Transverse Mixing

Question 1

Transverse mixing may be defined as the spreading of material perpendicular to the primary velocity. Explain for a wide river channel of uniform depth what data is required to determine the coefficient and how this may be obtained in practice.

Answer 1

Transverse mixing in a channel may be determined by employing the standard method of moments. This is developed from an assumption of Fickian type diffusion within the flow. Set up a constant injection of tracer and generate a steady plume downstream from the injection, see below. The peak concentration reduces and the transverse spread of the plume increases with distance downstream. Assuming a uniform channel cross-section with spatially constant transverse mixing coefficients, ε_y then the transverse distribution of the tracer concentration should approximate to a Gaussian distribution. Assumption of Fickian type mixing allows the transverse mixing coefficient to be estimated from ε_y = u/2.dσ_y²/dx. To employ this equation, the increase in variance with distance dσ_y²/dx and the area mean velocity is required. The area mean velocity, u may be obtained from velocities measured directly within the flow and the transverse spatial distribution of tracer concentration must be measured at several cross-sections downstream from the injection point, prior to bank impingement. To obtain a good description of the tracer concentration distribution a minimum of approximately 40 points across the plume are required. From these distributions the transverse variance (2nd moment of area about the centroid divided by the area of the distribution) is determined. Knowing the distance from the injection point to the downstream cross-section (x) dσ_y²/dx can be obtained. It is usual to plot σ_y² against distance, x and determine the best fit line through the data.

Question 2

Briefly describe what additional measurements would be required and how the technique may be adapted for channels where there is a significant variation in transverse depth of flow.

Answer 2

If channel cross-section is not uniform, either change in width or change in shape, then the recorded the concentration vs transverse distance data should be re-plotted as concentration vs relative discharge (q).

To do this requires both flow depth & velocity variation with y (transverse) to be measured. This the transverse mixing willl be determined as the increase in dσ_q²/dx and the mixing coefficient, ε_y must be adjusted accordingly.

Question 3

Briefly discus the major limitations of these predictions.

Answer 3

Major limitation is the assumption that the plume enters the channel with no momentum, i.e. there is no jet mixing and that all the spreading is due to velocity shear and turbulent fluctuations that can be described by a Gaussian profile prediction.

Other significant assumptions are:

• Spatially constant transverse mixing coefficients
• Perfect reflection at channel boundaries
• Primary velocity distribution is uniform

Question 4

Briefly explain, with the aid of sketches, why the assumption of a spatially uniform transverse mixing coefficient may be inappropriate and explain the processes that lead to spatial variations in mixing between the main channel and flood plain for over-bank channel river flows.

Answer 4

The lower mixing in the shallow slow flowing water away from the main channel means high concentrations can persist for a considerable distance downstream of the outfall

The complex flow is caused by the rapid reduction in depth in the transverse direction as the floodplain is approached from the main channel of the river. There is a shallow depth of slowly moving water on the flood plain and faster flowing deeper water in the main channel. Momentum is transferred from one region to another in the form of large scale horizontal eddies that extend into both the main river channel and onto the floodplain (horizontal shear layer). This is strongly dependent on the ratio of the floodplain flow depth to main channel flow depth, defined as the relative depth, with Knight & Shiono (1996) suggesting that the maximum interaction occurs at relative depths of between 0.1 and 0.3.

Question 5

Briefly explain, with the aid of sketches, why the assumption of a spatially uniform transverse mixing coefficient may be inappropriate and explain the processes that lead to spatial variations in on-offshore mixing in the near-shore coastal zone.

Answer 5

Studies show that for straight uniform channels under idealised laboratory conditions, the transverse mixing coefficient can be evaluated approximately from k_y = 0.134hu* where k_y is the transverse mixing coefficient, h the depth of flow and u* the bed shear velocity ≈ u_bar/10.

Mixing caused by wave activity could comprise a combination of; mixing generated by the production of turbulence due to (tidal or oblique waves driven) breaking wave activity, mixing generated by the oscillatory flow over the bed and a shear dispersion caused by the flow in the on-off shore direction.

Wave breaking leads to a longshore current which is proportional to d^3/2 and provides a local increase in turbulence.

Question 6

A simplified numerical solution used to describe the transverse mixing of soluble material in open channel flow assuming no spatial variation in the transverse mixing coefficient.

Answer 6

k_y = u/2.dσ²/dx

Question 7

Explain how vertical mixing influences the on-offshore mixing

Answer 7

Flow towards the shore at the top layer and flow in the opposite direction near the bed (undertow) causes secondary circulation turbulence and a shear dispersion. This causes additional vertical mixing that contibutes to the overall mixing process.

Question 8

Tranverse Mixing coefficient range around bends

Answer 8

1.0 < k_y/du*<3

(u* ≈ u_bar/10)

Question 9

Field ranges of transverse mixing coefficients

Answer 9

0. 15 < k_y/hu*< 0.3
1. 3 < k_y​/hu*< 1.0
2. 0 < k_y​/hu*< 3.0