Production of Atmospheric Turbulence

Turbulence and mixing behind wind turbines (Christian Steiness)

Production of turbulence Dissipation of turbulence How do we characterize turbulence – statistically Turbulence – how energy and mass is exchange (energy, mass, GHGs, pollutants, water)

Today’s learning objectives

• Describe how, when and where is turbulence produced in the atmospheric boundary layer.
• Explain where the energy for turbulent motions comes from.
• Explain whether the characteristics of a turbulent flow depend on the processes causing them.

Midterm (iClicker)

Will today’s material be on the mid-term?

• A - Yes
• B - No

Reynold’s number (Re)

• There is a critical point in fluid flow when well-behaved laminar flow breaks down into turbulent ‘chaos’.
• Depends on ratio of inertial forces (associated with horizontal motion in fluids) to the viscosity of the fluid:

\[ Re = \small\frac{ud}{\upsilon} \]

• u - characteristic velocity of the fluid
• d - characteristic length scale
• \(\upsilon\) - kinematic viscosity (includes density)

Laminar boundary layer

Turbulent boundary layer (Los Alamos Natl. Laboratory)

Reynold’s number (Re)

A dimensionless threshold to define whether the flow is turbulent or laminar.

• Conceptually, \(Re\) is defined as the ratio between inertial and viscous forces (inertial forces are associated with the mean motion of the fluid and the viscous forces are associated with frictional shear stress.)

• The threshold for the onset of turbulence is about 2000. I.e., turbulence is associated with large Reynolds numbers. Most atmospheric flows are turbulent (Re >> 2000) as \(\upsilon\) is on the order of 10^{-5} m^2 s^{-1}, d is 100s of meters and u is on the order of 0.1 to 10 m s^{-1}.

  • In other words, the characteristic length scale of the flow is much larger than the thickness of the laminar boundary layer

• \(Re\) for wind blowing 0.25 m s^{-1} over a 10 cm leaf is 250.

Reynold’s number - Examples

Daytime atmospheric boundary layer (ABL):

\(Re = \small\frac{ud}{\upsilon}=\small\frac{10 ms^{-1} * 1000m}{1.51^{-5}m^{2}s^{-1}} = 6.6*10^8\)

Model in a wind tunnel:

\(Re = \small\frac{ud}{\upsilon}=\small\frac{1.5 ms^{-1} * 0.3m}{1.51^{-5}m^{2}s^{-1}} = 3.0*10^4\)

• \(Re_{crit} \approx 10^6\) in real atmospheric flows, \(Re_{crit} \approx 2 * 10^4\) in engineering flows.
• But Re is of limited value in real atmospheric flows because of role of buoyancy (that increases or decreases the amount of turbulence).

inertial forces are dominant over the viscous forces.

Production of turbulence

• Turbulence in the ABL is a ‘mixture’ of mechanical (forced) and thermal (free) convection.

• Mechanical turbulence in the ABL is caused by instabilities arising from strong mean velocity gradients, which in turn are caused by surface skin or form drag (obstacles) or shear flow.

Wind Shear. Wind shear is the change in wind direction and/or wind speed over a specific horizontal or vertical distance. When the change in wind speed and direction is pronounced, quite severe turbulence can be expected. Shear Produces Turbulence: Turbulence is an instability generated by shear. The stronger the shear, the stronger the turbulence.

The maximum turbulence level occurs at the positions of maximum shear. The shear in the boundary layer decreases moving away from the bed, and as a result the turbulence intensity also decreases.

Skin drag

In micrometeorology and climatology we encounter fully adjusted wall-bounded flows with the surface as the lower boundary condition.

A strong velocity gradient is caused due to the no-slip condition at solid boundaries - skin drag. This velocity gradient becomes unstable when reaching higher Re.

H. Tennekes and J. L. Lumley (1972): A first course in turbulence. Massachusetts Institute of Technology.

Skin drag (or skin friction): the frictional force caused by the movement of air over a surface Every solid body, regardless of how smooth its surface is, it has roughness or imperfections or surface irregularities. When a fluid flow past the surface of any solid, it experiences resistance against the direction of the flow. This friction is called skin friction drag.

Form drag

Obstacles in the flow (trees, houses, etc.) cause separation and pressure differences giving eddies in the wake.

