Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Environmental Fluid Mechanics

ä Scope of Environmental Fluid Mechanics ä Transport Processes ä molecular diffusion ä turbulent diffusion (detour into turbulence) ä advection ä Turbulent Diffusion + Advection ä Jet and Plumes ä Scope of Environmental Fluid Mechanics ä Transport Processes ä molecular diffusion ä turbulent diffusion (detour into turbulence) ä advection ä Turbulent Diffusion + Advection ä Jet and Plumes 

Sources ä * Mixing in Inland and Coastal Waters. Hugo B. Fisher, E. John List, Robert C. Y. Koh, Jörg Imberger, and Norman H. Brooks Academic Press, New York. ä Fluid Mechanics. Victor L. Streeter and E. Benjamin Wylie Eighth edition, McGraw- Hill Book Company, New York. ä A First Course in Turbulence. H. Tennekes and J. L. Lumley MIT Press, Cambridge. ä * Mixing in Inland and Coastal Waters. Hugo B. Fisher, E. John List, Robert C. Y. Koh, Jörg Imberger, and Norman H. Brooks Academic Press, New York. ä Fluid Mechanics. Victor L. Streeter and E. Benjamin Wylie Eighth edition, McGraw- Hill Book Company, New York. ä A First Course in Turbulence. H. Tennekes and J. L. Lumley MIT Press, Cambridge.

Environmental Fluid Mechanics ä Motion and mixing of fluids in the environment ä Interested in the substances and properties transported by the fluid ä Examples ä wastewater discharge into stream, estuary, or ocean ä junction of two rivers ä smokestack discharge into atmosphere ä contaminant spill into ocean or river ä mixing of salt and fresh water in estuaries ä mixing of warm and cold water in lakes ä Motion and mixing of fluids in the environment ä Interested in the substances and properties transported by the fluid ä Examples ä wastewater discharge into stream, estuary, or ocean ä junction of two rivers ä smokestack discharge into atmosphere ä contaminant spill into ocean or river ä mixing of salt and fresh water in estuaries ä mixing of warm and cold water in lakes

Water as transportation... ä Water transports substances and properties Physical Heat Turbidity Color Suspended Solids Chemical Salinity Dissolved oxygen Dissolved solids Metals Pesticides BOD pH Biological Fish eggs Protozoa Bacteria Viruses

Hydrologic Transport Processes* ä Advection: Transport by an ________ ________, as in a river or coastal waters. ä Convection: Vertical transport induced by _________ ________, such as the flow over a heated plate, or below a chilled water surface in a lake. ä Diffusion (Molecular): The scattering of particles by random molecular motions. ä Diffusion (Turbulent): The random scattering of particles by turbulent motion. ä Advection: Transport by an ________ ________, as in a river or coastal waters. ä Convection: Vertical transport induced by _________ ________, such as the flow over a heated plate, or below a chilled water surface in a lake. ä Diffusion (Molecular): The scattering of particles by random molecular motions. ä Diffusion (Turbulent): The random scattering of particles by turbulent motion. imposed current hydrostatic instability

Hydrologic Transport Processes* ä Dispersion: The scattering of particles or a cloud of contaminants by the combined effects of ______ and transverse _________ ä Mixing: Diffusion or dispersion as described above; turbulent diffusion in buoyant jets and plumes; any process which causes one parcel of water to be mingled with or diluted by another. ä Dispersion: The scattering of particles or a cloud of contaminants by the combined effects of ______ and transverse _________ ä Mixing: Diffusion or dispersion as described above; turbulent diffusion in buoyant jets and plumes; any process which causes one parcel of water to be mingled with or diluted by another. shear diffusion

Molecular Diffusion ä Diffusion of particles (e.g. molecules of a substance) by random motion due to molecular ______ ______ ä Fick’s law of diffusion ä Empirical description ä Mass flux is proportional to ________ of mass concentration ä Diffusion of particles (e.g. molecules of a substance) by random motion due to molecular ______ ______ ä Fick’s law of diffusion ä Empirical description ä Mass flux is proportional to ________ of mass concentration kinetic energy gradient

Fick’s law D m : coefficient of molecular diffusion C: concentration (e.g. mg/liter) J: mass flux Does the gradient cause the diffusion? _____ NO! =

Coefficient of Molecular Diffusion ä D = f(solvent, solute, temperature) Gas molecules have much more kinetic energy (higher velocity) and greater distance between molecules and thus diffusion in air is higher than diffusion in water. Carrying FluidSoluteD m (cm 2 /s) H 2 OO 2 2.4x10 -5 H 2 ONaCl1.545x10 -5 H 2 OC 6 H 12 O x10 -5 Air H Air O AirC O

