Fluid Flow in Rivers

Fluid Flow in RiversOutline1. Flow uniformity and steadiness2. Newtonian fluids3. Laminar and turbulent flow4. Mixing-length concept5. Turbulent boundary layer6. Mean boundary shear stress7. Velocity distribution and the “Law of the Wall”8. Depth-averaged velocity

Flow in Rivers

• Rivers are nonuniform, unsteady, Newtonian, hydraulically rough turbulent flows

(Bridge, 2003)

Uniform Flow: 0xAU

Nonuniform Flow: 0xAU

Open Channel Flow

UAUdwQ

Steady Flow: 0

tAU

Unsteady Flow: 0tAU

Open Channel Flow

(Bridge, 2003)

Fluid Viscosity

(Bridge, 2003)

Types of fluids

Molecular viscosity is independent of the magnitude of shear (rivers)

Molecular viscosity is independent of the magnitude of shear once yield strength is exceeded (mud and lava flows)

Molecular viscosity is dependent on shear and it has a yield strength (paint)

(Hornberger et al., 1998)

Laminar flow, Re < 2000 (viscous forces dominate)

Turbulent flow, Re > 4000 (turbulent forces dominate)

Ud

forces viscousforcesturbulent Re

Large masses of fluid “eddies” are being transported

Types Flows

Laminar and Turbulent Flow

• For 1 m deep flow:Re = 300, U ~ 0.0003 m/sRe = 3000, U ~ 0.003 m/s

• Susquehanna River near Waverly, PAd ~ 2 m, U ~ 1 m/s, Re ~ 2,000,000

Ud

forces viscousforcesturbulent Re

(Bridge, 2003)

Turbulent velocity time series

Turbulent Velocities

01

01

1d1

1

2

1

10

n

iirms

n

ii

i

n

ii

T

uun

uu

uun

u

uuu

un

tuT

u

(Bridge, 2003)

22

yulvu

Reynolds Stress

Mixing length concept

l: the vertical distance over which the momentum of a fluid parcel is changed (mixing length)

(Bridge, 2003)

Turbulent boundary layer

(Bridge, 2003)

Viscous sublayer streaks

(Allen, 1985; Bridge, 2003)

Turbulent “bursting” process

~70% of all turbulence in open channel flows is due to the bursting process

(Julien, 1998)DU

g*Re

Grain roughness effects on turbulent boundary layer

(Falco, 1977)

Large-scale Turbulent Motions

(Nakagawa & Nezu, 1981)

(Ferguson et al., 1996)

(Belanger et al. 2000)

(Bridge, 2003)

gdS 0

Boundary Shear Stress

dy 10

Conservation of downstream momentum:Impelling force (downstream component of weight of water) = resistive force

sin''

xpwu

yy vis

“Law of the Wall”

0*

* ln1 ;yy

uu

yu

yu

0* u

• Derived from Reynolds-averaged Navier-Stokes equations (Gomez, 2006)– Nearly universal use

• Important assumptions:– Prandtl’s mixing length

theory– Zero slope (?!)– Constant shear stress (no

vertical gradient; ?!)

0neg.(Taylor, 1921; Prandtl, 1925, 1926)

yuLwu

dd''

0

ln y (m)

u (m/s)

slope

interceptln

slope:data regression From

ln1

wall" theof Law" theofn Applicatio

0

*

0*

y

u

yy

uu

regression

ln y (m)

u (m/s)

slope

interceptln

slope:data regression From

ln1

wall" theof Law" theofn Applicatio

0

*

0*

y

u

yy

uu

regression

Utility of the “Law of the Wall”

0

0

2*0

2.30height roughnessvelocity -zero

yky

u

s

Distance (m)

4 5 6 7 8

WS

E (m

)

-0.03

-0.02

-0.01

0.00

S=0.0026u*=0.051 m/s

Depth-slope productFor a mobile upper stage plane bed (mm-scale bedwaves), the boundary is essentially flat. Thus, the mean shear stress determined using a Reynolds stress projection, the law of the wall, and the depth-slope product (relative water surface elevation; WSE) are nearly identical (from Bennett et al., 1998).

u'v'

0.0 0.5 1.0 1.5 2.0 2.5

y/d

0.0

0.2

0.4

0.6

0.8

1.0

u*=0.045 m/s

Reynolds stress

U (m/s)

0.5 0.6 0.7 0.8 0.9 1.0 1.1

ln y

-6.0

-5.0

-4.0

-3.0

-2.0u*=0.045 m/s=0.33ks=2.0 mm

Law of the wall

RS LW DS

(P

a)

0.0

0.5

1.0

1.5

2.0

2.5

Distance from Reattachment (m)

0.0 0.1 0.2 0.3 0.4 0.5

Hei

ght (

m)

0.00

0.02

0.04

0.06

Dune Profile

Law of the wallNear-bed Reynolds stress

Crest

Trough

Reynolds Stress vs. Velocity Gradient Shear Stress over a Fixed Dune

Depth Averaged Velocity

i=4

tsmeasuremen ofnumber theis ;;0, :where

1

2

:form Specific

d1 :form General

: velocityflow integrated-Depth

1100

1

1

1

1

0

nuudyuy

yud

U

uuuyyy

yud

U

nnn

n

iii

iii

iii

d

y

y3,u3

y4,u4

344 yyy

2344 uuu

u (m/s)

y (m)

Fluid Flow and Stream Restoration

• Conforms to physics• Based on steady, uniform flow conditions, there

are three (3) methods to determine near-bed shear stress

• Distributions of velocity should be predictable

Fluid Flow in Rivers

Conclusions1. River flow is unsteady, non-uniform, turbulent,

and hydraulically rough2. River flow can be treated as a boundary layer,

and its distribution of velocity can be determined

3. Turbulence is derived from bursting process4. Bed shear stress can be determined from bulk

hydraulic parameters and velocity profiles