Flow and turbulence around a cube

Flow around building models placed in wind tunnel visualized using Laser Light Sheet (Institute of Industrial Science, University of Tokyo)

Form Drag: the force on an object due to the deceleration of the air flow Even before the air reaches the obstacle, it begins to react because of the pressure build-up ahead. The bulk of the flow is forcibly displaced up and over the barrier. Immediately above the barrier the streamlines are forced to converge as the same mass of air attempts to ‘squeeze’ over, causing an acceleration or jet, but once over it is able to expand again and decelerates accordingly. but the fluid cannot immediately react to fill it. The flow therefore separates from the barrier’s surface and its organization breaks down into a much more turbulent condition in the flow pressure or wake zone which extends downwind from the barrier. Immediately behind the barrier, the pressure is low and thus tends to suck air into a semi-stationary lee eddy or vortex. This part of the wake is known as the cavity zone, The large eddy structure is dissipated into the smaller turbulent eddies of the wake zone, before finally settling down and resuming conditions similar to those of the upwind flow. Within the wake the separation of the flow lessens the force of the wind on the found and the near surface wind speed decreases – this is the sheltering effect.

FORM DRAG: The drag which is created because of the shape of the solid body (or aircraft) and area of cross section, is called form drag.

Mechanical (forced) convection caused by an obstacle

Size of eddies scales with size of obstacles.

Even before the air reaches the obstacle, it begins to react because of the pressure build-up ahead. The bulk of the flow is forcibly displaced up and over the barrier. Immediately above the barrier the streamlines are forced to converge as the same mass of air attempts to ‘squeeze’ over, causing an acceleration or jet, but once over it is able to expand again and decelerates accordingly. but the fluid cannot immediately react to fill it. The flow therefore separates from the barrier’s surface and its organization breaks down into a much more turbulent condition in the flow pressure or wake zone which extends downwind from the barrier. Immediately behind the barrier, the pressure is low and thus tends to suck air into a semi-stationary lee eddy or vortex. This part of the wake is known as the cavity zone, The large eddy structure is dissipated into the smaller turbulent eddies of the wake zone, before finally settling down and resuming conditions similar to those of the upwind flow. Within the wake the separation of the flow lessens the force of the wind on the found and the near surface wind speed decreases – this is the sheltering effect.

Air flow around buildings

T.R. Oke (1987): ‘Boundary Layer Climates’ 2nd Edition.

Air is displaced upward, around the sides (wind tunnels) and then pushed downwards in the lee side (downwind) of the building. On the lee side of the building, a zone of lower pressure causes vortex formation behind it which are essentially turbulent winds that can cause building damage and strong winds that may knock pedestrians off their feet.  Often, the urban layout is structured in such a way that winds speeds are often abrupt and turbulent as a result of frictional drag. Venturi effect, high winds between buildings Creates complex flow patterns 

http://leonardthegeographerunit3.blogspot.com/2012/03/uhis-urban-heat-islands-winds.html

Practical Application: Aviation

Airplanes wings work by exploiting pressure differentials due to flow separation to produce lift!

Clouds over high-rise buildings in Panama City Beach, Florida, USA. Wind is from the right (ocean). (Photo courtesy of J. R. Hott, Panhandle Helicopters).

Simulation of turbulence above a city block using a numerical model (Courtesy of M. Parlange / M. Giometto, UBC)

Mechanical (forced) convection due to shear

Direct numerical simulation of turbulence in a Kelvin Helmholtz instability

Flow moving in different directions – i.e. shear

Kelvin Helmholtz waves in BC

These can produce interesting cloud features in areas of complex terrain.

• They even make it on to local news, like this example from November 2023

Mechanical (forced) convection

We can have three different scenarios that create turbulence mechanically

Friction and drag at the surface – and that layers with different speeds

Thermal (free) convection

Thermal (free) convection

• Surface heating → density differences in air → convective exchange due to buoyancy.
• Thermal turbulence production requires continual input of heat which is converted to turbulent kinetic energy.

T.R. Oke (1987): ‘Boundary Layer Climates’ 2nd Edition.

