Numerical Analysis Numerical Analysis of Secondary-Flow of Secondary-Flow around a Spur Dike around a Spur Dike

using a Three-Dimensional using a Three-Dimensional Free Water Surface LES ModelFree Water Surface LES Model

R. Akahori & M.W. SchmeeckleR. Akahori & M.W. Schmeeckle

Department of Geography, Arizona State UniversityDepartment of Geography, Arizona State University

Rapid Channel Expansion

• Produces recirculation eddy that may Produces recirculation eddy that may trap sedimenttrap sediment

--ecological and recreational ecological and recreational importantimportant

• Spur dikes are employed to trap Spur dikes are employed to trap sediment and inhibit lateral sediment and inhibit lateral migration of the channelmigration of the channel

• Natural rapid channel expansions Natural rapid channel expansions often occur downstream of tributary/ often occur downstream of tributary/ mass wasting inputs or fixed bedrock mass wasting inputs or fixed bedrock constrictionsconstrictions

Prof. S. Ikeda

Example of channel expansion.1Example of channel expansion.1

Spur-dike in Toyohira RiverSpur-dike in Toyohira River

Example of channel expansion.2

Channel expansion by Debris fans in the Colorado River

Eddy

Fence

Reattachment Zone

Features of Recirculation Eddy

Features of Recirculation Eddy

• Eddy Recirculation zoneEddy Recirculation zone• Eddy Fence (Free Shear Layer)Eddy Fence (Free Shear Layer)• Reattachment zoneReattachment zone

- intense - intense large scale large scale turbulenceturbulence

Sediment trapSediment trap

Eddy recirculation zoneEddy recirculation zone

Reattachment zoneReattachment zone

Eddy FenceEddy Fence

Previous Numerical Approaches

• 2-D depth averaged model2-D depth averaged model --Can’t treat secondary flowCan’t treat secondary flow

--Relies on adjusting unknown lateral diffusionRelies on adjusting unknown lateral diffusion

• 2-D/ Quasi 3-D models w/ secondary flow 2-D/ Quasi 3-D models w/ secondary flow based on streamline curvaturebased on streamline curvature --Unknown secondary flow structure inUnknown secondary flow structure in recirculation recirculation

zones w/ complex topographyzones w/ complex topography

• RANS 3-D model (e.g. k-RANS 3-D model (e.g. k-εε)) --Cannot capture time variability in the reattachment Cannot capture time variability in the reattachment

zonezone - -especially large scale turbulence produced especially large scale turbulence produced along the free shear layeralong the free shear layer

Previous Numerical Approaches

• SomeSome models assume hydrostatic models assume hydrostatic pressurepressure --Flow at the point of separation and reattachment Flow at the point of separation and reattachment

zone have large advective accelerations in the zone have large advective accelerations in the vertical momentum equationvertical momentum equation

• Most models assume either a rigid lid Most models assume either a rigid lid or no time-variance of the water or no time-variance of the water surfacesurface --Time variance of the water surface is critical to Time variance of the water surface is critical to

accurately model large scale turbulenceaccurately model large scale turbulence

--Adequacy of assuming a rigid lid has not been Adequacy of assuming a rigid lid has not been provenproven

Employed Features

• Full 3-dimensional equations (non-Full 3-dimensional equations (non-hydrostatic)hydrostatic)

• Large Eddy Simulation (LES) Large Eddy Simulation (LES) turbulent model – no time-averagingturbulent model – no time-averaging

• Body Fitted Coordinates (BFC) Body Fitted Coordinates (BFC) system and Moving Grid system – system and Moving Grid system – free water surfacefree water surface

Full 3D Equations

0

ixiu

ig

jxjxiu

jx

juiu

ix

P

tiu

2

1

Continuity EquationContinuity Equation

Momentum EquationsMomentum Equations(Navier-Stokes Equation)(Navier-Stokes Equation)

in which,in which, x xii or or xxjj = = xx, , yy, , zz, and, and u uii or or uujj = = uu, , vv, , ww

Large Eddy Simulation (LES)-1

• N-S equations are spatially-filtered, N-S equations are spatially-filtered, NOT time- or ensemble-averaged.NOT time- or ensemble-averaged.

