CE 344 - Topic 2.4 - Spring 2003 - February 16, 2003 9:06 pm2003/02/16 · CE 344 - Topic 2.4 -...

CE 344 - Topic 2.4 - Spring 2003 - February 16, 2003 9:06 pm

p. 2.4.1

2.4 VELOCITY DISTRIBUTIONS IN OPEN CHANNELS

• In our analysis of steady uniform flow we neglected the effects of the depth and/or later-ally varying velocity distribution.

Derivation of Velocity Distribution in a Wide Open Channel with Steady Uniform Flow

Governing Equations

• Instead of assuming depth averaged flow, we will derive a more detailed expression forflow and examine flow structure over the vertical.

• Thus we average our equations over turbulent time/space scale and no longeraverage over the vertical.

• Thus we must apply the Reynold’s equations.

• We examine steady uniform flow in a wide channel:


p. 2.4.2

• Note that we have oriented the x-axis with the channel bottom. This requires adjustingthe gravity term in the Reynolds equations as follows:

(2.4.1)

(2.4.2)

(2.4.3)

• We make the following assumptions:

• defines the free surface

• Flow is steady:

• Flow is uniform in the x-direction: and .

• No flow in the y direction:

• All flow conditions are uniform in y:

• Continuity can be used to show that the z-component of velocity is zero:

DuDt------- g θ0

1ρ---

∂p∂x------

∂∂x-----

µρ---

∂u∂x------ u'u'–� �

� � ∂∂y-----

µρ---

∂u∂y------ u'v'–� �

� � ∂∂z-----

µρ---

∂u∂z------ u'w'–� �

� �+ + +–sin=

DvDt-------

1ρ---

∂p∂x------–

∂∂x-----

µρ---

∂v∂x----- v'u'–� �

� � ∂∂y-----

µρ---

∂v∂y----- v'v'–� �

� � ∂∂z-----

µρ---

∂v∂z----- v'w'–� �

� �+ + +=

DwDt-------- g– θ0

1ρ---

∂p∂z------

∂∂x-----

µρ---

∂w∂x------- w'u'–� �

� � ∂∂y-----

µρ---

∂w∂y------- w'v'–� �

� � ∂∂z-----

µρ---

∂w∂z------- w'w'–� �

� �+ + +–cos=

z d0=

∂∂t---- 0→

u z( ) ∂u∂x------ 0=

v 0=

∂∂y----- 0→

w 0=


p. 2.4.3

• With the simplifying assumptions, the Reynold’s equations reduce to

(2.4.4)

(2.4.5)

(2.4.6)

• In order to simplify the turbulent cross correlations in the Reynold’s equations, we applythe Prandtl-Boussinesq turbulence model which indicates:

(2.4.7)

• This is a component of our constitutive model.

• This model transfers momentum through shear gradients.

• It does not transfer momentum through layers with equal turbulent time-averagedvelocities.

0 g θ01ρ---

∂p∂x------

∂∂x----- u'u'–( ) ∂

∂y----- u'v'–( ) ∂

∂z-----

µρ---

∂u∂z------ u'w'–� �

� �+ + +–sin=

0∂

∂x----- v'u'–( ) ∂

∂y----- v'v'–( ) ∂

∂z----- v'w'–( )+ +=

0 g– θ01ρ---

∂p∂z------

∂∂x----- w'u'–( ) ∂

∂y----- w'v'–( ) ∂

∂z----- w'w'–( )+ + +–cos=

τijt

ρ----- u'iu'j–

µt

ρ----

∂ui

∂xj-------

∂uj

∂xi-------+� �

� �= =


p. 2.4.4

• The turbulent viscosity is now dependent on the intensity of turbulence. Thisdepends on factors such as:

• Distance from the wall

• Flow intensity and gradients.

