From Mountain Clouds to River Plumes: Observations and modeling of turbulent mixing in stratified flows

Mixing between fluids of different properties goes on all around us every day (e.g., think of stirring milk into your coffee or smoke billowing from a chimney).  In many flows of engineering relevance, velocity differences between fluid bodies generate turbulent motions that, in turn, greatly enhance the mixing process; small-scale chaotic eddies that characterize the turbulence are much more effective than molecular diffusion at mixing fluid properties such as momentum, heat, salinity, sediment load, or pollution concentration.

When fluid bodies are of different densities the effects of gravity weigh heavily on the turbulent mixing process (pun intended).  For example, when warm air escapes from a chimney it accelerates upward in a turbulent billow because it is lighter (less dense) than the cooler air around it, and gravity acts to drive turbulence through buoyant convection.  On the other hand, in a stably-stratified lake, gravity acts to suppress turbulence at the thermocline where lighter, warm water overlies heavier cold water.

The signatures of turbulent mixing in stratified flows are perhaps most obvious in the sky above us.  Clouds provide a convenient flow visualization method!  An anvil-shaped thunder head reveals convectively-generated turbulence in an unstably-stratified environment, whereas rolling billows – think of the sky in van Gogh’s Starry Night – indicate shear-generated turbulence in a stably-stratified environment.  A great example of the latter type of cloud was recently observed outside Alden's Fort Collins office (Figure 1).  

lee-wave-clouds

Figure 1: Lenticular cloud bands forming at the crests of atmospheric gravity waves in the lee of the Rocky Mountain Front Range (flow is toward camera).

The two parallel cloud bands seen in the photograph are occurring at the crests of atmospheric waves that are occurring in the lee of Colorado’s Front Range Mountains (wind is coming toward you in the picture).  Flow over the mountains disturbs the stably-stratified atmosphere and generates a train of gravity waves similar to surface waves in a ship's wake.  Low pressure at the wave crests causes water vapor to condense and form the "lenticular" cloud bands you see in the picture.  Air is actually moving through the clouds rather than the clouds moving with the air!  A good cross-section schematic of mountain lee waves is show in Figure 2 from Durran (2013).

 

lee-wave-cloud-schematicFigure 2: Schematic of mountain lee waves and lenticular clouds (from Durran, 2013).  Flow is from left to right.

While the lee waves themselves are not breaking into turbulence, it appears that a crosswind acting perpendicular to the main flow is doing something interesting to the cloud bands.  See those curling billows in the photograph (close up picture in Figure 3)?  Those are caused by shear between the lighter air above the cloud and the heavier air below.  The fancy name for these shear-driven billows is Kelvin Helmholtz instabilities.  "K-H" instabilities can be observed in many natural flows with density stratification and are common in thermally-stratified flows of oceans, lakes, and rivers.  As the billows roll up, they lift heavy fluid up and push light fluid down – working against gravity.  Eventually, the coherent billows collapse into smaller-scale, chaotic, turbulent motions that mix the two fluid bodies.

lee-wave-clouds-with-shear

Figure 3: Zoomed photograph of Kelvin Helmholtz instabilities due to shear generated by a cross wind.

The computational fluid dynamics (CFD) models used at Alden can capture K-H instabilities as demonstrated in Figure 4 which shows a snapshot from a simple two-dimensional simulation of warm, lighter water (red) moving across cold heavy water (blue).  In many cases of engineering relevance, however, domain size and complexity often preclude the resolution needed to explicitly model the flow structures such as K-H billows that ultimately drive mixing.  Instead, numerical models rely on assumptions about what’s going on at the unresolved scales of the turbulence.  These assumptions form the basis for turbulence models that approximate the mixing process.  

Kelvin-Helmholtz-Instabilities-CFD

Figure 4: Kelvin Helmholtz instabilities occurring on a density and shear interface between warm water (red) and cold water (blue) as captured in a highly-resolved (Δx = 0.50 cm) two-dimensional simulation.  Each billow is approximately 10 cm tall.

Because no model is perfect, keys to a successful modeling effort include calibration and validation.  A promising approach to calibration and validation using field data comes in the form of airborne thermal imaging.  Alden is currently developing a drone-mounted infrared camera system that will provide overhead snapshots of mixing of water bodies of different temperatures – an application of focus being thermal plumes discharged from power plants.  IR images taken recently from a power plant discharging cooling water into a major river are shown in Figure 5.  IR imagery provides a valuable check on the lateral mixing predicted by a given CFD model and serves as yet another tool in Alden’s arsenal for understanding and solving mixing-related problems.

Infrared-image-discharge-plume

Figure 5: Aerial infrared image taken by Alden of a thermal plume discharged from a power plant on a major river.  

Flow is from top to bottom of the image.  Note the complex structure of the mixing/shear line and dilution of the plume in the downstream direction.

References:

Durran, D.R. (2013, Jan. 29)  Trapped lee waves over the western U.S.  

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