Thermal turbulence is caused by mean velocity gradients between warm air plumes rising up in the ABL which in turn are caused by differential surface heating. Development of thermals when there is little to no wind Best spot for thermals are relatively dry areas The hot air forms a flattened bubble until the instability becomes sufficient to cause It to start to rise. Where upon it contracts, becomes more spherical and begins to rise. The thermal grows in size as it rises due to the entrainment of surrounding air. Buoyancy: force exerted on an object that is wholly or partly immersed in a fluid. Buoyancy is caused by differences in pressure acting on opposite sides of an object immersed in a static fluid.

Intense Convection > Thunderstorm

Thermally produced turbulence - free convection

• Size of eddies scales are restricted by the height of the atmospheric boundary layer (ABL)
• The height of the ABL depends on the profile of potential temperature which in turn is controlled by turbulent mixing

Snapshot of the temperature field in a direct numerical simulation of convection in the ABL, Figure by Peter Sullivan (NCAR/MMM) and H. Jonker (Delft University, Netherlands)

But during a sunny day over land, the convective boundary layer starts as a very thin layer near the ground, and gets thicker and stronger as the day progresses as thermals of warm air rise from the ground. 

The temperature that a sample of air attains if reduced to a pressure of 1000 millibars without receiving or losing heat to the environment The temperature that an unsaturated parcel of dry air would have if brought adiabatically and reversibly from its initial state to a standard pressure, p0, typically 100 kPa.

Thermally produced turbulence - free convection

Lidar image is a vertical slice through the atmospheric boundary layer, where polluted airis being carried upward from the surface by thermals Source

These thermals often carry air pollutants upward (making them visible to lidars, see orange colours in the figure below, where vertical axis is height above ground and the horizontal axis is time). The diameter of these thermals averages about 750 m in this example, but they can be larger when the turbulent boundary layer is thicker.

Measuring thermal turbulence on an aircraft

Thermal (free) convection

Variations across the landscape

Surface temperature cross section obtained from a radiation thermometer mounted on an aircraft (i.e Lup). Sharp changes in temperature across the landscape The vertical structure shows the development of a series of boundary layers, some warmer and some cooler than the bulk air temperature. At the upwind edge of each new surface, and internal BL forms. Hence the structures appear as hot or cool plumes. The heated plumes being unstable have more vigorous vertical development than those from the cooler, more stable lake or irrigated land.

Mechanical vs. thermal turbulence

W. P. Lowry, P. P. Lowry (1989)

Thermal convection and suppression

• Unstable: Mechanical (forced) and thermal (free) convection.
• Neutral: Only mechanical (forced) convection.
• Stable: Mechanical (forced) convection and thermal suppression.

T.R. Oke (1987): ‘Boundary Layer Climates’ 2nd Edition.

Conservation of energy - The Turbulent Kinetic Energy (TKE) budget

Mechanical turbulence in the ABL is caused by instabilities arising from strong mean velocity gradients, which in turn are caused by surface skin or form drag (obstacles) or shear flow. Thermal turbulence is caused by mean velocity gradients between warm air plumes rising up which in turn are caused by differential surface heating. At the smallest eddy sizes, the cascade of energy is dissipated into heat by molecular viscosity. Turbulent kinetic energy is one of the most important variables in micrometeorology. It’s a measure of the intensity of turbulence. Mechanical turbulence production requires continual supply of kinetic energy (mean wind) while thermal turbulence requires continual supply of a sensible heat flux (surface heating). TKE can be produced by fluid shear, friction or buoyancy. Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the Kolmogorov scale.

Turbulence (iClicker)

Parcel of cool air moving over a 5 m tall tree canopy at 10 m s^-1 on a hot, sunny day would experience:

• A - Mechanical Convection
• B - Thermal Convection
• C - Both!

Take home points

• Turbulence in the ABL is a ‘mixture’ of mechanical (forced) and thermal (free) convection.
• Mechanical turbulence in the ABL is caused by instabilities arising from strong mean velocity gradients, which in turn are caused by surface skin or form drag (obstacles) or shear flow.
• Thermal turbulence is caused by mean velocity gradients between warm air plumes rising up in the ABL which in turn are caused by differential surface heating.
• Mechanical turbulence production requires continual supply of kinetic energy (mean wind) while thermal turbulence requires continual supply of a sensible heat flux (surface heating).