Eddies, larger than grid scale, are Eddies, larger than grid scale, are directly calculated by spatially-directly calculated by spatially-filtered N-S equations.filtered N-S equations.

Eddies, smaller than grid scale Eddies, smaller than grid scale (sub-grid scale: SGS), are (sub-grid scale: SGS), are parameterized.parameterized.

uu = = uu (spatially filtered) + (spatially filtered) + u’u’ (fluctuation (fluctuation))

Large Eddy Simulation (LES)-2

• Smagorinsky model is the simplest Smagorinsky model is the simplest model for SGS closure.model for SGS closure.

ig

jxjxiu

jx

ijjuiu

ix

P

tiu

2

1

Spatially-filteredSpatially-filteredMomentum EquationsMomentum Equations(Navier-Stokes Equations)(Navier-Stokes Equations)

qijij

RijC

ijLq

ijjuiu

juiu

ij

3

2

3

2

'','', jiijjijiji uuuuuuuuuuijR

ijC

ijL

Additional terms afterAdditional terms afterSpatial filteringSpatial filtering

Additional termAdditional term

ijtSqijij

RijC

ijL 2

3

1,0

2/12,

2

1,2

ijS

ijSS

ix

ju

jxiu

ijSS

sC

t

Smagorinsky ModelSmagorinsky Model

Body Fitted Coordinates (BFC)

• Body Fitted Coordinates (BFC) is Body Fitted Coordinates (BFC) is employed to fit the grid to arbitrarily employed to fit the grid to arbitrarily shaped boundaryshaped boundary

Cartesian Coordinates SystemCartesian Coordinates System BFC Coordinates SystemBFC Coordinates System

Moving Grid System• Moving Grid system enables the Moving Grid system enables the

model to trace temporally changing model to trace temporally changing free water surface boundary. free water surface boundary.

z

y

x

t

zyxt

zyxt

zyxt

zyxt

Operators for coordinates Operators for coordinates transformation into a combined transformation into a combined system of the BFC and a moving grid system of the BFC and a moving grid systemsystem

Example of combination of Example of combination of BFC and moving grid BFC and moving grid systemsystem

Existing Spur-dike Experiment

• Calculation results compared with existing Calculation results compared with existing experimental results (Muneta and Shimizu, 1994).experimental results (Muneta and Shimizu, 1994).

DischargeDischarge 18701870 cmcm33/sec/secSlopeSlope 1/10001/1000Channel lengthChannel length 700700 cmcmChannel widthChannel width 4040 cmcmDDownstream ownstream depthdepth 77 cmcmSpur dike lengthSpur dike length 2020 cmcmSpur dike widthSpur dike width 44 cmcm

Results (particle tracing)

• Particle tracing Particle tracing method is applied.method is applied.

• Recirculation is Recirculation is clearly observed.clearly observed.

• Reattachment point Reattachment point is varying over time.is varying over time.

• Jet-like stream can Jet-like stream can be seen beside a be seen beside a spur-dike, and it has spur-dike, and it has strong secondary-strong secondary-flow.flow.

Time: from 100(sec) to 170(sec)Time: from 100(sec) to 170(sec)

Results (vorticity: z-axis)

• Intense horizontal Intense horizontal vorticity generated vorticity generated in the free-shear in the free-shear layer (eddy fence)layer (eddy fence)

• Horizonatal eddies Horizonatal eddies are intermittently are intermittently shed into the shed into the reattachment zonereattachment zone

x

z

y

Time: from 100(sec) to 170(sec)Time: from 100(sec) to 170(sec)

Results (vorticity: x-axis)

• Stable counter-Stable counter-clockwise vorticity clockwise vorticity exists beside a exists beside a spur-dike.spur-dike.