Similarity of Transport Mechanisms: ( Mass, Momentum, Heat ) Shear Stress (Momentum transport/area) Newton’s law of viscosity coefficient of viscosity coefficient of momentum diffusivity

Similarity of Transport Mechanisms: ( Mass, Momentum, Heat) Fourier’s law of heat transport coefficient of conductivity coefficient of heat diffusivity specific heat/volume

Combination of Mass Transport & Mass Conservation dx A x change in mass = net transport (in - out) COnly diffusion, no advection

Governing Equation for 1-D mass transport by diffusion Can be generalized to 2 and 3 dimensions If D m constant Mass conservation Fick’s 1 st Law

Diffusion Fick's first law Fick's second law What does it look like a short time later? x x C C x x C C

Solutions to diffusion of slug ä Fundamental Solution - response to the introduction of slug of mass M A x note:

Solution to 1-D problem         distance (cm) concentration (g/mL)   1 s 10 s A D m = x cm 2 /s M = 1 g A = 1 cm s

Lateral Distribution of Slug ä Example: Find the distance from the center of the plume where the concentration is 10% of the maximum (as a function of time). Solution ?

Molecular Diffusion Example ä How long does it take for a slug of sugar to spread so that the concentration 10 cm away is 10% of the maximum? dissolved sugar layer 10 cm

Diffusion in the Environment ä How long would it take for the same glucose concentration gradient to disperse in Fall Creek? glucose Fall Creek (idealized) Flow Turbulence!!!!

Mean and Variation* ä Fluctuations and irregularities in hydrologic systems are just as important as the mean flows for pollutant analysis! ä The mean flows provide the advection, the fluctuations (turbulence) provide the mixing. ä Fluctuations and irregularities in hydrologic systems are just as important as the mean flows for pollutant analysis! ä The mean flows provide the advection, the fluctuations (turbulence) provide the mixing.

Turbulence ä A characteristic of the ____. (contrast with diffusion) ä How can we characterize turbulence? ä intensity of the velocity fluctuations ä size of the fluctuations (length scale) ä A characteristic of the ____. (contrast with diffusion) ä How can we characterize turbulence? ä intensity of the velocity fluctuations ä size of the fluctuations (length scale) t mean velocity instantaneous velocity fluctuation flow

Turbulence: Size of the Fluctuations or Eddies ä Eddies must be smaller than the physical dimension of the flow ä Generally the largest eddies are of similar size to the smallest dimension of the flow ä Examples of turbulence length scales ä rivers: _________ ä pipes: __________ ä A spectrum of eddy sizes ä Eddies must be smaller than the physical dimension of the flow ä Generally the largest eddies are of similar size to the smallest dimension of the flow ä Examples of turbulence length scales ä rivers: _________ ä pipes: __________ ä A spectrum of eddy sizes depth diameter

Turbulence: Flow Instability ä In turbulent flow (high Reynolds number) the force leading to stability (viscosity) is small relative to the force leading to instability (inertia). ä Any disturbance in the flow results in large scale motions superimposed on the mean flow. ä Some of the kinetic energy of the flow is transferred to these large scale motions (eddies). (__________) ä Large scale instabilities gradually lose kinetic energy to smaller scale motions. ä The kinetic energy of the smallest eddies is dissipated by _________ resistance and turned into ______. ä In turbulent flow (high Reynolds number) the force leading to stability (viscosity) is small relative to the force leading to instability (inertia). ä Any disturbance in the flow results in large scale motions superimposed on the mean flow. ä Some of the kinetic energy of the flow is transferred to these large scale motions (eddies). (__________) ä Large scale instabilities gradually lose kinetic energy to smaller scale motions. ä The kinetic energy of the smallest eddies is dissipated by _________ resistance and turned into ______. head loss! viscous heat

Turbulent Diffusion ä Mechanism of turbulent diffusion is different than the mechanism of molecular diffusion, but the effect is similar. ä Scale of the motion generating turbulent diffusion is much larger than for molecular diffusion! ä Mechanism of turbulent diffusion is different than the mechanism of molecular diffusion, but the effect is similar. ä Scale of the motion generating turbulent diffusion is much larger than for molecular diffusion! Turbulent diffusion coefficient C = time averaged concentration (no fluctuations) If plume size >> eddy size

Turbulent Diffusion ä Example: grid generated turbulence. Vortex shedding from grid will generate turbulence in the cup. Sugar will spread much faster. How fast? Recall molecular diffusion - (3 weeks)! dissolved sugar layer 10 cm Grid