• Thus, looking at each cross correlation term:

(2.4.8)

(2.4.9)

(2.4.10)

(2.4.11)

(2.4.12)

(2.4.13)

u'u'–µt

ρ---- 2

∂u∂x------� �

� � 0= =

u'v'–µt

ρ----

∂u∂y------

∂v∂x-----+� �

� � 0= =

u'w'–µt

ρ----

∂u∂z------

∂w∂x-------+� �

� � µt

ρ----

∂u∂z------= =

v'v'–µt

ρ---- 2

∂v∂y-----� �

� � 0= =

v'w'–µt

ρ----

∂v∂z-----

∂w∂y-------+� �

� � 0= =

w'w'–µt

ρ---- 2

∂w∂z-------� �

� � 0= =


p. 2.4.5

• Substituting these terms into our reduced form of the Reynolds equation:

(2.4.14)

(2.4.15)

(2.4.16)

• We must now complete the closure of our constitutive model by establishing a relation-ship for .

• There are a wide variety of models for . Some examples are:

• The shear velocity - wall distance model: Turbulent viscosity is correlated to propor-tioned distance from the wall and bottom stress acts as a measure of turbulenceintensity.

(2.4.17)

where = Von Karman constant = 0.40 for clean water (2.4.18)

z = distance from the bottom (2.4.19)

(2.4.20)

0 g θ01ρ---

∂p∂x------

∂∂z-----

µρ---

∂u∂z------

µt

ρ----

∂u∂z------+� �

� �+–sin=

0 0=

0 g– θ01ρ---

∂p∂z------

∂∂x-----

µt

ρ----

∂u∂z------� �

� �+–cos=

µt

µt

µt ρκzu*=

κ

u*

τ0

ρ-----≡ shear velocity where τ0 bottom stress= =


p. 2.4.6

• Prandtl Model: Turbulent viscosity is correlated to proportioned distance from thewall squared and gradient in velocity acts as a measure of turbulence intensity.

(2.4.21)

• Selecting the first model for simplicity, our reduced Reynold’s equations become:

(2.4.22)

(2.4.23)

Solving for p using the Hydrostatic Pressure Equation and Simplifying the x-direction Reynolds Equation

• Noting that the turbulent viscous term in the z-direction momentum equation is a func-tion of z only, this equation reduces to the pure hydrostatic equation:

(2.4.24)

u

µt ρκ2z

2 ∂u∂z------=

0 g θ01ρ---

∂p∂x------

∂∂z-----

µρ---

∂u∂z------ κzu*

∂u∂z------+� �

� �+–sin=

0 g– θ01ρ---

∂p∂z------

∂∂x----- κzu*

∂u∂z------� �

� �+–cos=

1ρ---

∂p∂z------– g θ0cos=


p. 2.4.7

• Integrating the hydrostatic equation between the surface where (assumingconstant atmospheric pressure) and a point under the free surface at height z:

(2.4.25)

• This leads to the pressure equation over the vertical:

(2.4.26)

• Taking the x-derivative and noting that = constant:

(2.4.27)

• Substituting into our x-direction Reynold’s equation and noting that :

(2.4.28)

z d0= p 0=

∂p

p

p 0=

� g θ0cos zd

z

z d0=

�–=

p g θ0 d0 z–( )cos=

d0

∂p∂x------ 0=

S0 θ0sin≅

gS0∂∂z-----

µρ---

∂u∂z------ κzu*

∂u∂z------+� �

� �=–


p. 2.4.8

Solving the z-direction Reynolds Equation for Stress and Reynolds Averaged Velocity

• We now assume that turbulent momentum transfer is significantly greater than molec-ular momentum transfer:

(2.4.29)

• This is valid through-out the water column except within the viscous sublayer.

• The viscous sublayer is very small compared to and we will neglect it since itseffect is small.

• This simplification does imply that the does not equal to zero at z = 0.

