In this paper, we simulate intermittent turbulence (also known as bursting
events) in stably stratified open-channel flows using direct numerical
simulation. Clear signatures of this intriguing phenomenon are observed for a
range of stabilities. However, the spatiotemporal characteristics of
intermittency are found to be strongly stability dependent. In general, the
bursting events are much more frequent near the bottom wall than in the
upper-channel region. A steady coexistence of laminar and turbulent flows is
detected at various horizontal planes in very stable cases. This spatially
intermittent pattern is found to propagate downstream and strongly correlate
with the temporal evolution of intermittency. Lastly, a long standing
hypothesis by

The term “intermittency” has different
connotations in various scientific communities. Even among the turbulence
researchers, it does not have a unique definition.

The term intermittency is used in two distinct (but not independent) aspects of turbulent flows. The first one is the so-called external intermittency. It is associated with what is called here partially turbulent flows, specifically with the strongly irregular and convoluted structure and random movement of the “boundary” between the turbulent and nonturbulent fluid. The second aspect is the so-called small scale, internal or intrinsic intermittency. It is usually associated with the tendency to spatial and temporal localization of the “fine” or small scale structure(s) of turbulent flows.

There is extensive literature on small-scale intermittency. A wide range of conceptual models (e.g., multifractal model) have been proposed to explain various traits of intermittency. Chapter 8 ofIntermittent time series of vertical velocity observed in field and
laboratory experiments.

Quantification of spatiotemporal characteristics of intermittent
turbulence is of considerable importance from practical
standpoints. For example, it has been shown that the turbulent bursts
can cause an unusually high concentration of surface layer ozone by
transporting it from higher altitudes in the ABL

Given its ubiquitous occurrence and practical importance, much effort
has been devoted to investigate the characteristics and generation
mechanism of intermittent turbulence. In the recent years, various
mesoscale atmospheric phenomena, including (but not limited to)
low-level jets (LLJ)

Several decades ago,

At present, LES is one of the most efficient computational techniques
available for high Reynolds number turbulent flow
simulations. However, its applicability to simulate strongly
stratified flows has remained an unsettled issue

During the past decades, several DNS studies focused on the turbulence
laminarization problem under stably stratified conditions

Since the coexistence of laminar and turbulent flows has been observed
in the papers by

The structure of this paper is as follows: in Sect.

In this paper, the incompressible turbulent flow in a plane open channel is
studied under stably stratified conditions with three friction Reynolds
numbers (

According to

Under the Boussinesq approximation, the governing Navier–Stokes and the
scalar transport equations are presented in
Eqs. (

The DNS solver used here is based on an open-source CFD package called
OpenFOAM. OpenFOAM has been well verified for a number of scientific and
engineering applications

Illustration of the computational domain, where

In order to obtain initial conditions for the simulations, coarse-resolution
cases are run for about 200 non-dimensional time units (i.e.,

Summary of the simulation configurations used in this study.

Summary of different grid resolutions used for DNS studies of
neutral and stratified channel flows.

According to

Vertical distributions of variables with different

According to the linear stability theory in

Before investigating the spatiotemporal features of intermittency, it is
informative to document the influence of stable stratification on the overall
flow structure in the open channel by analyzing the mean flow and turbulence
statistics profiles. Vertical distributions of mean streamwise velocity,
temperature, and turbulent momentum and heat fluxes with different

The acceleration of mean velocity is caused by the reduced vertical mixing
due to stable stratification, which can be clearly seen in
Fig.

The temperature profiles are also modified by the stratification. With
increasing

One can see that the Nusselt number equals the non-dimensional temperature
gradient at the bottom wall. For the neutral stratification case
(

Under the influence of the stable stratification, the turbulent
heat flux is significantly suppressed, which can be seen in
Fig.

In summary, the stable stratification significantly reduces the transport and mixing of turbulent flows in the vertical direction as would be expected. Therefore, in the following subsections, the vertical velocity field will be primarily utilized to probe if the simulated flow becomes laminarized or intermittent with high stratification levels.

Time series of vertical velocity in the channel with
different

To analyze the temporal characteristics of intermittent turbulence, several
probe points are placed in the channel to monitor the time variations of the
stratified turbulent flows. Time series of vertical velocity in the channel
with different

Time series of streamwise velocity and temperature in the channel with

Time series of streamwise velocity and temperature in the channel at

As were depicted in Figs.

Time height plots of square of instantaneous
vertical
velocity (

In order to quantitatively analyze the vertical distribution of
turbulent intermittency, the turbulence enhancement index (TEI) is
used to detect the turbulent bursting events. The TEI is defined as
the ratio of increase of vertical velocity variance
(

Vertical distribution of turbulent bursting events in the channel is shown in
Fig.