Reynolds Analogy ä In turbulent processes all properties are exchanged at the same rate ä momentum ä heat ä mass ä Diffusion is a function of path length and velocity ä Molecular diffusion: ________ between molecules, _________ of molecules ä Turbulent diffusion: _____ of eddies, velocity of fluid in eddy relative to mean flow ä In turbulent processes all properties are exchanged at the same rate ä momentum ä heat ä mass ä Diffusion is a function of path length and velocity ä Molecular diffusion: ________ between molecules, _________ of molecules ä Turbulent diffusion: _____ of eddies, velocity of fluid in eddy relative to mean flow Why? distance velocity size All transported by fluid at rate >> molecular transfer

Magnitude of Turbulent Diffusion in a River ä for order of magnitude estimate we need to ä estimate velocity fluctuations - u’ ä estimate size of eddies - __ ä for order of magnitude estimate we need to ä estimate velocity fluctuations - u’ ä estimate size of eddies - __ where u’ = standard deviation of the velocity (often called root mean square or rms velocity) We could measure u’ directly or estimate it! l=d Bottom shear u’  u * = shear velocity = Force balance (depth of river)

Manning Eq. (SI) units assume n of 0.03 wide channel so R h = __ Velocity fluctuations in rivers are typically ­ _____ Magnitude of RMS Velocity (u’) in a River ä Example: moderately sloped river ä Susquehanna at Binghamton ä S = ä d = 1 m = 100 cm ä Example: moderately sloped river ä Susquehanna at Binghamton ä S = ä d = 1 m = 100 cm 0.1V d d

Magnitude of Turbulent Diffusion in a River   ­ 0.2 for straight channels   ­ 0.5 ± 0.2 for natural rivers ä Example: moderately sloped river ä Susquehanna at Binghamton ä S = ä d = 1 m = 100 cm   ­ 0.2 for straight channels   ­ 0.5 ± 0.2 for natural rivers ä Example: moderately sloped river ä Susquehanna at Binghamton ä S = ä d = 1 m = 100 cm Recall D m is approximately cm 2 /s D t is 7 orders of magnitude larger than D m ! Remember D t is property of fluid _____. flow

Summary ä Water as transportation medium ä Similarity of transport processes ä Fundamental equation describing diffusion ä Mechanisms of mixing ä molecular diffusion ä turbulent diffusion ä Reynolds analogy ä Estimates of the magnitude of turbulent diffusion in rivers ä Water as transportation medium ä Similarity of transport processes ä Fundamental equation describing diffusion ä Mechanisms of mixing ä molecular diffusion ä turbulent diffusion ä Reynolds analogy ä Estimates of the magnitude of turbulent diffusion in rivers

Advection and Turbulent Diffusion: Passive Plume in River x U x x x=0 U y d dx Assume complete mixing in the vertical direction! Side view Top view Concentration gradients in x are small.

Pure diffusionAdvective diffusion instantaneous continuous release x_____ t_____ M_____ A ____= ____ C(x,t)_____ Passive Plume in River Correspondence Table y y x/U C(y,x)

Passive Plume in River x x x=0 y

Example: Turbulent Diffusion in the Susquehanna (1) ä Wastewater containing 20 mg/L COD (chemical oxygen demand) is discharged at 0.5 m 3 /s into the center of Susquehanna River at Binghamton. How wide is the plume (defined by 10% of the centerline concentration) as a function of distance downstream and what is the centerline concentration? d = 1 m D t =160 cm 2 /s = m 2 /s U = 1 m/s Solution

Example: Turbulent Diffusion in the Susquehanna (2) ä Narrow plume ä Dilution by factor of 10 in 120 meters ä Our solution does not apply in the region close to the source (_______) ä size of plume must be greater than eddy size for equation to be applicable ä maximum concentration can not exceed discharge pipe concentration! ä Narrow plume ä Dilution by factor of 10 in 120 meters ä Our solution does not apply in the region close to the source (_______) ä size of plume must be greater than eddy size for equation to be applicable ä maximum concentration can not exceed discharge pipe concentration! distance downstream (m) Plume width (m) C y=0 (mg/L) width (m) Concentration (mg/L) nearfield

River isn’t infinitely wide! ä Region 1 ä mixing in vertical direction ä plume __ largest eddies ä point source: 3-D problem ä Region 2 ä river width __ plume __ river depth ä line source: 2-D problem ä Region 3 ä plume development is affected by river banks ä image sources ä Region 4 ä river is completely mixed ä plane source ä Region 1 ä mixing in vertical direction ä plume __ largest eddies ä point source: 3-D problem ä Region 2 ä river width __ plume __ river depth ä line source: 2-D problem ä Region 3 ä plume development is affected by river banks ä image sources ä Region 4 ä river is completely mixed ä plane source < < > > > >

Region 3: Image Sources Significant reflection x ______ source real source image 0 0

Sampling time ä If we sample for less time than it takes for a large eddy to rotate then our average value may not be a good average ä Need an estimate of the integral time scale (t I ). ä If we sample for less time than it takes for a large eddy to rotate then our average value may not be a good average ä Need an estimate of the integral time scale (t I ).