µρ---

∂u∂z------ κzu*

∂u∂z------«

d0

u z( )


p. 2.4.9

• Thus we will solve the differential equation:

(2.4.30)

• We note that by the way we defined our constitutive relationships (including neglectingthe viscous sublayer), shear stress is computed as:

(2.4.31)

• We note that at the free surface, there is no shear stress:

(2.4.32)

• We can integrate our d.e. (2.4.30) between a point z and the free surface:

(2.4.33)

gS0∂∂z----- κzu*

∂u∂z------� �

� �=–

τzx

ρ------ κzu*

∂u∂z------≡

τzx

ρ------

z d0=

κzu*∂u∂z------

z d0=

0= =

gS0 zd

z

z d0=

�– ∂ κzu*∂u∂z------� �

� �

κzu*∂u∂z------

κzu*∂u∂z------� �

� �z d0=

0=

�=


p. 2.4.10

• Completing the integration and switching the order of the terms:

(2.4.34)

• Noting that is defined in (2.4.32), stress over the water column varies as:

(2.4.35)

• Thus stress varies linearly between zero at the free surface and having a maximum valueat the bottom.

κzu*∂u∂z------⋅ gS0 d0 z–( )=

τzx

τzx

ρ------ gS0 d0 z–( )=


p. 2.4.11

• We can re-arrange our simplified d.e. (2.4.34)

(2.4.36)

• Now integrate (2.4.36) between some location at the bottom where and somepoint z.

(2.4.37)

(2.4.38)

(2.4.39)

(2.4.40)

∂u∂z------

gS0

κu*---------

d0

z----- 1–� �� =

zb u 0=

∂u

u 0=

u

�gS0

κu*---------

d0

z----- 1–� �� zd

zb

z

�=

ugS0

κu*--------- d0 zln z–[ ]zb

z=

ugS0

κu*--------- d0 zln z– d0 zbln– zb+[ ]=

ugS0

κu*--------- d0

zzb----� �� ln z zb+–=


p. 2.4.12

Computing a Depth Averaged Velocity in Terms of Maximum Channel Velocity

• Let’s now average over the vertical:

(2.4.41)

(2.4.42)

(2.4.43)

(2.4.44)

(2.4.45)

(2.4.46)

u

u1d0----- u zd

zb

d0

�=

u1d0-----

gS0

κu*--------- d0ln

zzb----� �� z– zb+

� ��

zd

zb

d0

�=

u1d0-----

gS0

κu*--------- d0 zln d0 zbln z–– zb+{ } zd

zb

d0

�=

u1d0-----

gS0

κu*--------- d0 z zln z–( ) d0 zb( )ln z–

z2

2----– zbz+

zb

d0

=

u1d0-----

gS0

κu*--------- z d0 zln d0– d0 zbln– z–

z2--- zb+ +

� ��

zb

d0

=

u1d0-----

gS0

κu*--------- z d0

zzb----� �� ln z– zb d0–

z2---+ +

� ��

zb

d0

=


p. 2.4.13

• Noting from equation (2.4.40) that

(2.4.47)

• Substituting

(2.4.48)

(2.4.49)

• Note that and :

(2.4.50)

ugS0

κu*---------� �� -------------- d0

zzb----ln z– zb+=

u1d0-----

gS0

κu*--------- z

u z( )gS0

κu*---------� �� -------------- zd0–

z2

2----+

zb

d0

=

u1d0-----

gS0

κu*--------- d0

u d0( )gS0

κu*---------� �� -------------- d0

2–

d02

2------- zb

u zb( )gS0

κu*---------� �� --------------– zbd0

zb2

2----–+ +=

u d0( ) umax= u zb( ) 0=

u1d0-----

gS0

κu*--------- d0

umax

gS0

κu*---------

-----------d0

2

2-------– zbd0

zb2

2----–+=


p. 2.4.14

• Further reworking the previous equation:

(2.4.51)

• Re-arranging this equation:

(2.4.52)

• Since for

(2.4.53)

• This lead us to relations between the maximum profile velocity and the depth averagedvelocity:

(2.4.54)