• Large-scale vorticity Large-scale vorticity generated in the generated in the reattachment zonereattachment zone

x

z

y

Time: from 100(sec) to 170(sec)Time: from 100(sec) to 170(sec)

Results (vorticity: y-axis)

• Large scale vorticity Large scale vorticity generated at the bed generated at the bed and in the and in the reattachment zone reattachment zone

• Near-bed flow in jet Near-bed flow in jet has strong vorticity.has strong vorticity.

x

z

y

Time: from 100(sec) to 170(sec)Time: from 100(sec) to 170(sec)

Turbulence Intensity (u’u’)

• Turbulence intensity of Turbulence intensity of downstream velocitydownstream velocity

Strong turbulence exists in and downstream of the reattachment zone

Turbulence Intensity (v’v’)

• Turbulence intensity of Turbulence intensity of cross-stream velocitycross-stream velocity

Strong turbulence exists in the middle of the reattachment zone

Turbulence Intensity (w’w’)

• Turbulence intensity of Turbulence intensity of vertical velocityvertical velocity

Turbulence intensity is respectively small also has a peak in the zone of reattachment

Water Surface

• Time-averaged Time-averaged calculated water calculated water surface level surface level seems reasonable.seems reasonable.

• Fluctuation is intense Fluctuation is intense in front of a spur-dike, in front of a spur-dike, and downstream after and downstream after a recirculation area. a recirculation area.

Comparison (depth-averaged)

• Depth-averaged Depth-averaged recirculation recirculation eddy is observed eddy is observed both in both in experiment and experiment and calculation calculation results.results.

• Calculated result Calculated result shows good shows good agreement. agreement.

Comparison (cross-stream velocity)

• Results show Results show similar tendency similar tendency near the bottom.near the bottom.

• But near the But near the surface, surface, calculation calculation results show results show stronger stronger leftward flow leftward flow than the than the experiment.experiment. dot: experiment

line: calculation

Lateral velocity at cross-stream sections

Cross-sectional Vectors

• Strong secondary-flow is produced as flow is Strong secondary-flow is produced as flow is accelerated around the spur-dike, and it accelerated around the spur-dike, and it remains far downstream.remains far downstream.

Conclusion

• Intense horizontal vorticity (z-Intense horizontal vorticity (z-axis) generated but confined to axis) generated but confined to the eddy fence (free shear layer)the eddy fence (free shear layer)

• Large-scale intense turbulence Large-scale intense turbulence generated in the reattachment generated in the reattachment zonezone

• The point of reattachment varies The point of reattachment varies substantially over timesubstantially over time

Morphodynamics of Eddies

• Sediment trapping occurs in the zone of Sediment trapping occurs in the zone of reattachment with generation of large-reattachment with generation of large-scale turbulent eddies and temporal scale turbulent eddies and temporal changes in the point of reattachmentchanges in the point of reattachment

• LLittle sediment can cross the eddy fence ittle sediment can cross the eddy fence because horizontal vorticity confined to because horizontal vorticity confined to narrow shear zonenarrow shear zone

• NoNo mean mean inward secondary flow near inward secondary flow near the bed means that sediment is not the bed means that sediment is not trapped by secondary flowtrapped by secondary flow

– –only large-scale turbulent motiononly large-scale turbulent motion

The ENDThe END

Other Features

• Boundary friction from the law of the wall at Boundary friction from the law of the wall at the bed and bank boundariesthe bed and bank boundaries

• Slip condition at water surface.Slip condition at water surface.

• Convection terms are calculated by the Convection terms are calculated by the Cubic-Interpolated Propagation (CIP) Cubic-Interpolated Propagation (CIP) scheme.scheme.

• Water surface level is implicitly calculated Water surface level is implicitly calculated by the kinematic boundary condition.by the kinematic boundary condition.

Calculation Conditions

• Calculation conditions are following.Calculation conditions are following.

Total calculation timeTotal calculation time 117700 secsecTime stepTime step 0.0010.001 secsecGrid number Grid number (streamwise)(streamwise) 175175Grid number Grid number (crossstream)(crossstream) 2020Grid number Grid number (vertical)(vertical) 1010

• At the upstream end, discharge is given.At the upstream end, discharge is given.• At the downstream end, water depth is given.At the downstream end, water depth is given.