Pollutant Mixing in Open Channel Flow: Objectives ä Characterize turbulent flow ä integral velocity ä integral length ä “but it doesn’t look turbulent” ä Apply the advective dispersion equation ä Measure the dispersion coefficient ä Characterize turbulent flow ä integral velocity ä integral length ä “but it doesn’t look turbulent” ä Apply the advective dispersion equation ä Measure the dispersion coefficient

Experiment description ä Flume (laboratory river) ä 46 cm wide, 7 m long, variable depth ä tap water supply (1.8 L/s) ä Plume ä sodium chloride (to increase conductivity) ä red dye #40 (for qualitative observations) ä discharged by peristaltic pump through single port ä Flume (laboratory river) ä 46 cm wide, 7 m long, variable depth ä tap water supply (1.8 L/s) ä Plume ä sodium chloride (to increase conductivity) ä red dye #40 (for qualitative observations) ä discharged by peristaltic pump through single port

Conductivity probe Platinum electrodes

The Instrument pH/ion/con ductivity meter positioning system controller vertical (z) slide horizontal (y) slide conductivity probe zxy horizontal (x) tracks

Plume in a Flume Coming up... ä Environmental Fluid Mechanics ä Apply the advective dispersion equation ä Discuss turbulent dispersion ä Quantitative analysis ä Estimate the dispersion coefficient ä Compare model and data ä Qualitative observations ä Environmental Fluid Mechanics ä Apply the advective dispersion equation ä Discuss turbulent dispersion ä Quantitative analysis ä Estimate the dispersion coefficient ä Compare model and data ä Qualitative observations

Passive Plume in Turbulent Flow: Theory C concentration E y dispersion coefficient Uadvective velocity Mrate of mass input ddepth of water xdistance in direction of flow from source ydistance from plume centerline

Quantitative Analysis ä Estimate the dispersion coefficient from the centerline concentration in region 2 ä Compare the measured dispersion coefficient with “rule of thumb” estimates ä Compare measured concentration profiles with theoretical predictions ä Estimate the dispersion coefficient from the centerline concentration in region 2 ä Compare the measured dispersion coefficient with “rule of thumb” estimates ä Compare measured concentration profiles with theoretical predictions

Dispersion Coefficient (E y ) Measurements Concentration at center of plume Estimate the dispersion coefficient at each x position

Dispersion Coefficient (E y ) Measurements x (cm) NaCl (mg/L) Plume Centerline Concentrations

Centerline concentration: Vertical Mixing Region 1 Incomplete vertical mixing x (cm) NaCl (mg/L) Measured with conductivity probe

Dispersion Coefficient (E y ) “rule of thumb” Expectations Integral length Integral velocity Ud Plume Transects

Qualitative Observations ä Depth of flow ä Objects in flow ä Momentum of discharge ä Depth of flow ä Objects in flow ä Momentum of discharge

Effect of Depth with Constant Flow 2.5 cm depth 5 cm depth 10 cm depth

Kármán Vortex Shedding Strouhal number Reynolds number Frequency of eddy shedding Distance between eddies DU 4.3D

Kármán Vortex Street L 8 cm diameter cylinder at side of port 8 cm diameter cylinder downstream of port Control (no objects in flow)

Summary ä Mixing of a passive plume in a river is controlled by the large scale river turbulence ä The largest scale of turbulence is roughly equal to the smallest dimension of the flow (in this case the depth of the river) ä Instantaneous measurements of velocity and concentrations vary with time in a turbulent environment ä The solution to the advective dispersion equation is a time averaged solution ä Mixing of a passive plume in a river is controlled by the large scale river turbulence ä The largest scale of turbulence is roughly equal to the smallest dimension of the flow (in this case the depth of the river) ä Instantaneous measurements of velocity and concentrations vary with time in a turbulent environment ä The solution to the advective dispersion equation is a time averaged solution

Solution: Lateral Distribution of Slug =1

Susquehanna: Plume Width in meters 10% of centerline

Susquehanna: Centerline Concentration

Plume in River

Plume contraction!

Plume transect 50 cm from source y (cm) NaCl (mg/L)

Plume transect 100 cm from source

Plume transect 200 cm from source

Plume transect 300 cm from source

Steady-Uniform Flow: Force Balance  W  W sin  xx a b c d Shear force Energy grade line Hydraulic grade line Shear force =________ W cos  Wetted perimeter = __ Gravitational force = ________ Hydraulic radius Relationship between shear and velocity? ______________  o P  x P  A  x sin  Turbulence