(2.4.55)

u umax1d0-----

gS0

κu*---------

d02

2-------– zbd0

zb2

2----–++=

u umax12---

gS0

κu*---------

d0 zb–( )d0

---------------------2

–=

d0 zb»

d0 zb–( )2

d0-----------------------

d02

d0----- d0≅ ≅

u umax12---

gS0d0

κu*--------------⋅–=

umax u12---

gS0d0

κu*--------------+=


p. 2.4.15

Relating Channel Velocity to Channel Depth Averaged Velocity

• We note that we could have also integrated our d.e. (2.4.36) between the surface and z as

(2.4.56)

(2.4.57)

(2.4.58)

(2.4.59)

(2.4.60)

∂u

u

umax

�gS0

κu*---------

d0

z----- 1–� �� zd

z

d0

�=

umax u–gS0

κu*--------- d0lnz z–[ ]z

d0=

umax u–gS0

κu*--------- d0lnd0 d0– d0lnz– z+[ ]=

u umax–gS0

κu*--------- d0lnz d0lnd0– z– d0+[ ]=

u umax

gS0

κu*--------- d0ln

zd0-----� �� z– d0++=


p. 2.4.16

• Now let’s substitute in our relationship for and from (2.4.55) into (2.4.60):

(2.4.61)

(2.4.62)

(2.4.63)

Deviation in Reynolds Averaged Velocity from the Depth Averaged Velocity Normalized by the Depth Averaged Velocity

• Now let’s work out a relationship for the deviation in velocity from the mean relative tothe mean:

(2.4.64)

(2.4.65)

umax u

u z( ) u12---

gS0d0

κu*--------------

gS0

κu*--------- d0ln

zd0-----� �� z– d0++ +=

u z( ) ugS0

κu*---------

d0

2----- d+

0ln

zd0-----� �� z– d0++=

u z( ) ugS0

κu*---------

32---d0 d+

0ln

zd0-----� �� z–+=

u z( ) u z( ) u–=

uu---

u u–u

------------uu--- 1–= =


p. 2.4.17

• Substituting in for our relationship (2.4.63) between and :

(2.4.66)

• This leads to an equation for the velocity deviation from the depth averaged velocitynormalized by the depth averaged velocity:

(2.4.67)

Deviation in Reynolds Averaged Velocity from the Depth Averaged Velocity Normalized to the Depth Averaged Velocity Using only Channel Parameters

• Now, we would like to relate the shear velocity to the bottom stress. We note fromequation (2.4.35)

(2.4.68)

• Evaluating this equation at the bottom:

(2.4.69)

u u

uu---

uu---

gS0

uκu*------------

32---d0 d0

zd0-----� �� z–ln+ 1–+=

uu---

gS0d0

uκu*--------------

32---

zd0-----� �� z

d0-----–ln+=

u*

τzx

ρ------ gS0 d0 z–( )=

τzx

ρ------

z zb=

gS0 d0 zb–( )=


p. 2.4.18

• However we note that

(2.4.70)

• Furthermore we can readily approximate:

(2.4.71)

• Thus (2.4.69) simplifies to:

(2.4.72)

• Using the definition for shear velocity, this implies that:

(2.4.73)

τzx

ρ------

z zb=

τ0

ρ-----≅

d0 zb d0≅–

τ0

ρ----- gS0d0=

u*

τ0

ρ----- gS0d0= =


p. 2.4.19

• Furthermore, we previously related bottom stress to using the constitutive relation-ship.

(2.4.74)

• Noting the definition for shear velocity

(2.4.75)

• Using equation (2.4.73), we can substitute in for . Thus:

(2.4.76)

• Now we can solve for depth averaged velocity in terms of the channel slope, depth andfriction factor:

(2.4.77)

u

τ0

ρ-----

fDW

8--------- u

2=

u*2 fDW

8--------- u

2=

u* gS0d0=

gS0d0

fDW

8--------- u

2=

u8gS0d0

fDW------------------=


p. 2.4.20

• Substituting in for and into our relationship for , equation (2.4.67):

(2.4.78)

• This simplifies to our final relationship for the ratio in terms of water depth and fric-

tion factor:

(2.4.79)

u u*uu---

uu---

gS0do

8gS0d0

fDW------------------κ gS0d0

---------------------------------------------32---

zd0-----� �� z

d0-----–ln+=

uu---

u z( )u z( )----------

1κ---

fDW

8---------

32---

zd0-----� �� z

d0-----–ln+=


p. 2.4.21

• Plotting versus for , , and :

• These friction values correspond to low, medium and high friction in an openchannel.

