use the framework for the radiatively heated equilibrium state flow described by Williamson et al. (2015). A schematic of the flow is shown in figure 1.…”

“…Williamson et al. (2015) define a non-dimensional bulk stability parameter, where is a bulk Obukhov length scale defined as Here is gravitational acceleration and the coefficient of thermal expansion, which relates fluid density to temperature through . This formulation of the Obukhov length scale results from the current context in which the heat flux into the domain takes the form of a volumetric heat source given in (2.1), which leads to Combining this with (2.8) gives an alternative expression for , which has the same form as a friction Richardson number.…”

“…In previous work Williamson et al. (2015) studied the statistically steady state reached by a turbulent open channel flow subjected to radiative heating. The radiative heating was modelled using a volumetric heat source following the Beer–Lambert law.…”

Destratification of thermally stratified turbulent open-channel flow by surface cooling

“…Williamson et al (2015) show that the statistically steady heated equilibrium state can be defined in terms of a friction Reynolds number , Prandtl number , a stability parameter and a dimensionless attenuation coefficient , defined as Here, is the friction velocity at the channel bottom, the kinematic viscosity, the thermal diffusivity, the coefficient of thermal expansion, gravitational acceleration and a characteristic temperature scale defined as where is a characteristic volumetric heat source associated with the radiative forcing.…”

LES mesh resolution requirements for particledriven gravity currents

“…The time step varies to maintain the value of the Courant-Lewy-Friedrich number ( i C L t F u x    ) within the range 0.15-0.25, and convergence is assured at each time step by ensuring a velocity divergence of less than 10 -6 . The code was originally designed for the efficient modelling of buoyancydriven convective flows, and has been applied to the modelling of natural [25] and mixed convection [26], and to the LES modelling of wind turbines [27].…”