Using observational data based on the CASES-99 experiment,

As was mentioned in Sect.

Vertical distribution of turbulent bursting events with different

Instantaneous contours of

Instantaneous contours of

Instantaneous contours of

The aforementioned results show that, for

Instantaneous contours of

In the vertical direction, the spatially intermittent patterns can vary under
different flow conditions. For example, in Couette flow, the spatial patterns
can extend from the bottom to the top walls at low Reynolds numbers, whereas
at high Reynolds numbers, these patterns can be confined to each wall, and
uncorrelated

Additionally, by increasing the

Instantaneous contours of

The aforementioned results clearly show that a coexistence of laminar and
turbulent flows can be found in the stratified open-channel flows, and the
spatial characteristic may vary with respect to vertical location. In order
to quantify the turbulence coverage with respect to different

Vertical distribution of

Vertical distribution of turbulent fraction (

An intriguing phenomenon in stratified flows is the internal gravity wave. As
reported in

Instantaneous contours of temperature at the mid-spanwise plane (not
to scale,

As reported in

Profiles of mean velocity (

Profiles of mean velocity (

Now, using Figs.

In stratified shear flows, the gradient Richardson number
(

Vertical distribution of gradient Richardson number
(

As a first step, we analyze the mean profiles of

Next, we discuss the overall impact of

Distribution of

In addition to the impact of

Instantaneous contour of

Similarly, we also plot the profiles of plane-averaged

Profiles of plane-averaged

Time-series of

In addition to investigate the spatial-correlation between

The above analyses mainly focused on a relatively low Reynolds number case
(

Time series of vertical velocity in the channel.

Time series of streamwise velocity in the channel.

Vertical distribution of turbulent bursting events in the channel. The turbulent bursting events are calculated within 10 non-dimensional time.

Time series of

In this paper, direct numerical simulation is utilized to simulate open-channel flows under stably stratified conditions. The spatiotemporal characteristics of intermittent turbulence are discussed, and several conclusions are made as summarized below.

Under stably stratified conditions, the vertical transport of turbulent
momentum and heat fluxes is suppressed and the mean flow and turbulence
statistics profiles are significantly modified. For the weakly stratified
case with

Stable coexistence of laminar and turbulent flows is found at various
horizontal planes. For the case with

Strong correlation between the spatial and temporal intermittency is observed
in the channel. The laminar–turbulent bands are found to propagate
downstream with a uniform convection velocity, and the time series signal
alternates between turbulent and laminar states when the corresponding
laminar–turbulent bands pass though the probe points. This prominent
correlation, observed in the direct numerical simulation (DNS) results,
corroborates the observation-based analyses in

The instantaneous gradient Richardson number (

It is evident that some of the findings of the present work have striking
similarities with results gleaned from atmospheric boundary layer (ABL)
flows. However, given the Reynolds number disparity, these similarities
should be considered with caution. In the near future, we will explore the
strengths and weaknesses of a suite of large-eddy simulation subgrid-scale
(LES–SGS) models in capturing low Reynolds number intermittency by using our
DNS-based results as benchmark. Several contemporary LES-SGS models assume
local balance of turbulent kinetic energy (TKE) production with its
dissipation. We speculate that these models might not fare well in this
exercise. On the other hand, we expect the SGS models with dynamic TKE
formulations and energy backscatter options

Our future work will also involve characterization of intermittent turbulence
from a complex systems perspective. Over the years, a number of simple
dynamical systems have been reported to exhibit intermittency

In this Appendix, we first compare the simulation results produced by the
current DNS solver (OF) against the DNS results from

Next, we also compare our results with those obtained by a spectral method in
order to ascertain that the grid resolution utilized in the present DNS study
is sufficient. For this purpose, we use the well-cited DNS results (neutrally
stratified case) from

Although there are some differences between the results from the current DNS
solver and those from NS97 and MKM99, this level of discrepancy is not
unexpected from different DNS solvers (for example, see the code verification
in

Comparisons of variables for neutral and stratified flows,

Comparisons of variables for neutral flows,

In this appendix, we evaluate the influence of simulation configurations,
e.g., grid resolution, domain size, on the spatiotemporal characteristics of
intermittency. Two additional simulations are conducted for the

Influence of simulation configurations on the spatiotemporal
characteristics of intermittency (

As mentioned in Sect.

Time series of

The authors acknowledge financial support received from the Department of Defense (AFOSR grant under award number FA9550-12-1-0449) and the National Science Foundation (grant AGS-1122315). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Department of Defense or the National Science Foundation. The authors also acknowledge computational resources obtained from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (grant number ACI-1053575). In addition, the authors would like to thank Christopher G. Nunalee, Kiliyanpilakkil V. Praju and the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. Edited by: J. M. Redondo Reviewed by: two anonymous referees