• From this plot we note:

• That the maximum velocities exceed the mean by 4%, 9% and 14% respectively.

• The logarithmic profile of the velocity distributions.

• The depth averaged velocity approximation appears to be an excellent estimate ofthe actual depth varying velocity except very near the bottom.

• The higher friction, the more that the velocity distribution deviates from the mean .

u z( )u z( )----------

zd0----- f 0.01= f 0.04= f 0.10=

uu z( )

u


p. 2.4.22

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.20

0.2

0.4

0.6

0.8

1


p. 2.4.23

Practical Velocity Distribution Function in Wide Open Channels

• The velocity distribution for fully developed turbulent open channel flow is givenapproximately by Prandtl’s power law:

(2.4.80)

where d = depth of flow (2.4.81)

• The value of N depends on boundary friction and cross section shape.

• N ranges from 4 (for shallow water in wide rough channels) up to 12 (for smoothnarrow channels).

• For uniform equilibrium flows:

(2.4.82)

where (2.4.83)

• Most commonly used values are N=6 or N=7.

u z( )umax-----------

zd---� ��

1N----

=

N κ 8fDW---------=

κ 0.4=


p. 2.4.24

Implications of Depth Varying Velocity Profiles for the Shallow Water Equations

• Let us examine the effect of z variability on the shallow water equations.

• For the depth averaged continuity equation, the depth variability has been averaged out.Thus we can simply use the equation as is:

(2.4.84)

• For the depth averaged momentum equations, the dispersion terms appeared:

(2.4.85)

(2.4.86)

∂n∂t------

∂ uH( )∂x

---------------∂ vH( )

∂y---------------+ + 0=

∂u∂t------ u

∂u∂x------ v

∂u∂y------+ + g

∂η∂x------–

1H----

∂∂x-----

τxxt m⁄

ρ---------- u

2–

� ��

z1H----

∂∂y-----

τyxt m⁄

ρ---------- uv–� ��

z1H----

τxs

ρ----

1H----

τxb

ρ-----–+d

h–

η

�+d

h–

η

�+=

∂v∂t----- u

∂v∂x----- v

∂v∂y-----+ + g

∂η∂y------–

1H----

∂∂x-----

τxyt m⁄

ρ---------- uv–� ��

z1H----

∂∂y-----

τyyt m⁄

ρ---------- v

2–

� ��

z1H----

τys

ρ----

1H----

τyb

ρ-----–+d

h–

η

�+d

h–

η

�+=


p. 2.4.25

• We note that for steady uni-directional steady uniform open channel flow

(2.4.87)

(2.4.88)

• Thus it is clear that for uniform flow the dispersion portions of the lateral/dispersionterms drop out.

(2.4.89)

(2.4.90)

(2.4.91)

(2.4.92)

u u z( ) only=

v 0=

1H----

∂∂x----- u

2–( ) zd

h–

η

� 0=

1H----

∂∂y----- uv–( ) zd

h–

η

� 0=

1H----

∂∂x----- uv–( ) zd

h–

η

� 0=

1H----

∂∂y----- v

2–( ) zd

h–

η

� 0=


p. 2.4.26

• In fact we can also demonstrate that the lateral diffusion terms drop out assuming aPrandtl-Boussinesq-type closure (e.g., see our derivation of the vertical velocity profilein a channel).

• It makes sense that the lateral diffusion/dispersion terms drop out for a uniform flow inthe x-direction (despite ) since all these terms originate from the advectiveterms. Since the advective terms are zero in a uniform flow, the associated diffusion/dispersion terms will be zero.

• For non-uniform flows, the advective terms and thus the lateral diffusion/dispersionterms will become a portion of the momentum balance.

• Their importance is definitely correlated to the importance of the advective terms.

• Due to the logarithmic nature of the depth-varying component of the velocity profile,, the dispersion terms are often not significant.

• If these terms are significant, they can be explicitly computed by either assuming orcalculating a vertical velocity profile. This is what is done in sophisticated, state-of-the-art shallow water equation codes.

u u z( )=

u


p. 2.4.27

Implication of Depth-Varying Velocity Profiles for the Bernoulli Equation

• We must adjust the kinetic energy term in the total energy when using depth averaged orcross sectional mean flow as the variable.

(2.4.93)

• The true mean velocity head is found by adjusting this term by:

(2.4.94)

• For the power law velocity distribution, the kinetic energy coefficient equals

(2.4.95)

• We note that for a typical value of N = 7,

(2.4.96)

HEpγ--- z

u2

2g------+ +=

α

u3

Ad

A�

u3A

---------------=

1N----

α N 1+( )3

N2

N 3+( )-------------------------=

α 1.045=


p. 2.4.28

• Note that this concept can be extended to channels with widely varying flow propertiesand that the averaging can be accomplished on a sectional basis:

(2.4.97)

(2.4.98)

um

u1A1 u2A2 u3A3+ +

A1 A2 A3+ +------------------------------------------------=

αu1

3A1 u2

3A2 u3

3A3+ +

um A1 A2 A3+ +( )------------------------------------------------=


p. 2.4.29

• For all cases, when using depth averaged or cross sectionally averaged flow, the totalenergy becomes:

(2.4.99)

• We note that for uniform flow, , and therefore the kinetic energy terms do notcome into play.

• This makes sense since the advective acceleration terms are zero and the advectiveterms are the origin of the kinetic energy terms.

• Thus for uniform flow, the correction factor becomes an inconsequential point.

• For non-uniform flow, we must compute or assume a velocity profile and estimate avalue of .

• Often the variation in is small compared to uncertainties such as frictional resistance.

HEpγ--- z α u

2

2g------+ +=

u1 u2=

α

α

α


p. 2.4.30

Implication of Depth-Varying Velocity Profiles for Momentum Conservation for Fi-nite Control Volumes

• We must adjust the momentum flux term in the conservation of momentum applied to afinite control volume.

(2.4.100)

• For a channel, the momentum flux term simplified to

(2.4.101)

• We can use depth averaged velocity in this expression by computing an adjustmentcoefficient:

(2.4.102)

∂∂t---- Uρ V U ρU n⋅ Ad( )

cs��+d�

cv�� T A Bρ Vd�

cv��+d

cs��=

u ρu( ) Ad

cs��

u

β

ρu2

Ad

A�

ρu2A

-------------------=


p. 2.4.31

• For the power law velocity distribution, the momentum correction coefficient equals

(2.4.103)

• We note that for a typical value of N = 7

(2.4.104)

• The momentum flux term for an open channel can be computed as

(2.4.105)

1N----

β N 1+( )2

N N 2+( )----------------------=

β 1.016=

u ρu( ) Ad

cs�� ρQ β2u2 β1u1–( )=


p. 2.4.32

• Again we note that for uniform flow, and therefore the momentum flux termsdo not come into play.

• Again, this makes sense since the momentum flux terms are related to the advectiveacceleration terms (they advect momentum). Since the advective terms for auniform flow are zero, the momentum flux terms in the surface integral must bezero.

• Thus, for uniform flow, the momentum flux correction factor, , does not come intoplay.

• For non-uniform flow, we must compute or assume a velocity profile and estimate avalue of .

• Often the variation in is small compared to uncertainties such as frictional resistance.

u1 u2=

β

β

β

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CE 344 - Topic 2.4 - Spring 2003 - February 16, 2003 9:06 pm2003/02/16 · CE 344 - Topic 2.4 -...

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